ICL 7053: Curriculum Materials

For your INITIAL POST, respond to the following questions:

To what extent do the Modules align with the information provided in the Progressions for Grade 8? If perfect alignment is a 10 and no alignment is a 1, what rating would you give to the Modules?
If you were teaching Grade 8, would you implement these Modules “as is” or would you supplement/replace the curriculum materials? Why? What would you change?

You are required to respond to AT LEAST TWO other students. In your response, address the following questions:

Do you agree with the student’s analysis of the alignment between the Eureka modules and the Progressions? Why?
Do you agree with the student’s perspective on supplementing/replacing in the Eureka modules?

Progressions for the Common Core State
Standards in Mathematics (draft)
c©The Common Core Standards Writing Team

2 July 201

Suggested citation:
Common Core Standards Writing Team. (2013, March
1). Progressions for the Common Core State Stan-
dards in Mathematics (draft). Grade 8, High School,
Functions. Tucson, AZ: Institute for Mathematics and
Education, University of Arizona.
For updates and more information about the
Progressions, see http://ime.math.arizona.edu/
progressions.
For discussion of the Progressions and related top-
ics, see the Tools for the Common Core blog: http:
//commoncoretools.me.

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http://ime.math.arizona.edu/progressions

http://commoncoretools.me

commoncoretools.wordpress.com

Grade 8, High School,
Functions*
Overview
Functions describe situations in which one quantity is determined
by another. The area of a circle, for example, is a function of its ra-
dius. When describing relationships between quantities, the defin-
ing characteristic of a function is that the input value determines the
output value or, equivalently, that the output value depends upon the
input value.

The mathematical meaning of function is quite different from
some common uses of the word, as in, “One function of the liver
is to remove toxins from the body,” or “The party will be held in the
function room at the community center.” The mathematical meaning
of function is close, however, to some uses in everyday language.
For example, a teacher might say, “Your grade in this class is a
function of the effort you put into it.” A doctor might say, “Some ill-
nesses are a function of stress.” Or a meteorologist might say, “After
a volcano eruption, the path of the ash plume is a function of wind
and weather.” In these examples, the meaning of “function” is close
to its mathematical meaning.

In some situations where two quantities are related, each ca

be viewed as a function of the other. For example, in the context of
rectangles of fixed perimeter, the length can be viewed as depending
upon the width or vice versa. In some of these cases, a problem
context may suggest which one quantity to choose as the input
variable.

*The study of functions occupies a large part of a student’s high school career,
and this document does not treat in detail all of the material studied. Rather it
gives some general guidance about ways to treat the material and ways to tie it
together. It notes key connections among standards, points out cognitive difficulties
and pedagogical solutions, and gives more detail on particularly knotty areas of the
mathematics.

The high school standards specify the mathematics that all students should study
in order to be college and career ready. Additional material corresponding to (+)
standards, mathematics that students should learn in order to take advanced courses
such as calculus, advanced statistics, or discrete mathematics, is indicated by plus
signs in the left margin.

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Undergraduate mathematics may involve functions of more than
one variable. The area of a rectangle, for example, can be viewed as
a function of two variables: its width and length. But in high school
mathematics the study of functions focuses primarily on real-valued
functions of a single real variable, which is to say that both the
input and output values are real numbers. One exception is in high
school geometry, where geometric transformations are considered to
be functions.• For example, a translation T, which moves the plane • G-CO.2 . . . [D]escribe transformations as functions that take

points in the plane as inputs and give other points as outputs. . . .3 units to the right and 2 units up might be represented by T :
px, yq ÞÑ px � 3, y� 2q.
Sequences and functions Patterns are sequences, and sequences
are functions with a domain consisting of whole numbers. How-
ever, in many elementary patterning activities, the input values are
not given explicitly. In high school, students learn to use an inde

The problem with patterns3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Students are asked to continue the pattern 2, 4, 6, 8, . . . . Here
are some legitimate responses:

• Cody: I am thinking of a “plus 2 pattern,” so it continues
10, 12, 14, 16, . . . .

• Ali: I am thinking of a repeating pattern, so it continues 2,
4, 6, 8, 2, 4, 6, 8, . . . .

• Suri: I am thinking of the units digit in the multiples of 2,
so it continues 0, 2, 4, 6, 8, 0, 2, . . . .

• Erica: If gpnq is any polynomial, then
fpnq � 2n� pn� 1qpn� 2qpn� 3qpn� 4qgpnq
describes a continuation of this sequence.

• Zach: I am thinking of that high school cheer, “Who do
we appreciate?”

Because the task provides no structure, all of these answers
must be considered correct. Without any structure, continuing
the pattern is simply speculation—a guessing game. Because
there are infinitely many ways to continue a sequence,
patterning problems should provide enough structure so that the
sequence is well defined.

to indicate which term is being discussed. In the example in the
margin, Erica handles this issue by deciding that the term 2 would
correspond to an index value of 1. Then the terms 4, 6, and 8 would
correspond to input values of 2, 3, and 4, respectively. Erica could
have decided that the term 2 would correspond to a different index
value, such as 0. The resulting formula would have been different,
but the (unindexed) sequence would have been the same.
Functions and Modeling In modeling situations, knowledge of the
context and statistics are sometimes used together to find a func-
tion defined by an algebraic expression that best fits an observed
relationship between quantities. (Here “best” is assessed informally,
see the Modeling Progression and high school Statistics and Prob-
ability Progression for further discussion and examples.) Then the
algebraic expressions can be used to interpolate (i.e., approximate
or predict function values between and among the collected data
values) and to extrapolate (i.e., to approximate or predict function
values beyond the collected data values). One must always ask
whether such approximations are reasonable in the context.

In school mathematics, functional relationships are often given
by algebraic expressions. For example, f pnq � n2 for n ¥ 1 gives
the nth square number. But in many modeling situations, such as
the temperature at Boston’s Logan Airport as a function of time,
algebraic expressions may not be suitable.
Functions and Algebra See the Algebra Progression for a discus-
sion of the connection and distinctions between functions, on the one
hand, and algebra and equation solving, on the other. Perhaps the
most productive connection is that solving equations can be seen as
finding the intersections of graphs of functions.A-REI.1

A-REI.11 Explain why the x-coordinates of the points where the
graphs of the equations y � fpxq and y � gpxq intersect are the
solutions of the equation fpxq � gpxq; find the solutions approx-
imately, e.g., using technology to graph the functions, make ta-
bles of values, or find successive approximations. Include cases
where fpxq and/or gpxq are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.K–7 foundations for functions Before they learn the term “func-

tion,” students begin to gain experience with functions in elementary

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grades. In Kindergarten, they use patterns with numbers such as
the 5� n pattern to learn particular additions and subtractions.

A trickle of pattern standards in Grades 4 and 5 continues the
preparation for functions.4.OA.5, 5.OA.3 Note that in both these stan-

4.OA.5Generate a number or shape pattern that follows a given
rule. Identify apparent features of the pattern that were not ex-
plicit in the rule itself.

5.OA.3Generate two numerical patterns using two given rules.
Identify apparent relationships between corresponding terms.
Form ordered pairs consisting of corresponding terms from the
two patterns, and graph the ordered pairs on a coordinate plane.

dards a rule is explicitly given. Traditional pattern activities, where
students are asked to continue a pattern through observation, are
not a mathematical topic, and do not appear in the Standards in
their own right.1

The Grade 4–5 pattern standards expand to the domain of Ratios
Experiences with functions before Grade 83Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Kindergarten Operations and Algebraic Thinking

fpnq � 5� n
Grade 3 Operations and Algebraic Thinking

1� 9 �

2� 9 � 2� p10� 1q � p2� 10q � p2� 1q � 20� 2 � 1

3� 9 � 3� p10� 1q � p3� 10q � p3� 1q � 30� 3 � 27,
fpnq � 9� n � 10� n� n

Grade 4 Geometric Measurement
feet 0 1 2 3

inches 0 12 24
fptq � 12t

Grade 6 Ratios and Proportional Relationships

d meters 3 6 9 12 15 3
2 1

2 4

t seconds 2 4 6 8 10 1

4
3

8
3fptq � 3

2 t

and Proportional Relationships in Grades 6–7. In Grade 6, as they
work with collections of equivalent ratios, students gain experience
with tables and graphs, and correspondences between them. They
attend to numerical regularities in table entries and corresponding
geometrical regularities in their graphical representations as plotted
points.MP.8 In Grade 7, students recognize and represent an impor-

MP.8 “Mathematically proficient students notice if calculations are
repeated, and look both for general methods and for shortcuts.”

tant type of regularity in these numerical tables—the multiplicative
relationship between each pair of values—by equations of the form
y � cx , identifying c as the constant of proportionality in equations
and other representations7.RP.2 (see the Ratios and Proportional Re-
lationships Progression).

The notion of a function is introduced in Grade 8. Linear functions
are a major focus, but note that students are also expected to give
examples of functions that are not linear.8.F.3 In high school, students
deepen their understanding of the notion of function, expanding their
repertoire to include quadratic and exponential functions, and in-
creasing their understanding of correspondences between geomet-
ric transformations of graphs of functions and algebraic transforma-
tions of the associated equations.F-BF.3 The trigonometric functions

F-BF.3 Identify the effect on the graph of replacing fpxq by fpxq�
k , kfpxq, fpkxq, and fpx�kq for specific values of k (both positive
and negative); find the value of k given the graphs. Experiment
with cases and illustrate an explanation of the effects on the graph
using technology.

are another important class of functions. In high school, students
study trigonometric ratios in right triangles.G-SRT.6 Understanding

G-SRT.6 Understand that by similarity, side ratios in right trian-
gles are properties of the angles in the triangle, leading to defini-
tions of trigonometric ratios for acute angles.

radian measure of an angle as arc length on the unit circle enables
students to build on their understanding of trigonometric ratios as-
sociated with acute angles, and to explain how these ratios extend
to trigonometric functions whose domains are included in the real
numbers.

The (+) standards for the conceptual categories of Geometry and
Functions detail further trigonometry addressed to students who
intend to take advanced mathematics courses such as calculus. This
includes the Law of Sines and Law of Cosines, as well as further
study of the values and properties of trigonometric functions.

1This does not exclude activities where patterns are used to support other stan-
dards, as long as the case can be made that they do so.

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Grade 8
Define, evaluate, and compare functions Since the elementary
grades, students have been describing patterns and expressing re-

8.F.1Understand that a function is a rule that assigns to each
input exactly one output. The graph of a function is the set of
ordered pairs consisting of an input and the corresponding out-
put.Function notation is not required in Grade 8.

8.F.2Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in tables, or
by verbal descriptions).

MP.1 “Mathematically proficient students can explain correspon-
dences between equations, verbal descriptions, tables, and
graphs or draw diagrams of important features and relationships,
graph data, and search for regularity or trends.”

lationships between quantities. These ideas become semi-formal in
Grade 8 with the introduction of the concept of function: a rule that
assigns to each input exactly one output.8.F.1 Formal language, such
as domain and range, and function notation may be postponed until
high school.

Building on their earlier experiences with graphs and tables in
Grades 6 and 7, students a routine of exploring functional relation-
ships algebraically, graphically, numerically in tables, and through
verbal descriptions.8.F.2 They explain correspondences between equa-
tions, verbal descriptions, tables, and graphs (MP.1). Repeated rea-
soning about entries in tables or points on graphs results in equa-
tions for functional relationships (MP.8). To develop flexibility in
interpreting and translating among these various representations,
students compare two functions represented in different ways, as
illustrated by “Battery Charging” in the margin.

Battery Charging 3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Sam wants to take his MP3 player and his video game player on
a car trip. An hour before they plan to leave, he realized that he
forgot to charge the batteries last night. At that point, he plugged
in both devices so they can charge as long as possible before
they leave.

Sam knows that his MP3 player has 40% of its battery life left
and that the battery charges by an additional 12 percentage
points every 15 minutes.

His video game player is new, so Sam doesn’t know how fast it is
charging but he recorded the battery charge for the first 30
minutes after he plugged it in.

time charging in minutes 0 10 20 30
percent player battery charged 20 32 44 5

1. If Sam’s family leaves as planned, what percent of the
battery will be charged for each of the two devices when
they leave?

2. How much time would Sam need to charge the battery
100% on both devices?

Task from Illustrative Mathematics. For solutions and discussion,
see illustrativemathematics.org/illustrations/641.

The main focus in Grade 8 is linear functions, those of the form
y � mx � b, where m and b are constants.8.F.3 Students learn to

8.F.3Interpret the equation y � mx�b as defining a linear func-
tion, whose graph is a straight line; give examples of functions
that are not linear.

recognize linearity in a table: when constant differences between
input values produce constant differences between output values.
And they can use the constant rate of change appropriately in a
verbal description of a context.

The proof that y � mx � b is also the equation of a line, and
hence that the graph of a linear function is a line, is an important
piece of reasoning connecting algebra with geometry in Grade 8.
See the Expressions and Equations Progression.
Connection to Algebra and Geometry In high school, after stu-
dents have become fluent with geometric transformations and have
worked with similarity, another connection between algebra and
geometry can be made in the context of linear functions.

The figure in the margin shows a “slope triangle” with one red
side formed by the vertical intercept and the point on the line with
x-coordinate equal to 1. The larger triangle is formed from the inter- Dilation of a “slope triangle”3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

b
p1, b�mq

px, b�mxq

m
x

cept and a point with arbitrary x-coordinate. A dilation with center
at the vertical intercept and scale factor x takes the slope triangle
to the larger triangle, because it takes lines to parallel lines.G-SRT.1a

G-SRT.1a A dilation takes a line not passing through the center
of the dilation to a parallel line, and leaves a line passing through
the center unchanged.

Thus the larger triangle is similar to the slope triangle,G-SRT.2 and

G-SRT.2 Given two figures, use the definition of similarity in
terms of similarity transformations to decide if they are similar;
explain using similarity transformations the meaning of similarity
for triangles as the equality of all corresponding pairs of angles
and the proportionality of all corresponding pairs of sides.

so the height of the larger triangle is mx , and the coordinates of the
general point on the triangle are px, b�mxq. Which is to say that
the point satisfies the equation y � b�mx .

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Use functions to model relationships between quantities When
using functions to model a linear relationship between quantities,
students learn to determine the rate of change of the function, which
is the slope of the line that is its graph. They can read (or compute
or approximate) the rate of change from a table or a graph, and they
can interpret the rate of change in context.8.F.4

8.F.4Construct a function to model a linear relationship between
two quantities. Determine the rate of change and initial value of
the function from a description of a relationship or from two px, yq
values, including reading these from a table or from a graph. In-
terpret the rate of change and initial value of a linear function in
terms of the situation it models, and in terms of its graph or a
table of values.

Graphs are ubiquitous in the study of functions, but it is impor-
tant to distinguish a function from its graph. For example, a linear
function does not have a slope but the graph of a non-vertical line
has a slope.• • The slope of a vertical line is undefined and the slope of a

horizontal line is 0. Either of these cases might be considered “no
slope.” Thus, the phrase “no slope” should be avoided because it
is ambiguous and “non-existent slope” and “slope of 0” should be
distinguished from each other.

Within the class of linear functions, students learn that some are
proportional relationships and some are not. Functions of the form
y � mx � b are proportional relationships exactly when b � 0,
so that y is proportional to x . Graphically, a linear function is a
proportional relationship if its graph goes through the origin.

To understand relationships between quantities, it is often help-
ful to describe the relationships qualitatively, paying attention to
the general shape of the graph without concern for specific numer-
ical values.8.F.5 The standard approach proceeds from left to right,

8.F.5Describe qualitatively the functional relationship between
two quantities by analyzing a graph (e.g., where the function is
increasing or decreasing, linear or nonlinear). Sketch a graph
that exhibits the qualitative features of a function that has been
described verbally.

describing what happens to the output as the input value increases.
For example, pianist Chris Donnelly describes the relationship be-
tween creativity and structure via a graph.

•

The qualitative description might be as follows: “As the input
value (structure) increases, the output (creativity) increases quickly
at first and gradually slowing down. As input (structure) continues to
increase, the output (creativity) reaches a maximum and then starts
decreasing, slowly at first, and gradually faster.” Thus, from the
graph alone, one can infer Donnelly’s point that there is an optimal
amount of structure that produces maximum creativity. With little
structure or with too much structure, in contrast, creativity is low.
Connection to Statistics and Probability In Grade 8, students plot
bivariate data in the coordinate plane (by hand or electronically)
and use linear functions to analyze the relationship between two
paired variables.8.SP.2 See the Grades 6–8 Statistics and Probability

8.SP.2Know that straight lines are widely used to model relation-
ships between two quantitative variables. For scatter plots that
suggest a linear association, informally fit a straight line, and in-
formally assess the model fit by judging the closeness of the data
points to the line.

Progression.
In high school, students take a deeper look at bivariate data,

making use of their expanded repertoire of functions in modeling
associations between two variables. See the sections on bivariate
data and interpreting linear models in the High School Statistics
and Probability Progression.

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High School
The high school standards on functions are organized into four groups:
Interpreting Functions (F-IF); Building Functions (F-BF); Linear, Qua-
dratic and Exponential Models (F-LE); and Trigonometric Functions
(F-TF). The organization of the first two groups under mathemati-
cal practices rather than types of function is an important aspect
of the Standards: students should develop ways of thinking that
are general and allow them to approach any type of function, work
with it, and understand how it behaves, rather than see each func-
tion as a completely different animal in the bestiary. For example,
they should see linear and exponential functions as arising out of
structurally similar growth principles; they should see quadratic,
polynomial, and rational functions as belonging to the same sys-
tem (helped along by the unified study in the Algebra category of
Arithmetic with Polynomials and Rational Expressions).
Interpreting Functions
Understand the concept of a function and use function notation
Building on semi-formal notions of functions from Grade 8, students
in high school begin to use formal notation and language for func-
tions. Now the input/output relationship is a correspondence be-
tween two sets: the domain and the range.F-IF.1 The domain is the

F-IF.1 Understand that a function from one set (called the do-
main) to another set (called the range) assigns to each element
of the domain exactly one element of the range. If f is a function
and x is an element of its domain, then fpxq denotes the output
of f corresponding to the input x. The graph of f is the graph of
the equation y � fpxq.

set of input values, and the range is the set of output values. A key
advantage of function notation is that the correspondence is built
into the notation. For example, f p5q is shorthand for “the output
value of f when the input value is 5.”

Students sometimes interpret the parentheses in function no-
tation as indicating multiplication. Because they might have seen

Interpreting the Graph3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Use the graph (for example, by marking specific points) to
illustrate the statements in (a)–(d). If possible, label the
coordinates of any points you draw.

(a) fp0q � 2

(b) fp�3q � fp3q � fp9q � 0

Task from Illustrative Mathematics. For solutions and discussion,
see at illustrativemathematics.org/illustrations/636.

numerical expressions like 3p4q, meaning 3 times 4, students can in-
terpret f pxq as f times x . This can lead to false generalizations of the
distributive property, such replacing f px� 3q with f pxq� f p3q. Work
with correspondences between values of the function represented in
function notation and their location on the graph of f can help stu-

MP.1 “Mathematically proficient students can explain correspon-
dences between equations, verbal descriptions, tables, and
graphs. . . .”

dents avoid this misinterpretation of the symbols (see “Interpreting
the Graph” in margin).

Although it is common to say “the function f pxq,” the notation
f pxq refers to a single output value when the input value is x . To
talk about the function as a whole, write f , or perhaps “the function f ,
where f pxq � 3x�4.” The x is merely a placeholder, so f ptq � 3t�4
describes exactly the same function.

Later, students can make interpretations like those in the follow-
ing table:

Expression Interpretation
fpa� 2q The output when the input is 2 greater than a
fpaq � 3 3 more than the output when the input is a
2fpxq � 5 5 more than twice the output of f when the input is x
fpbq � fpaq The change in output when the input changes from a to b

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Notice that a common preoccupation of high school mathematics, The square root function3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Since the equation x2 � 9 has two solutions, x � �3, students
might think incorrectly that

?
9 � �3. However, if we want

?x
to be a function of x, we need to choose one of these square
roots. The square root function, gpxq � ?x, is defined to be the
positive square root of x for any positive x.

distinguishing functions from relations, is not in the Standards. Time
normally spent on exercises involving the vertical line test, or search-
ing lists of ordered pairs to find two with the same x-coordinate
and different y-coordinate, can be reallocated elsewhere. Indeed,
the vertical line test is problematic, because it makes it difficult to
discuss questions such as “Is x a function of y?” (an important ques-
tion for students thinking about inverse functions) using a graph
in which x-coordinates are on the horizontal axis. The essential
question when investigating functions is: “Does each element of the
domain correspond to exactly one element in the range?” The mar-
gin shows a discussion of the square root function oriented around
this question.

To promote fluency with function notation, students interpret
function notation in contexts.F-IF.2 For example, if h is a function F-IF.2 Use function notation, evaluate functions for inputs in their

domains, and interpret statements that use function notation in
terms of a context.

MP.2 “Mathematically proficient students . . . [have] the ability to
contextualize, to pause as needed during the manipulation pro-
cess in order to probe into the referents for the symbols involved.”

Cell Phones3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Let fptq be the number of people, in millions, who own cell
phones t years after 1990. Explain the meaning of the following
statements.

(a) fp10q � 100.3
(b) fpaq �

(c) fp20q � b
(d) n � fptq
Task from Illustrative Mathematics. For solutions and discussion,
see illustrativemathematics.org/illustrations/634.

that relates Kristin’s height in inches to her age in years, then the
statement hp7q � 49 means, “When Kristin was 7 years old, she was
49 inches tall.” The value of hp12q is the answer to “How tall was
Kristin when she was 12 years old.” And the solution of hpxq � 60
is the answer to “How old was Kristin when she was 60 inches tall?”
See also “Cell Phones” in the margin.

Sometimes, especially in real-world contexts, there is no expres-
sion (or closed formula) for a function. In those cases, it is common to
use a graph or a table of values to (partially) represent the function.

A sequence is a function whose domain is a subset of the integers.F-IF.3

Sequences as functions3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

n fpnq
1 4
2 7
3

2 4

f(n)

F-IF.3 Recognize that sequences are functions, sometimes de-
fined recursively, whose domain is a subset of the integers.

In fact, many patterns explored in grades K–8 can be considered se-
quences. For example, the sequence 4, 7, 10, 13, 16, . . . might be de-
scribed as a “plus 3 pattern” because terms are computed by adding
3 to the previous term. To show how the sequence can be consid-
ered a function, we need an index that indicates which term of the
sequence we are talking about, and which serves as an input value
to the function. Deciding that the 4 corresponds to an index value of
1, we make a table showing the correspondence, as in the margin.
The sequence can be described recursively by the rule f p1q � 4,
f pn � 1q � f pnq � 3 for n ¥ 2. Notice that the recursive definition
requires both a starting value and a rule for computing subsequent
terms. The sequence can also be described with the closed formula
f pnq � 3n � 1, for integers n ¥ 1. Notice that the domain is in-
cluded as part of the description. A graph of the sequence consists
of discrete points, because the specification does not indicate what
happens “between the dots.”

In courses that address material corresponding to the plus stan-

dards, students may use subscript notation for sequences.+

Interpret functions that arise in applications in terms of the con-
text Functions are often described and understood in terms of their
behavior.F-IF.4 Over what input values is it increasing, decreasing, F-IF.4 For a function that models a relationship between two

quantities, interpret key features of graphs and tables in terms
of the quantities, and sketch graphs showing key features given
a verbal description of the relationship.

or constant? For what input values is the output value positive,

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negative, or 0? What happens to the output when the input value F-IF.5 Relate the domain of a function to its graph and, where
applicable, to the quantitative relationship it describes.gets very large in magnitude? Graphs become very useful represen-

tations for understanding and comparing functions because these
“behaviors” are often easy to see in the graphs of functions (see
“Warming and Cooling” in the margin). Graphs and contexts are op-

Warming and Cooling3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

The figure shows the graph of T , the temperature (in degrees
Fahrenheit) over one particular 20-hour period in Santa Elena as
a function of time t.

(a) Estimate T p14q.
(b) If t � 0 corresponds to midnight, interpret what we mean by

T p14q in words.

(d) When was the temperature decreasing?

(e) If Anya wants to go for a two-hour hike and return before the
temperature gets over 80 degrees, when should she leave?

Task from Illustrative Mathematics. For solutions and discussion,
see illustrativemathematics.org/illustrations/639.

portunities to talk about the notion of the domain of a function (for an
illustration, go to illustrativemathematics.org/illustrations/631).F-IF.5

Graphs help us reason about rates of change. Students learned
in Grade 8 that the rate of change of a linear function is equal to the
slope of the line that is its graph.8.EE.5 And because the slope of a line

8.EE.5Graph proportional relationships, interpreting the unit rate
as the slope of the graph. Compare two different proportional
relationships represented in different ways.

is constant, that is, between any two points it is the same8.EE.6 (see

8.EE.6Use similar triangles to explain why the slope m is the
same between any two distinct points on a non-vertical line in the
coordinate plane; derive the equation y � mx for a line through
the origin and the equation y � mx�b for a line intercepting the
vertical axis at b.

the Expressions and Equations Progression), “the rate of change”
has an unambiguous meaning for a linear function. For nonlinear
functions, however, rates of change are not constant, and so we talk
about average rates of change over an interval.F-IF.6

F-IF.6 Calculate and interpret the average rate of change of a
function (presented symbolically or as a table) over a specified
interval. Estimate the rate of change from a graph.

For example, for the function gpxq � x2, the average rate of
change from x � 2 to x � 5 is

gp5q � gp2q
5� 2

� 25� 4

5� 2
�

3
� 7.

This is the slope of the line from p2, 4q to p5, 25q on the graph of g.
And if g is interpreted as returning the area of a square of side x ,
then this calculation means that over this interval the area changes,
on average, 7 square units for each unit increase in the side length
of the square.

F-IF.7 Graph functions expressed symbolically and show key fea-
tures of the graph, by hand in simple cases and using technology
for more complicated cases.

a Graph linear and quadratic functions and show intercepts,
maxima, and minima.

b Graph square root, cube root, and piecewise-defined
functions, including step functions and absolute value
functions.

c Graph polynomial functions, identifying zeros when suit-
able factorizations are available, and showing end behav-
ior.

d (+) Graph rational functions, identifying zeros and asymp-
totes when suitable factorizations are available, and
showing end behavior.

e Graph exponential and logarithmic functions, showing in-
tercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.

Analyze functions using different representations Functions are
often studied and understood as families, and students should spend
time studying functions within a family, varying parameters to de-
velop an understanding of how the parameters affect the graph of a
function and its key features.F-IF.7

Within a family, the functions often have commonalities in the
shapes of their graphs and in the kinds of features that are impor-
tant for identifying functions more precisely within a family. This
standard indicates which function families should be in students’
repertoires, detailing which features are required for several key
families. It is an overarching standard that covers the entire range
of a student’s high school experience; in this part of the progression
we merely indicate some guidelines for how it should be treated.

First, linear and exponential functions (and to a lesser extent
quadratic functions) receive extensive treatment and comparison in
a dedicated group of standards, Linear and Exponential Models.
Thus, those function families should receive the bulk of the atten-
tion related to this standard. Second, all students are expected to
develop fluency with linear, quadratic, and exponential functions, in-
cluding the ability to graph them by hand. Finally, in most of the
other function families, students are expected to graph simple cases
without technology, and more complex ones with technology.

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Consistent with the practice of looking for and making use of
structure (MP.7), students should also develop the practice of writing
expressions for functions in ways that reveal the key features of the
function.F-IF.8 F-IF.8 Write a function defined by an expression in different but

equivalent forms to reveal and explain different properties of the
function.

a Use the process of factoring and completing the square in
a quadratic function to show zeros, extreme values, and
symmetry of the graph, and interpret these in terms of a
context.

b Use the properties of exponents to interpret expressions
for exponential functions.

Quadratic functions provide a rich playground for developing this
ability, since the three principal forms for a quadratic expression
(expanded, factored, and completed square) each give insight into
different aspects of the function. However, there is a danger that

Which Expression? 3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

Which of the following could be an expression for the function
whose graph is shown below? Explain.

(a) px � 12q2 � 4 (b) �px � 2q2 � 1
(c) px � 18q2 � 40 (d) px � 10q2 �

(e) �4px � 2qpx � 3q (f) px � 4qpx � 6q
(g) px � 12qp�x � 18q (h) p20� xqp30� xq

Task from Illustrative Mathematics. For solutions and discussion,
see illustrativemathematics.org/illustrations/640.

working with these different forms becomes an exercise in picking
numbers out of an expression. For example, students often arrive
at college talking about “minus b over 2a method” for finding the
vertex of the graph of a quadratic function. To avoid this problem it
is useful to give students tasks such as “Which Expression?” in the
margin, where they must read both the graphs and the expression
and choose for themselves which parts of each correspond.F-IF.9

F-IF.9 Compare properties of two functions each represented in
a different way (algebraically, graphically, numerically in tables, or
by verbal descriptions).

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Building Functions
The previous group of standards focuses on interpreting functions
given by expressions, graphs, or tables. The Building Functions
group focuses on building functions to model relationships, and
building new functions from existing functions. A-CED.2 Create equations in two or more variables to represent

relationships between quantities; graph equations on coordinate
axes with labels and scales.Note: Inverse of a function and composition of a function with

its inverse are among the plus standards. The following discussion
describes in detail what is required for students to grasp these se-
curely. Because of the subtleties and pitfalls involved, it is strongly
recommended that these topics be included only in optional courses. F-BF.1a Write a function that describes a relationship between

two quantities.

a Determine an explicit expression, a recursive process, or

steps for calculation from a context.Build a function that models a relationship between two quanti-
ties This cluster of standards is very closely related to the algebra
standard on writing equations in two variables.A-CED.2 Indeed, that
algebra standard might well be met by a curriculum in the same
unit as this cluster. Although students will eventually study vari-
ous families of functions, it is useful for them to have experiences of
building functions from scratch, without the aid of a host of special
recipes, by grappling with a concrete context for clues.F-BF.1a For

Lake Algae3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

On June 1, a fast growing species of algae is accidentally
introduced into a lake in a city park. It starts to grow and cover
the surface of the lake in such a way that the area covered by
the algae doubles every day. If it continues to grow unabated,
the lake will be totally covered and the fish in the lake will
suffocate. At the rate it is growing, this will happen on June 30.

(a) When will the lake be covered half-way?

(b) On June 26, a pedestrian who walks by the lake every day
warns that the lake will be completely covered soon. Her
friend just laughs. Why might her friend be skeptical of the
warning?

(c) On June 29, a clean-up crew arrives at the lake and
removes almost all of the algae. When they are done, only
1% of the surface is covered with algae. How well does this
solve the problem of the algae in the lake?

(d) Write an equation that represents the percentage of the
surface area of the lake that is covered in algae as a function
of time (in days) that passes since the algae was introduced
into the lake if the cleanup crew does not come on June 29.

Task from Illustrative Mathematics. For solutions and discussion,
see illustrativemathematics.org/illustrations/533.

example, in “Lake Algae” in the margin, a solution for part (a) might
involve noting that if the lake is completely covered with algae on
June 30, then half of its surface will be covered on June 29 because
the area covered doubles each day. This might be expressed in a
table:

date 29 30
percent covered 50 100

Finding a solution for part (b) might start from the table above.
Repeatedly using the information that the algae doubles each day:
one divides the amount for June 29 by 2, then divides the amount
for June 28 by 2, then divides the amount for June 27 by 2. This
repeated reasoning (MP.8) might be suggested by the table:

date 26 27 28 29 30
percent covered 1

16 � 100 1
8 � 100 1

4 � 100 1
2 � 100 1 � 100

Some students might express the action of repeatedly dividing by
2 by writing the table entries for surface area as a product of 100
and a power of 1

2 or 2, making use of structure (MP.7) by using an
exponential expression. Or they might express this action with a
recursively defined function, e.g., if t is a number between 2 and
30, and f ptq gives the amount of surface covered on June t , then
f pt � 1q � 1

2 f ptq.The Algebra Progression discusses the difference between a
function and an expression. Not all functions are given by expres-

Drug Dosage3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

A student strained her knee in an intramural volleyball game,
and her doctor has prescribed an anti-inflammatory drug to
reduce the swelling. She is to take two 220-milligram tablets
every 8 hours for 10 days. Her kidneys filter 60% of this drug
from her body every 8 hours. How much of the drug is in her
system after 24 hours?

Task from High School Mathematics at Work: Essays and
Examples for the Education of All Students, 1998, National
Academies Press. For discussion of the task, see
http://www.nap.edu/openbook/0309063531/html/80.html.

sions, and in many situations it is natural to use a function defined
recursively. Calculating mortgage payment and drug dosages are
typical cases where recursively defined functions are useful (see
“Drug Dosage” in the margin).

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Modeling contexts also provide a natural place for students to
start building functions with simpler functions as components.F-BF.1bc F-BF.1 Write a function that describes a relationship between

two quantities.

b Combine standard function types using arithmetic opera-
tions.

c (+) Compose functions.

Situations of cooling or heating involve functions which approach a
limiting value according to a decaying exponential function. Thus,
if the ambient room temperature is 700 Fahrenheit and a cup of tea
is made with boiling water at a temperature of 2120 Fahrenheit, a
student can express the function describing the temperature as a
function of time using the constant function f ptq � 70 to represent
the ambient room temperature and the exponentially decaying func-
tion gptq � 142e�kt to represent the decaying difference between
the temperature of the tea and the temperature of the room, leading
to a function of the form F-BF.2 Write arithmetic and geometric sequences both recur-

sively and with an explicit formula, use them to model situations,
and translate between the two forms.T ptq � 70� 142e�kt .

Students might determine the constant k experimentally.
In contexts where change occurs at discrete intervals (such as

payments of interest on a bank balance) or where the input vari-
able is a whole number (for example the number of a pattern in a
sequence of patterns), the functions chosen will be sequences. In
preparation for the deeper study of linear and exponential functions,
students can study arithmetic sequences (which are linear functions)
and geometric sequences (which are exponential functions).F-BF.2
This is a good point at which to start making the distinction be-
tween additive and multiplicative changes.
Build new functions from existing functions With a basis of ex-
periences in building specific functions from scratch, students start

Transforming Functions3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

The figure shows the graph of a function f whose domain is the
interval �2 ¤ x ¤ 2.

(a) In (i)–(iii), sketch the graph of the given function and
compare with the graph of f . Explain what you see.

(i) gpxq � fpxq � 2

(ii) hpxq � �fpxq
(iii) ppxq � fpx � 2q

(b) The points labelled Q,O,P on the graph of f have
coordinates

Q � p�2� 0.509q, O � p0,�0.4q, P � p2, 1.309q.
What are the coordinates of the points corresponding to
P,O,Q on the graphs of g, h, and p?

Task from Illustrative Mathematics. For solutions and discussion,
see illustrativemathematics.org/illustrations/742.

to develop a notion of naturally occurring families of functions that
deserve particular attention. It is possible to harden the curricu-
lum too soon around these families, before students have enough
experience to get a feel for the effects of different parameters. Stu-
dents can start getting that feel by playing around with the effect
on the graph of simple algebraic transformations of the input and
output variables.F-BF.3 Quadratic and absolute value functions are

good contexts for getting a sense of the effects of many of these
transformations, but eventually students need to understand these
ideas abstractly and be able to talk about them for any function f .

Students may find the effect of adding a constant to the input
variable to be counterintuitive, because the effect on the graph ap-
pears to be the opposite to the transformation on the variable, e.g.,
the graph of y � f px � 2q is a horizontal translation of the graph
of y � f pxq �2 units along the x-axis rather than in the opposite
direction. In part (b) of “Transforming Functions” in the margin, ask-
ing students to talk through the positions of the points in terms of
function values can help.•

• The graphs of linear functions are especially complicated with
respect to adding a constant to the input variable because its
effect can be seen as one of many different translations. For
example, the graph of y � 2px � 3q can be seen as a horizontal
translation of the graph of y � 2x. But, thinking of it as y � 2x�6
it can also be seen as a vertical translation that moves the graph 6
units. And, it can also be seen as a translation in other directions,
e.g., as suggested by y � 2px � 3� cq � 2c.

The concepts of even and odd functions are useful for noticing
symmetry. A function f is called an even function if f p�xq � f pxq
for all x in its domain and an odd function if f p�xq � �f pxq for
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all x in its domain. To understand the names of these concepts,
consider that polynomial functions are even exactly when all terms
are of even degree and odd exactly when all terms are of odd degree.
With some grounding in polynomial functions, students can reason
that lots of functions are neither even nor odd.

Students can show from the definitions that the sum of two even
functions is even and the sum of two odd functions is odd, and they
can interpret these results graphically.

An interesting fact

Suppose f is a function with a domain of all real numbers.
Define g and h as follows:

gpxq � fpxq � fp�xq
2

and hpxq � fpxq � fp�xq
2

Then fpxq � gpxq � hpxq, g is even, and h is odd. (Students
may use the definitions to verify these claims.) Thus, any
function defined on the real numbers can be expressed as the
sum of an even and an odd function.

When it comes to inverse functions,F-BF.4a the expectations are

F-BF.4a Find inverse functions.
a Solve an equation of the form fpxq � c for a simple func-

tion f that has an inverse and write an expression for the
inverse.

modest, requiring only that students solve equations of the form
f pxq � c. The point is to provide an informal sense of determining
the input when the output is known. Much of this work can be done
with specific values of c. Eventually, some generality is warranted.
For example, if f pxq � 2×3, then solving f pxq � c leads to x �
pc{2q1{3, which is the general formula for finding an input from a
specific output, c, for this function, f .

At this point, students need neither the notation nor the formal
language of inverse functions, but only the idea of “going backwards”
from output to input. This can be interpreted for a table and graph
of the function under examination. Correspondences between equa-
tions giving specific values of the functions, table entries, and points
on the graph can be noted (MP.1). And although not required in the
standard, it is reasonable to include, for comparison, a few examples
where the input cannot be uniquely determined from the output. For
example, if gpxq � x2, then gpxq � 5 has two solutions, x � �?5.

For some advanced mathematics courses, students will need a+
formal sense of inverse functions, which requires careful develop-+
ment. For example, as students begin formal study, they can easily+
believe that “inverse functions” are a new family of functions, sim-+
ilar to linear functions and exponential functions. To help students

A joke

Teacher: Are these two functions inverses?
Student: Um, the first one is and the second one isn’t.

What does this student misunderstand about inverse functions?

+
develop the instinct that “inverse” is a relationship between two+
functions, the recurring questions should be “What is the inverse of+
this function?” and “Does this function have an inverse?” The fo-+
cus should be on “inverses of functions” rather than a new type of+
function.+

Discussions of the language and notation for inverse functions+
can help to provide students a sense of what the adjective “inverse”+
means and mention that a function which has an inverse is known+
as an “invertible function.”+

The function Ipxq � x is sometimes called the identity function+
because it assigns each number to itself. It behaves with respect+
to composition of functions the way the multiplicative identity, 1,+
behaves with multiplication of real numbers and the way that the+
identity matrix behaves with matrix multiplication. If f is any func-+
tion (defined on the real numbers), this analogy can be expressed+
symbolically as f � I � f � I � f , and it can be verified as follows:+

f � Ipxq � f pIpxqq � f pxq and I � f pxq � Ipf pxqq � f pxq
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Suppose f denotes a function with an inverse whose domain is+
the real numbers and a is a nonzero real number (which thus has a+
multiplicative inverse), and B is an invertible matrix. The following+
table compares the concept of inverse function with the concepts of+
multiplicative inverse and inverse matrix:+

Equation Interpretation
f�1 � f � I � f � f�1 The composition of f�1 with f is the identity function
a�1 � a � 1 � a � a�1 The product of a�1 and a is the multiplicative identity
B�1 � B � I � B � B�1 The product of B�1 and B is the identity matrix

In other words, where a�1 means the inverse of a with respect+
to multiplication, f�1 means the inverse of f with respect to func-+
tion composition. Thus, when students interpret the notation f�1pxq+
incorrectly to mean 1{f pxq, the guidance they need is that the mean-+
ing of the “exponent” in f�1 is about function composition, not about+
multiplication.

A note on notation3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

In the expression sin2 x, the superscript denotes exponentiation.
In sin�1 x, the superscript denotes inverse with respect to
composition of functions rather than with respect to
multiplication. Despite the similar look, these superscripts act in
different ways. The 2 acts as an exponent but the �1 does not.
Both notations, however, allow the expression to be written
without the parentheses that would be needed otherwise.

Another convention that allows parentheses to be omitted is the
use of sinax rather than sinpaxq. Thus, some expressions built
from trigonometric functions may written in ways that look quite
different to students, but differ only in the use or omission of
parentheses.

+
Students do not need to develop the abstract sense of identity+

and inverse detailed in this table. Nonetheless, these perspectives+
can inform the language and conversation in the classroom as stu-+
dents verify by composition (in both directions) that given functions+
are inverses of each other.F-BF.4b. Furthermore, students can con-

F-BF.4b(+) Verify by composition that one function is the inverse
of another.

+
tinue to refine their informal “going backwards” notions, as they con-+
sider inverses of functions given by graphs or tables.F-BF.4c In this

F-BF.4c(+) Read values of an inverse function from a graph or a
table, given that the function has an inverse.

+
work, students can gain a sense that “going backwards” interchanges+
the input and output and therefore the stereotypical roles of the let-+
ters x and y. And they can reason why the graph of y � f�1pxq will+
be the reflection across the line y � x of the graph of y � f pxq.+

Suppose gpxq � px � 3q2. From the graph, it can be seen that+
gpxq � c will have two solutions for any c ¡ 0. (This draws on+
the understanding that solutions of gpxq � c are x-coordinates of+
points that lie on both the graphs of g and y � c.) Thus, to create an+
invertible function,F-BF.4d we must restrict the domain of g so that F-BF.4d(+) Produce an invertible function from a non-invertible

function by restricting the domain.
+

every range value corresponds to exactly one domain value. One+
possibility is to restrict the domain of g to x ¥ 3, as illustrated by+
the solid purple curve in the graph on the left.•

•

g x( ) = x 3( )2 10

h x( ) = 3 + x
g x( ) = x 3( )2

+
When solving px � 3q2 � c, we get x � 3� ?c, illustrating that+

positive values of c will yield two solutions x for the unrestricted+
function. With the restriction, 3� ?c is not in the domain. Thus, x �+
3� ?c, which corresponds to choosing the solid curve and ignoring+
the dotted portion. The inverse function, then, is hpcq � 3� ?c, for+
c ¥ 0.+

We check that h is the inverse of (restricted) g as follows:
gphpxqq � g �3� ?x� � �p3� ?xq � 3

�2 � p?xq2 � x, x ¥ 0

hpgpxqq � h �px � 3q2� � 3�
a
px � 3q2 � 3�px�3q � x, x ¥ 3.

The first verification requires that x ¥ 0 so that x is in the domain of+
h. The second verification requires that x ¥ 3 so that x is in the do-+
main of (restricted) g. This allows apx � 3q2 to be written without+

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the square root symbol as px � 3q.• The rightmost graph shows the • In general,
a
px � 3q2 � |x � 3|. If x were restricted to the

dotted portion of the graph (i.e., x ¤ 3), the corresponding ex-
pression could have been written as �px � 3q or 3� x.

+
graph of h. Students can draw on their work with transformations+
in Grades 7 and 8,8.G.3 possibly augmented by plotting points such 8.G.3Describe the effect of dilations, translations, rotations, and

reflections on two-dimensional figures using coordinates.

+
as (0,3) and (3,0), to perceive the graph of h as the reflection of the+
graph of g across the line y � x .+

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Linear and Exponential Models
Construct and compare linear and exponential models and solve
problems Distinguishing between situations that can be modeled
with linear functions and with exponential functionsF-LE.1a turns on

F-LE.1a Prove that linear functions grow by equal differences
over equal intervals, and that exponential functions grow by equal
factors over equal intervals.understanding their rates of growth and looking for indications of

these types of growth rates (MP.7). One indicator of these growth
rates is differences over equal intervals, given, for example, in a
table of values drawn from the situation—with the understanding
that such a table may only approximate the situation (MP.4).

To prove that a linear function grows by equal differences over
equal intervals,F-LE.1b students draw on the understanding devel- F-LE.1b Recognize situations in which one quantity changes at

a constant rate per unit interval relative to another.oped in Grade 8 that the ratio of the rise and run for any two dis-
tinct points on a line is the same (see the Expressions and Equations
Progression) and recast it in terms of function inputs and outputs.
An interval can be seen as determining two points on the line whose
inputs (x-coordinates) occur at the boundaries of the intervals. The
equal intervals can be seen as the runs for two pairs of points. Be-
cause these runs have equal length and the ratio of rise to run is
the same for any pair of distinct points, the differences of the cor-
responding outputs (the rises) are the same. These differences are
the growth of the function over each interval.

In the process of this proof, students note the correspondence
between rise and run on a graph and symbolic expressions for dif-
ferences of inputs and outputs (MP.1). Using such expressions has
the advantage that the analogous proof showing that exponential
functions grow by equal factors over equal intervals begins in an
analogous way with expressions for differences of inputs and out-
puts.

The process of going from linear or exponential functions to ta-
bles can go in the opposite direction. Given sufficient information,
e.g., a table of values together with information about the type of
relationship represented,F-LE.4 students construct the appropriate

F-LE.4 For exponential models, express as a logarithm the solu-
tion to abct � d where a, c, and d are numbers and the base b
is 2, 10, or e; evaluate the logarithm using technology.function. For example, students might be given the information that

the table below shows inputs and outputs of an exponential function,
and asked to write an expression for the function.

Input Output
0 5
8 33

For most students, the logarithm of x is merely shorthand for a
number that is the solution of an exponential equation in x .F-LE.4 F-LE.4 For exponential models, express as a logarithm the solu-

tion to abct � d where a, c, and d are numbers and the base b
is 2, 10, or e; evaluate the logarithm using technology.Students in advanced mathematics courses such as calculus,+

however, need to understand logarithms as functions—and as in-+
verses of exponential functions.F-BF.5 They should be able to explain F-BF.5(+) Understand the inverse relationship between expo-

nents and logarithms and use this relationship to solve problems
involving logarithms and exponents.

+
identities such as logbpbxq � x and blogb x � x as well as the laws+
of logarithms, such as logpabq � loga� log b. In doing so, students+
can think of the logarithms as unknown exponents in expressions+
with base 10 (e.g. loga answers the question “Ten to the what+

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equals a?”) and use the properties of exponents,N-RN.1 building on+
the understanding of exponents that began in Grade 8.8.EE.1

N-RN.1 Explain how the definition of the meaning of rational ex-
ponents follows from extending the properties of integer expo-
nents to those values, allowing for a notation for radicals in terms
of rational exponents.

8.EE.1Know and apply the properties of integer exponents to
generate equivalent numerical expressions.

Interpret expressions for functions in terms of the situation they
model Students may build a function to model a situation, using
parameters from that situation. In these cases, interpreting expres-
sions for a linear or exponential function in terms of the situation
it modelsF-LE.5 is often just a matter of remembering how the func- F-LE.5 Interpret the parameters in a linear or exponential func-

tion in terms of a context.tion was constructed. However, interpreting expressions may be
less straightforward for students when they are given an algebraic
expression for a function and a description of what the function is
intended to model.

For example, in doing the task “Illegal Fish” in the margin, stu-
dents may need to rely on their understanding of a function as de-
termining an output for a given input to answer the question “Find
b if you know the lake contains 33 fish after eight weeks.”

Illegal Fish3Grade3ThemeaningoffractionsInGrades1and2,studentsusefractionlanguagetodescribepartitionsofshapesintoequalshares.2.G.3In2.G.3Partitioncirclesandrectanglesintotwo,three,orfourequalshares,describethesharesusingthewordshalves,thirds,halfof,athirdof,etc.,anddescribethewholeastwohalves,threethirds,fourfourths.Recognizethatequalsharesofidenticalwholesneednothavethesameshape.Grade3theystarttodeveloptheideaofafractionmoreformally,buildingontheideaofpartitioningawholeintoequalparts.Thewholecanbeacollectionofobjects,ashapesuchasacircleorrect-angle,alinesegment,oranyfiniteentitysusceptibletosubdivisionandmeasurement.Thewholeasacollectionofobjects!Ifthewholeisacollectionof4bunnies,thenonebunnyis14ofthewholeand3bunniesis34ofthewhole.Grade3studentsstartwtihunitfractions(fractionswithnumer-ator1).Theseareformedbydividingawholeintoequalpartsandtakingonepart,e.g.,ifawholeisdividedinto4equalpartstheneachpartis14ofthewhole,and4copiesofthatpartmakethewhole.Next,studentsbuildfractionsfromunitfractions,seeingthenumer-ator3of34assayingthat34iswhatyougetbyputting3ofthe14’stogether.3.NF.1Anyfractioncanbereadthisway,andinparticular3.NF.1Understandafraction1�asthequantityformedby1partwhenawholeispartitionedinto�equalparts;understandafrac-tion��asthequantityformedby�partsofsize1�.thereisnoneedtointroducetheconceptsof“properfraction”and“improperfraction”initially;53iswhatonegetsbycombining5partstogetherwhenthewholeisdividedinto3equalparts.Twoimportantaspectsoffractionsprovideopportunitiesforthemathematicalpracticeofattendingtoprecision(MP6):•Specifyingthewhole.TheimportanceofspecifyingthewholeWithoutspecifyingthewholeitisnotreasonabletoaskwhatfractionisrepresentedbytheshadedarea.Iftheleftsquareisthewhole,itrepresentsthefraction32;iftheentirerectangleisthewhole,itrepresents34.•Explainingwhatismeantby“equalparts.”Initially,studentscanuseanintuitivenotionofcongruence(“samesizeandsameshape”)toexplainwhythepartsareequal,e.g.,whentheydivideasquareintofourequalsquaresorfourequalrectangles.Arearepresentationsof14Ineachrepresentationthesquareisthewhole.Thetwosquaresontheleftaredividedintofourpartsthathavethesamesizeandshape,andsothesamearea.Inthethreesquaresontheright,theshadedareais14ofthewholearea,eventhoughitisnoteasilyseenasonepartoutofadivisionintofourpartsofthesameshapeandsize.Studentscometounderstandamoreprecisemeaningfor“equalparts”as“partswithequalmeasurement.”Forexample,whenarulerisdividedintohalvesorquartersofaninch,theyseethateachsubdivisionhasthesamelength.Inareamodelstheyreasonabouttheareaofashadedregiontodecidewhatfractionofthewholeitrepresents(MP3).Thegoalisforstudentstoseeunitfractionsasthebasicbuildingblocksoffractions,inthesamesensethatthenumber1isthebasicbuildingblockofthewholenumbers;justaseverywholenumberisobtainedbycombiningasufficientnumberof1s,everyfractionisobtainedbycombiningasufficientnumberofunitfractions.ThenumberlineOnthenumberline,thewholeistheunitinterval,thatis,theintervalfrom0to1,measuredbylength.Iteratingthiswholetotherightmarksoffthewholenumbers,sothattheintervalsbetweenconsecutivewholenumbers,from0to1,1to2,2to3,etc.,areallofthesamelength,asshown.Studentsmightthinkofthenumberlineasaninfiniteruler.Thenumberline0123456etc.Toconstructaunitfractiononthenumberline,e.g.13,studentsdividetheunitintervalinto3intervalsofequallengthandrecognizethateachhaslength13.Theylocatethenumber13onthenumberDraft,5/29/2011,commentatcommoncoretools.wordpress.com.

A fisherman illegally introduces some fish into a lake, and they
quickly propagate. The growth of the population of this new
species (within a period of a few years) is modeled by
Ppxq � 5bx , where x is the time in weeks following the
introduction and b is a positive unknown base.

(a) Exactly how many fish did the fisherman release into the
lake?

(b) Find b if you know the lake contains 33 fish after eight
weeks. Show step-by-step work.

(c) Instead, now suppose that Ppxq � 5bx and b � 2. What is
the weekly percent growth rate in this case? What does this
mean in every-day language?

Task from Illustrative Mathematics. For solutions and discussion,
see illustrativemathematics.org/illustrations/579.

See the linear and exponential model section of the Modeling
Progression for an example of an interpretation of the intersection
of a linear and an exponential function in terms of the situation that
is being modeled.

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Trigonometric Functions
Students begin their study of trigonometry with right triangles.G-SRT.6 G-SRT.6 Understand that by similarity, side ratios in right trian-

gles are properties of the angles in the triangle, leading to defini-
tions of trigonometric ratios for acute angles.Right triangle trigonometry is concerned with ratios of sides of right

triangles, allowing functions of angle measures to be defined in
terms of these ratios.• This limits the angles considered to those • Traditionally, trigonometry concerns “ratios.” Note, however,

that according to the usage of the Ratio and Proportional Rela-
tionships Progression, that these would be called the “value of the
ratio.” In high school, students’ understanding of ratio may now be
sophisticated enough to allow the traditional “ratio” to be used for
“value of the ratio” in the traditional manner. Likewise, angles are
carefully distinguished from their measurements when students
are learning about measuring angles in Grades 4 and 5. In high
school, students’ understanding of angle measure may now allow
angles to be referred to by their measures.

between 0� and 90�. This section briefly outlines some considera-
tions involved in extending the domains of the trigonometric functions
within the real numbers.

Traditionally, trigonometry includes six functions (sine, cosine,
tangent, cotangent, secant, cosecant). Because the second three
may be expressed as reciprocals of the first three, this progression
discusses only the first three.
Extend the domain of trigonometric functions using the unit circle
After study of trigonometric ratios in right triangles, students expand
the types of angles considered. Students learn, by similarity, that
the radian measure of an angle can be defined as the quotient of arc
length to radius.G-C.5 As a quotient of two lengths, therefore, radian

G-C.5 Derive using similarity the fact that the length of the arc
intercepted by an angle is proportional to the radius, and define
the radian measure of the angle as the constant of proportionality;
derive the formula for the area of a sector.measure is “dimensionless.” That is why the “unit” is often omitted

when measuring angles in radians.
In calculus, the benefits of radian measure become plentiful, lead-

ing, for example, to simple formulas for derivatives and integrals of
trigonometric functions. Before calculus, there are two key benefits
of using radians rather than degrees: G-SRT.5 Use congruence and similarity criteria for triangles to

solve problems and to prove relationships in geometric figures.

G-SRT.9(+) Derive the formula A � 1{2ab sinpCq for the area of
a triangle by drawing an auxiliary line from a vertex perpendicular
to the opposite side.

F-TF.1 Understand radian measure of an angle as the length of
the arc on the unit circle subtended by the angle.

• arclength is simply rθ, and
• sinθ � θ for small θ.
Steps to extending the domain of trigonometric functions and

introduction of radian measurement may include:
• Extending consideration of trigonometric ratios from right tri-

angles to obtuse triangles. This may occur in the context of
solving problems about geometric figures.G-SRT.5G-SRT.9 See
the Geometry Progression.
• Associating the degree measure of an angle with the length of

the arc it subtends on the unit circle,F-TF.1 as described below.
With the help of a diagram, students mark the intended angle, •

θ, measured counterclockwise from the positive ray of the x-axis.•

• Note that this convention for measurement is consistent with
conventions for measuring angles with protractors that students
learned in Grade 4. The protractor is placed so that the initial
side of the angle lies on the 0�-mark. For the angles of positive
measure (such as the angles considered in Grade 4), the termi-
nal side of the angle is located by a clockwise rotation. See the
Geometric Measurement Progression.

They identify the coordinates x and y; draw a reference triangle;
and then use their knowledge of right triangle trigonometry. In
particular, sinθ � y{1 � y, cosθ � x{1 � x , and tanθ � y{x .
(Note the simplicity afforded by using a circle of radius 1.) This way,
students can compute values of any of the trigonometric functions,
being careful to note the signs of x and y. In the figure as drawn
in the second quadrant, for example, x is negative and y is positive,

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which implies that sinθ is positive and cosθ and tanθ are both
negative.

The next step is sometimes called “unwrapping the unit circle.”
On a fresh set of axes, the angle θ is plotted along the horizontal axis
and one of the trigonometric functions is plotted along the vertical
axis. Dynamic presentations with shadows can help considerably,
and the point should be that students notice the periodicity of the
functions, caused by the repeated rotation about the origin, regularly
reflecting on the grounding in right triangle trigonometry.

With the help of the special right triangles, 30�-60�-90� and 45�-+
45�-90�, for which the quotients of sides can be computed using the+
Pythagorean Theorem,8.G.7 the values of the trigonometric functions

8.G.7Apply the Pythagorean Theorem to determine unknown
side lengths in right triangles in real-world and mathematical
problems in two and three dimensions.

+
can be computed for the angles π{3, π{4, and π{6 as well as their+
multiples.F-TF.3 For advanced mathematics, students need to develop F-TF.3(+) Use special triangles to determine geometrically the

values of sine, cosine, tangent for π{3, π{4 and π{6, and use the
unit circle to express the values of sine, cosines, and tangent for
π � x, π � x, and 2π � x in terms of their values for x, where x
is any real number.

+
fluency with the trigonometric functions of these special angles to+
support fluency with the “unwrapping of the unit circle” to create+
and graph the trigonometric functions.+

Building on their understanding of geometric transformations,G-CO.7
G-CO.7 Use the definition of congruence in terms of rigid mo-
tions to show that two triangles are congruent if and only if cor-
responding pairs of sides and corresponding pairs of angles are
congruent.

•

0.5

1 1

1
y

-y

θ
θ

(x, -y)

(x, y)

+
either directly or via the side-angle-side congruence criterion,G-CO.8

G-CO.8 Explain how the criteria for triangle congruence (ASA,
SAS, and SSS) follow from the definition of congruence in terms
of rigid motions.

+
students see that, compared to the reference triangle with angle θ,+
an angle of �θ will produce a congruent reference triangle that is its+
reflection across the x-axis. They can then reason that sinp�θq �+
�y � � sinpθq, so sine is an odd function. Similarly, cosp�θq � x �+
cospθq, so cosine is an even function.F-TF.4 Some additional work is

F-TF.4(+) Use the unit circle to explain symmetry (odd and even)
and periodicity of trigonometric functions.

+
required to verify that these relationships hold for values of θ outside+
the first quadrant.+

The same sorts of pictures can be used to argue that the trigono-+
metric functions are periodic. For example, for any integer n, sinpθ�+
2nπq � sinpθq because angles that differ by a multiple of 2π have+
the same terminal side and thus the same coordinates x and y.+

Model periodic phenomena with trigonometric functions Now that
students are equipped with trigonometric functions, they can model
some periodic phenomena that occur in the real world. For stu-
dents who do not continue into advanced mathematics, this is the
culmination of their study of trigonometric functions.

The tangent function is not often useful for modeling periodic
phenomena because tan x is undefined for x �

2 � kπ , where k is
an integer. Because the graphs of sine and cosine have the same
shape (each is a horizontal translation of the other), either suffices
to model simple periodic phenomena.F-TF.5 A function is described

F-TF.5 Choose trigonometric functions to model periodic phe-
nomena with specified amplitude, frequency, and midline.

as sinusoidal or is called a sinusoid if it has the same shape as
the sine graph, i.e. has the form f ptq � A � B sinpCt � Dq. Many
real-world phenomena can be approximated by sinusoids, including
sound waves, oscillation on a spring, the motion of a pendulum, tides,
and phases of the moon. Some students will learn in college that
sinusoids are used as building blocks to approximate any periodic
waveform.

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Because sin t oscillates between �1 and 1, A � B sinpCt � Dq
will oscillate between A�B and A�B. Thus, y � A is the midline,
and B is the amplitude of the sinusoid. Students can obtain the
frequency of f : the period of sint is 2π , so (knowing the effect of
multiplying t by C ) the period of sinCt is 2π{C , and the frequency
is its reciprocal. When modeling, students need to have the sense
that C affects the frequency and that C and D together produce
a phase shift, but getting these correct might involve technological
support, except in simple cases.

Frequency vs. period

For a sinusoid, the frequency is often measured in cycles per
time unit, thus the period is often measured in time unit per
cycle. For reasoning about a context, it is common to choose
whichever is greater numerically.

For example, students might be asked to model the length of
the day in Columbus, Ohio. Day length as a function of date is
approximately sinusoidal, varying from about 9 hours, 20 minutes
on December 21 to about 15 hours on June 21. The average of
the maximum and minimum gives the value for the midline, and the
amplitude is half the difference. So A � 12.17, and B � 2.83. With
some support, students can determine that for the period to be 365
days (per cycle) (or the frequency to be 1

365 cycles/day), C � 2π{365,
and if day 0 corresponds to March 21, no phase shift is needed, so
D � 0. Thus,

f ptq � 12.17� 2.83 sin
�
2πt
365

From the graph, students can see that the period is indeed 365 days, •

Length of Day (hrs), Columbus, OH

21 Mar 20 Jun 20 Sep 20 Dec 21 Mar21‐Mar 20‐Jun 20‐Sep 20‐Dec 21‐Mar

as desired, as it takes one year to complete the cycle. They can also
see that two days are approximately 14 hours long, which is to say
that f ptq � 14 has two solutions over a domain of one year, and
they might use graphing or spreadsheet technology to determine
that May 1 and August 10 are the closest such days. Students can
also see that f ptq � 9 has no solutions, which makes sense because
9 hours, 20 minutes is the minimum length of day.

Students who take advanced mathematics will need additional+
fluency with transformations of trigonometric functions, including+
changes in frequency and phase shifts.+

Based on plenty of experience solving equations of the form+
f ptq � c graphically, students of advanced mathematics should be+
able to see that such equations will have an infinite number of solu-+
tions when f is a trigonometric function. Furthermore, they should+
have had experience of restricting the domain of a function so that it+
has an inverse. For trigonometric functions, a common approach to+
restricting the domain is to choose an interval on which the function+
is always increasing or always decreasing.F-TF.6 The obvious choice F-TF.6(+) Understand that restricting a trigonometric function to

a domain on which it is always increasing or always decreasing
allows its inverse to be constructed.

+
for sinpxq is the interval �π

2 ¤ x ¤ π
2 , shown as the solid part of the+

graph. This yields a function θ � sin�1pxq with domain �1 ¤ x ¤ 1

•

π π

π 3π

2π

f x( ) = sin x( )
+

and range �π
2 ¤ θ ¤ π

2 .+
Inverses of trigonometric functions can be used in solving equa-

tions in modeling contexts.F-TF.7 For example, in the length of day

F-TF.7(+) Use inverse functions to solve trigonometric equations
that arise in modeling contexts; evaluate the solutions using tech-
nology, and interpret them in terms of the context.

context, students can use inverse trig functions to determine days
with particular lengths. This amounts to solving f ptq � d for t , which

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yields
t � 365

2π sin�1

�d� 12.17
2.83

Using d � 14 and a calculator (in radian mode), they can compute+
that t � 40.85, which is closest to May 1. Finding the other solution+
is a bit of a challenge, but the graph indicates that it should occur+
just as many days before midyear (day 182.5) as May 1 occurs after+
day 0. So the other solution is t � 182.5� 40.85 � 141.65, which is+
closest to August 10.+

Prove and apply trigonometric identities For the cases illustrated
by the diagram (in which the terminal side of angle θ does not lie on •
an axis) and the definitions of sinθ and cosθ, students can reason
that, in any quadrant, the lengths of the legs of the reference triangle
are |x| and |y|. It then follows from the Pythagorean Theorem that
|x|2 � |y|2 � 1. Because |a|2 � a2 for any real number a, this
equation can be written x2 � y2 � 1. Because x � cosθ and
y � sinθ, the equation can be written as sin2pθq � cos2pθq � 1.
When the terminal side of angle θ does lie on an axis, then one of x
or y is 0 and the other is 1 or �1 and the equation still holds. This
argument proves what is known as the Pythagorean identityF-TF.8 F-TF.8 Prove the Pythagorean identity sin2pθq � cos2pθq � 1

and use it to find sinpθq, cospθq, or tanpθq given sinpθq, cospθq,
or tanpθq and the quadrant of the angle.because it is essentially a restatement of the Pythagorean Theorem

for a right triangle of hypotenuse 1.
With this identity and the value of one of the trigonometric

functions for a given angle, students can find the values of the
other functions for that angle, as long as they know the quad-
rant in which the angle lies. For example, if sinpθq � 0.6 and θ
lies in the second quadrant, then cos2pθq � 1 � 0.62 � 0.64, so
cospθq � �?0.64 � �0.8. Because cosine is negative in the sec-
ond quadrant, it follows that cospθq � 0.8, and therefore tanpθq �
sinpθq{ cospθq � 0.6{p�0.8q � �0.75.

Students in advanced mathematics courses can prove and use+
other trigonometric identities, including the addition and subtrac-+
tion formulas.F-TF.9 If students have already represented complex F-TF.9(+) Prove the addition and subtraction formulas for sine,

cosine, and tangent and use them to solve problems.
+

numbers on the complex planeN-CN.4 and developed the geomet- N-CN.4(+) Represent complex numbers on the complex plane
in rectangular and polar form (including real and imaginary num-
bers), and explain why the rectangular and polar forms of a given
complex number represent the same number.

+
ric interpretation of their multiplication,N-CN.5 then the the product

N-CN.5(+) Represent addition, subtraction, multiplication, and
conjugation of complex numbers geometrically on the complex
plane; use properties of this representation for computation.

+
pcosα � i sinαqpcosβ � i sinβq can be used in deriving the addition+
formulas for cosine and sine. Subtraction and double angle formulas+
can follow from these.+

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8
G R A D E

New York State Common Core

Mathematics Curriculum
GRADE 8 • MODULE 5

Module 5: Examples of Functions from Geometry

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Table of Contents1

Examples of Functions from Geometry
Module Overview ……………………………………………………………………………………………………………………………….. 2

Topic A: Functions (8.F.A.1, 8.F.A.2, 8.F.A.3) ………………………………………………………………………………………….. 9

Lesson 1: The Concept of a Function ………………………………………………………………………………………… 11

Lesson 2: Formal Definition of a Function …………………………………………………………………………………. 22

Lesson 3: Linear Functions and Proportionality …………………………………………………………………………. 36

Lesson 4: More Examples of Functions …………………………………………………………………………………….. 50

Lesson 5: Graphs of Functions and Equations ……………………………………………………………………………. 61

Lesson 6: Graphs of Linear Functions and Rate of Change ………………………………………………………….. 78

Lesson 7: Comparing Linear Functions and Graphs …………………………………………………………………….. 91

Lesson 8: Graphs of Simple Nonlinear Functions ……………………………………………………………………… 104

Topic B: Volume (8.G.C.9) …………………………………………………………………………………………………………………. 115

Lesson 9: Examples of Functions from Geometry …………………………………………………………………….. 116

Lesson 10: Volumes of Familiar Solids—Cones and Cylinders ……………………………………………………. 130

Lesson 11: Volume of a Sphere ……………………………………………………………………………………………… 143

End-of-Module Assessment and Rubric ……………………………………………………………………………………………… 155
Topics A–B (assessment 1 day, return 1 day, remediation or further applications 2 days)

1Each lesson is ONE day, and ONE day is considered a 45-minute period.

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8•5 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Module 5: Examples of Functions from Geometry

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Grade 8 • Module 5

Examples of Functions from Geometry

OVERVIEW
In Module 5, Topic A, students learn the concept of a function and why functions are necessary for describing
geometric concepts and occurrences in everyday life. The module begins by explaining the important role
functions play in making predictions. For example, if an object is dropped, a function allows us to determine
its height at a specific time. To this point, student work has relied on assumptions of constant rates; here,
students are given data that show that objects do not always travel at a constant speed. Once the concept of
a function is explained, a formal definition of function is provided. A function is defined as an assignment to
each input, exactly one output (8.F.A.1). Students learn that the assignment of some functions can be
described by a mathematical rule or formula. With the concept and definition firmly in place, students begin
to work with functions in real-world contexts. For example, students relate constant speed and other
proportional relationships (8.EE.B.5) to linear functions. Next, students consider functions of discrete and
continuous rates and understand the difference between the two. For example, students are asked to
explain why they can write a cost function for a book, but they cannot input 2.6 into the function and get an
accurate cost as the output.

Students apply their knowledge of linear equations and their graphs from Module 4 (8.EE.B.5, 8.EE.B.6) to
graphs of linear functions. Students know that the definition of a graph of a function is the set of ordered
pairs consisting of an input and the corresponding output (8.F.A.1). Students relate a function to an input-
output machine: a number or piece of data, known as the input, goes into the machine, and a number or
piece of data, known as the output, comes out of the machine. In Module 4, students learned that a linear
equation graphs as a line and that all lines are graphs of linear equations. In Module 5, students inspect the
rate of change of linear functions and conclude that the rate of change is the slope of the graph of a line.
They learn to interpret the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 (8.EE.B.6) as defining a linear function whose graph is a line
(8.F.A.3). Students also gain some experience with nonlinear functions, specifically by compiling and graphing
a set of ordered pairs and then by identifying the graph as something other than a straight line.

Once students understand the graph of a function, they begin comparing two functions represented in
different ways (8.EE.C.8), similar to comparing proportional relationships in Module 4. For example, students
are presented with the graph of a function and a table of values that represent a function and are asked to
determine which function has the greater rate of change (8.F.A.2). Students are also presented with
functions in the form of an algebraic equation or written description. In each case, students examine the
average rate of change and know that the one with the greater rate of change must overtake the other at
some point.

In Topic B, students use their knowledge of volume from previous grade levels (5.MD.C.3, 5.MD.C.5) to learn
the volume formulas for cones, cylinders, and spheres (8.G.C.9). First, students are reminded of what they
already know about volume, that volume is always a positive number that describes the hollowed-out portion
of a solid figure that can be filled with water. Next, students use what they learned about the area of circles

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8•5 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Module 5: Examples of Functions from Geometry

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

(7.G.B.4) to determine the volume formulas for cones and cylinders. In each case, physical models are used
to explain the formulas, beginning with a cylinder seen as a stack of circular disks that provide the height of
the cylinder. Students consider the total area of the disks in three dimensions, understanding it as volume of
a cylinder. Next, students make predictions about the volume of a cone that has the same dimensions as a
cylinder. A demonstration shows students that the volume of a cone is one-third the volume of a cylinder
with the same dimensions, a fact that will be proved in Module 7. Next, students compare the volume of a
sphere to its circumscribing cylinder (i.e., the cylinder of dimensions that touches the sphere at points but
does not cut off any part of it). Students learn that the formula for the volume of a sphere is two-thirds the
volume of the cylinder that fits tightly around it. Students extend what they learned in Grade 7 (7.G.B.6)
about how to solve real-world and mathematical problems related to volume from simple solids to include
problems that require the formulas for cones, cylinders, and spheres.

Focus Standards
Define, evaluate, and compare functions.2

8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph
of a function is the set of ordered pairs consisting of an input and the corresponding
output.3

8.F.A.2 Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a linear
function represented by a table of values and a linear function represented by an algebraic
expression, determine which function has the greater rate of change.

8.F.A.3 Interpret the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 as defining a linear function, whose graph is a straight
line; give examples of functions that are not linear. For example, the function 𝐴𝐴 = 𝑠𝑠2 giving
the area of a square as a function of its side length is not linear because its graph contains
the points (1, 1), (2, 4) and (3, 9) which are not on a straight line.

Solve real-world and mathematical problems involving volume of cylinders, cones, and
spheres.

8.G.C.94 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.

2Linear and nonlinear functions are compared in this module using linear equations and area/volume formulas as examples.
3Function notation is not required in Grade 8.
4Solutions that introduce irrational numbers are not introduced until Module 7.

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Module 5: Examples of Functions from Geometry

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Foundational Standards
Geometric measurement: Understand concepts of volume and relate volume to
multiplication and to addition.

5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume
measurement.

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of
volume, and can be used to measure volume.

b. A solid figure which can be packed without gaps or overlaps using 𝑛𝑛 unit cubes is said to
have a volume of 𝑛𝑛 cubic units.

5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing
it with unit cubes, and show that the volume is the same as would be found by
multiplying the edge lengths, equivalently by multiplying the height by the area of the
base. Represent threefold whole-number products as volumes, e.g., to represent the
associative property of multiplication.

b. Apply the formulas 𝑉𝑉 = 𝑙𝑙 × 𝑤𝑤 × ℎ and 𝑉𝑉 = 𝑏𝑏 × ℎ for rectangular prisms to find
volumes of right rectangular prisms with whole–number edge lengths in the context of
solving real world and mathematical problems.

c. Recognize volume as additive. Find volume of solid figures composed of two non-
overlapping right rectangular prisms by adding the volumes of the non-overlapping
parts, applying this technique to real world problems.

Solve real-life and mathematical problems involving angle measure, area, surface area, and
volume.

7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve
problems; give an informal derivation of the relationship between the circumference and
area of a circle.

7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of
two-and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes,
and right prisms.

Understand the connections between proportional relationships, lines, and linear equations.

8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways. For
example, compare a distance-time graph to a distance-time equation to determine which of
two moving objects has greater speed.

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Module 5: Examples of Functions from Geometry

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8.EE.B.6 Use similar triangles to explain why the slope 𝑚𝑚 is the same between any two distinct points
on a non-vertical line in the coordinate plane; derive the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 for a line through
the origin and the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 for a line intercepting the vertical axis at 𝑏𝑏.

Analyze and solve linear equations and pairs of simultaneous linear equations.

8.EE.C.7 Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many
solutions, or no solutions. Show which of these possibilities is the case by successively
transforming the given equation into simpler forms, until an equivalent equation of the
form 𝑚𝑚 = 𝑎𝑎, 𝑎𝑎 = 𝑎𝑎, or 𝑎𝑎 = 𝑏𝑏 results (where 𝑎𝑎 and 𝑏𝑏 are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and collecting
like terms.

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection
satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations. Solve simple cases by inspection. For example,
3𝑚𝑚 + 2𝑦𝑦 = 5 and 3𝑚𝑚 + 2𝑦𝑦 = 6 have no solution because 3𝑚𝑚 + 2𝑦𝑦 cannot
simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two
variables. For example, given coordinates for two pairs of points, determine whether
the line through the first pair of points intersects the line through the second pair.

Focus Standards for Mathematical Practice
MP.2 Reason abstractly or quantitatively. Students examine, interpret, and represent functions

symbolically. They make sense of quantities and their relationships in problem situations.
For example, students make sense of values as they relate to the total cost of items
purchased or a phone bill based on usage in a particular time interval. Students use what
they know about rate of change to distinguish between linear and nonlinear functions.
Further, students contextualize information gained from the comparison of two functions.

MP.6 Attend to precision. Students use notation related to functions, in general, as well as
notation related to volume formulas. Students are expected to clearly state the meaning of
the symbols used in order to communicate effectively and precisely to others. Students
attend to precision when they interpret data generated by functions. They know when
claims are false; for example, calculating the height of an object after it falls for −2 seconds.
Students also understand that a table of values is an incomplete representation of a
continuous function, as an infinite number of values can be found for a function.

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Module 5: Examples of Functions from Geometry

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MP.8 Look for and express regularity in repeated reasoning. Students use repeated
computations to determine equations from graphs or tables. While focused on the details
of a specific pair of numbers related to the input and output of a function, students maintain
oversight of the process. As students develop equations from graphs or tables, they
evaluate the reasonableness of their equation as they ensure that the desired output is a
function of the given input.

Terminology

New or Recently Introduced Terms

 Cone8.G.9 (Let 𝐵𝐵 be a polygonal region or a disk in a plane 𝐸𝐸, and 𝑉𝑉 be a point not in 𝐸𝐸. The cone with
base 𝐵𝐵 and vertex 𝑉𝑉 is the union of all segments 𝑉𝑉𝑉𝑉 for all points 𝑉𝑉 in 𝐵𝐵.
A cone is named by its base. If the base is a polygonal region, then the cone is usually called a
pyramid. For example, a cone with a triangular region for its base is called a triangular pyramid. A
cone with a circular region for its base whose vertex lies on the perpendicular line to the base that
passes through the center of the circle is called a right circular cone. This cone is the one usually
shown in elementary and middle school textbooks. The name of it is often shortened to just cone.)

 Cylinder (Let 𝐸𝐸 and 𝐸𝐸′ be two parallel planes, let 𝐵𝐵 be a polygonal region or a disk in the plane 𝐸𝐸,
and let 𝐿𝐿 be a line which intersects 𝐸𝐸 and 𝐸𝐸′ but not 𝐵𝐵. At each point 𝑉𝑉 of 𝐵𝐵, consider the segment
𝑉𝑉𝑉𝑉′ parallel to line 𝐿𝐿, joining 𝑉𝑉 to a point 𝑉𝑉′ of the plane 𝐸𝐸′. The union of all these segments is called
a cylinder with base 𝐵𝐵. The regions 𝐵𝐵 and 𝐵𝐵′ are called the base faces (or just bases) of the prism.
A cylinder is named by its base. If the base is a polygonal region, then the cylinder is usually called a
prism. For example, a cylinder with a triangular region for its base is called a triangular prism. A
cylinder with a circular region for its base that is defined by a line that is perpendicular to the base is
called a right circular cylinder. This cylinder is the one usually shown in elementary and middle
school textbooks, where the name is often shortened to just cylinder.)
Equation Form of a Linear Function (description) (The equation form of a linear function is an
equation of the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏, where the number 𝑚𝑚 is called the rate of change of the linear
function and the number 𝑏𝑏 is called the initial value of the linear function. To calculate the output
named by the dependent variable 𝑦𝑦, an input is substituted into the independent variable 𝑚𝑚 and
evaluated.)

 Function (description) (A function is a correspondence between a set (whose elements are called
inputs) and another set (whose elements are called outputs) such that each input corresponds to one
and only one output. The correspondence is often given as a rule: the output is a number found by
substituting an input number into the variable of a one-variable expression and evaluating.
For example, a proportional relationship is a special type of function whose output is always given by
multiplying the input number by another number (the constant of proportionality).)

 Graph of a Linear Function (The graph of a linear function represented by the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏
is the set of ordered pairs (𝑚𝑚,𝑦𝑦) for inputs 𝑚𝑚 and outputs 𝑦𝑦 that make the equation true.
When the graph of a linear function is a line (i.e., the set of inputs is all real numbers), then 𝑚𝑚 is the
slope of the line and 𝑏𝑏 is the 𝑦𝑦-intercept of the line.)

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 Lateral Edge and Face of a Prism (Suppose the base 𝐵𝐵 of a prism is a polygonal region and 𝑉𝑉𝑖𝑖 is a
vertex of 𝐵𝐵. Let 𝑉𝑉𝑖𝑖′ be the corresponding point in 𝐵𝐵′ such that 𝑉𝑉𝑖𝑖𝑉𝑉𝑖𝑖′ is parallel to the line 𝐿𝐿 defining
the prism. The segment 𝑉𝑉𝑖𝑖𝑉𝑉𝑖𝑖′ is called a lateral edge of the prism. If 𝑉𝑉𝑖𝑖𝑉𝑉𝑖𝑖+1 is a base edge of the base
𝐵𝐵 (a side of 𝐵𝐵), and 𝐹𝐹 is the union of all segments 𝑉𝑉𝑉𝑉′ parallel to line 𝐿𝐿 for which points 𝑉𝑉 are in
segment 𝑉𝑉𝑖𝑖𝑉𝑉𝑖𝑖+1 and points 𝑉𝑉′ are in 𝐵𝐵′, then 𝐹𝐹 is a lateral face of the prism. It can be shown that a
lateral face of a prism is always a region enclosed by a parallelogram.)

 Lateral Edge and Face of a Pyramid8.G.9 (Suppose the base 𝐵𝐵 of a pyramid with vertex 𝑉𝑉 is a
polygonal region and 𝑉𝑉𝑖𝑖 is a vertex of 𝐵𝐵. The segment 𝑉𝑉𝑖𝑖𝑉𝑉 is called a lateral edge of the pyramid. If
𝑉𝑉𝑖𝑖𝑉𝑉𝑖𝑖+1 is a base edge of the base 𝐵𝐵 (a side of 𝐵𝐵), and 𝐹𝐹 is the union of all segments 𝑉𝑉𝑉𝑉 for all points
𝑉𝑉 in the segment 𝑉𝑉𝑖𝑖𝑉𝑉𝑖𝑖+1, then 𝐹𝐹 is called a lateral face of the pyramid. It can be shown that the face
of a pyramid is always a triangular region.)

 Linear Function (description) (A linear function is a function whose inputs and outputs are real
numbers such that each output is given by substituting an input into a linear expression and
evaluating.

 Solid Sphere or Ball (Given a point 𝐶𝐶 in the 3-dimensional space and a number 𝑟𝑟 > 0, the solid
sphere (or ball) with center 𝐶𝐶 and radius 𝑟𝑟 is the set of all points in space whose distance from the
point 𝐶𝐶 is less than or equal to 𝑟𝑟.)

 Sphere (Given a point 𝐶𝐶 in the 3-dimensional space and a number 𝑟𝑟 > 0, the sphere with center 𝐶𝐶
and radius 𝑟𝑟 is the set of all points in space that are distance 𝑟𝑟 from the point 𝐶𝐶.)

Familiar Terms and Symbols5

 Area
 Linear equation
 Nonlinear equation
 Rate of change
 Solids
 Volume

Suggested Tools and Representations
 3D solids: cones, cylinders, and spheres

5These are terms and symbols students have seen previously.

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Module 5: Examples of Functions from Geometry

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Rapid White Board Exchanges
Implementing a RWBE requires that each student be provided with a personal white board, a white board
marker, and a means of erasing his work. An economic choice for these materials is to place sheets of card
stock inside sheet protectors to use as the personal white boards and to cut sheets of felt into small squares
to use as erasers.

A RWBE consists of a sequence of 10 to 20 problems on a specific topic or skill that starts out with a relatively
simple problem and progressively gets more difficult. The teacher should prepare the problems in a way that
allows her to reveal them to the class one at a time. A flip chart or PowerPoint presentation can be used, or
the teacher can write the problems on the board and either cover some with paper or simply write only one
problem on the board at a time.

The teacher reveals, and possibly reads aloud, the first problem in the list and announces, “Go.” Students
work the problem on their personal white boards as quickly as possible and hold their work up for their
teacher to see their answers as soon as they have the answer ready. The teacher gives immediate feedback
to each student, pointing and/or making eye contact with the student and responding with an affirmation for
correct work such as, “Good job!”, “Yes!”, or “Correct!”, or responding with guidance for incorrect work such
as, “Look again,” “Try again,” “Check your work.” In the case of the RWBE, it is not recommended that the
feedback include the name of the student receiving the feedback.

If many students have struggled to get the answer correct, go through the solution of that problem as a class
before moving on to the next problem in the sequence. Fluency in the skill has been established when the
class is able to go through each problem in quick succession without pausing to go through the solution of
each problem individually. If only one or two students have not been able to successfully complete a
problem, it is appropriate to move the class forward to the next problem without further delay; in this case
find a time to provide remediation to that student before the next fluency exercise on this skill is given.

Assessment Summary

Assessment Type Administered Format Standards Addressed

End-of-Module
Assessment Task After Topic B Constructed response with rubric 8.F.A.1, 8.F.A.2, 8.F.A.3,

8.G.C.9

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8
G R A D E

New York State Common Core

Mathematics Curriculum
GRADE 8 • MODULE 5

Topic A: Functions

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Topic A

Functions

8.F.A.1, 8.F.A.2, 8.F.A.3

Focus Standards: 8.F.A.1 Understand that a function is a rule that assigns to each input exactly one
output. The graph of a function is the set of ordered pairs consisting of an
input and the corresponding output.

8.F.A.2 Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a linear function represented by a table of values and a linear
function represented by an algebraic expression, determine which function has
the greater rate of change.

8.F.A.3 Interpret the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 as defining a linear function, whose graph
is a straight line; give examples of functions that are not linear. For example,
the function 𝐴𝐴 = 𝑠𝑠2 giving the area of a square as a function of its side length is
not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which
are not on a straight line.

Instructional Days: 8

Lesson 1: The Concept of a Function (P)1

Lesson 2: Formal Definition of a Function (S)

Lesson 3: Linear Functions and Proportionality (P)

Lesson 4: More Examples of Functions (P)

Lesson 5: Graphs of Functions and Equations (E)

Lesson 6: Graphs of Linear Functions and Rate of Change (S)

Lesson 7: Comparing Linear Functions and Graphs (E)

Lesson 8: Graphs of Simple Nonlinear Functions (E)

1Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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8•5 Topic A NYS COMMON CORE MATHEMATICS CURRICULUM

Topic A: Functions

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Lesson 1 relies on students’ understanding of constant rate, a skill developed in previous grade levels and
reviewed in Module 4 (6.RP.A.3b, 7.RP.A.2). Students are confronted with the fact that the concept of
constant rate, which requires the assumption that a moving object travels at a constant speed, cannot be
applied to all moving objects. Students examine a graph and a table that demonstrate the nonlinear effect of
gravity on a falling object. This example provides the reasoning for the need of functions. In Lesson 2,
students continue their investigation of time and distance data for a falling object and learn that the scenario
can be expressed by a formula. Students are introduced to the terms input and output and learn that a
function assigns to each input exactly one output. Though students do not learn the traditional “vertical-line
test,” students know that the graph of a function is the set of ordered pairs consisting of an input and the
corresponding output. Students also learn that not all functions can be expressed by a formula, but when
they are, the function rule allows us to make predictions about the world around us. For example, with
respect to the falling object, the function allows us to predict the height of the object for any given time
interval.

In Lesson 3, constant rate is revisited as it applies to the concept of linear functions and proportionality in
general. Lesson 4 introduces students to the fact that not all rates are continuous. That is, a cost function for
the cost of a book can be written, yet the cost of 3.6 books cannot realistically be found. Students are also
introduced to functions that do not use numbers at all, as in a function where the input is a card from a
standard deck, and the output is the suit.

Lesson 5 is when students begin graphing functions of two variables. Students graph linear and nonlinear
functions, and the guiding question of the lesson, “Why not just look at graphs of equations in two variables?”
is answered because not all graphs of equations are graphs of functions. Students continue their work on
graphs of linear functions in Lesson 6. In this lesson, students investigate the rate of change of functions and
conclude that the rate of change for linear functions is the slope of the graph. In other words, this lesson
solidifies the fact that the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 defines a linear function whose graph is a straight line.

With the knowledge that the graph of a linear function is a straight line, students begin to compare properties
of two functions that are expressed in different ways in Lesson 7. One example of this relates to a
comparison of phone plans. Students are provided a graph of a function for one plan and an equation of a
function that represents another plan. In other situations, students are presented with functions that are
expressed algebraically, graphically, and numerically in tables, or are described verbally. Students must use
the information provided to answer questions about the rate of change of each function. In Lesson 8,
students work with simple nonlinear functions of area and volume and their graphs.

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Lesson 1: The Concept of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 1

Lesson 1: The Concept of a Function

Student Outcomes

 Students analyze a nonlinear data set.
 Students realize that an assumption of a constant rate of motion is not appropriate for all situations.

Lesson Notes
Using up-to-date data can add new elements of life to a lesson for students. If time permits, consider gathering
examples of data from the Internet, and use that data as examples throughout this topic.

Much of the discussion in this module is based on parts from the following sources:

H. Wu, Introduction to School Algebra, http://math.berkeley.edu/~wu/Algebrasummary

H. Wu, Teaching Geometry in Grade 8 and High School According to the Common Core Standards,

https://math.berkeley.edu/~wu/CCSS-Geometry 1

Classwork

Discussion (4 minutes)

 In the last module we focused on situations that were worked with two varying quantities, one changing with
respect to the other according to some constant rate of change factor. Consequently, each situation could be
analyzed with a linear equation. Such a formula then gave us the means to determine the value of the one
quantity given a specific value of the other.

 There are many situations, however, for which assuming a constant rate of change relationship between two
quantities is not appropriate. Consequently, there is no linear equation to describe their relationship. If we
are fortunate, we may be able to find a mathematical equation describing their relationship nonetheless.

 Even if this is not possible, we may be able to use data from the situation to see a relationship of some kind
between the values of one quantity and their matching values of the second quantity.

 Mathematicians call any clearly-described rule that assigns to each value of one quantity a single value of a
second quantity a function. Functions could be described by words (“Assign to whole number its first digit,”
for example), by a table of values or by a graph (a graph or table of your height at different times of your life,
for example) or, if we are lucky, by a formula. (For instance, the formula 𝑦𝑦 = 2𝑥𝑥 describes the rule “Assign to
each value 𝑥𝑥 double its value.”)

 All becomes clear as this topic progresses. We start by looking at one curious set of data.

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http://math.berkeley.edu/%7Ewu/Algebrasummary

https://math.berkeley.edu/%7Ewu/CCSS-Geometry_1

Lesson 1: The Concept of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 1

Example 1 (7 minutes)

This example is used to point out that, contrary to much of the previous work in assuming a constant rate relationship
exists, students cannot, in general, assume this is always the case. Encourage students to make sense of the problem
and attempt to solve it on their own. The goal is for students to develop an understanding of the subtleties predicting
data values.

Example 1

Suppose a moving object travels 𝟐𝟐𝟐𝟐𝟐𝟐 feet in 𝟒𝟒 seconds. Assume that the object travels at a constant speed, that is, the
motion of the object can be described by a linear equation. Write a linear equation in two variables to represent the
situation, and use the equation to predict how far the object has moved at the four times shown.

Number of seconds in
motion

(𝒙𝒙)

Distance traveled in feet
(𝒚𝒚)

𝟏𝟏 𝟐𝟐𝟒𝟒

𝟐𝟐 𝟏𝟏𝟐𝟐𝟏𝟏

𝟑𝟑 𝟏𝟏𝟏𝟏𝟐𝟐

𝟒𝟒 𝟐𝟐𝟐𝟐𝟐𝟐

 Suppose a moving object travels 256 feet in 4 seconds. Assume that the object travels at a constant speed,
that is, the motion of the object can be described by a linear equation. Write a linear equation in two variables
to represent the situation, and use the equation to predict how far the object has moved at the four times
shown.

 Let 𝑥𝑥 represent the time it takes to travel 𝑦𝑦 feet.
256

4
=
𝑦𝑦
𝑥𝑥

𝑦𝑦 =
256

4
𝑥𝑥

𝑦𝑦 = 64𝑥𝑥

 What are some of the predictions that this equation allows us to make?

 After one second, or when 𝑥𝑥 = 1, the distance traveled is 64 feet.

Accept any reasonable predictions that students make.

 Use your equation to complete the table.

 What is the average speed of the moving object from 0 to 3 seconds?

 The average speed is 64 feet per second. We know that the object has a constant rate of change;
therefore, we expect the average speed to be the same over any time interval.

Example 2 (15 minutes)

 Suppose I now reveal that the object is a stone being dropped from a height of 256 feet. It takes exactly 4
seconds for the stone to hit the ground. Do you think we can assume constant speed in this situation? Is our
linear equation describing the situation still valid?

MP.1

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 1

Example 2

The object, a stone, is dropped from a height of 𝟐𝟐𝟐𝟐𝟐𝟐 feet. It takes exactly 𝟒𝟒 seconds for the stone to hit the ground. How
far does the stone drop in the first 𝟑𝟑 seconds? What about the last 𝟑𝟑 seconds? Can we assume constant speed in this
situation? That is, can this situation be expressed using a linear equation?

Number of seconds
(𝒙𝒙)

Distance traveled in feet
(𝒚𝒚)

𝟏𝟏 𝟏𝟏𝟐𝟐

𝟐𝟐 𝟐𝟐𝟒𝟒

𝟑𝟑 𝟏𝟏𝟒𝟒𝟒𝟒

𝟒𝟒 𝟐𝟐𝟐𝟐𝟐𝟐

Provide students time to discuss this in pairs. Lead a discussion in which students share their thoughts with the class. It
is likely they will say the motion of a falling object is linear and that the work conducted in the previous example is
appropriate.

 If this is a linear situation, then we predict that the stone drops 192 feet in the first 3 seconds.

Now consider viewing the 10-second “ball drop” video at the following link:
http://www.youtube.com/watch?v=KrX zLuwOvc. Consider showing it more than once.

 If we were to slow the video down and record the distance the ball dropped after each second, here is the data
we would obtain:

Have students record the data in the table of Example 2.

 Was the linear equation developed in Example 1 appropriate after all?

Students who thought the stone was traveling at constant speed should realize that the predictions were not accurate
for this situation. Guide their thinking using the discussion points below.

 According to the data, how many feet did the stone drop in 3 seconds?

 The stone dropped 144 feet.

 How can that be? It must be that our initial assumption of constant rate was incorrect.

What predictions can we make now?

 After one second, 𝑥𝑥 = 1; the stone dropped 16 feet, etc.

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 Let’s make a prediction based on a value of 𝑥𝑥 that is not listed in the table. How far did the stone drop in the
first 3.5 seconds? What have we done in the past to figure something like this out?

 We wrote a proportion using the known times and distances.

Allow students time to work with proportions. Encourage them to use more than one pair of data values to determine
an answer. Some students might suggest they cannot use proportions for this work as they have just ascertained that
there is not a constant rate of change. Acknowledge this. The work with proportions some students do will indeed
confirm this.

 Sample student work:

Let 𝑥𝑥 be the distance, in feet, the stone drops in 3.5 seconds.

16
1

=
𝑥𝑥

3.5

𝑥𝑥 = 56

64
2

=
𝑥𝑥

3.5

2𝑥𝑥 = 224

𝑥𝑥 = 112

144
3

=
𝑥𝑥

3.5

3𝑥𝑥 = 504

𝑥𝑥 = 168

 Is it reasonable that the stone would drop 56 feet in 3.5 seconds? Explain.

 No, it is not reasonable. Our data shows that after 2 seconds, the stone has already dropped 64 feet.
Therefore, it is impossible that it could have only dropped 56 feet in 3.5 seconds.

 What about 112 feet in 3.5 seconds? How reasonable is that answer? Explain.

 The answer of 112 feet in 3.5 seconds is not reasonable either. The data shows that the stone dropped
144 feet in 3 seconds.

 What about 168 feet in 3.5 seconds? What do you think about that answer? Explain.

 That answer is the most likely because at least it is greater than the recorded 144 feet in 3 seconds.

 What makes you think that the work done with a third proportion will give us a correct answer when the first
two did not? Can we rely on this method for determining an answer?

 This does not seem to be a reliable method. If we had only done one computation and not evaluated
the reasonableness of our answer, we would have been wrong.

 What this means is that the table we used does not tell the whole story about the falling stone. Suppose, by
repeating the experiment and gathering more data of the motion, we obtained the following table:

Number of seconds (𝒙𝒙) Distance traveled in feet (𝒚𝒚)

0.5 4

1 16

1.5 36

2 64

2.5 100

3 144

3.5 196

4 256

 Were any of the predictions we made about the distance dropped during the first 3.5 seconds correct?

MP.3

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Have a discussion with students about why we want to make predictions at all. Students should recognize that making
predictions helps us make sense of the world around us. Some scientific discoveries began with a prediction, then an
experiment to prove or disprove the prediction, and then were followed by some conclusion.

 Now it is clear that none of our answers for the distance traveled in 3.5 seconds were correct. In fact, the
stone dropped 196 feet in the first 3.5 seconds. Does the table on the previous page capture the motion of
the stone completely? Explain?

 No. There are intervals of time between those in the table. For example, the distance it drops in 1.6
seconds is not represented.

 If we were to record the data for every 0.1 second that passed, would that be enough to capture the motion of
the stone?
 No. There would still be intervals of time not represented. For example, 1.61 seconds.

 To tell the whole story, we would need information about where the stone is after the first 𝑡𝑡 seconds for every
𝑡𝑡 satisfying 0 ≤ 𝑡𝑡 ≤ 4.

 Nonetheless, for each value 𝑡𝑡 for 0 ≤ 𝑡𝑡 ≤ 4, there is some specific value to assign to that value: the distance
the stone fell during the first 𝑡𝑡 seconds. That it, there is some rule that assigns to each value 𝑡𝑡 between
0 ≤ 𝑡𝑡 ≤ 4 a real number. We have an example of a function. (And, at present, we don’t have a mathematical
formula describing that function.)

Some students, however, might observe at this point the data does seem to be following the formula 𝑦𝑦 = 16𝑥𝑥2.

Exercises 1–6 (10 minutes)

Students complete Exercises 1–6 in pairs or small groups.

Exercises 1–6

Use the table to answer Exercises 1–5.

Number of seconds (𝒙𝒙) Distance traveled in feet (𝒚𝒚)

𝟎𝟎.𝟐𝟐 𝟒𝟒

𝟏𝟏 𝟏𝟏𝟐𝟐

𝟏𝟏.𝟐𝟐 𝟑𝟑𝟐𝟐

𝟐𝟐 𝟐𝟐𝟒𝟒

𝟐𝟐.𝟐𝟐 𝟏𝟏𝟎𝟎𝟎𝟎

𝟑𝟑 𝟏𝟏𝟒𝟒𝟒𝟒

𝟑𝟑.𝟐𝟐 𝟏𝟏𝟏𝟏𝟐𝟐

𝟒𝟒 𝟐𝟐𝟐𝟐𝟐𝟐

1. Name two predictions you can make from this table.

Sample student responses:

After 𝟐𝟐 seconds, the object traveled 𝟐𝟐𝟒𝟒 feet. After 𝟑𝟑.𝟐𝟐 seconds, the object traveled 𝟏𝟏𝟏𝟏𝟐𝟐 feet.

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2. Name a prediction that would require more information.

Sample student response:

We would need more information to predict the distance traveled after 𝟑𝟑.𝟕𝟕𝟐𝟐 seconds.

3. What is the average speed of the object between 𝟎𝟎 and 𝟑𝟑 seconds? How does this compare to the average speed
calculated over the same interval in Example 1?

𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 𝐒𝐒𝐒𝐒𝐀𝐀𝐀𝐀𝐒𝐒 =
𝐒𝐒𝐝𝐝𝐝𝐝𝐝𝐝𝐀𝐀𝐝𝐝𝐝𝐝𝐀𝐀 𝐝𝐝𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐭𝐭𝐀𝐀𝐒𝐒 𝐨𝐨𝐀𝐀𝐀𝐀𝐀𝐀 𝐀𝐀 𝐀𝐀𝐝𝐝𝐀𝐀𝐀𝐀𝐝𝐝 𝐝𝐝𝐝𝐝𝐭𝐭𝐀𝐀 𝐝𝐝𝐝𝐝𝐝𝐝𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐭𝐭

𝐝𝐝𝐝𝐝𝐭𝐭𝐀𝐀 𝐝𝐝𝐝𝐝𝐝𝐝𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐭𝐭

The average speed is 𝟒𝟒𝟏𝟏 feet per second:
𝟏𝟏𝟒𝟒𝟒𝟒
𝟑𝟑

= 𝟒𝟒𝟏𝟏. This is different from the average speed calculated in Example

1. In Example 1, the average speed over an interval of 𝟑𝟑 seconds was 𝟐𝟐𝟒𝟒 feet per second.

4. Take a closer look at the data for the falling stone by answering the questions below.

a. How many feet did the stone drop between 𝟎𝟎 and 𝟏𝟏 second?

The stone dropped 𝟏𝟏𝟐𝟐 feet between 𝟎𝟎 and 𝟏𝟏 second.

b. How many feet did the stone drop between 𝟏𝟏 and 𝟐𝟐 seconds?

The stone dropped 𝟒𝟒𝟏𝟏 feet between 𝟏𝟏 and 𝟐𝟐 seconds.

c. How many feet did the stone drop between 𝟐𝟐 and 𝟑𝟑 seconds?

The stone dropped 𝟏𝟏𝟎𝟎 feet between 𝟐𝟐 and 𝟑𝟑 seconds.

d. How many feet did the stone drop between 𝟑𝟑 and 𝟒𝟒 seconds?

The stone dropped 𝟏𝟏𝟏𝟏𝟐𝟐 feet between 𝟑𝟑 and 𝟒𝟒 seconds.

e. Compare the distances the stone dropped from one time interval to the next. What do you notice?

Over each interval, the difference in the distance was 𝟑𝟑𝟐𝟐 feet. For example, 𝟏𝟏𝟐𝟐+ 𝟑𝟑𝟐𝟐 = 𝟒𝟒𝟏𝟏, 𝟒𝟒𝟏𝟏 + 𝟑𝟑𝟐𝟐 = 𝟏𝟏𝟎𝟎,
and 𝟏𝟏𝟎𝟎 + 𝟑𝟑𝟐𝟐 = 𝟏𝟏𝟏𝟏𝟐𝟐.

5. What is the average speed of the stone in each interval 𝟎𝟎.𝟐𝟐 second? For example, the average speed over the
interval from 𝟑𝟑.𝟐𝟐 seconds to 𝟒𝟒 seconds is

𝐒𝐒𝐝𝐝𝐝𝐝𝐝𝐝𝐀𝐀𝐝𝐝𝐝𝐝𝐀𝐀 𝐝𝐝𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐭𝐭𝐀𝐀𝐒𝐒 𝐨𝐨𝐀𝐀𝐀𝐀𝐀𝐀 𝐀𝐀 𝐀𝐀𝐝𝐝𝐀𝐀𝐀𝐀𝐝𝐝 𝐝𝐝𝐝𝐝𝐭𝐭𝐀𝐀 𝐝𝐝𝐝𝐝𝐝𝐝𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐭𝐭
𝐝𝐝𝐝𝐝𝐭𝐭𝐀𝐀 𝐝𝐝𝐝𝐝𝐝𝐝𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐭𝐭

=
𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟏𝟏𝟏𝟏𝟐𝟐
𝟒𝟒 − 𝟑𝟑.𝟐𝟐

=
𝟐𝟐𝟎𝟎
𝟎𝟎.𝟐𝟐

= 𝟏𝟏𝟐𝟐𝟎𝟎;𝟏𝟏𝟐𝟐𝟎𝟎 𝐟𝐟𝐀𝐀𝐀𝐀𝐝𝐝 𝐒𝐒𝐀𝐀𝐀𝐀 𝐝𝐝𝐀𝐀𝐝𝐝𝐨𝐨𝐝𝐝𝐒𝐒

Repeat this process for every half-second interval. Then, answer the question that follows.

a. Interval between 𝟎𝟎 and 𝟎𝟎.𝟐𝟐 second:
𝟒𝟒
𝟎𝟎.𝟐𝟐

= 𝟏𝟏;𝟏𝟏 feet per second

b. Interval between 𝟎𝟎.𝟐𝟐 and 𝟏𝟏 second:
𝟏𝟏𝟐𝟐
𝟎𝟎.𝟐𝟐

= 𝟐𝟐𝟒𝟒;𝟐𝟐𝟒𝟒 feet per second

c. Interval between 𝟏𝟏 and 𝟏𝟏.𝟐𝟐 seconds:
𝟐𝟐𝟎𝟎
𝟎𝟎.𝟐𝟐

= 𝟒𝟒𝟎𝟎;𝟒𝟒𝟎𝟎 feet per second

d. Interval between 𝟏𝟏.𝟐𝟐 and 𝟐𝟐 seconds:
𝟐𝟐𝟏𝟏
𝟎𝟎.𝟐𝟐

= 𝟐𝟐𝟐𝟐;𝟐𝟐𝟐𝟐 feet per second

MP.2

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e. Interval between 𝟐𝟐 and 𝟐𝟐.𝟐𝟐 seconds:
𝟑𝟑𝟐𝟐
𝟎𝟎.𝟐𝟐

= 𝟕𝟕𝟐𝟐;𝟕𝟕𝟐𝟐 feet per second

f. Interval between 𝟐𝟐.𝟐𝟐 and 𝟑𝟑 seconds:
𝟒𝟒𝟒𝟒
𝟎𝟎.𝟐𝟐

= 𝟏𝟏𝟏𝟏;𝟏𝟏𝟏𝟏 feet per second

g. Interval between 𝟑𝟑 and 𝟑𝟑.𝟐𝟐 seconds:
𝟐𝟐𝟐𝟐
𝟎𝟎.𝟐𝟐

= 𝟏𝟏𝟎𝟎𝟒𝟒;𝟏𝟏𝟎𝟎𝟒𝟒 feet per second

h. Compare the average speed between each time interval. What do you notice?

Over each interval, there is an increase in the average speed of 𝟏𝟏𝟐𝟐 feet per second. For example,

𝟏𝟏 + 𝟏𝟏𝟐𝟐 = 𝟐𝟐𝟒𝟒, 𝟐𝟐𝟒𝟒+ 𝟏𝟏𝟐𝟐 = 𝟒𝟒𝟎𝟎, 𝟒𝟒𝟎𝟎+ 𝟏𝟏𝟐𝟐 = 𝟐𝟐𝟐𝟐, and so on.

6. Is there any pattern to the data of the falling stone? Record your thoughts below.

Time of interval in seconds
(𝒕𝒕) 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒

Distance stone fell in feet
(𝒚𝒚) 𝟏𝟏𝟐𝟐 𝟐𝟐𝟒𝟒 𝟏𝟏𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐𝟐𝟐

Accept any reasonable patterns that students notice as long as they can justify their claim. In the next lesson,
students learn that 𝒚𝒚 = 𝟏𝟏𝟐𝟐𝒕𝒕𝟐𝟐.

Each distance has 𝟏𝟏𝟐𝟐 as a factor. For example, 𝟏𝟏𝟐𝟐 = 𝟏𝟏(𝟏𝟏𝟐𝟐), 𝟐𝟐𝟒𝟒 = 𝟒𝟒(𝟏𝟏𝟐𝟐), 𝟏𝟏𝟒𝟒𝟒𝟒 = 𝟏𝟏(𝟏𝟏𝟐𝟐), and 𝟐𝟐𝟐𝟐𝟐𝟐 = 𝟏𝟏𝟐𝟐(𝟏𝟏𝟐𝟐).

Closing (4 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 We know that we cannot always assume motion is given at a constant rate.

Exit Ticket (5 minutes)

Lesson Summary

A function is a rule that assigns to each value of one quantity a single value of a second quantity Even though we
might not have a formula for that rule, we see that functions do arise in real-life situations.

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Name Date

Lesson 1: The Concept of a Function

Exit Ticket

A ball is bouncing across the school yard. It hits the ground at (0,0) and bounces up and lands at (1,0) and bounces
again. The graph shows only one bounce.

a. Identify the height of the ball at the following values of 𝑡𝑡: 0, 0.25, 0.5, 0.75, 1.

b. What is the average speed of the ball over the first 0.25 seconds? What is the average speed of the ball over

the next 0.25 seconds (from 0.25 to 0.5 seconds)?

c. Is the height of the ball changing at a constant rate?

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Exit Ticket Sample Solutions

A ball is bouncing across the school yard. It hits the ground at (𝟎𝟎,𝟎𝟎) and bounces up and lands at (𝟏𝟏,𝟎𝟎) and bounces
again. The graph shows only one bounce.

a. Identify the height of the ball at the following time values: 𝟎𝟎, 𝟎𝟎.𝟐𝟐𝟐𝟐, 𝟎𝟎.𝟐𝟐, 𝟎𝟎.𝟕𝟕𝟐𝟐, 𝟏𝟏.

When 𝒕𝒕 = 𝟎𝟎, the height of the ball is 𝟎𝟎 feet above the ground. It has just hit the ground.

When 𝒕𝒕 = 𝟎𝟎.𝟐𝟐𝟐𝟐, the height of the ball is 𝟑𝟑 feet above the ground.

When 𝒕𝒕 = 𝟎𝟎.𝟐𝟐, the height of the ball is 𝟒𝟒 feet above the ground.

When 𝒕𝒕 = 𝟎𝟎.𝟕𝟕𝟐𝟐, the height of the ball is 𝟑𝟑 feet above the ground.

When 𝒕𝒕 = 𝟏𝟏, the height of the ball is 𝟎𝟎 feet above the ground. It has hit the ground again.

b. What is the average speed of the ball over the first 𝟎𝟎.𝟐𝟐𝟐𝟐 seconds? What is the average speed of the ball over
the next 𝟎𝟎.𝟐𝟐𝟐𝟐 seconds (from 𝟎𝟎.𝟐𝟐𝟐𝟐 to 𝟎𝟎.𝟐𝟐 seconds)?

=
𝟑𝟑 − 𝟎𝟎

𝟎𝟎.𝟐𝟐𝟐𝟐 − 𝟎𝟎
=

𝟑𝟑
𝟎𝟎.𝟐𝟐𝟐𝟐

= 𝟏𝟏𝟐𝟐;𝟏𝟏𝟐𝟐 feet per second

=
𝟒𝟒 − 𝟑𝟑

𝟎𝟎.𝟐𝟐−.𝟐𝟐𝟐𝟐
=

𝟏𝟏
𝟎𝟎.𝟐𝟐𝟐𝟐

= 𝟒𝟒;𝟒𝟒 feet per second

c. Is the height of the ball changing at a constant rate?

No, it is not. If the ball were traveling at a constant rate, the average speed would be the same over any time
interval.

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Problem Set Sample Solutions

A ball is thrown across the field from point 𝑨𝑨 to point 𝑩𝑩. It hits the ground at point 𝑩𝑩. The path of the ball is shown in the
diagram below. The 𝒙𝒙-axis shows the horizontal distance the ball travels in feet, and the 𝒚𝒚-axis shows the height of the
ball in feet. Use the diagram to complete parts (a)–(g).

a. Suppose point 𝑨𝑨 is approximately 𝟐𝟐 feet above ground and that at time 𝒕𝒕 = 𝟎𝟎 the ball is at point 𝑨𝑨. Suppose

the length of 𝑶𝑶𝑩𝑩 is approximately 𝟏𝟏𝟏𝟏 feet. Include this information on the diagram.

Information is noted on the diagram in red.

b. Suppose that after 𝟏𝟏 second, the ball is at its highest point of 𝟐𝟐𝟐𝟐 feet (above point 𝑪𝑪) and has traveled a
horizontal distance of 𝟒𝟒𝟒𝟒 feet. What are the approximate coordinates of the ball at the following values of 𝒕𝒕:
𝟎𝟎.𝟐𝟐𝟐𝟐, 𝟎𝟎.𝟐𝟐, 𝟎𝟎.𝟕𝟕𝟐𝟐, 𝟏𝟏, 𝟏𝟏.𝟐𝟐𝟐𝟐, 𝟏𝟏.𝟐𝟐, 𝟏𝟏.𝟕𝟕𝟐𝟐, and 𝟐𝟐.

Most answers will vary because students are approximating the coordinates. The coordinates that must be
correct because enough information was provided are denoted by a *.

At 𝒕𝒕 = 𝟎𝟎.𝟐𝟐𝟐𝟐, the coordinates are approximately (𝟏𝟏𝟏𝟏,𝟏𝟏𝟎𝟎).

At 𝒕𝒕 = 𝟎𝟎.𝟐𝟐, the coordinates are approximately (𝟐𝟐𝟐𝟐,𝟏𝟏𝟏𝟏).

At 𝒕𝒕 = 𝟎𝟎.𝟕𝟕𝟐𝟐, the coordinates are approximately (𝟑𝟑𝟑𝟑,𝟐𝟐𝟎𝟎).

*At 𝒕𝒕 = 𝟏𝟏, the coordinates are approximately (𝟒𝟒𝟒𝟒,𝟐𝟐𝟐𝟐).

At 𝒕𝒕 = 𝟏𝟏.𝟐𝟐𝟐𝟐, the coordinates are approximately (𝟐𝟐𝟐𝟐,𝟏𝟏𝟏𝟏).

At 𝒕𝒕 = 𝟏𝟏.𝟐𝟐, the coordinates are approximately (𝟐𝟐𝟐𝟐,𝟏𝟏𝟒𝟒).

At 𝒕𝒕 = 𝟏𝟏.𝟕𝟕𝟐𝟐, the coordinates are approximately (𝟕𝟕𝟕𝟕,𝟏𝟏).

*At 𝒕𝒕 = 𝟐𝟐, the coordinates are approximately (𝟏𝟏𝟏𝟏,𝟎𝟎).

c. Use your answer from part (b) to write two predictions.

Sample predictions:

At a distance of 𝟒𝟒𝟒𝟒 feet from where the ball was thrown, it is 𝟐𝟐𝟐𝟐 feet in the air. At a distance of 𝟐𝟐𝟐𝟐 feet from
where the ball was thrown, it is 𝟏𝟏𝟒𝟒 feet in the air.

d. What is happening to the ball when it has coordinates (𝟏𝟏𝟏𝟏,𝟎𝟎)?

At point (𝟏𝟏𝟏𝟏,𝟎𝟎), the ball has traveled for 𝟐𝟐 seconds and has hit the ground at a distance of 𝟏𝟏𝟏𝟏 feet from
where the ball began.

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Lesson 1: The Concept of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 1

e. Why do you think the ball is at point (𝟎𝟎,𝟐𝟐) when 𝒕𝒕 = 𝟎𝟎? In other words, why isn’t the height of the ball 𝟎𝟎?

The ball is thrown from point 𝑨𝑨 to point 𝑩𝑩. The fact that the ball is at a height of 𝟐𝟐 feet means that the
person throwing it must have released the ball from a height of 𝟐𝟐 feet.

f. Does the graph allow us to make predictions about the height of the ball at all points?

While we cannot predict exactly, the graph allows us to make approximate predictions of the height for any
value of horizontal distance we choose.

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Lesson 2: Formal Definition of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 2

Lesson 2: Formal Definition of a Function

Student Outcomes

 Students refine their understanding of the definition of a function.
 Students recognize that some, but not all, functions can be described by an equation between two variables.

Lesson Notes
A function is a correspondence between a set (whose elements are called inputs) and another set (whose elements are
called outputs) such that each input corresponds to one and only one output. The correspondence is often given as a
rule (e.g., the output is a number found by substituting an input number into the variable of a one-variable expression
and evaluating). Students develop here their intuitive definition of a function as a rule that assigns to each element of
one set of objects one, and only one, element from a second set of objects. Refinement of this definition, function
notation, and detailed attention to the domain and range of functions are all left to the high school work of standards
F-IF.A.1 and F-IF.B.5.

We begin this lesson by looking at a troublesome set of data values.

Classwork

Opening (3 minutes)

 Shown below is the table from Example 2 of the last lesson and another table of values for the alleged motion
of a second moving object. Make some comments about any troublesome features you observe in the second
table of values. Does the first table of data have these troubles too?

Number of
seconds (𝒙𝒙)

Distance traveled
in feet (𝒚𝒚)

Number of
seconds (𝒙𝒙)

Distance traveled
in feet (𝒚𝒚)

0.5 4 0.5 4
1 16 1 4

1.5 36 1 36
2 64 2 64

2.5 100 2.5 80
3 144 3 99

3.5 196 3 196
4 256 4 256

Allow students to share their thoughts about the differences between the two tables. Then proceed with the discussion
that follows.

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Lesson 2: Formal Definition of a Function

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Discussion (8 minutes)

 Consider the object following the motion described on the left table. How far did it travel during the first
second?
 After 1 second, the object traveled 16 feet.

 Consider the object following the motion described in the right table. How far did it travel during the first
second?

 It is unclear. After 1 second, the table indicates that the object traveled 4 feet and it also indicates that
it traveled 36 feet.

 Which of the two tables above allows us to make predictions with some accuracy? Explain.
 The table on the left seems like it would be more accurate. The table on the right gives two completely

different distances for the stone after 1 second. We cannot make an accurate prediction because after
1 second, the stone may either be 4 feet from where it started or 36 feet.

 In the last lesson we defined a function to be a rule that assigns to each value of one quantity one, and only
one, value of a second quantity. The right table does not follow this definition: it is assigning two different
values, 4 feet and 36 feet, to the same time of 1 second. For the sake of meaningful discussion in a real-world
situation this is problematic.

 Let’s formalize this idea of assignment for the example of a falling stone from the last lesson. It seems more
natural to use the symbol 𝐷𝐷, for distance, for the function that assigns to each time the distance the object has
fallen by that time. So here 𝐷𝐷 is a rule that assigns to each number 𝑡𝑡 (with 0 ≤ 𝑡𝑡 ≤ 4) another number, the
distance of the fall of the stone in 𝑡𝑡 seconds. Here is the table from the last lesson.

Number of
seconds (𝒕𝒕)

Distance traveled
in feet (𝑫𝑫)

0.5 4
1 16

1.5 36
2 64

2.5 100
3 144

3.5 196
4 256

 We can interpret this table explicitly as a function rule:

𝐷𝐷 assigns the value 4 to the value 0.5.
𝐷𝐷 assigns the value 16 to the value 1.
𝐷𝐷 assigns the value 36 to the value 1.5.
𝐷𝐷 assigns the value 64 to the value 2.
𝐷𝐷 assigns the value 100 to the value 2.5.
𝐷𝐷 assigns the value 144 to the value 3.
𝐷𝐷 assigns the value 196 to the value 3.5.
𝐷𝐷 assigns the value 256 to the value 4.

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Lesson 2: Formal Definition of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 2

 If you like, you can think of this as an input–output machine. That is, we put in a number for the time (the
input), and out comes another number (the output) that tells us the distance traveled in feet up to that time.

 With the example of the falling stone, what are we inputting?

 The input would be the time, in seconds, between 0 and 4 seconds.
 What is the output?

 The output is the distance, in feet, the stone traveled up to that time.

 If we input 3 into the machine, what is the output?

 The output is 144.

 If we input 1.5 into the machine, what is the output?

 The output is 36.
 Of course, with this particular machine, we are limited to inputs in the range of 0 to 4 because we are

inputting times 𝑡𝑡 during which the stone was falling.

 We are lucky with the function 𝐷𝐷: Sir Isaac Newton (1643–1727) studied the motion of objects falling under
gravity and established a formula for their motion. It is given by 𝐷𝐷 = 16𝑡𝑡2, that is the distance traveled over
time interval 𝑡𝑡 is 16𝑡𝑡2. We can see that it fits our data values. Not all functions have equations describing
them.

Time in seconds

1 2 3 4

Distance stone fell by that
time in feet

16 64 144 256

 Functions can be represented in a variety of ways. At this point, we have seen the function that describes the
distance traveled by the stone pictorially (from Lesson 1, Example 2), as a table of values, and as a rule
described in words or as a mathematical equation.

Scaffolding:

Highlighting the components of
the words input and output and
exploring how the words
describe related concepts
would be useful.

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Lesson 2: Formal Definition of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 2

Exercise 1 (5 minutes)

Have students verify that 𝐷𝐷 = 16𝑡𝑡2 does indeed match the data values of Example 1 by completing this next exercise.
To expedite the verification, allow the use of calculators.

Exercises 1–5

1. Let 𝑫𝑫 be the distance traveled in time 𝒕𝒕. Use the equation 𝑫𝑫 = 𝟏𝟏𝟏𝟏𝒕𝒕𝟐𝟐 to calculate the distance the stone dropped for
the given time 𝒕𝒕.

Time in seconds
𝟎𝟎.𝟓𝟓 𝟏𝟏 𝟏𝟏.𝟓𝟓 𝟐𝟐 𝟐𝟐.𝟓𝟓 𝟑𝟑 𝟑𝟑.𝟓𝟓 𝟒𝟒

Distance stone fell in feet
by that time 𝟒𝟒 𝟏𝟏𝟏𝟏 𝟑𝟑𝟏𝟏 𝟏𝟏𝟒𝟒 𝟏𝟏𝟎𝟎𝟎𝟎 𝟏𝟏𝟒𝟒𝟒𝟒 𝟏𝟏𝟏𝟏𝟏𝟏 𝟐𝟐𝟓𝟓𝟏𝟏

a. Are the distances you calculated equal to the table from Lesson 1?

Yes

b. Does the function 𝑫𝑫 = 𝟏𝟏𝟏𝟏𝒕𝒕𝟐𝟐 accurately represent the distance the stone fell after a given time 𝒕𝒕? In other
words, does the function described by this rule assign to 𝒕𝒕 the correct distance? Explain.

Yes, the function accurately represents the distance the stone fell after the given time interval. Each
computation using the function resulted in the correct distance. Therefore, the function assigns to 𝒕𝒕 the
correct distance.

Discussion (10 minutes)

 Being able to write a formula for the function has superb implications—it is predictive. That is, we can predict
what will happen each time a stone is released from a height of 256 feet. The equation describing the
function makes it possible for us to know exactly how many feet the stone will fall for a time 𝑡𝑡 as long as we
select a 𝑡𝑡 so that 0 ≤ 𝑡𝑡 ≤ 4.

 Not every function can be expressed as a formula, however. For example, consider the function 𝐻𝐻 which
assigns to each moment since you were born your height at that time. This is a function (Can you have two
different heights at the same moment?), but it is very unlikely that there is a formula detailing your height over
time.

 A function is a rule that assigns to each value of one quantity exactly one value of a second quantity. A
function is a correspondence between a set of inputs and a set of outputs such that each input corresponds to
one and only one output.

Note: Sometimes the phrase exactly one is used instead of one and only one. Both phrases mean the same thing; that is,
an input with no corresponding output is unacceptable, and an input corresponding to several outputs is also
unacceptable.

 Let’s examine the definition of function more closely: For every input, there is one and only one output. Can
you think of why the phrase one and only one (or exactly one) must be included in the definition?

 We don’t want an input-output machine that gives different output each time you put in the same
input.

MP.6

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Lesson 2: Formal Definition of a Function

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 Most of the time in Grade 8, the correspondence is given by a rule, which can also be considered a set of
instructions used to determine the output for any given input. For example, a common rule is to substitute a
number into the variable of a one-variable expression and evaluating. When a function is given by such a rule
or formula, we often say that function is a rule that assigns to each input exactly one output.

 Is it clear that our function 𝐷𝐷, the rule that assigns to each time 𝑡𝑡 satisfying 0 ≤ 𝑡𝑡 ≤ 4 the distance the object
has fallen by that time, satisfies this condition of being a function?

Provide time for students to consider the phrase. Allow them to talk in pairs or small groups perhaps and then share
their thoughts with the class. Use the question below, if necessary. Then resume the discussion.

 Using our stone-dropping example, if 𝐷𝐷 assigns 64 to 2—that is, the function assigns 64 feet to the time 2
seconds—would it be possible for 𝐷𝐷 to assign 65 to 2 as well? Explain.
 It would not be possible for 𝐷𝐷 to assign 64 and 65 to 2. The reason is that we are talking about a stone

dropping. How could the stone drop 64 feet in 2 seconds and 65 feet in 2 seconds? The stone cannot
be in two places at once.

 When given a formula for a function, we need to be careful of its context. For example, with our falling stone
we have the formula 𝐷𝐷 = 16𝑡𝑡2 describing the function. This formula holds for all values of time 𝑡𝑡 with
0 ≤ 𝑡𝑡 ≤ 4. But it is also possible to put the value 𝑡𝑡 = −2 into this formula and compute a supposed value of
𝐷𝐷:

𝐷𝐷 = 16(−2)2
= 16(4)

= 64

Does this mean that for the two seconds before the stone was dropped it had fallen 64 feet? Of course not.
We could also compute, for 𝑡𝑡 = 5:

𝐷𝐷 = 16(5)2
= 16(25)
= 400

 What is wrong with this statement?

 It would mean that the stone dropped 400 feet in 5 seconds, but the stone was dropped from a height
of 256 feet. It makes no sense.

 To summarize, a function is a rule that assigns to each value of one quantity (an input) exactly one value to a
second quantity (the matching output). Additionally, we should always consider the context when working
with a function to make sure our answers makes sense: If a function is described by a formula, then we can
only consider values to insert into that formula relevant to the context.

MP.6

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Lesson 2: Formal Definition of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 2

Exercises 2–5 (10 minutes)

Students work independently to complete Exercises 2–5.

2. Can the table shown below represent values of a function? Explain.

Input
(𝒙𝒙) 𝟏𝟏 𝟑𝟑 𝟓𝟓 𝟓𝟓 𝟏𝟏

Output
(𝒚𝒚) 𝟕𝟕 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟐𝟐𝟎𝟎 𝟐𝟐𝟐𝟐

No, the table cannot represent a function because the input of 𝟓𝟓 has two different outputs. Functions assign only
one output to each input.

3. Can the table shown below represent values of a function? Explain.

Input
(𝒙𝒙) 𝟎𝟎.𝟓𝟓 𝟕𝟕 𝟕𝟕 𝟏𝟏𝟐𝟐 𝟏𝟏𝟓𝟓

Output
(𝒚𝒚) 𝟏𝟏 𝟏𝟏𝟓𝟓 𝟏𝟏𝟎𝟎 𝟐𝟐𝟑𝟑 𝟑𝟑𝟎𝟎

No, the table cannot represent a function because the input of 𝟕𝟕 has two different outputs. Functions assign only
one output to each input.

4. Can the table shown below represent values of a function? Explain.

Input
(𝒙𝒙) 𝟏𝟏𝟎𝟎 𝟐𝟐𝟎𝟎 𝟓𝟓𝟎𝟎 𝟕𝟕𝟓𝟓 𝟏𝟏𝟎𝟎

Output
(𝒚𝒚) 𝟑𝟑𝟐𝟐 𝟑𝟑𝟐𝟐 𝟏𝟏𝟓𝟓𝟏𝟏 𝟐𝟐𝟒𝟒𝟎𝟎 𝟐𝟐𝟐𝟐𝟐𝟐

Yes, the table can represent a function. Even though there are two outputs that are the same, each input has only
one output.

5. It takes Josephine 𝟑𝟑𝟒𝟒 minutes to complete her homework assignment of 𝟏𝟏𝟎𝟎 problems. If we assume that she works
at a constant rate, we can describe the situation using a function.

a. Predict how many problems Josephine can complete in 𝟐𝟐𝟓𝟓 minutes.

Answers will vary.

b. Write the two-variable linear equation that represents Josephine’s constant rate of work.

Let 𝒚𝒚 be the number of problems she can complete in 𝒙𝒙 minutes.

𝟏𝟏𝟎𝟎
𝟑𝟑𝟒𝟒

=
𝒚𝒚
𝒙𝒙

𝒚𝒚 =
𝟏𝟏𝟎𝟎
𝟑𝟑𝟒𝟒

𝒙𝒙

𝒚𝒚 =
𝟓𝟓
𝟏𝟏𝟕𝟕

𝒙𝒙

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Lesson 2: Formal Definition of a Function

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c. Use the equation you wrote in part (b) as the formula for the function to complete the table below. Round
your answers to the hundredths place.

Time taken to
complete problems

(𝒙𝒙)
𝟓𝟓 𝟏𝟏𝟎𝟎 𝟏𝟏𝟓𝟓 𝟐𝟐𝟎𝟎 𝟐𝟐𝟓𝟓

Number of problems
completed

(𝒚𝒚)
𝟏𝟏.𝟒𝟒𝟕𝟕 𝟐𝟐.𝟏𝟏𝟒𝟒 𝟒𝟒.𝟒𝟒𝟏𝟏 𝟓𝟓.𝟐𝟐𝟐𝟐 𝟕𝟕.𝟑𝟑𝟓𝟓

After 𝟓𝟓 minutes, Josephine was able to complete 𝟏𝟏.𝟒𝟒𝟕𝟕 problems, which means that she was able to complete
𝟏𝟏 problem, then get about halfway through the next problem.

d. Compare your prediction from part (a) to the number you found in the table above.

Answers will vary.

e. Use the formula from part (b) to compute the number of problems completed when 𝒙𝒙 = −𝟕𝟕. Does your
answer make sense? Explain.

𝒚𝒚 =
𝟓𝟓
𝟏𝟏𝟕𝟕

(−𝟕𝟕)

= −𝟐𝟐.𝟎𝟎𝟏𝟏

No, the answer does not make sense in terms of the situation. The answer means that Josephine can
complete −𝟐𝟐.𝟎𝟎𝟏𝟏 problems in −𝟕𝟕 minutes. This obviously does not make sense.

f. For this problem, we assumed that Josephine worked at a constant rate. Do you think that is a reasonable
assumption for this situation? Explain.

It does not seem reasonable to assume constant rate for this situation. Just because Josephine was able to
complete 𝟏𝟏𝟎𝟎 problems in 𝟑𝟑𝟒𝟒 minutes does not necessarily mean she spent the exact same amount of time on
each problem. For example, it may have taken her 𝟐𝟐𝟎𝟎 minutes to do 𝟏𝟏 problem and then 𝟏𝟏𝟒𝟒 minutes total to
finish the remaining 𝟏𝟏 problems.

Closing (4 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 We know that a function is a rule that assigns to each value of one quantity (an input) exactly one value of a
second quantity (its matching output).

 Not every function can be described by a mathematical formula.
 If we can describe a function by a mathematical formula, we must still be careful of context. For example,

asking for the distance a stone drops in −2 seconds is meaningless.

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Lesson 2: Formal Definition of a Function

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Exit Ticket (5 minutes)

Lesson Summary

A function is a correspondence between a set (whose elements are called inputs) and another set (whose elements
are called outputs) such that each input corresponds to one and only one output.

Sometimes the phrase exactly one output is used instead of one and only one output in the definition of function
(they mean the same thing). Either way, it is this fact, that there is one and only one output for each input, which
makes functions predictive when modeling real life situations.

Furthermore, the correspondence in a function is often given by a rule (or formula). For example, the output is
equal to the number found by substituting an input number into the variable of a one-variable expression and
evaluating.

Functions are sometimes described as an input–output machine. For example, given a function 𝑫𝑫, the input is time
𝒕𝒕, and the output is the distance traveled in 𝒕𝒕 seconds.

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Lesson 2: Formal Definition of a Function

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Name Date

Lesson 2: Formal Definition of a Function

Exit Ticket

1. Can the table shown below represent values of a function? Explain.

Input
(𝒙𝒙)

10 20 30 40 50

Output
(𝒚𝒚) 32 64 96 64 32

2. Kelly can tune 4 cars in 3 hours. If we assume he works at a constant rate, we can describe the situation using a
function.

a. Write the function that represents Kelly’s constant rate of work.

b. Use the function you wrote in part (a) as the formula for the function to complete the table below. Round
your answers to the hundredths place.

Time spent
tuning cars (𝒙𝒙)

2 3 4 6 7

Number of cars
tuned up (𝒚𝒚)

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Lesson 2: Formal Definition of a Function

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c. Kelly works 8 hours per day. According to this work, how many cars will he finish tuning at the end of a shift?

d. For this problem, we assumed that Kelly worked at a constant rate. Do you think that is a reasonable

assumption for this situation? Explain.

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Lesson 2: Formal Definition of a Function

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Exit Ticket Sample Solutions

1. Can the table shown below represent values of a function? Explain.

Input
(𝒙𝒙) 𝟏𝟏𝟎𝟎 𝟐𝟐𝟎𝟎 𝟑𝟑𝟎𝟎 𝟒𝟒𝟎𝟎 𝟓𝟓𝟎𝟎

Output
(𝒚𝒚) 𝟑𝟑𝟐𝟐 𝟏𝟏𝟒𝟒 𝟏𝟏𝟏𝟏 𝟏𝟏𝟒𝟒 𝟑𝟑𝟐𝟐

Yes, the table can represent a function. Each input has exactly one output.

2. Kelly can tune 𝟒𝟒 cars in 𝟑𝟑 hours. If we assume he works at a constant rate, we can describe the situation using a
function.

a. Write the function that represents Kelly’s constant rate of work.

Let 𝒚𝒚 represent the number of cars Kelly can tune up in 𝒙𝒙 hours; then

𝒚𝒚
𝒙𝒙

=
𝟒𝟒
𝟑𝟑

𝒚𝒚 =
𝟒𝟒
𝟑𝟑
𝒙𝒙

b. Use the function you wrote in part (a) as the formula for the function to complete the table below. Round
your answers to the hundredths place.

Time spent tuning
cars (𝒙𝒙) 𝟐𝟐 𝟑𝟑 𝟒𝟒 𝟏𝟏 𝟕𝟕

Number of cars
tuned up (𝒚𝒚) 𝟐𝟐.𝟏𝟏𝟕𝟕 𝟒𝟒 𝟓𝟓.𝟑𝟑𝟑𝟑 𝟐𝟐 𝟏𝟏.𝟑𝟑𝟑𝟑

c. Kelly works 𝟐𝟐 hours per day. According to this work, how many cars will he finish tuning at the end of a shift?

Using the function, Kelly will tune up 𝟏𝟏𝟎𝟎.𝟏𝟏𝟕𝟕 cars at the end of his shift. That means he will finish tuning up
𝟏𝟏𝟎𝟎 cars and begin tuning up the 𝟏𝟏𝟏𝟏th car.

d. For this problem, we assumed that Kelly worked at a constant rate. Do you think that is a reasonable
assumption for this situation? Explain.

No, it does not seem reasonable to assume a constant rate for this situation. Just because Kelly tuned up 𝟒𝟒
cars in 𝟑𝟑 hours does not mean he spent the exact same amount of time on each car. One car could have taken
𝟏𝟏 hour, while the other three could have taken 𝟐𝟐 hours total.

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Lesson 2: Formal Definition of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 2

Problem Set Sample Solutions

1. The table below represents the number of minutes Francisco spends at the gym each day for a week. Does the data
shown below represent values of a function? Explain.

Day
(𝒙𝒙) 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒 𝟓𝟓 𝟏𝟏 𝟕𝟕

Time in minutes
(𝒚𝒚) 𝟑𝟑𝟓𝟓 𝟒𝟒𝟓𝟓 𝟑𝟑𝟎𝟎 𝟒𝟒𝟓𝟓 𝟑𝟑𝟓𝟓 𝟎𝟎 𝟎𝟎

Yes, the table can represent a function because each input has a unique output. For example, on day 𝟏𝟏, Francisco
was at the gym for 𝟑𝟑𝟓𝟓 minutes.

2. Can the table shown below represent values of a function? Explain.

Input
(𝒙𝒙) 𝟏𝟏 𝟐𝟐 𝟕𝟕 𝟐𝟐 𝟏𝟏

Output
(𝒚𝒚) 𝟏𝟏𝟏𝟏 𝟏𝟏𝟓𝟓 𝟏𝟏𝟏𝟏 𝟐𝟐𝟒𝟒 𝟐𝟐𝟐𝟐

No, the table cannot represent a function because the input of 𝟏𝟏 has two different outputs, and so does the input of
𝟐𝟐. Functions assign only one output to each input.

3. Olivia examined the table of values shown below and stated that a possible rule to describe this function could be
𝒚𝒚 = −𝟐𝟐𝒙𝒙 + 𝟏𝟏. Is she correct? Explain.

Input
(𝒙𝒙) −𝟒𝟒 𝟎𝟎 𝟒𝟒 𝟐𝟐 𝟏𝟏𝟐𝟐 𝟏𝟏𝟏𝟏 𝟐𝟐𝟎𝟎 𝟐𝟐𝟒𝟒

Output
(𝒚𝒚) 𝟏𝟏𝟕𝟕 𝟏𝟏 𝟏𝟏 −𝟕𝟕 −𝟏𝟏𝟓𝟓 −𝟐𝟐𝟑𝟑 −𝟑𝟑𝟏𝟏 −𝟑𝟑𝟏𝟏

Yes, Olivia is correct. When the rule is used with each input, the value of the output is exactly what is shown in the
table. Therefore, the rule for this function could well be 𝒚𝒚 = −𝟐𝟐𝒙𝒙 + 𝟏𝟏.

4. Peter said that the set of data in part (a) describes a function, but the set of data in part (b) does not. Do you agree?
Explain why or why not.

Input
(𝒙𝒙) 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒 𝟓𝟓 𝟏𝟏 𝟕𝟕 𝟐𝟐

Output
(𝒚𝒚) 𝟐𝟐 𝟏𝟏𝟎𝟎 𝟑𝟑𝟐𝟐 𝟏𝟏 𝟏𝟏𝟎𝟎 𝟐𝟐𝟕𝟕 𝟏𝟏𝟓𝟓𝟏𝟏 𝟒𝟒

Input
(𝒙𝒙) −𝟏𝟏 −𝟏𝟏𝟓𝟓 −𝟏𝟏 −𝟑𝟑 −𝟐𝟐 −𝟑𝟑 𝟐𝟐 𝟏𝟏

Output
(𝒚𝒚) 𝟎𝟎 −𝟏𝟏 𝟐𝟐 𝟏𝟏𝟒𝟒 𝟏𝟏 𝟐𝟐 𝟏𝟏𝟏𝟏 𝟒𝟒𝟏𝟏

Peter is correct. The table in part (a) fits the definition of a function. That is, there is exactly one output for each
input. The table in part (b) cannot be a function. The input −𝟑𝟑 has two outputs, 𝟏𝟏𝟒𝟒 and 𝟐𝟐. This contradicts the
definition of a function; therefore, it is not a function.

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Lesson 2: Formal Definition of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 2

5. A function can be described by the rule 𝒚𝒚 = 𝒙𝒙𝟐𝟐 + 𝟒𝟒. Determine the corresponding output for each given input.

Input
(𝒙𝒙) −𝟑𝟑 −𝟐𝟐 −𝟏𝟏 𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒

Output
(𝒚𝒚) 𝟏𝟏𝟑𝟑 𝟐𝟐 𝟓𝟓 𝟒𝟒 𝟓𝟓 𝟐𝟐 𝟏𝟏𝟑𝟑 𝟐𝟐𝟎𝟎

6. Examine the data in the table below. The inputs and outputs represent a situation where constant rate can be
assumed. Determine the rule that describes the function.

Input
(𝒙𝒙) −𝟏𝟏 𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒 𝟓𝟓 𝟏𝟏

Output
(𝒚𝒚) 𝟑𝟑 𝟐𝟐 𝟏𝟏𝟑𝟑 𝟏𝟏𝟐𝟐 𝟐𝟐𝟑𝟑 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟐𝟐

The rule that describes this function is 𝒚𝒚 = 𝟓𝟓𝒙𝒙 + 𝟐𝟐.

7. Examine the data in the table below. The inputs represent the number of bags of candy purchased, and the outputs
represent the cost. Determine the cost of one bag of candy, assuming the price per bag is the same no matter how
much candy is purchased. Then, complete the table.

Bags of
Candy

(𝒙𝒙)
𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒 𝟓𝟓 𝟏𝟏 𝟕𝟕 𝟐𝟐

Cost in
Dollars

(𝒚𝒚)
𝟏𝟏.𝟐𝟐𝟓𝟓 𝟐𝟐.𝟓𝟓𝟎𝟎 𝟑𝟑.𝟕𝟕𝟓𝟓 𝟓𝟓.𝟎𝟎𝟎𝟎 𝟏𝟏.𝟐𝟐𝟓𝟓 𝟕𝟕.𝟓𝟓𝟎𝟎 𝟐𝟐.𝟕𝟕𝟓𝟓 𝟏𝟏𝟎𝟎.𝟎𝟎𝟎𝟎

a. Write the rule that describes the function.

𝒚𝒚 = 𝟏𝟏.𝟐𝟐𝟓𝟓𝒙𝒙

b. Can you determine the value of the output for an input of 𝒙𝒙 = −𝟒𝟒? If so, what is it?

When 𝒙𝒙 = −𝟒𝟒, the output is −𝟓𝟓.

c. Does an input of −𝟒𝟒 make sense in this situation? Explain.

No, an input of −𝟒𝟒 does not make sense for the situation. It would mean −𝟒𝟒 bags of candy. You cannot
purchase −𝟒𝟒 bags of candy.

8. Each and every day a local grocery store sells 𝟐𝟐 pounds of bananas for $𝟏𝟏.𝟎𝟎𝟎𝟎. Can the cost of 2 pounds of bananas
be represented as a function of the day of the week? Explain.

Yes, this situation can be represented by a function. Assign to each day of the week the value $1.00.

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Lesson 2: Formal Definition of a Function

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 2

9. Write a brief explanation to a classmate who was absent today about why the table in part (a) is a function and the
table in part (b) is not.

Input
(𝒙𝒙) −𝟏𝟏 −𝟐𝟐 −𝟑𝟑 −𝟒𝟒 𝟒𝟒 𝟑𝟑 𝟐𝟐 𝟏𝟏

Output
(𝒚𝒚) 𝟐𝟐𝟏𝟏 𝟏𝟏𝟎𝟎𝟎𝟎 𝟑𝟑𝟐𝟐𝟎𝟎 𝟒𝟒𝟎𝟎𝟎𝟎 𝟒𝟒𝟎𝟎𝟎𝟎 𝟑𝟑𝟐𝟐𝟎𝟎 𝟏𝟏𝟎𝟎𝟎𝟎 𝟐𝟐𝟏𝟏

Input
(𝒙𝒙) 𝟏𝟏 𝟏𝟏 −𝟏𝟏 −𝟐𝟐 𝟏𝟏 −𝟏𝟏𝟎𝟎 𝟐𝟐 𝟏𝟏𝟒𝟒

Output
(𝒚𝒚) 𝟐𝟐 𝟏𝟏 −𝟒𝟒𝟕𝟕 −𝟐𝟐 𝟏𝟏𝟏𝟏 −𝟐𝟐 𝟏𝟏𝟓𝟓 𝟑𝟑𝟏𝟏

The table in part (a) is a function because each input has exactly one output. This is different from the information in
the table in part (b). Notice that the input of 𝟏𝟏 has been assigned two different values. The input of 𝟏𝟏 is assigned 𝟐𝟐
and 𝟏𝟏𝟏𝟏. Because the input of 𝟏𝟏 has more than one output, this table cannot represent a function.

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Lesson 3: Linear Functions and Proportionality

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 3

Scaffolding:
In addition to explanations
about functions, it may be
useful for students to have a
series of structured
experiences with real-world
objects and data to reinforce
their understanding of a
function. An example is
experimenting with different
numbers of batches of a given
recipe; students can observe
the effect of the number of
batches on quantities of
various ingredients.

Lesson 3: Linear Functions and Proportionality

Student Outcomes

 Students realize that linear equations of the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 can be seen as rules defining functions
(appropriately called linear functions).

 Students explore examples of linear functions.

Classwork

Example 1 (7 minutes)

Example 1

In the last lesson, we looked at several tables of values showing the inputs and outputs of functions. For instance, one
table showed the costs of purchasing different numbers of bags of candy:

Bags of candy
(𝒙𝒙) 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒 𝟓𝟓 𝟔𝟔 𝟕𝟕 𝟖𝟖

Cost in Dollars
(𝒚𝒚) 𝟏𝟏.𝟐𝟐𝟓𝟓 𝟐𝟐.𝟓𝟓𝟓𝟓 𝟑𝟑.𝟕𝟕𝟓𝟓 𝟓𝟓.𝟓𝟓𝟓𝟓 𝟔𝟔.𝟐𝟐𝟓𝟓 𝟕𝟕.𝟓𝟓𝟓𝟓 𝟖𝟖.𝟕𝟕𝟓𝟓 𝟏𝟏𝟓𝟓.𝟓𝟓𝟓𝟓

 What do you think a linear function is?

 A linear function is likely a function with a rule described by a linear equation. Specifically, the rate of
change in the situation being described is constant, and the graph of the equation is a line.

 Do you think this is a linear function? Justify your answer.

 Yes, this is a linear function because there is a proportional relationship:
10.00

8
= 1.25; $1.25 per each bag of candy

5.00
4

= 1.25; $1.25 per each bag of candy

2.50
2

= 1.25; $1.25 per each bag of candy

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Lesson 3: Linear Functions and Proportionality

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 3

 The total cost is increasing at a rate of $1.25 with each bag of candy. Further justification comes from
the graph of the data shown below.

 A linear function is a function with a rule that can be described by a linear equation. That is, if we use 𝑚𝑚 to

denote an input of the function and 𝑦𝑦 its matching output, then the function is linear if the rule for the
function can be described by the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 for some numbers 𝑚𝑚 and 𝑏𝑏.

 What rule or equation describes our cost function for bags of candy?

 The rule that represents the function is then 𝑦𝑦 = 1.25𝑚𝑚.
 Notice that the constant 𝑚𝑚 is 1.25, which is the cost of one bag of candy, and the constant 𝑏𝑏 is 0. Also notice

that the constant 𝑚𝑚 was found by calculating the unit rate for a bag of candy.

 No matter the value of 𝑚𝑚 chosen, as long as 𝑚𝑚 is a nonnegative integer, the rule 𝑦𝑦 = 1.25𝑚𝑚 gives the cost of
purchasing that many bags of candy. The total cost of candy is a function of the number of bags purchased.

 Why must we set 𝑚𝑚 as a nonnegative integer for this function?
 Since 𝑚𝑚 represents the number of bags of candy, it does not make sense that there would be a negative

number of bags. It is also unlikely that we might be allowed to buy fractional bags of candy, and so we
require 𝑚𝑚 to be a whole number.

 Would you say that the table represents all possible inputs and outputs? Explain.

 No, it does not represent all possible inputs and outputs. Someone can purchase more than 8 bags of
candy, and inputs greater than 8 are not represented by this table (unless the store has a limit on the
number of bags one may purchase, perhaps).

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Lesson 3: Linear Functions and Proportionality

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 3

 As a matter of precision we say that “this function has the above table of values” instead of “the table above
represents a function” because not all values of the function might be represented by the table. Any rule that
describes a function usually applies to all of the possible values of a function. For example, in our candy
example, we can determine the cost for 9 bags of candy using our equation for the function even though no
column is shown in the table for when 𝑚𝑚 is 9. However, if for some reason our interest was in a function with
only the input values 1, 2, 3, 4, 5, 6, 7, and 8, then our table gives complete information about the function and
so fully represents the function.

Example 2 (4 minutes)

Example 2

Walter walks at a constant speed of 𝟖𝟖 miles every 𝟐𝟐 hours. Describe a linear function for the number of miles he walks in
𝒙𝒙 hours. What is a reasonable range of 𝒙𝒙-values for this function?

 Consider the following rate problem: Walter walks at a constant speed of 8 miles every 2 hours. Describe a
linear function for the number of miles he walks in 𝑚𝑚 hours. What is a reasonable range of 𝑚𝑚-values for this
function?

 Walter’s average speed of walking 8 miles is
8
2

= 4, or 4 miles per hour.

 We have 𝑦𝑦 = 4𝑚𝑚, where 𝑦𝑦 is the distance walked in 𝑚𝑚 hours. It seems
reasonable to say that 𝑚𝑚 is any real number between 0 and 20, perhaps? (Might
there be a cap on the number of hours he walks? Perhaps we are counting up
the number of miles he walks over a lifetime?)

 In the last example, the total cost of candy was a function of the number of bags
purchased. Describe the function in this walking example.

 The distance that Walter travels is a function of the number of hours he spends walking.

 What limitations did we put on 𝑚𝑚?
We must insist that 𝑚𝑚 ≥ 0. Since 𝑚𝑚 represents the time Walter walks, then it makes sense that he would
walk for a positive amount of time or no time at all.

 Since 𝑚𝑚 is positive, then we know that the distance 𝑦𝑦 will also be positive.

Example 3 (4 minutes)

Example 3

Veronica runs at a constant speed. The distance she runs is a function of the time she spends running. The function has
the table of values shown below.

Time in minutes
(𝒙𝒙) 𝟖𝟖 𝟏𝟏𝟔𝟔 𝟐𝟐𝟒𝟒 𝟑𝟑𝟐𝟐

Distance run in miles
(𝒚𝒚) 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝟒𝟒

Scaffolding:
As the language becomes more
abstract, it can be useful to use
visuals and even pantomime
situations related to speed,
rate, etc.

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Lesson 3: Linear Functions and Proportionality

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 3

 Since Veronica runs at a constant speed, we know that her average speed over any time interval will be the
same. Therefore, Veronica’s distance function is a linear function. Write the equation that describes her
distance function.

 The function that represents Veronica’s distance is described by the equation 𝑦𝑦 = 1
8 𝑚𝑚, where 𝑦𝑦 is the

distance in miles Veronica runs in 𝑚𝑚 minutes and 𝑚𝑚,𝑦𝑦 ≥ 0.

 Describe the function in terms of distance and time.

 The distance that Veronica runs is a function of the number of minutes she spends running.

Example 4 (5 minutes)

Example 4

Water flows from a faucet into a bathtub at the constant rate of 𝟕𝟕 gallons of water pouring out every 𝟐𝟐 minutes. The
bathtub is initially empty, and its plug is in. Determine the rule that describes the volume of water in the tub as a
function of time. If the tub can hold 𝟓𝟓𝟓𝟓 gallons of water, how long will it take to fill the tub?

The rate of water flow is 𝟕𝟕
𝟐𝟐

, or 𝟑𝟑.𝟓𝟓 gallons per minute. Then the rule that describes the volume of water in the tub as a

function of time is 𝒚𝒚 = 𝟑𝟑.𝟓𝟓𝒙𝒙, where 𝒚𝒚 is the volume of water, and 𝒙𝒙 is the number of minutes the faucet has been on.

To find the time when 𝒚𝒚 = 𝟓𝟓𝟓𝟓, we need to look at the equation 𝟓𝟓𝟓𝟓 = 𝟑𝟑.𝟓𝟓𝒙𝒙. This gives 𝒙𝒙 = 𝟓𝟓𝟓𝟓
𝟑𝟑.𝟓𝟓 = 𝟏𝟏𝟒𝟒.𝟐𝟐𝟖𝟖𝟓𝟓𝟕𝟕… ≈ 𝟏𝟏𝟒𝟒 . It

will take about 𝟏𝟏𝟒𝟒 minutes to fill the tub.

 Assume that the faucet is filling a bathtub that can hold 50 gallons of water. How long will it take the faucet to
fill the tub?

 Since we want the total volume to be 50 gallons, then
50 = 3.5𝑚𝑚
50
3.5

= 𝑚𝑚

14.2857 … = 𝑚𝑚

14 ≈ 𝑚𝑚

It will take about 14 minutes to fill a tub that has a volume of 50 gallons.

Now assume that you are filling the same 𝟓𝟓𝟓𝟓-gallon bathtub with water flowing in at the constant rate of 𝟑𝟑.𝟓𝟓 gallons per
minute, but there were initially 8 gallons of water in the tub. Will it still take about 𝟏𝟏𝟒𝟒 minutes to fill the tub?

No. It will take less time because there is already some water in the tub.

 What now is the appropriate equation describing the volume of water in the tub as a function of time?

 If 𝑦𝑦 is the volume of water that flows from the faucet, and 𝑚𝑚 is the number of minutes the faucet has
been on, then 𝑦𝑦 = 3.5𝑚𝑚 + 8.

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 How long will it take to fill the tub according to this equation?

 Since we still want the total volume of the tub to be 50 gallons, then:

50 = 3.5𝑚𝑚 + 8
42 = 3.5𝑚𝑚
12 = 𝑚𝑚

It will take 12 minutes for the faucet to fill a 50-gallon tub when 8 gallons are already in it.

(Be aware that some students may observe that we can use the previous function rule 𝑦𝑦 = 3.5𝑚𝑚 to answer this
question by noting that we need to add only 42 more gallons to the tub. This will lead directly to the equation
42 = 3.5𝑚𝑚.)

 Generate a table of values for this function:

Time in minutes
(𝒙𝒙) 𝟓𝟓 𝟑𝟑 𝟔𝟔 𝟗𝟗 𝟏𝟏𝟐𝟐

Total volume in tub in gallons
(𝒚𝒚) 𝟖𝟖 𝟏𝟏𝟖𝟖.𝟓𝟓 𝟐𝟐𝟗𝟗 𝟑𝟑𝟗𝟗.𝟓𝟓 𝟓𝟓𝟓𝟓

Example 5 (7 minutes)

Example 5

Water flows from a faucet at a constant rate. Assume that 𝟔𝟔 gallons of water are already in a tub by the time we notice
the faucet is on. This information is recorded in the first column of the table below. The other columns show how many
gallons of water are in the tub at different numbers of minutes since we noticed the running faucet.

Time in minutes
(𝒙𝒙) 𝟓𝟓 𝟑𝟑 𝟓𝟓 𝟗𝟗

Total volume in tub in gallons
(𝒚𝒚) 𝟔𝟔 𝟗𝟗.𝟔𝟔 𝟏𝟏𝟐𝟐 𝟏𝟏𝟔𝟔.𝟖𝟖

 After 3 minutes pass, there are 9.6 gallons in the tub. How much water flowed from the faucet in those 3
minutes? Explain.

 Since there were already 6 gallons in the tub, after 3 minutes an additional 3.6 gallons filled the tub.

 Use this information to determine the rate of water flow.

 In 3 minutes, 3.6 gallons were added to the tub, then
3.6
3

= 1.2, and the faucet fills the tub at a rate of

1.2 gallons per minute.

 Verify that the rate of water flow is correct using the other values in the table.

 Sample student work:
5(1.2) = 6, and since 6 gallons were already in the tub, the total volume in the tub is 12 gallons.

9(1.2) = 10.8, and since 6 gallons were already in the tub, the total volume in the tub is 16.8 gallons.

 Write an equation that describes the volume of water in the tub as a function of time.

 The volume function that represents the rate of water flow from the faucet is 𝑦𝑦 = 1.2𝑚𝑚 + 6, where 𝑦𝑦 is
the volume of water in the tub, and 𝑚𝑚 is the number of minutes that have passed since we first noticed
the faucet being on.

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 For how many minutes was the faucet on before we noticed it? Explain.

Since 6 gallons were in the tub by the time we noticed the faucet was on, we need to determine how
many minutes it takes for 6 gallons to flow into the tub. The columns for 𝑚𝑚 = 0 and 𝑚𝑚 = 5 in the table
show that six gallons of water pour in the tub over a five-minute period. The faucet was on for 5
minutes before we noticed it.

Exercises 1–3 (10 minutes)

Students complete Exercises 1–3 independently or in pairs.

Exercises 1–3

1. Hana claims she mows lawns at a constant rate. The table below shows the area of lawn she can mow over
different time periods.

Number of minutes
(𝒙𝒙) 𝟓𝟓 𝟐𝟐𝟓𝟓 𝟑𝟑𝟓𝟓 𝟓𝟓𝟓𝟓

Area mowed in square feet
(𝒚𝒚) 𝟑𝟑𝟔𝟔 𝟏𝟏𝟒𝟒𝟒𝟒 𝟐𝟐𝟏𝟏𝟔𝟔 𝟑𝟑𝟔𝟔𝟓𝟓

a. Is the data presented consistent with the claim that the area mowed is a linear function of time?

Sample responses:

Linear functions have a constant rate of change. When we compare the rates at each interval of time, they
will be equal to the same constant.

When the data is graphed on the coordinate plane, it appears to make a line.

b. Describe in words the function in terms of area mowed and time.

The total area mowed is a function of the number of minutes spent mowing.

c. At what rate does Hana mow lawns over a 𝟓𝟓-minute period?

𝟑𝟑𝟔𝟔
𝟓𝟓

= 𝟕𝟕.𝟐𝟐

The rate is 𝟕𝟕.𝟐𝟐 square feet per minute.

d. At what rate does Hana mow lawns over a 𝟐𝟐𝟓𝟓-minute period?

𝟏𝟏𝟒𝟒𝟒𝟒
𝟐𝟐𝟓𝟓

= 𝟕𝟕.𝟐𝟐

The rate is 𝟕𝟕.𝟐𝟐 square feet per minute.

e. At what rate does Hana mow lawns over a 𝟑𝟑𝟓𝟓-minute period?

𝟐𝟐𝟏𝟏𝟔𝟔
𝟑𝟑𝟓𝟓

= 𝟕𝟕.𝟐𝟐

The rate is 𝟕𝟕.𝟐𝟐 square feet per minute.

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f. At what rate does Hana mow lawns over a 𝟓𝟓𝟓𝟓-minute period?

𝟑𝟑𝟔𝟔𝟓𝟓
𝟓𝟓𝟓𝟓

= 𝟕𝟕.𝟐𝟐

The rate is 𝟕𝟕.𝟐𝟐 square feet per minute.

g. Write the equation that describes the area mowed, 𝒚𝒚, in square feet, as a linear function of time, 𝒙𝒙, in
minutes.

𝒚𝒚 = 𝟕𝟕.𝟐𝟐𝒙𝒙

h. Describe any limitations on the possible values of 𝒙𝒙 and 𝒚𝒚.

Both 𝒙𝒙 and 𝒚𝒚 must be positive numbers. The symbol 𝒙𝒙 represents time spent mowing, which means it should
be positive. Similarly, 𝒚𝒚 represents the area mowed, which should also be positive.

i. What number does the function assign to 𝒙𝒙 = 𝟐𝟐𝟒𝟒? That is, what area of lawn can be mowed in 𝟐𝟐𝟒𝟒 minutes?

𝒚𝒚 = 𝟕𝟕.𝟐𝟐(𝟐𝟐𝟒𝟒)
𝒚𝒚 = 𝟏𝟏𝟕𝟕𝟐𝟐.𝟖𝟖

In 𝟐𝟐𝟒𝟒 minutes, an area of 𝟏𝟏𝟕𝟕𝟐𝟐.𝟖𝟖 square feet can be mowed.

j. According to this work, how many minutes would it take to mow an area of 𝟒𝟒𝟓𝟓𝟓𝟓 square feet?

𝟒𝟒𝟓𝟓𝟓𝟓 = 𝟕𝟕.𝟐𝟐𝒙𝒙
𝟒𝟒𝟓𝟓𝟓𝟓
𝟕𝟕.𝟐𝟐

= 𝒙𝒙

𝟓𝟓𝟓𝟓.𝟓𝟓𝟓𝟓𝟓𝟓… = 𝒙𝒙
𝟓𝟓𝟔𝟔 ≈ 𝒙𝒙

It would take about 𝟓𝟓𝟔𝟔 minutes to mow an area of 𝟒𝟒𝟓𝟓𝟓𝟓 square feet.

2. A linear function has the table of values below. The information in the table shows the total volume of water, in
gallons, that flows from a hose as a function of time, the number of minutes the hose has been running.

Time in minutes
(𝒙𝒙) 𝟏𝟏𝟓𝟓 𝟐𝟐𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟓𝟓

Total volume of water in gallons
(𝒚𝒚) 𝟒𝟒𝟒𝟒 𝟏𝟏𝟏𝟏𝟓𝟓 𝟐𝟐𝟐𝟐𝟓𝟓 𝟑𝟑𝟓𝟓𝟖𝟖

a. Describe the function in terms of volume and time.

The total volume of water that flows from a hose is a function of the number of minutes the hose is left on.

b. Write the rule for the volume of water in gallons, 𝒚𝒚, as a linear function of time, 𝒙𝒙, given in minutes.

𝒚𝒚 =
𝟒𝟒𝟒𝟒
𝟏𝟏𝟓𝟓

𝒙𝒙

𝒚𝒚 = 𝟒𝟒.𝟒𝟒𝒙𝒙

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c. What number does the function assign to 𝟐𝟐𝟓𝟓𝟓𝟓? That is, how many gallons of water flow from the hose
during a period of 𝟐𝟐𝟓𝟓𝟓𝟓 minutes?

𝒚𝒚 = 𝟒𝟒.𝟒𝟒(𝟐𝟐𝟓𝟓𝟓𝟓)
𝒚𝒚 = 𝟏𝟏 𝟏𝟏𝟓𝟓𝟓𝟓

In 𝟐𝟐𝟓𝟓𝟓𝟓 minutes, 𝟏𝟏,𝟏𝟏𝟓𝟓𝟓𝟓 gallons of water flow from the hose.

d. The average swimming pool holds about 𝟏𝟏𝟕𝟕,𝟑𝟑𝟓𝟓𝟓𝟓 gallons of water. Suppose such a pool has already
been filled one quarter of its volume. Write an equation that describes the volume of water in the pool
if, at time 𝟓𝟓 minutes, we use the hose described above to start filling the pool.

𝟏𝟏
𝟒𝟒

(𝟏𝟏𝟕𝟕 𝟑𝟑𝟓𝟓𝟓𝟓) = 𝟒𝟒 𝟑𝟑𝟐𝟐𝟓𝟓

𝒚𝒚 = 𝟒𝟒.𝟒𝟒𝒙𝒙 + 𝟒𝟒 𝟑𝟑𝟐𝟐𝟓𝟓

e. Approximately how many hours will it take to finish filling the pool?

𝟏𝟏𝟕𝟕 𝟑𝟑𝟓𝟓𝟓𝟓 = 𝟒𝟒.𝟒𝟒𝒙𝒙 + 𝟒𝟒 𝟑𝟑𝟐𝟐𝟓𝟓
𝟏𝟏𝟐𝟐 𝟗𝟗𝟕𝟕𝟓𝟓 = 𝟒𝟒.𝟒𝟒𝒙𝒙
𝟏𝟏𝟐𝟐 𝟗𝟗𝟕𝟕𝟓𝟓
𝟒𝟒.𝟒𝟒

= 𝒙𝒙

𝟐𝟐 𝟗𝟗𝟒𝟒𝟖𝟖.𝟖𝟖𝟔𝟔𝟑𝟑𝟔𝟔… = 𝒙𝒙
𝟐𝟐 𝟗𝟗𝟒𝟒𝟗𝟗 ≈ 𝒙𝒙

𝟐𝟐 𝟗𝟗𝟒𝟒𝟗𝟗
𝟔𝟔𝟓𝟓

= 𝟒𝟒𝟗𝟗.𝟏𝟏𝟓𝟓

It will take about 𝟒𝟒𝟗𝟗 hours to fill the pool with the hose.

3. Recall that a linear function can be described by a rule in the form of 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃, where 𝒎𝒎 and 𝒃𝒃 are constants. A
particular linear function has the table of values below.

Input
(𝒙𝒙) 𝟓𝟓 𝟒𝟒 𝟏𝟏𝟓𝟓 𝟏𝟏𝟏𝟏 𝟏𝟏𝟓𝟓 𝟐𝟐𝟓𝟓 𝟐𝟐𝟑𝟑

Output
(𝒚𝒚) 𝟒𝟒 𝟐𝟐𝟒𝟒 𝟓𝟓𝟒𝟒 𝟓𝟓𝟗𝟗 𝟕𝟕𝟗𝟗 𝟏𝟏𝟓𝟓𝟒𝟒 𝟏𝟏𝟏𝟏𝟗𝟗

a. What is the equation that describes the function?

𝒚𝒚 = 𝟓𝟓𝒙𝒙 + 𝟒𝟒

b. Complete the table using the rule.

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Lesson Summary

A linear equation 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃 describes a rule for a function. We call any function defined by a linear equation a
linear function.

Problems involving a constant rate of change or a proportional relationship can be described by linear functions.

Closing (4 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 We know that a linear function is a function whose rule can be described by a linear equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏
with 𝑚𝑚 and 𝑏𝑏 as constants.

 We know that problems involving constant rates and proportional relationships can be described by linear
functions.

Exit Ticket (4 minutes)

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Name Date

Lesson 3: Linear Functions and Proportionality

Exit Ticket

The information in the table shows the number of pages a student can read in a certain book as a function of time in
minutes spent reading. Assume a constant rate of reading.

Time in minutes
(𝒙𝒙) 2 6 11 20

Total number of pages read in a certain book
(𝒚𝒚) 7 21 38.5 70

a. Write the equation that describes the total number of pages read, 𝑦𝑦, as a linear function of the number of
minutes, 𝑚𝑚, spent reading.

b. How many pages can be read in 45 minutes?

c. A certain book has 396 pages. The student has already read
3
8

of the pages and now picks up the book again at

time 𝑚𝑚 = 0 minutes. Write the equation that describes the total number of pages of the book read as a
function of the number of minutes of further reading.

d. Approximately how much time, in minutes, will it take to finish reading the book?

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Exit Ticket Sample Solutions

The information in the table shows the number of pages a student can read in a certain book as a function of time in
minutes spent reading. Assume a constant rate of reading.

Time in minutes
(𝒙𝒙) 𝟐𝟐 𝟔𝟔 𝟏𝟏𝟏𝟏 𝟐𝟐𝟓𝟓

Total number of pages read in a certain book
(𝒚𝒚) 𝟕𝟕 𝟐𝟐𝟏𝟏 𝟑𝟑𝟖𝟖.𝟓𝟓 𝟕𝟕𝟓𝟓

a. Write the equation that describes the total number of pages read, 𝒚𝒚, as a linear function of the number of
minutes, 𝒙𝒙, spent reading.

𝒚𝒚 =
𝟕𝟕
𝟐𝟐
𝒙𝒙

𝒚𝒚 = 𝟑𝟑.𝟓𝟓𝒙𝒙

b. How many pages can be read in 𝟒𝟒𝟓𝟓 minutes?

𝒚𝒚 = 𝟑𝟑.𝟓𝟓(𝟒𝟒𝟓𝟓)
𝒚𝒚 = 𝟏𝟏𝟓𝟓𝟕𝟕.𝟓𝟓

In 𝟒𝟒𝟓𝟓 minutes, the student can read 𝟏𝟏𝟓𝟓𝟕𝟕.𝟓𝟓 pages.

c. A certain book has 𝟑𝟑𝟗𝟗𝟔𝟔 pages. The student has already read
𝟑𝟑
𝟖𝟖

of the pages and now picks up the book again

at time 𝒙𝒙 = 𝟓𝟓 minutes. Write the equation that describes the total number of pages of the book read as a
function of the number of minutes of further reading.

𝟑𝟑
𝟖𝟖

(𝟑𝟑𝟗𝟗𝟔𝟔) = 𝟏𝟏𝟒𝟒𝟖𝟖.𝟓𝟓

𝒚𝒚 = 𝟑𝟑.𝟓𝟓𝒙𝒙 + 𝟏𝟏𝟒𝟒𝟖𝟖.𝟓𝟓

d. Approximately how much time, in minutes, will it take to finish reading the book?

𝟑𝟑𝟗𝟗𝟔𝟔 = 𝟑𝟑.𝟓𝟓𝒙𝒙 + 𝟏𝟏𝟒𝟒𝟖𝟖.𝟓𝟓
𝟐𝟐𝟒𝟒𝟕𝟕.𝟓𝟓 = 𝟑𝟑.𝟓𝟓𝒙𝒙
𝟐𝟐𝟒𝟒𝟕𝟕.𝟓𝟓
𝟑𝟑.𝟓𝟓

= 𝒙𝒙

𝟕𝟕𝟓𝟓.𝟕𝟕𝟏𝟏𝟒𝟒𝟐𝟐𝟖𝟖𝟓𝟓𝟕𝟕𝟏𝟏… = 𝒙𝒙
𝟕𝟕𝟏𝟏 ≈ 𝒙𝒙

It will take about 𝟕𝟕𝟏𝟏 minutes to finish reading the book.

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Problem Set Sample Solutions

1. A food bank distributes cans of vegetables every Saturday. The following table shows the total number of cans they
have distributed since the beginning of the year. Assume that this total is a linear function of the number of weeks
that have passed.

Number of weeks
(𝒙𝒙) 𝟏𝟏 𝟏𝟏𝟐𝟐 𝟐𝟐𝟓𝟓 𝟒𝟒𝟓𝟓

Number of cans of vegetables distributed
(𝒚𝒚) 𝟏𝟏𝟖𝟖𝟓𝟓 𝟐𝟐,𝟏𝟏𝟔𝟔𝟓𝟓 𝟑𝟑,𝟔𝟔𝟓𝟓𝟓𝟓 𝟖𝟖,𝟏𝟏𝟓𝟓𝟓𝟓

a. Describe the function being considered in words.

The total number of cans handed out is a function of the number of weeks that pass.

b. Write the linear equation that describes the total number of cans handed out, 𝒚𝒚, in terms of the number of
weeks, 𝒙𝒙, that have passed.

𝒚𝒚 =
𝟏𝟏𝟖𝟖𝟓𝟓
𝟏𝟏

𝒙𝒙

𝒚𝒚 = 𝟏𝟏𝟖𝟖𝟓𝟓𝒙𝒙

c. Assume that the food bank wants to distribute 𝟐𝟐𝟓𝟓,𝟓𝟓𝟓𝟓𝟓𝟓 cans of vegetables. How long will it take them to
meet that goal?

𝟐𝟐𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓 = 𝟏𝟏𝟖𝟖𝟓𝟓𝒙𝒙
𝟐𝟐𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟖𝟖𝟓𝟓

= 𝒙𝒙

𝟏𝟏𝟏𝟏𝟏𝟏.𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏… = 𝒙𝒙
𝟏𝟏𝟏𝟏𝟏𝟏 ≈ 𝒙𝒙

It will take about 𝟏𝟏𝟏𝟏𝟏𝟏 weeks to distribute 𝟐𝟐𝟓𝟓,𝟓𝟓𝟓𝟓𝟓𝟓 cans of vegetables, or about 𝟐𝟐 years.

d. The manager had forgotten to record that they had distributed 𝟑𝟑𝟓𝟓,𝟓𝟓𝟓𝟓𝟓𝟓 cans on January 1. Write an adjusted
linear equation to reflect this forgotten information.

𝒚𝒚 = 𝟏𝟏𝟖𝟖𝟓𝟓𝒙𝒙 + 𝟑𝟑𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓

e. Using your function in part (d), determine how long in years it will take the food bank to hand out 𝟖𝟖𝟓𝟓,𝟓𝟓𝟓𝟓𝟓𝟓
cans of vegetables.

𝟖𝟖𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓 = 𝟏𝟏𝟖𝟖𝟓𝟓𝒙𝒙 + 𝟑𝟑𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓
𝟒𝟒𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓 = 𝟏𝟏𝟖𝟖𝟓𝟓𝒙𝒙
𝟒𝟒𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟖𝟖𝟓𝟓

= 𝒙𝒙

𝟐𝟐𝟓𝟓𝟓𝟓 = 𝒙𝒙

To determine the number of years:

𝟐𝟐𝟓𝟓𝟓𝟓
𝟓𝟓𝟐𝟐

= 𝟒𝟒.𝟖𝟖𝟓𝟓𝟕𝟕𝟔𝟔… ≈ 𝟒𝟒.𝟖𝟖

It will take about 𝟒𝟒.𝟖𝟖 years to distribute 𝟖𝟖𝟓𝟓,𝟓𝟓𝟓𝟓𝟓𝟓 cans of vegetables.

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Lesson 3: Linear Functions and Proportionality

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2. A linear function has the table of values below. It gives the number of miles a plane travels over a given number of
hours while flying at a constant speed.

Number of hours traveled
(𝒙𝒙) 𝟐𝟐.𝟓𝟓 𝟒𝟒 𝟒𝟒.𝟐𝟐

Distance in miles
(𝒚𝒚) 𝟏𝟏,𝟓𝟓𝟔𝟔𝟐𝟐.𝟓𝟓 𝟏𝟏,𝟕𝟕𝟓𝟓𝟓𝟓 𝟏𝟏,𝟕𝟕𝟖𝟖𝟓𝟓

a. Describe in words the function given in this problem.

The total distance traveled is a function of the number of hours spent flying.

b. Write the equation that gives the distance traveled, 𝒚𝒚, in miles, as a linear function of the number of hours, 𝒙𝒙,
spent flying.

𝒚𝒚 =
𝟏𝟏 𝟓𝟓𝟔𝟔𝟐𝟐.𝟓𝟓
𝟐𝟐.𝟓𝟓

𝒙𝒙

𝒚𝒚 = 𝟒𝟒𝟐𝟐𝟓𝟓𝒙𝒙

c. Assume that the airplane is making a trip from New York to Los Angeles, which is a journey of approximately
𝟐𝟐,𝟒𝟒𝟕𝟕𝟓𝟓 miles. How long will it take the airplane to get to Los Angeles?

𝟐𝟐 𝟒𝟒𝟕𝟕𝟓𝟓 = 𝟒𝟒𝟐𝟐𝟓𝟓𝒙𝒙
𝟐𝟐 𝟒𝟒𝟕𝟕𝟓𝟓
𝟒𝟒𝟐𝟐𝟓𝟓

= 𝒙𝒙

𝟓𝟓.𝟖𝟖𝟐𝟐𝟑𝟑𝟓𝟓𝟐𝟐… = 𝒙𝒙
𝟓𝟓.𝟖𝟖 ≈ 𝒙𝒙

It will take about 𝟓𝟓.𝟖𝟖 hours for the airplane to fly 𝟐𝟐,𝟒𝟒𝟕𝟕𝟓𝟓 miles.

d. If the airplane flies for 𝟖𝟖 hours, how many miles will it cover?

𝒚𝒚 = 𝟒𝟒𝟐𝟐𝟓𝟓(𝟖𝟖)
𝒚𝒚 = 𝟑𝟑 𝟒𝟒𝟓𝟓𝟓𝟓

The airplane would travel 𝟑𝟑,𝟒𝟒𝟓𝟓𝟓𝟓 miles in 𝟖𝟖 hours.

3. A linear function has the table of values below. It gives the number of miles a car travels over a given number of
hours.

Number of hours traveled
(𝒙𝒙) 𝟑𝟑.𝟓𝟓 𝟑𝟑.𝟕𝟕𝟓𝟓 𝟒𝟒 𝟒𝟒.𝟐𝟐𝟓𝟓

Distance in miles
(𝒚𝒚) 𝟐𝟐𝟓𝟓𝟑𝟑 𝟐𝟐𝟏𝟏𝟕𝟕.𝟓𝟓 𝟐𝟐𝟑𝟑𝟐𝟐 𝟐𝟐𝟒𝟒𝟔𝟔.𝟓𝟓

a. Describe in words the function given.

The total distance traveled is a function of the number of hours spent traveling.

b. Write the equation that gives the distance traveled, in miles, as a linear function of the number of hours spent
driving.

𝒚𝒚 =
𝟐𝟐𝟓𝟓𝟑𝟑
𝟑𝟑.𝟓𝟓

𝒙𝒙

𝒚𝒚 = 𝟓𝟓𝟖𝟖𝒙𝒙

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Lesson 3: Linear Functions and Proportionality

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c. Assume that the person driving the car is going on a road trip to reach a location 𝟓𝟓𝟓𝟓𝟓𝟓 miles from her starting
point. How long will it take the person to get to the destination?

𝟓𝟓𝟓𝟓𝟓𝟓 = 𝟓𝟓𝟖𝟖𝒙𝒙
𝟓𝟓𝟓𝟓𝟓𝟓
𝟓𝟓𝟖𝟖

= 𝒙𝒙

𝟖𝟖.𝟔𝟔𝟐𝟐𝟓𝟓𝟔𝟔… = 𝒙𝒙
𝟖𝟖.𝟔𝟔 ≈ 𝒙𝒙

It will take about 𝟖𝟖.𝟔𝟔 hours to travel 𝟓𝟓𝟓𝟓𝟓𝟓 miles.

4. A particular linear function has the table of values below.

Input
(𝒙𝒙) 𝟐𝟐 𝟑𝟑 𝟖𝟖 𝟏𝟏𝟏𝟏 𝟏𝟏𝟓𝟓 𝟐𝟐𝟓𝟓 𝟐𝟐𝟑𝟑

Output
(𝒚𝒚) 𝟕𝟕 𝟏𝟏𝟓𝟓 𝟐𝟐𝟓𝟓 𝟑𝟑𝟒𝟒 𝟒𝟒𝟔𝟔 𝟔𝟔𝟏𝟏 𝟕𝟕𝟓𝟓

a. What is the equation that describes the function?

𝒚𝒚 = 𝟑𝟑𝒙𝒙 + 𝟏𝟏

b. Complete the table using the rule.

5. A particular linear function has the table of values below.

Input
(𝒙𝒙) 𝟓𝟓 𝟓𝟓 𝟖𝟖 𝟏𝟏𝟑𝟑 𝟏𝟏𝟓𝟓 𝟏𝟏𝟖𝟖 𝟐𝟐𝟏𝟏

Output
(𝒚𝒚) 𝟔𝟔 𝟏𝟏𝟏𝟏 𝟏𝟏𝟒𝟒 𝟏𝟏𝟗𝟗 𝟐𝟐𝟏𝟏 𝟐𝟐𝟒𝟒 𝟐𝟐𝟕𝟕

a. What is the rule that describes the function?

𝒚𝒚 = 𝒙𝒙 + 𝟔𝟔

b. Complete the table using the rule.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

Lesson 4: More Examples of Functions

Student Outcomes

 Students classify functions as either discrete or not discrete.

Classwork

Discussion (5 minutes)

 Consider two functions we discussed in previous lessons: the function that assigns to each count of candy bags
the cost of purchasing that many bags (Table A) and the function that assigns to each time value between 0
and 4 seconds the distance a falling stone has dropped up to that time (Table B).

Table A:

Bags of candy
(𝒙𝒙) 1 2 3 4 5 6 7 8

Cost in Dollars
(𝒚𝒚)

1.25 2.50 3.75 5.00 6.25 7.50 8.75 10.00

Table B:

Number of
seconds

(𝒙𝒙)
0.5 1 1.5 2 2.5 3 3.5 4

Distance
traveled in feet

(𝒚𝒚)
4 16 36 64 100 144 196 256

 How do the two functions differ in the types of inputs they each admit?

Solicit answers from students, and then continue with the discussion below.

 We described the first function, Table A, with the rule 𝑦𝑦 = 1.25𝑥𝑥 for 𝑥𝑥 ≥ 0.
 Why did we restrict 𝑥𝑥 to numbers equal to or greater than 0?

 We restricted 𝑥𝑥 to numbers equal to or greater than 0 because you cannot purchase a negative number
of bags of candy.

 If we further assume that only a whole number of bags can be sold (that one cannot purchase only a portion of
a bag’s contents), then we need to be more precise about our restrictions on permissible inputs for the
function. Specifically, we must say that 𝑥𝑥 is a nonnegative integer: 0, 1, 2, 3, etc.

 We described the second function, Table B, with the rule 𝑦𝑦 = 16𝑥𝑥2. Does this function require the same
restrictions on its inputs as the previous problem? Explain.

 We should state that 𝑥𝑥 must be a positive number because 𝑥𝑥 represents the amount of time traveled.
But we do not need to say that 𝑥𝑥 must be an integer: intervals of time need not be in whole seconds as
fractional counts of seconds are meaningful.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

 The word discrete in English means individually separate or distinct. If a function admits only individually
separate input values (like whole number counts of candy bags, for example), then we say we have a discrete
function. If a function admits, over a range of values, any input value within that range (all fractional values
too, for example), then we have a function that is not discrete. Functions that describe motion, for example,
are typically not discrete.

Example 1 (6 minutes)

Practice the definitions with this example.

Example 1

Classify each of the functions described below as either discrete or not discrete.

a) The function that assigns to each whole number the cost of buying that many cans of beans in a particular grocery
store.

b) The function that assigns to each time of day one Wednesday the temperature of Sammy’s fever at that time.

c) The function that assigns to each real number its first digit.

d) The function that assigns to each day in the year 2015 my height at noon that day.

e) The function that assigns to each moment in the year 2015 my height at that moment.

f) The function that assigns to each color the first letter of the name of that color.

g) The function that assigns the number 𝟐𝟐𝟐𝟐 to each and every real number between 𝟐𝟐𝟐𝟐 and 𝟐𝟐𝟐𝟐.𝟔𝟔.

h) The function that assigns the word YES to every yes/no question.

i) The function that assigns to each height directly above the North Pole the temperature of the air at that height right
at this very moment.

a) Discrete b) Not discrete c) Not discrete d) Discrete e) Not discrete f) Discrete g) Not discrete

h) Discrete i) Not discrete

Example (2 minutes)

 Let’s revisit a problem that we examined in the last lesson.

Example 2

Water flows from a faucet into a bathtub at a constant rate of 𝟕𝟕 gallons of water every 𝟐𝟐 minutes Regard the volume of
water accumulated in the tub as a function of the number of minutes the faucet has be on. Is this function discrete or not
discrete?

 Assuming the tub is initially empty, we determined last lesson that the volume of water in the tub is given by
𝑦𝑦 = 3.5𝑥𝑥, where 𝑦𝑦 is the volume of water in gallons, and 𝑥𝑥 is the number of minutes the faucet has been on.

 What limitations are there on 𝑥𝑥 and 𝑦𝑦?

 Both 𝑥𝑥 and 𝑦𝑦 should be positive numbers because they represent time and volume.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

 Would this function be considered discrete or not discrete? Explain.

 This function is not discrete because we can assign any positive number to 𝑥𝑥, not just positive integers.

Example 3 (8 minutes)

This is a more complicated example of a problem leading to a non-discrete function.

Example 3

You have just been served freshly made soup that is so hot that it cannot be eaten. You measure
the temperature of the soup, and it is 𝟐𝟐𝟐𝟐𝟐𝟐°𝐅𝐅. Since 𝟐𝟐𝟐𝟐𝟐𝟐°𝐅𝐅 is boiling, there is no way it can safely
be eaten yet. One minute after receiving the soup, the temperature has dropped to 𝟐𝟐𝟐𝟐𝟐𝟐°𝐅𝐅. If
you assume that the rate at which the soup cools is constant, write an equation that would
describe the temperature of the soup over time.

The temperature of the soup dropped 𝟕𝟕°𝐅𝐅 in one minute. Assuming the cooling continues at the
same rate, then if 𝒚𝒚 is the temperature of the soup after 𝒙𝒙 minutes, then, 𝒚𝒚 = 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟕𝟕𝒙𝒙.

 We want to know how long it will be before the temperature of the soup is at a more tolerable temperature of
147°F. The difference in temperature from 210°F to 147°F is 63°F. For what number 𝑥𝑥 will our function have
the value 147?
 147 = 210 − 7𝑥𝑥; then 7𝑥𝑥 = 63, and so 𝑥𝑥 = 9.

 Curious whether or not you are correct in assuming the cooling rate of the soup is constant, you decide to
measure the temperature of the soup each minute after its arrival to you. Here’s the data you obtain:

Time Temperature in
Fahrenheit

after 2 minutes 196
after 3 minutes 190
after 4 minutes 184
after 5 minutes 178
after 6 minutes 173
after 7 minutes 168
after 8 minutes 163
after 9 minutes 158

Our function led us to believe that after 9 minutes the soup would be safe to eat. The data in the table shows
that it is still too hot.

 What do you notice about the change in temperature from one minute to the next?

 For the first few minutes, minute 2 to minute 5, the temperature decreased 6°F each minute. From
minute 5 to minute 9, the temperature decreased just 5°F each minute.

 Since the rate of cooling at each minute is not constant, this function is said to be a nonlinear function.

 Sir Isaac Newton not only studied the motion of objects under gravity but also studied the rates of cooling of
heated objects. He found that they do not cool at constant rates and that the functions that describe their
temperature over time are indeed far from linear. (In fact, Newton’s theory establishes that the temperature

of soup at time 𝑥𝑥 minutes would actually be given by the formula 𝑦𝑦 = 70 + 140 �133
140 �

𝑥𝑥
.)

Scaffolding:
The more real you can make
this, the better. Consider
having a cooling cup of soup,
coffee, or tea with a digital
thermometer available for
students to observe.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

Example 4 (6 minutes)

Example 4

Consider the function that assigns to each of nine baseball players, numbered 1 through 9, his height. The data for this
function is given below. Call the function 𝑮𝑮.

Player Number Height

𝟐𝟐 𝟓𝟓′𝟐𝟐𝟐𝟐′′
𝟐𝟐 𝟓𝟓′𝟒𝟒′′
𝟐𝟐 𝟓𝟓′𝟗𝟗′′
𝟒𝟒 𝟓𝟓′𝟔𝟔′′
𝟓𝟓 𝟔𝟔′𝟐𝟐′′
𝟔𝟔 𝟔𝟔′𝟖𝟖′′
𝟕𝟕 𝟓𝟓′𝟗𝟗′′
𝟖𝟖 𝟓𝟓′𝟐𝟐𝟐𝟐′′
𝟗𝟗 𝟔𝟔′𝟐𝟐′′

 What output does 𝐺𝐺 assign to the input 2?

 The function 𝐺𝐺 assigns the height 5′4′′ to the player 2.

 Could the function 𝐺𝐺 simultaneously assign a second, different output to player 2? Explain.

 No. The function assigns height to a particular player. There is no way that a player can have two
different heights.

 It is not clear if there is a formula for this function. (And even if there were, it is not clear that it would be
meaningful since who is labeled player 1, player 2, and so on is probably arbitrary.) In general, we can hope to
have formulas for functions, but in reality we cannot expect to find them. (People would love to have a
formula that explains and predicts the stock market, for example.)

 Can we classify this function as discrete or not discrete? Explain.
 This function would be described as discrete because the inputs are particular players.

Exercises 1–3 (10 minutes)

Exercises 1–3

1. At a certain school, each bus in its fleet of buses can transport 35 students. Let 𝑩𝑩 be the function that assigns to
each count of students the number of buses needed to transport that many students on a field trip.

When Jinpyo thought about matters, he drew the following table of values and wrote the formula 𝑩𝑩 = 𝒙𝒙
𝟐𝟐𝟓𝟓. Here 𝒙𝒙 is

the count of students, and 𝑩𝑩 is the number of buses needed to transport that many students. He concluded that 𝑩𝑩
is a linear function.

Number of students
(𝒙𝒙) 𝟐𝟐𝟓𝟓 𝟕𝟕𝟐𝟐 𝟐𝟐𝟐𝟐𝟓𝟓 𝟐𝟐𝟒𝟒𝟐𝟐

Number of buses
(𝑩𝑩) 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟒𝟒

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

Alicia looked at Jinpyo’s work and saw no errors with his arithmetic. But she said that the function is not actually
linear.

a. Alicia is right. Explain why 𝑩𝑩 is not a linear function.

For 𝟐𝟐𝟔𝟔 students, say, we’ll need two buses—an extra bus for the extra student. In fact, for 𝟐𝟐𝟔𝟔,𝟐𝟐𝟕𝟕, …, up to
𝟕𝟕𝟐𝟐 students, the function 𝑩𝑩 assigns the same value 𝟐𝟐. For 𝟕𝟕𝟐𝟐,𝟕𝟕𝟐𝟐, …, up to 𝟐𝟐𝟐𝟐𝟓𝟓, it assigns the value 𝟐𝟐. There
is not a constant rate of increase of the buses needed, and so the function is not linear.

b. Is 𝑩𝑩 a discrete function?

It is a discrete function.

2. A linear function has the table of values below. It gives the costs of purchasing certain numbers of movie tickets.

Number of tickets
(𝒙𝒙) 𝟐𝟐 𝟔𝟔 𝟗𝟗 𝟐𝟐𝟐𝟐

Total cost in
dollars

(𝒚𝒚)
𝟐𝟐𝟕𝟕.𝟕𝟕𝟓𝟓 𝟓𝟓𝟓𝟓.𝟓𝟓𝟐𝟐 𝟖𝟖𝟐𝟐.𝟐𝟐𝟓𝟓 𝟐𝟐𝟐𝟐𝟐𝟐.𝟐𝟐𝟐𝟐

a. Write the linear function that represents the total cost, 𝒚𝒚, for 𝒙𝒙 tickets purchased.

𝒚𝒚 =
𝟐𝟐𝟕𝟕.𝟕𝟕𝟓𝟓
𝟐𝟐

𝒙𝒙

𝒚𝒚 = 𝟗𝟗.𝟐𝟐𝟓𝟓𝒙𝒙

b. Is the function discrete? Explain.

The function is discrete. You cannot have half of a movie ticket; therefore, it must be a whole number of
tickets, which means it is discrete.

c. What number does the function assign to 𝟒𝟒? What do the question and your answer mean?

It is asking us to determine the cost of buying 𝟒𝟒 tickets. The function assigns 𝟐𝟐𝟕𝟕 to 𝟒𝟒. The answer means that
𝟒𝟒 tickets will cost $𝟐𝟐𝟕𝟕.𝟐𝟐𝟐𝟐.

3. A function produces the following table of values.

Input Output

Banana B

Cat C

Flippant F

Oops O

Slushy S

a. Make a guess as to the rule this function follows. Each input is a word from the English language.

This function assigns to each word its first letter.

b. Is this function discrete?

It is discrete.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

Closing (4 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 We have classified functions as either discrete or not discrete.
 Discrete functions admit only individually separate input values (such as whole numbers of students, or words

of the English language). Functions that are not discrete admit any input value within a range of values
(fractional values, for example).

 Functions that describe motion or smooth changes over time, for example, are typically not discrete.

Exit Ticket (4 minutes)

Lesson Summary

Functions are classified as either discrete or not discrete.

Discrete functions admit only individually separate input values (such as whole numbers of students, or words of
the English language). Functions that are not discrete admit any input value within a range of values (fractional
values, for example).

Functions that describe motion or smooth changes over time, for example, are typically not discrete.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

Name Date

Lesson 4: More Examples of Functions

Exit Ticket

1. The table below shows the costs of purchasing certain numbers of tablets. We can assume that the total cost is a

linear function of the number of tablets purchased.

Number of tablets
(𝒙𝒙) 17 22 25

Total cost in dollars
(𝒚𝒚) 10,183.00 13,178.00 14,975.00

a. Write an equation that describes the total cost, 𝑦𝑦, as a linear function of the number, 𝑥𝑥, of tablets purchased.

b. Is the function discrete? Explain.

c. What number does the function assign to 7? Explain.

2. A function 𝐶𝐶 assigns to each word in the English language the number of letters in that word. For example, 𝐶𝐶
assigns the number 6 to the word action.

a. Give an example of an input to which 𝐶𝐶 would assign the value 3.

b. Is 𝐶𝐶 a discrete function? Explain.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

Exit Ticket Sample Solutions

1. The table below shows the costs of purchasing certain numbers of tablets. We can assume that the total cost is a
linear function of the number of tablets purchased.

Number of tablets
(𝒙𝒙) 𝟐𝟐𝟕𝟕 𝟐𝟐𝟐𝟐 𝟐𝟐𝟓𝟓

Total cost in dollars
(𝒚𝒚) 𝟐𝟐𝟐𝟐,𝟐𝟐𝟖𝟖𝟐𝟐.𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐,𝟐𝟐𝟕𝟕𝟖𝟖.𝟐𝟐𝟐𝟐 𝟐𝟐𝟒𝟒,𝟗𝟗𝟕𝟕𝟓𝟓.𝟐𝟐𝟐𝟐

a. Write an equation that described the total cost, 𝒚𝒚, as a linear function of the number, 𝒙𝒙, of tablets purchased.

𝒚𝒚 =
𝟐𝟐𝟐𝟐,𝟐𝟐𝟖𝟖𝟐𝟐
𝟐𝟐𝟕𝟕

𝒙𝒙

𝒚𝒚 = 𝟓𝟓𝟗𝟗𝟗𝟗𝒙𝒙

b. Is the function discrete? Explain.

The function is discrete. You cannot have half of a tablet; therefore, it must be a whole number of tablets,
which means it is discrete.

c. What number does the function assign to 𝟕𝟕? Explain.

The function assigns 𝟒𝟒,𝟐𝟐𝟗𝟗𝟐𝟐 to 𝟕𝟕, which means that the cost of 𝟕𝟕 tablets would be $𝟒𝟒,𝟐𝟐𝟗𝟗𝟐𝟐.𝟐𝟐𝟐𝟐.

2. A function 𝑪𝑪 assigns to each word in the English language the number of letters in that word. For example, 𝑪𝑪 assigns
the number 𝟔𝟔 to the word action.

a. Give an example of an input to which 𝑪𝑪 would assign the value 𝟐𝟐.

Any three-letter word will do.

b. Is 𝑪𝑪 a discrete function? Explain.

The function is discrete. The input is a word in the English language, therefore it must be an entire word, not
part of one, which means it is discrete.

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Lesson 4: More Examples of Functions

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Problem Set Sample Solutions

1. The costs of purchasing certain volumes of gasoline are shown below. We can assume that there is a linear
relationship between 𝒙𝒙, the number of gallons purchased, and 𝒚𝒚, the cost of purchasing that many gallons.

Number of gallons
(𝒙𝒙) 𝟓𝟓.𝟒𝟒 𝟔𝟔 𝟐𝟐𝟓𝟓 𝟐𝟐𝟕𝟕

Total cost in dollars
(𝒚𝒚) 𝟐𝟐𝟗𝟗.𝟕𝟕𝟐𝟐 𝟐𝟐𝟐𝟐.𝟗𝟗𝟐𝟐 𝟓𝟓𝟒𝟒.𝟕𝟕𝟓𝟓 𝟔𝟔𝟐𝟐.𝟐𝟐𝟓𝟓

a. Write an equation that describes 𝒚𝒚 as a linear function of 𝒙𝒙.

𝒚𝒚 = 𝟐𝟐.𝟔𝟔𝟓𝟓𝒙𝒙

b. Are there any restrictions on the values 𝒙𝒙 and 𝒚𝒚 can adopt?

Both 𝒙𝒙 and 𝒚𝒚 must be positive rational numbers.

c. Is the function discrete?

The function is not discrete.

d. What number does the linear function assign to 𝟐𝟐𝟐𝟐? Explain what your answer means.

𝒚𝒚 = 𝟐𝟐.𝟔𝟔𝟓𝟓(𝟐𝟐𝟐𝟐)
𝒚𝒚 = 𝟕𝟕𝟐𝟐

The function assigns 𝟕𝟕𝟐𝟐 to 𝟐𝟐𝟐𝟐. It means that if 𝟐𝟐𝟐𝟐 gallons of gas are purchased, it will cost $𝟕𝟕𝟐𝟐.𝟐𝟐𝟐𝟐.

2. A function has the table of values below. Examine the information in the table to answer the questions below.

Input Output

one 𝟐𝟐

two 𝟐𝟐

three 𝟓𝟓

four 𝟒𝟒

five 𝟒𝟒

six 𝟐𝟐

seven 𝟓𝟓

a. Describe the function.

The function assigns those particular numbers to those particular seven words. We don’t know if the function
accepts any more inputs and what it might assign to those additional inputs. (Though it does seem
compelling to say that this function assigns to each positive whole number the count of letters in the name of
that whole number.)

b. What number would the function assign to the word eleven?

We do not have enough information to tell. We are not even sure if eleven is considered a valid input for this
function.

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Lesson 4: More Examples of Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 4

3. The table shows the distances covered over certain counts of hours traveled by a driver driving a car at a constant
speed.

Number of hours driven
(𝒙𝒙) 𝟐𝟐 𝟒𝟒 𝟓𝟓 𝟔𝟔

Total miles driven
(𝒚𝒚) 𝟐𝟐𝟒𝟒𝟐𝟐 𝟐𝟐𝟖𝟖𝟖𝟖 𝟐𝟐𝟐𝟐𝟓𝟓 𝟐𝟐𝟖𝟖𝟐𝟐

a. Write an equation that describes 𝒚𝒚, the number of miles covered, as a linear function of 𝒙𝒙, number of hours
driven.

𝒚𝒚 =
𝟐𝟐𝟒𝟒𝟐𝟐
𝟐𝟐

𝒙𝒙

𝒚𝒚 = 𝟒𝟒𝟕𝟕𝒙𝒙

b. Are there any restrictions on the value 𝒙𝒙 and 𝒚𝒚 can adopt?

Both 𝒙𝒙 and 𝒚𝒚 must be positive rational numbers.

c. Is the function discrete?

The function is not discrete.

d. What number does the function assign to 𝟖𝟖? Explain what your answer means.

𝒚𝒚 = 𝟒𝟒𝟕𝟕(𝟖𝟖)
𝒚𝒚 = 𝟐𝟐𝟕𝟕𝟔𝟔

The function assigns 𝟐𝟐𝟕𝟕𝟔𝟔 to 𝟖𝟖. The answer means that 𝟐𝟐𝟕𝟕𝟔𝟔 miles are driven in 𝟖𝟖 hours.

e. Use the function to determine how much time it would take to drive 𝟓𝟓𝟐𝟐𝟐𝟐 miles.

𝟓𝟓𝟐𝟐𝟐𝟐 = 𝟒𝟒𝟕𝟕𝒙𝒙
𝟓𝟓𝟐𝟐𝟐𝟐
𝟒𝟒𝟕𝟕

= 𝒙𝒙

𝟐𝟐𝟐𝟐.𝟔𝟔𝟐𝟐𝟖𝟖𝟐𝟐𝟗𝟗… = 𝒙𝒙
𝟐𝟐𝟐𝟐.𝟔𝟔 ≈ 𝒙𝒙

It would take about 𝟐𝟐𝟐𝟐.𝟔𝟔 hours to drive 𝟓𝟓𝟐𝟐𝟐𝟐 miles.

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Lesson 4: More Examples of Functions

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4. Consider the function that assigns to each time of a particular day the air temperature at a specific location in
Ithaca, NY. The following table shows the values of this function at some specific times.

12:00 noon 𝟗𝟗𝟐𝟐°𝐅𝐅

1:00 p.m. 𝟗𝟗𝟐𝟐.𝟓𝟓°𝐅𝐅

2:00 p.m. 𝟖𝟖𝟗𝟗°𝐅𝐅

4:00 p.m. 𝟖𝟖𝟔𝟔°𝐅𝐅

8:00 p.m. 𝟖𝟖𝟐𝟐°𝐅𝐅

a. Let 𝒚𝒚 represent the air temperature at time 𝒙𝒙 hours past noon. Verify that the data in the table satisfies the
linear equation 𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓𝒙𝒙.

At 12:00, 𝟐𝟐 hours have passed since 12:00; then, 𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓(𝟐𝟐) = 𝟗𝟗𝟐𝟐.

At 1:00, 𝟐𝟐 hour has passed since 12:00; then, 𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓(𝟐𝟐) = 𝟗𝟗𝟐𝟐.𝟓𝟓.

At 2:00, 𝟐𝟐 hours have passed since 12:00; then, 𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓(𝟐𝟐) = 𝟖𝟖𝟗𝟗.

At 4:00, 𝟒𝟒 hours have passed since 12:00; then, 𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓(𝟒𝟒) = 𝟖𝟖𝟔𝟔.

At 8:00, 𝟖𝟖 hours have passed since 12:00; then, 𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓(𝟖𝟖) = 𝟖𝟖𝟐𝟐.

b. Are there any restrictions on the types of values 𝒙𝒙 and 𝒚𝒚 can adopt?

The input is a particular time of the day, and 𝒚𝒚 is the temperature. The input cannot be negative but could be
intervals that are fractions of an hour. The output could potentially be negative because it can get that cold.

c. Is the function discrete?

The function is not discrete.

d. According to the linear function of part (a), what will the air temperature be at 5:30 p.m.?

At 5:30, 𝟓𝟓.𝟓𝟓 hours have passed since 12:00; then 𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓(𝟓𝟓.𝟓𝟓) = 𝟖𝟖𝟐𝟐.𝟕𝟕𝟓𝟓.

The temperature at 5:30 will be 𝟖𝟖𝟐𝟐.𝟕𝟕𝟓𝟓°𝐅𝐅.

e. Is it reasonable to assume that this linear function could be used to predict the temperature for 10:00 a.m.
the following day or a temperature at any time on a day next week? Give specific examples in your
explanation.

No. There is no reason to expect this function to be linear. Temperature typically fluctuates and will, for
certain, rise at some point.

We can show that our model for temperature is definitely wrong by looking at the predicted temperature one
week (168 hours) later:

𝒚𝒚 = 𝟗𝟗𝟐𝟐 − 𝟐𝟐.𝟓𝟓(𝟐𝟐𝟔𝟔𝟖𝟖)

𝒚𝒚 = −𝟐𝟐𝟔𝟔𝟐𝟐.

This is an absurd prediction.

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Lesson 5: Graphs of Functions and Equations

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 5

Lesson 5: Graphs of Functions and Equations

Student Outcomes

 Students define the graph of a numerical function to be the set of all points (𝑥𝑥,𝑦𝑦) with 𝑥𝑥 an input of the
function and 𝑦𝑦 its matching output.

 Students realize that if a numerical function can be described by an equation, then the graph of the function
precisely matches the graph of the equation.

Classwork

Exploratory Challenge/Exercises 1–3 (15 minutes)

Students work independently or in pairs to complete Exercises 1–3.

Exploratory Challenge/Exercises 1–3

1. The distance that Giselle can run is a function of the amount of time she spends running. Giselle runs 𝟑𝟑 miles in 𝟐𝟐𝟐𝟐
minutes. Assume she runs at a constant rate.

a. Write an equation in two variables that represents her distance run, 𝒚𝒚, as a function of the time, 𝒙𝒙, she
spends running.

𝟑𝟑
𝟐𝟐𝟐𝟐

=
𝒚𝒚
𝒙𝒙

𝒚𝒚 =
𝟐𝟐
𝟕𝟕
𝒙𝒙

b. Use the equation you wrote in part (a) to determine how many miles Giselle can run in 𝟐𝟐𝟏𝟏 minutes.

𝒚𝒚 =
𝟐𝟐
𝟕𝟕

(𝟐𝟐𝟏𝟏)

𝒚𝒚 = 𝟐𝟐
Giselle can run 𝟐𝟐 miles in 𝟐𝟐𝟏𝟏 minutes.

c. Use the equation you wrote in part (a) to determine how many miles Giselle can run in 𝟐𝟐𝟐𝟐 minutes.

𝒚𝒚 =
𝟐𝟐
𝟕𝟕

(𝟐𝟐𝟐𝟐)

𝒚𝒚 = 𝟏𝟏
Giselle can run 𝟏𝟏 miles in 𝟐𝟐𝟐𝟐 minutes.

d. Use the equation you wrote in part (a) to determine how many miles Giselle can run in 𝟕𝟕 minutes.

𝒚𝒚 =
𝟐𝟐
𝟕𝟕

(𝟕𝟕)

𝒚𝒚 = 𝟐𝟐
Giselle can run 𝟐𝟐 mile in 𝟕𝟕 minutes.

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Lesson 5: Graphs of Functions and Equations

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 5

e. For a given input 𝒙𝒙 of the function, a time, the matching output of the function, 𝒚𝒚, is the distance Giselle ran
in that time. Write the inputs and outputs from parts (b)–(d) as ordered pairs, and plot them as points on a
coordinate plane.

(𝟐𝟐𝟏𝟏,𝟐𝟐), (𝟐𝟐𝟐𝟐,𝟏𝟏), (𝟕𝟕,𝟐𝟐)

f. What do you notice about the points you plotted?

The points appear to be in a line.

g. Is the function discrete?

The function is not discrete because we can find the distance Giselle runs for any given amount of time she
spends running.

h. Use the equation you wrote in part (a) to determine how many miles Giselle can run in 𝟑𝟑𝟔𝟔 minutes. Write
your answer as an ordered pair, as you did in part (e), and include the point on the graph. Is the point in a
place where you expected it to be? Explain.

𝒚𝒚 =
𝟐𝟐
𝟕𝟕

(𝟑𝟑𝟔𝟔)

𝒚𝒚 =
𝟑𝟑𝟔𝟔
𝟕𝟕

𝒚𝒚 = 𝟓𝟓
𝟐𝟐
𝟕𝟕

�𝟑𝟑𝟔𝟔,𝟓𝟓𝟐𝟐𝟕𝟕� The point is where I expected it to be because it is in line with the other points.

i. Assume you used the rule that describes the function to determine how many miles Giselle can run for any
given time and wrote each answer as an ordered pair. Where do you think these points would appear on the
graph?

I think all of the points would fall on a line.

j. What do you think the graph of all the possible input/output pairs would look like? Explain.

I know the graph will be a line as we can find all of the points that represent fractional intervals of time too.
We also know that Giselle runs at a constant rate, so we would expect that as the time she spends running
increases, the distance she can run will increase at the same rate.

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Lesson 5: Graphs of Functions and Equations

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 5

k. Connect the points you have graphed to make a line. Select a point on the graph that has integer
coordinates. Verify that this point has an output that the function would assign to the input.

Answers will vary. Sample student work:

The point (𝟏𝟏𝟐𝟐,𝟔𝟔) is a point on the graph.

𝒚𝒚 =
𝟐𝟐
𝟕𝟕
𝒙𝒙

𝟔𝟔 =
𝟐𝟐
𝟕𝟕

(𝟏𝟏𝟐𝟐)

𝟔𝟔 = 𝟔𝟔

The function assigns the output of 𝟔𝟔 to the input of 𝟏𝟏𝟐𝟐.

l. Sketch the graph of the equation 𝒚𝒚 = 𝟐𝟐
𝟕𝟕𝒙𝒙 using the same coordinate plane in part (e). What do you notice

about the graph of all the input/output pairs that describes Giselle’s constant rate of running and the graph

of the equation 𝒚𝒚 = 𝟐𝟐
𝟕𝟕𝒙𝒙?

The graphs of the equation and the function coincide completely.

2. Sketch the graph of the equation 𝒚𝒚 = 𝒙𝒙𝟐𝟐 for positive values of 𝒙𝒙. Organize your work using the table below, and
then answer the questions that follow.

𝒙𝒙 𝒚𝒚

𝟎𝟎 𝟎𝟎
𝟐𝟐 𝟐𝟐
𝟐𝟐 𝟏𝟏
𝟑𝟑 𝟗𝟗
𝟏𝟏 𝟐𝟐𝟔𝟔
𝟓𝟓 𝟐𝟐𝟓𝟓
𝟔𝟔 𝟑𝟑𝟔𝟔

a. Plot the ordered pairs on the coordinate plane.

b. What shape does the graph of the points appear to take?

It appears to take the shape of a curve.

c. Is this equation a linear equation? Explain.

No, the equation 𝒚𝒚 = 𝒙𝒙𝟐𝟐 is not a linear equation because the
exponent of 𝒙𝒙 is greater than 𝟐𝟐.

d. Consider the function that assigns to each square of side length 𝒔𝒔
units its area 𝑨𝑨 square units. Write an equation that describes this
function.

𝑨𝑨 = 𝒔𝒔𝟐𝟐

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e. What do you think the graph of all the input/output pairs (𝒔𝒔,𝑨𝑨) of this function will look like? Explain.

I think the graph of input/output pairs will look like the graph of the equation 𝒚𝒚 = 𝒙𝒙𝟐𝟐. The inputs and outputs
would match the solutions to the equation exactly. For the equation, the 𝒚𝒚 value is the square of the 𝒙𝒙 value.
For the function, the output is the square of the input.

f. Use the function you wrote in part (d) to determine the area of a square with side length 𝟐𝟐.𝟓𝟓 units. Write the
input and output as an ordered pair. Does this point appear to belong to the graph of 𝒚𝒚 = 𝒙𝒙𝟐𝟐?

𝑨𝑨 = (𝟐𝟐.𝟓𝟓)𝟐𝟐
𝑨𝑨 = 𝟔𝟔.𝟐𝟐𝟓𝟓

The area of the square is 𝟔𝟔.𝟐𝟐𝟓𝟓 units squared. (𝟐𝟐.𝟓𝟓,𝟔𝟔.𝟐𝟐𝟓𝟓) The point looks like it would belong to the graph
of 𝒚𝒚 = 𝒙𝒙𝟐𝟐; it looks like it would be on the curve that the shape of the graph is taking.

3. The number of devices a particular manufacturing company can produce is a function of the number of hours spent
making the devices. On average, 𝟏𝟏 devices are produced each hour. Assume that devices are produced at a
constant rate.

a. Write an equation in two variables that describes the number of devices, 𝒚𝒚, as a function of the time the
company spends making the devices, 𝒙𝒙.

𝟏𝟏
𝟐𝟐

=
𝒚𝒚
𝒙𝒙

𝒚𝒚 = 𝟏𝟏𝒙𝒙

b. Use the equation you wrote in part (a) to determine how many devices are produced in 𝟐𝟐 hours.

𝒚𝒚 = 𝟏𝟏(𝟐𝟐)
𝒚𝒚 = 𝟑𝟑𝟐𝟐

The company produces 𝟑𝟑𝟐𝟐 devices in 𝟐𝟐 hours.

c. Use the equation you wrote in part (a) to determine
how many devices are produced in 𝟔𝟔 hours.

𝒚𝒚 = 𝟏𝟏(𝟔𝟔)
𝒚𝒚 = 𝟐𝟐𝟏𝟏

The company produces 𝟐𝟐𝟏𝟏 devices in 𝟔𝟔 hours.

d. Use the equation you wrote in part (a) to determine
how many devices are produced in 𝟏𝟏 hours.

𝒚𝒚 = 𝟏𝟏(𝟏𝟏)
𝒚𝒚 = 𝟐𝟐𝟔𝟔

The company produces 𝟐𝟐𝟔𝟔 devices in 𝟏𝟏 hours.

e. The input of the function, 𝒙𝒙, is time, and the output
of the function, 𝒚𝒚, is the number of devices
produced. Write the inputs and outputs from parts
(b)–(d) as ordered pairs, and plot them as points on a
coordinate plane.

(𝟐𝟐,𝟑𝟑𝟐𝟐), (𝟔𝟔,𝟐𝟐𝟏𝟏), (𝟏𝟏,𝟐𝟐𝟔𝟔)

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Lesson 5: Graphs of Functions and Equations

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f. What shape does the graph of the points appear to take?

The points appear to be in a line.

g. Is the function discrete?

The function is not discrete because we can find the number of devices produced for any given time, including
fractions of an hour.

h. Use the equation you wrote in part (a) to determine how many devices are produced in 𝟐𝟐.𝟓𝟓 hours. Write
your answer as an ordered pair, as you did in part (e), and include the point on the graph. Is the point in a
place where you expected it to be? Explain.

𝒚𝒚 = 𝟏𝟏(𝟐𝟐.𝟓𝟓)
𝒚𝒚 = 𝟔𝟔

(𝟐𝟐.𝟓𝟓,𝟔𝟔) The point is where I expected it to be because it is in line with the other points.

i. Assume you used the equation that describes the function to determine how many devices are produced for
any given time and wrote each answer as an ordered pair. Where do you think these points would appear on
the graph?

I think all of the points would fall on a line.

j. What do you think the graph of all possible input/output pairs will look like? Explain.

I think the graph of this function will be a line. Since the rate is continuous, we can find all of the points that
represent fractional intervals of time. We also know that devices are produced at a constant rate, so we
would expect that as the time spent producing devices increases, the number of devices produced would
increase at the same rate.

k. Connect the points you have graphed to make a line. Select a point on the graph that has integer
coordinates. Verify that this point has an output that the function would assign to the input.

Answers will vary. Sample student work:

The point (𝟓𝟓,𝟐𝟐𝟎𝟎) is a point on the graph.

𝒚𝒚 = 𝟏𝟏𝒙𝒙
𝟐𝟐𝟎𝟎 = 𝟏𝟏(𝟓𝟓)
𝟐𝟐𝟎𝟎 = 𝟐𝟐𝟎𝟎

The function assigns the output of 𝟐𝟐𝟎𝟎 to the input of 𝟓𝟓.

l. Sketch the graph of the equation 𝒚𝒚 = 𝟏𝟏𝒙𝒙 using the same coordinate plane in part (e). What do you notice
about the graph of input/output pairs that describes the company’s constant rate of producing devices and
the graph of the equation 𝒚𝒚 = 𝟏𝟏𝒙𝒙?

The graphs of the equation and the function coincide completely.

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Discussion (10 minutes)

 What was the equation that described the function in Exercise 1, Giselle’s distance run over given time
intervals?

 The equation was 𝑦𝑦 = 1
7 𝑥𝑥.

 Given an input, how did you determine the output the function would assign?

 We used the equation. In place of 𝑥𝑥, we put the input. The number that was computed was the output.

 So each input and its matching output correspond to a pair of numbers (𝑥𝑥,𝑦𝑦) that makes the equation 𝑦𝑦 = 1
7 𝑥𝑥

a true number sentence?

 Yes

Give students a moment to make sense of this, verifying that each pair of input/output values in Exercise 1 is indeed a

pair of numbers (𝑥𝑥,𝑦𝑦) that make 𝑦𝑦 = 1
7 𝑥𝑥 a true statement.

 And suppose we have a pair of numbers (𝑥𝑥,𝑦𝑦) that make 𝑦𝑦 = 1
7 𝑥𝑥 a true statement with 𝑥𝑥 positive. If 𝑥𝑥 is an

input of the function, the number of minutes Giselle runs, would 𝑦𝑦 be its matching output, the distance she
covers?

 Yes. We computed the outputs precisely by following the equation 𝑦𝑦 = 1
7 𝑥𝑥. So 𝑦𝑦 will be the matching

output to 𝑥𝑥.

 So can we conclude that any pair of numbers (𝑥𝑥,𝑦𝑦) that make the equation 𝑦𝑦 = 1
7 𝑥𝑥 a true number statement

correspond to an input and its matching output for the function?

 Yes

 And, backward, any pair of numbers (𝑥𝑥,𝑦𝑦) that represent an input/output pair for the function is a pair of

numbers that make the equation 𝑦𝑦 = 1
7 𝑥𝑥 a true number statement?

 Yes

 Can we make similar conclusions about Exercise 3, the function that gives the devices built over a given
number of hours?

Give students time to verify that the conclusions about Exercise 3 are the same as the conclusions about Exercise 1.
Then continue with the discussion.

 The function in Exercise 3 is described by the equation 𝑦𝑦 = 4𝑥𝑥.
 We have that the ordered pairs (𝑥𝑥,𝑦𝑦) that make the equation 𝑦𝑦 = 4𝑥𝑥 a true number sentence precisely match

the ordered pairs (𝑥𝑥,𝑦𝑦) with 𝑥𝑥 an input of the function and 𝑦𝑦 its matching output.

 Recall, in previous work, we defined the graph of an equation to be the set of all ordered pairs (𝑥𝑥,𝑦𝑦) that
make the equation a true number sentence. Today we define the graph of a function to be the set of all the
ordered pairs (𝑥𝑥,𝑦𝑦) with 𝑥𝑥 an input of the function and 𝑦𝑦 its matching output.

 And our discussion today shows that if a function can be described by an equation, then the graph of the
function is precisely the same as the graph of the equation.

 It is sometimes possible to draw the graph of a function even if there is no obvious equation describing the
function. (Consider having students plot some points of the function that assigns to each positive whole
number its first digit, for example.)

MP.6

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 For Exercise 2, you began by graphing the equation 𝑦𝑦 = 𝑥𝑥2 for positive values of 𝑥𝑥. What was the shape of the
graph?

 It looked curved.

 The graph had a curve in it because it was not the graph of a linear equation. All linear equations graph as
lines. That is what we learned in Module 4. Since this equation was not linear, we should expect it to graph as
something other than a line.

 What did you notice about the ordered pairs of the equation 𝑦𝑦 = 𝑥𝑥2 and the inputs and corresponding outputs
for the function 𝐴𝐴 = 𝑠𝑠2?
 The ordered pairs were exactly the same for the equation and the function.

 What does that mean about the graphs of functions, even those that are not linear?

 It means that the graph of a function will be identical to the graph of an equation.

Exploratory Challenge/Exercise 4 (7 minutes)

Students work in pairs to complete Exercise 4.

Exploratory Challenge/Exercise 4

4. Examine the three graphs below. Which, if any, could represent the graph of a function? Explain why or why not for
each graph.

Graph 1:

This is the graph of a function. Each input is a real number 𝒙𝒙, and we see from the graph that there is an output 𝒚𝒚 to
associate with each such input. For example, the ordered pair (−𝟐𝟐,𝟏𝟏) on the line associates the output 𝟏𝟏 to the
input −𝟐𝟐.

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Graph 2:

This is not the graph of a function. The ordered pairs (𝟔𝟔,𝟏𝟏) and (𝟔𝟔,𝟔𝟔) show that for the input of 𝟔𝟔 there are two
different outputs, both 𝟏𝟏 and 𝟔𝟔. We do not have a function.

Graph 3:

This is the graph of a function. The ordered pairs (−𝟑𝟑,−𝟗𝟗), (−𝟐𝟐,−𝟏𝟏), (−𝟐𝟐,−𝟐𝟐), (𝟎𝟎,𝟎𝟎), (𝟐𝟐,−𝟐𝟐), (𝟐𝟐,−𝟏𝟏), and
(𝟑𝟑,−𝟗𝟗) represent inputs and their unique outputs.

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Discussion (3 minutes)

 The graph of a function is the set of all points (𝑥𝑥,𝑦𝑦) with 𝑥𝑥 an input for the function and 𝑦𝑦 its matching output.
How did you use this definition to determine which graphs, if any, were functions?
 By the definition of a function, we need each input to have only one output. On a graph, this means

there cannot be two different ordered pairs with the same 𝑥𝑥 value.

 Assume the following set of ordered pairs is from some graph. Could this be the graph of a function? Explain.
(3, 5), (4, 7), (3, 9), (5,−2)

 No, because the input of 3 has two different outputs. It does not fit the definition of a function.

 Assume the following set of ordered pairs is from some graph. Could this be the graph of a function? Explain.
(−1, 6), (−3, 8), (5, 10), (7, 6)

 Yes, it is possible as each input has a unique output. It satisfies the definition of a function so far.

 Which of the following four graphs are functions? Explain.

Graph 1:

Graph 2:

Graph 3:

Graph 4:

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 Graphs 1 and 4 are functions. Graphs 2 and 3 are not. Graphs 1 and 4 show that for each input of 𝑥𝑥,
there is a unique output of 𝑦𝑦. For Graph 2, the input of 𝑥𝑥 = 1 has two different outputs, 𝑦𝑦 = 0 and
𝑦𝑦 = 2, which means it cannot be a function. For Graph 3, it appears that each value of 𝑥𝑥 between −5
and −1, excluding −5 and −1, has two outputs, one on the lower half of the circle and one on the
upper half, which means it does not fit the definition of function.

Closing (5 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 The graph of a function is defined to be the set of all points (𝑥𝑥,𝑦𝑦) with 𝑥𝑥 an input for the function and 𝑦𝑦 its
matching output.

 If a function can be described by an equation, then the graph of the function matches the graph of the
equation (at least at points which correspond to valid inputs of the function).

 We can look at plots of points and determine if they could be the graphs of functions.

Exit Ticket (5 minutes)

Lesson Summary

The graph of a function is defined to be the set of all points (𝒙𝒙,𝒚𝒚) with 𝒙𝒙 an input for the function and 𝒚𝒚 its
matching output.

If a function can be described by an equation, then the graph of the function is the same as the graph of the
equation that represents it (at least at points which correspond to valid inputs of the function).

It is not possible for two different points in the plot of the graph of a function to have the same 𝒙𝒙-coordinate.

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Name Date

Lesson 5: Graphs of Functions and Equations

Exit Ticket

Water flows from a hose at a constant rate of 11 gallons every 4 minutes. The total amount of water that flows from the
hose is a function of the number of minutes you are observing the hose.

a. Write an equation in two variables that describes the amount of water, 𝑦𝑦, in gallons, that flows from the hose
as a function of the number of minutes, 𝑥𝑥, you observe it.

b. Use the equation you wrote in part (a) to determine the amount of water that flows from the hose during an

8-minute period, a 4-minute period, and a 2-minute period.

c. An input of the function, 𝑥𝑥, is time in minutes, and the
output of the function, 𝑦𝑦, is the amount of water that
flows out of the hose in gallons. Write the inputs and
outputs from part (b) as ordered pairs, and plot them as
points on the coordinate plane.

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Exit Ticket Sample Solutions

Water flows from a hose at a constant rate of 𝟐𝟐𝟐𝟐 gallons every 𝟏𝟏 minutes. The total amount of water that flows from the
hose is a function of the number of minutes you are observing the hose.

a. Write an equation in two variables that describes the amount of water, 𝒚𝒚, in gallons, that flows from the hose
as a function of the number of minutes, 𝒙𝒙, you observe it.

𝟐𝟐𝟐𝟐
𝟏𝟏

=
𝒚𝒚
𝒙𝒙

𝒚𝒚 =
𝟐𝟐𝟐𝟐
𝟏𝟏
𝒙𝒙

b. Use the equation you wrote in part (a) to determine the amount of water that flows from the hose during an

𝟐𝟐-minute period, a 𝟏𝟏-minute period, and a 𝟐𝟐-minute period.

𝒚𝒚 =
𝟐𝟐𝟐𝟐
𝟏𝟏

(𝟐𝟐)

𝒚𝒚 = 𝟐𝟐𝟐𝟐

In 𝟐𝟐 minutes, 𝟐𝟐𝟐𝟐 gallons of water flow out of the hose.

𝒚𝒚 =
𝟐𝟐𝟐𝟐
𝟏𝟏

(𝟏𝟏)

𝒚𝒚 = 𝟐𝟐𝟐𝟐

In 𝟏𝟏 minutes, 𝟐𝟐𝟐𝟐 gallons of water flow out of the hose.

𝒚𝒚 =
𝟐𝟐𝟐𝟐
𝟏𝟏

(𝟐𝟐)

𝒚𝒚 = 𝟓𝟓.𝟓𝟓

In 𝟐𝟐 minutes, 𝟓𝟓.𝟓𝟓 gallons of water flow out of the hose.

c. An input of the function, 𝒙𝒙, is time in
minutes, and the output of the function, 𝒚𝒚,
is the amount of water that flows out of
the hose in gallons. Write the inputs and
outputs from part (b) as ordered pairs, and
plot them as points on the coordinate
plane.

(𝟐𝟐,𝟐𝟐𝟐𝟐), (𝟏𝟏,𝟐𝟐𝟐𝟐), (𝟐𝟐,𝟓𝟓.𝟓𝟓)

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Problem Set Sample Solutions

1. The distance that Scott walks is a function of the time he spends walking. Scott can walk
𝟐𝟐
𝟐𝟐

mile every 𝟐𝟐 minutes.

Assume he walks at a constant rate.

a. Predict the shape of the graph of the function. Explain.

The graph of the function will likely be a line because a linear equation can describe Scott’s motion, and I
know that the graph of the function will be the same as the graph of the equation.

b. Write an equation to represent the distance that Scott can walk in miles, 𝒚𝒚, in 𝒙𝒙 minutes.

𝟎𝟎.𝟓𝟓
𝟐𝟐

=
𝒚𝒚
𝒙𝒙

𝒚𝒚 =
𝟎𝟎.𝟓𝟓
𝟐𝟐
𝒙𝒙

𝒚𝒚 =
𝟐𝟐
𝟐𝟐𝟔𝟔

𝒙𝒙

c. Use the equation you wrote in part (b) to determine how many miles Scott can walk in 𝟐𝟐𝟏𝟏 minutes.

𝒚𝒚 =
𝟐𝟐
𝟐𝟐𝟔𝟔

(𝟐𝟐𝟏𝟏)

𝒚𝒚 = 𝟐𝟐.𝟓𝟓

Scott can walk 𝟐𝟐.𝟓𝟓 miles in 𝟐𝟐𝟏𝟏 minutes.

d. Use the equation you wrote in part (b) to determine how many miles Scott can walk in 𝟐𝟐𝟐𝟐 minutes.

𝒚𝒚 =
𝟐𝟐
𝟐𝟐𝟔𝟔

(𝟐𝟐𝟐𝟐)

𝒚𝒚 =
𝟑𝟑
𝟏𝟏

Scott can walk 𝟎𝟎.𝟕𝟕𝟓𝟓 miles in 𝟐𝟐𝟐𝟐 minutes.

e. Use the equation you wrote in part (b) to determine how many miles Scott can walk in 𝟐𝟐𝟔𝟔 minutes.

𝒚𝒚 =
𝟐𝟐
𝟐𝟐𝟔𝟔

(𝟐𝟐𝟔𝟔)

𝒚𝒚 = 𝟐𝟐

Scott can walk 𝟐𝟐 mile in 𝟐𝟐𝟔𝟔 minutes.

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f. Write your inputs and corresponding outputs as ordered pairs, and then plot them on a coordinate plane.

(𝟐𝟐𝟏𝟏,𝟐𝟐.𝟓𝟓), (𝟐𝟐𝟐𝟐,𝟎𝟎.𝟕𝟕𝟓𝟓), (𝟐𝟐𝟔𝟔,𝟐𝟐)

g. What shape does the graph of the points appear to take? Does it match your prediction?

The points appear to be in a line. Yes, as I predicted, the graph of the function is a line.

h. Connect the points to make a line. What is the equation of the line?

It is the equation that described the function: 𝒚𝒚 = 𝟐𝟐
𝟐𝟐𝟔𝟔𝒙𝒙.

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2. Graph the equation 𝒚𝒚 = 𝒙𝒙𝟑𝟑 for positive values of 𝒙𝒙. Organize your work using the table below, and then answer the
questions that follow.

𝒙𝒙 𝒚𝒚

𝟎𝟎 𝟎𝟎

𝟎𝟎.𝟓𝟓 𝟎𝟎.𝟐𝟐𝟐𝟐𝟓𝟓

𝟐𝟐 𝟐𝟐

𝟐𝟐.𝟓𝟓 𝟑𝟑.𝟑𝟑𝟕𝟕𝟓𝟓

𝟐𝟐 𝟐𝟐

𝟐𝟐.𝟓𝟓 𝟐𝟐𝟓𝟓.𝟔𝟔𝟐𝟐𝟓𝟓

a. Plot the ordered pairs on the coordinate plane.

b. What shape does the graph of the points appear to
take?

It appears to take the shape of a curve.

c. Is this the graph of a linear function? Explain.

No, this is not the graph of a linear function. The
equation 𝒚𝒚 = 𝒙𝒙𝟑𝟑 is not a linear equation.

d. Consider the function that assigns to each positive real
number 𝒔𝒔 the volume 𝑽𝑽 of a cube with side length 𝒔𝒔
units. An equation that describes this function is 𝑽𝑽 =
𝒔𝒔𝟑𝟑. What do you think the graph of this function will
look like? Explain.

I think the graph of this function will look like the
graph of the equation 𝒚𝒚 = 𝒙𝒙𝟑𝟑. The inputs and outputs
would match the solutions to the equation exactly. For
the equation, the 𝒚𝒚-value is the cube of the 𝒙𝒙-value.
For the function, the output is the cube of the input.

e. Use the function in part (d) to determine the volume of a cube with side length of 𝟑𝟑 units. Write the input
and output as an ordered pair. Does this point appear to belong
to the graph of 𝒚𝒚 = 𝒙𝒙𝟑𝟑?

𝑽𝑽 = (𝟑𝟑)𝟑𝟑
𝑽𝑽 = 𝟐𝟐𝟕𝟕

(𝟑𝟑,𝟐𝟐𝟕𝟕) The point looks like it would belong to the graph of 𝒚𝒚 = 𝒙𝒙𝟑𝟑; it looks like it would be on the curve that
the shape of the graph is taking.

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Lesson 5: Graphs of Functions and Equations

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 5

3. Sketch the graph of the equation 𝒚𝒚 = 𝟐𝟐𝟐𝟐𝟎𝟎(𝒙𝒙 − 𝟐𝟐) for whole numbers. Organize your work using the table below,
and then answer the questions that follow.

𝒙𝒙 𝒚𝒚

𝟑𝟑 𝟐𝟐𝟐𝟐𝟎𝟎

𝟏𝟏 𝟑𝟑𝟔𝟔𝟎𝟎

𝟓𝟓 𝟓𝟓𝟏𝟏𝟎𝟎

𝟔𝟔 𝟕𝟕𝟐𝟐𝟎𝟎

a. Plot the ordered pairs on the coordinate plane.

b. What shape does the graph of the points appear to
take?

It appears to take the shape of a line.

c. Is this graph a graph of a function? How do you
know?

It appears to be a function because each input has
exactly one output.

d. Is this a linear equation? Explain.

Yes, 𝒚𝒚 = 𝟐𝟐𝟐𝟐𝟎𝟎(𝒙𝒙 − 𝟐𝟐) is a linear equation. It can be
rewritten as 𝒚𝒚 = 𝟐𝟐𝟐𝟐𝟎𝟎𝒙𝒙 − 𝟑𝟑𝟔𝟔𝟎𝟎.

e. The sum 𝑺𝑺 of interior angles, in degrees, of a polygon
with 𝒏𝒏 sides is given by 𝑺𝑺 = 𝟐𝟐𝟐𝟐𝟎𝟎(𝒏𝒏− 𝟐𝟐). If we take
this equation as defining 𝑺𝑺 as a function of 𝒏𝒏, how do
you think the graph of this 𝑺𝑺 will appear? Explain.

I think the graph of this function will look like the
graph of the equation 𝒚𝒚 = 𝟐𝟐𝟐𝟐𝟎𝟎(𝒙𝒙 − 𝟐𝟐). The inputs
and outputs would match the solutions to the
equation exactly.

f. Is this function discrete? Explain.

The function 𝑺𝑺 = 𝟐𝟐𝟐𝟐𝟎𝟎(𝒏𝒏− 𝟐𝟐) is discrete. The inputs are the number of sides, which are integers. The input,
𝒏𝒏, must be greater than 𝟐𝟐 since three sides is the smallest number of sides for a polygon.

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Lesson 5: Graphs of Functions and Equations

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 5

4. Examine the graph below. Could the graph represent the graph of a function? Explain why or why not.

This is not the graph of a function. The ordered
pairs (𝟐𝟐,𝟎𝟎) and (𝟐𝟐,−𝟐𝟐) show that for the input of
𝟐𝟐 there are two different outputs, both 𝟎𝟎 and −𝟐𝟐.
For that reason, this cannot be the graph of a
function because it does not fit the definition of a
function.

5. Examine the graph below. Could the graph represent the graph of a function? Explain why or why not.

This is not the graph of a function. The ordered
pairs (𝟐𝟐,−𝟐𝟐) and (𝟐𝟐,−𝟑𝟑) show that for the input of
𝟐𝟐 there are two different outputs, both −𝟐𝟐 and −𝟑𝟑.
Further, the ordered pairs (𝟓𝟓,−𝟑𝟑) and (𝟓𝟓,−𝟏𝟏) show
that for the input of 𝟓𝟓 there are two different
outputs, both −𝟑𝟑 and −𝟏𝟏. For these reasons, this
cannot be the graph of a function because it does
not fit the definition of a function.

6. Examine the graph below. Could the graph represent the graph of a function? Explain why or why not.

This is the graph of a function. The ordered pairs (−𝟐𝟐,−𝟏𝟏),
(−𝟐𝟐,−𝟑𝟑), (𝟎𝟎,−𝟐𝟐), (𝟐𝟐,−𝟐𝟐), (𝟐𝟐,𝟎𝟎), and (𝟑𝟑,𝟐𝟐) represent inputs
and their unique outputs. By definition, this is a function.

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

Lesson 6: Graphs of Linear Functions and Rate of Change

Student Outcomes

 Students deepen their understanding of linear functions.

Lesson Notes
This lesson contains a ten-minute fluency exercise that can occur at any time throughout this lesson. The exercise has
students look for and make use of structure while solving multi-step equations.

Classwork

Opening Exercise (5 minutes)

Opening Exercise

A function is said to be linear if the rule defining the function can be described by a linear equation.

Functions 𝟏𝟏, 𝟐𝟐, and 𝟑𝟑 have table-values as shown. Which of these functions appear to be linear? Justify your answers.

Input Output Input Output Input Output

𝟐𝟐 𝟓𝟓 𝟐𝟐 𝟒𝟒 𝟎𝟎 −𝟑𝟑

𝟒𝟒 𝟕𝟕 𝟑𝟑 𝟗𝟗 𝟏𝟏 𝟏𝟏

𝟓𝟓 𝟖𝟖 𝟒𝟒 𝟏𝟏𝟏𝟏 𝟐𝟐 𝟏𝟏

𝟖𝟖 𝟏𝟏𝟏𝟏 𝟓𝟓 𝟐𝟐𝟓𝟓 𝟑𝟑 𝟗𝟗

Lead a short discussion that allows students to share their conjectures and reasoning. Revisit the Opening Exercise at
the end of the discussion so students can verify if their conjectures were correct. Only the first function is a linear
function.

Discussion (15 minutes)

Ask students to summarize what they learned from the last lesson. Make sure they recall
that the graph of a numerical function is the set of ordered pairs (𝑥𝑥,𝑦𝑦) with 𝑥𝑥 an input of
the function and 𝑦𝑦 its corresponding output. Also, recall that the graph of a function is
identical to the graph of the equation that describes it (if there is one). Next, ask students
to recall what they know about rate of change and slope and to recall that the graph of a
linear equation (the set of pairs (𝑥𝑥, 𝑦𝑦) that make the equation a true number sentence) is
a straight line.

Scaffolding:
Students may need a brief
review of the terms related to
linear equations.

MP.1
&

MP.3

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

 Suppose a function can be described by an equation in the form of 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 and assigns the values shown
in the table below:

Input Output

2 5

3.5 8

4 9

4.5 10

 The graph of a linear equation is a line, and so the graph of this function will be a line. How do we compute
the slope of the graph of a line?
 To compute slope, we find the difference in 𝑦𝑦-values compared to the difference in 𝑥𝑥-values. We use

the following formula:

𝑚𝑚 =
𝑦𝑦1 − 𝑦𝑦2
𝑥𝑥1 − 𝑥𝑥2

 And what is the slope of the line associated with this data? Using the first two rows of the table we get:
5 − 8

2 − 3.5
=

−3
−1.5

= 2

 To check, calculate the rate of change between rows two and three and rows three and four as well.

 Sample student work:
8 − 9

3.5 − 4
=

−1
−0.5

= 2

or 9 − 10
4 − 4.5

=
−1
−0.5

= 2

 Does the claim that the function is linear seem reasonable?

 Yes, the rate of change between each pair of inputs and outputs does seem to be constant.

 Check one more time by computing the slope from one more pair.

 Sample student work:
5 − 10
2 − 4.5

=
−5
−2.5

= 2

or 5 − 9
2 − 4

=
−4
−2

= 2

 Can we now find the equation that describes the function? At this point, we expect the equation to be
described by 𝑦𝑦 = 2𝑥𝑥 + 𝑏𝑏 because we know the slope is 2. Since the function assigns 5 to 2, 8 to 3.5, and so
on, we can use that information to determine the value of 𝑏𝑏.

Using the assignment of 5 to 2 we see:

5 = 2(2) + 𝑏𝑏
5 = 4 + 𝑏𝑏
1 = 𝑏𝑏

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

 Now that we know that 𝑏𝑏 = 1, we can substitute into 𝑦𝑦 = 2𝑥𝑥 + 𝑏𝑏, which results in the equation 𝑦𝑦 = 2𝑥𝑥 + 1.
The equation that describes the function is 𝑦𝑦 = 2𝑥𝑥 + 1, and the function is a linear function. What would the
graph of this function look like?

 It would be a line because the rule that describes the function in the form of 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 is an equation
known to graph as a line.

 The following table represents the outputs that a function would assign to given inputs. We want to know if
the function is a linear function and, if so, what linear equation describes the function.

Input Output

−2 4

3 9

4.5 20.25

5 25

 How should we begin? How do we check if the function is linear?

 We need to inspect the rate of change between pairs of inputs and their corresponding outputs and see
if that value is constant.

 Compare at least three pairs of inputs and their corresponding outputs.

 Sample student work:

4 − 9
−2 − 3

=
−5
−5

= 1

4 − 25
−2 − 5

=
−21
−7

= 3

9 − 25
3 − 5

=
−16
−2

= 8

 What do you notice about the rate of change, and what does this mean about the function?

 The rate of change was different for each pair of inputs and outputs inspected, which means that it is
not a linear function.

 If this were a linear function, what would we expect to see?
 If this were a linear function, each inspection of the rate of change would result in the same number

(similar to what we saw in the last problem, in which each result was 2).

 We have enough evidence to conclude that this function is not a linear function. Would the graph of this
function be a line? Explain.

 No, the graph of this function would not be a line. Only linear functions, whose equations are in the
form of 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏, graph as lines. Since this function does not have a constant rate of change, it will
not graph as a line.

Exercise (5 minutes)

Students work independently or in pairs to complete the exercise. Make sure to discuss with the class the subtle points
made in parts (c) and (d) and their solutions.

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

Exercise

A function assigns to the inputs shown the corresponding outputs given in the table below.

Input Output

𝟏𝟏 𝟐𝟐

𝟐𝟐 −𝟏𝟏

𝟒𝟒 −𝟕𝟕

𝟏𝟏 −𝟏𝟏𝟑𝟑

a. Do you suspect the function is linear? Compute the rate of change of this data for at least three pairs of
inputs and their corresponding outputs.

𝟐𝟐 − (−𝟏𝟏)
𝟏𝟏 − 𝟐𝟐

=
𝟑𝟑
−𝟏𝟏

= −𝟑𝟑

−𝟕𝟕− (−𝟏𝟏𝟑𝟑)
𝟒𝟒 − 𝟏𝟏

=
𝟏𝟏
−𝟐𝟐

= −𝟑𝟑

𝟐𝟐 − (−𝟕𝟕)
𝟏𝟏 − 𝟒𝟒

=
𝟗𝟗
−𝟑𝟑

= −𝟑𝟑

Yes, the rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is
equal to −𝟑𝟑. Since the rate of change is the same, then I know it is a linear function.

b. What equation seems to describe the function?

Using the assignment of 𝟐𝟐 to 𝟏𝟏:

𝟐𝟐 = −𝟑𝟑(𝟏𝟏) + 𝒃𝒃
𝟐𝟐 = −𝟑𝟑+ 𝒃𝒃
𝟓𝟓 = 𝒃𝒃

The equation that seems to describe the function is 𝒚𝒚 = −𝟑𝟑𝟑𝟑 + 𝟓𝟓.

c. As you did not verify that the rate of change is constant across all input/output pairs, check that the equation
you found in part (a) does indeed produce the correct output for each of the four inputs 1, 2, 4, and 6.

For 𝟑𝟑 = 𝟏𝟏 we have 𝒚𝒚 = −𝟑𝟑(𝟏𝟏) + 𝟓𝟓 = 𝟐𝟐.

For 𝟑𝟑 = 𝟐𝟐 we have 𝒚𝒚 = −𝟑𝟑(𝟐𝟐) + 𝟓𝟓 = −𝟏𝟏.

For 𝟑𝟑 = 𝟒𝟒 we have 𝒚𝒚 = −𝟑𝟑(𝟒𝟒) + 𝟓𝟓 = −𝟕𝟕.

For 𝟑𝟑 = 𝟏𝟏 we have 𝒚𝒚 = −𝟑𝟑(𝟏𝟏) + 𝟓𝟓 = −𝟏𝟏𝟑𝟑.

These are correct.

d. What will the graph of the function look like? Explain.

The graph of the function will be a plot of four points lying on a common line. As we were not told about any
other inputs for this function, we must assume for now that there are only these four input values for the
function.

The four points lie on the line with equation 𝒚𝒚 = −𝟑𝟑𝟑𝟑 + 𝟓𝟓.

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

Closing (5 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 We know that if the rate of change for pairs of inputs and corresponding outputs for a function is the same for
all pairs, then the function is a linear function.

 We know that a linear function can be described by a linear equation 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏.

 We know that the graph of a linear function will be a set of points all lying on a common line. If the linear
function is discrete, then its graph will be a set of distinct collinear points. If the linear function is not discrete,
then its graph will be a full straight line or a portion of the line (as appropriate for the context of the problem).

Exit Ticket (5 minutes)

Fluency Exercise (10 minutes): Multi-Step Equations I

Rapid White Board Exchange (RWBE): In this exercise, students solve three sets of similar multi-step equations. Display
the equations one at a time. Each equation should be solved in less than one minute; however, students may need
slightly more time for the first set and less time for the next two sets if they notice the pattern. Consider having
students work on personal white boards, and have them show their solutions for each problem. The three sets of
equations and their answers are located at the end of the lesson. Refer to the Rapid White Board Exchanges section in
the Module Overview for directions to administer a RWBE.

Lesson Summary

If the rate of change for pairs of inputs and corresponding outputs for a function is the same for all pairs (constant),
then the function is a linear function. It can thus be described by a linear equation 𝒚𝒚 = 𝒎𝒎𝟑𝟑 + 𝒃𝒃.

The graph of a linear function will be a set of points contained in a line. If the linear function is discrete, then its
graph will be a set of distinct collinear points. If the linear function is not discrete, then its graph will be a full
straight line or a portion of the line (as appropriate for the context of the problem).

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

Name Date

Lesson 6: Graphs of Linear Functions and Rate of Change

Exit Ticket

1. Sylvie claims that a function with the table of inputs and outputs below is a linear function. Is she correct? Explain.

Input Output
−3 −25
2 10
5 31
8 54

2. A function assigns the inputs and corresponding outputs shown in the table to the
right.
a. Does the function appear to be linear? Check at least three pairs of inputs and

their corresponding outputs.

Input Output
−2 3
8 −2

10 −3
20 −8

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

b. Can you write a linear equation that describes the function?

c. What will the graph of the function look like? Explain.

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

Exit Ticket Sample Solutions

1. Sylvie claims that the function with the table of inputs and outputs is a linear function. Is she correct? Explain.

Input Output

−𝟑𝟑 −𝟐𝟐𝟓𝟓

𝟐𝟐 𝟏𝟏𝟎𝟎

𝟓𝟓 𝟑𝟑𝟏𝟏

𝟖𝟖 𝟓𝟓𝟒𝟒

−𝟐𝟐𝟓𝟓 − (𝟏𝟏𝟎𝟎)
−𝟑𝟑 − 𝟐𝟐

=
−𝟑𝟑𝟓𝟓
−𝟓𝟓

= 𝟕𝟕

𝟏𝟏𝟎𝟎 − 𝟑𝟑𝟏𝟏
𝟐𝟐 − 𝟓𝟓

=
−𝟐𝟐𝟏𝟏
−𝟑𝟑

= 𝟕𝟕

𝟑𝟑𝟏𝟏 − 𝟓𝟓𝟒𝟒
𝟓𝟓 − 𝟖𝟖

=
−𝟐𝟐𝟑𝟑
−𝟑𝟑

=
𝟐𝟐𝟑𝟑
𝟑𝟑

No, this is not a linear function. The rate of change was not the same for each pair of inputs and outputs inspected,
which means that it is not a linear function.

2. A function assigns the inputs and corresponding outputs shown in the table below.

a. Does the function appear to be linear? Check at least three pairs of inputs and their corresponding outputs.

𝟑𝟑 − (−𝟐𝟐)
−𝟐𝟐− 𝟖𝟖

=
𝟓𝟓

−𝟏𝟏𝟎𝟎
= −

𝟏𝟏
𝟐𝟐

−𝟐𝟐− (−𝟑𝟑)
𝟖𝟖 − 𝟏𝟏𝟎𝟎

=
𝟏𝟏
−𝟐𝟐

= −
𝟏𝟏
𝟐𝟐

−𝟑𝟑− (−𝟖𝟖)
𝟏𝟏𝟎𝟎 − 𝟐𝟐𝟎𝟎

=
𝟓𝟓

−𝟏𝟏𝟎𝟎
= −

𝟏𝟏
𝟐𝟐

Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is

equal to −𝟏𝟏
𝟐𝟐 . Since the rate of change is the same for at least these three examples, the function could well

be linear.

b. Can you write a linear equation that describes the function?

We suspect we have an equation of the form 𝒚𝒚 = −𝟏𝟏
𝟐𝟐𝟑𝟑 + 𝒃𝒃. Using the assignment of 𝟑𝟑 to −𝟐𝟐:

𝟑𝟑 = −
𝟏𝟏
𝟐𝟐

(−𝟐𝟐) + 𝒃𝒃

𝟑𝟑 = 𝟏𝟏 + 𝒃𝒃
𝟐𝟐 = 𝒃𝒃

The equation that describes the function might be 𝒚𝒚 = −𝟏𝟏
𝟐𝟐𝟑𝟑 + 𝟐𝟐.

Checking: When 𝟑𝟑 = −𝟐𝟐, we get 𝒚𝒚 = −𝟏𝟏
𝟐𝟐 (−𝟐𝟐) + 𝟐𝟐 = 𝟑𝟑. When 𝟑𝟑 = 𝟖𝟖, we get 𝒚𝒚 = −𝟏𝟏

𝟐𝟐 (𝟖𝟖) + 𝟐𝟐 = −𝟐𝟐. When

𝟑𝟑 = 𝟏𝟏𝟎𝟎, we get 𝒚𝒚 = −𝟏𝟏
𝟐𝟐 (𝟏𝟏𝟎𝟎) + 𝟐𝟐 = −𝟑𝟑. When 𝟑𝟑 = 𝟐𝟐𝟎𝟎, we get 𝒚𝒚 = −𝟏𝟏

𝟐𝟐 (𝟐𝟐𝟎𝟎) + 𝟐𝟐 = −𝟖𝟖.

It works.

c. What will the graph of the function look like? Explain.

The graph of the function will be four distinct points all lying in a line. (They all lie on the line with equation

𝒚𝒚 = −𝟏𝟏
𝟐𝟐𝟑𝟑 + 𝟐𝟐 ).

Input Output

−𝟐𝟐 𝟑𝟑

𝟖𝟖 −𝟐𝟐

𝟏𝟏𝟎𝟎 −𝟑𝟑

𝟐𝟐𝟎𝟎 −𝟖𝟖

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

Problem Set Sample Solutions

1. A function assigns to the inputs given the corresponding outputs shown in the table below.

Input Output

𝟑𝟑 𝟗𝟗
𝟗𝟗 𝟏𝟏𝟕𝟕
𝟏𝟏𝟐𝟐 𝟐𝟐𝟏𝟏
𝟏𝟏𝟓𝟓 𝟐𝟐𝟓𝟓

a. Does the function appear to be linear? Check at least three pairs of inputs and their corresponding outputs.

𝟗𝟗 − 𝟏𝟏𝟕𝟕
𝟑𝟑 − 𝟗𝟗

=
−𝟖𝟖
−𝟏𝟏

=
𝟒𝟒
𝟑𝟑

𝟏𝟏𝟕𝟕 − 𝟐𝟐𝟏𝟏
𝟗𝟗 − 𝟏𝟏𝟐𝟐

=
−𝟒𝟒
−𝟑𝟑

=
𝟒𝟒
𝟑𝟑

𝟐𝟐𝟏𝟏 − 𝟐𝟐𝟓𝟓
𝟏𝟏𝟐𝟐 − 𝟏𝟏𝟓𝟓

=
−𝟒𝟒
−𝟑𝟑

=
𝟒𝟒
𝟑𝟑

Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is

equal to
𝟒𝟒
𝟑𝟑

. Since the rate of change is the same, the function does appear to be linear.

b. Find a linear equation that describes the function.

Using the assignment of 𝟗𝟗 to 𝟑𝟑

𝟗𝟗 =
𝟒𝟒
𝟑𝟑

(𝟑𝟑) + 𝒃𝒃

𝟗𝟗 = 𝟒𝟒 + 𝒃𝒃
𝟓𝟓 = 𝒃𝒃

The equation that describes the function is 𝒚𝒚 = 𝟒𝟒
𝟑𝟑𝟑𝟑 + 𝟓𝟓. (We check that for each of the four inputs given, this

equation does indeed produce the correct matching output.)

c. What will the graph of the function look like? Explain.

The graph of the function will be four points in a row. They all lie on the line given by the equation

𝒚𝒚 = 𝟒𝟒
𝟑𝟑𝟑𝟑 + 𝟓𝟓.

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

2. A function assigns to the inputs given the corresponding outputs shown in the table below.

Input Output

−𝟏𝟏 𝟐𝟐
𝟎𝟎 𝟎𝟎
𝟏𝟏 𝟐𝟐
𝟐𝟐 𝟖𝟖
𝟑𝟑 𝟏𝟏𝟖𝟖

a. Is the function a linear function?

𝟐𝟐 − 𝟎𝟎
−𝟏𝟏− 𝟎𝟎

=
𝟐𝟐
−𝟏𝟏

= −𝟐𝟐

𝟎𝟎 − 𝟐𝟐
𝟎𝟎 − 𝟏𝟏

=
−𝟐𝟐
−𝟏𝟏

= 𝟐𝟐

No. The rate of change is not the same when I check the first two pairs of inputs and corresponding outputs.
All rates of change must be the same for all inputs and outputs for the function to be linear.

b. What equation describes the function?

I am not sure what equation describes the function. It is not a linear function.

3. A function assigns the inputs and corresponding outputs shown in the table below.

Input Output

𝟎𝟎.𝟐𝟐 𝟐𝟐
𝟎𝟎.𝟏𝟏 𝟏𝟏
𝟏𝟏.𝟓𝟓 𝟏𝟏𝟓𝟓
𝟐𝟐.𝟏𝟏 𝟐𝟐𝟏𝟏

a. Does the function appear to be linear? Check at least three pairs of inputs and their corresponding outputs..

𝟐𝟐 − 𝟏𝟏
𝟎𝟎.𝟐𝟐 − 𝟎𝟎.𝟏𝟏

=
−𝟒𝟒
−𝟎𝟎.𝟒𝟒

= 𝟏𝟏𝟎𝟎

𝟏𝟏 − 𝟏𝟏𝟓𝟓
𝟎𝟎.𝟏𝟏 − 𝟏𝟏.𝟓𝟓

=
−𝟗𝟗
−𝟎𝟎.𝟗𝟗

= 𝟏𝟏𝟎𝟎

𝟏𝟏𝟓𝟓 − 𝟐𝟐𝟏𝟏
𝟏𝟏.𝟓𝟓 − 𝟐𝟐.𝟏𝟏

=
−𝟏𝟏
−𝟎𝟎.𝟏𝟏

= 𝟏𝟏𝟎𝟎

Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is
equal to 𝟏𝟏𝟎𝟎. The function appears to be linear.

b. Find a linear equation that describes the function.

Using the assignment of 𝟐𝟐 to 𝟎𝟎.𝟐𝟐:

𝟐𝟐 = 𝟏𝟏𝟎𝟎(𝟎𝟎.𝟐𝟐) + 𝒃𝒃
𝟐𝟐 = 𝟐𝟐+ 𝒃𝒃
𝟎𝟎 = 𝒃𝒃

The equation that describes the function is 𝒚𝒚 = 𝟏𝟏𝟎𝟎𝟑𝟑 . It clearly fits the data presented in the table.

c. What will the graph of the function look like? Explain.

The graph will be four distinct points in a row. They all sit on the line given by the equation 𝒚𝒚 = 𝟏𝟏𝟎𝟎𝟑𝟑.

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

4. Martin says that you only need to check the first and last input and output values to determine if the function is
linear. Is he correct? Explain.

No, he is not correct. For example, consider the function with input and output values in this table.

Using the first and last input and output, the rate of change is

𝟗𝟗 − 𝟏𝟏𝟐𝟐
𝟏𝟏 − 𝟑𝟑

=
−𝟑𝟑
−𝟐𝟐

=
𝟑𝟑
𝟐𝟐

But when you use the first two inputs and outputs, the rate of change is

𝟗𝟗 − 𝟏𝟏𝟎𝟎
𝟏𝟏 − 𝟐𝟐

=
−𝟏𝟏
−𝟏𝟏

= 𝟏𝟏

Note to teacher: Accept any example where the rate of change is different for any two inputs and outputs.

5. Is the following graph a graph of a linear function? How would you determine if it is a linear function?

It appears to be a linear function. To check, I would organize the coordinates in an input and output table. Next, I
would check to see that all the rates of change are the same. If they are the same rates of change, I would use the
equation 𝒚𝒚 = 𝒎𝒎𝟑𝟑 + 𝒃𝒃 and one of the assignments to write an equation to solve for 𝒃𝒃. That information would allow
me to determine the equation that represents the function.

Input Output
𝟏𝟏 𝟗𝟗
𝟐𝟐 𝟏𝟏𝟎𝟎
𝟑𝟑 𝟏𝟏𝟐𝟐

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

6. A function assigns to the inputs given the corresponding outputs shown in the table below.

Input Output

−𝟏𝟏 −𝟏𝟏
−𝟓𝟓 −𝟓𝟓
−𝟒𝟒 −𝟒𝟒
−𝟐𝟐 −𝟐𝟐

a. Does the function appear to be a linear function?

−𝟏𝟏 − (−𝟓𝟓)
−𝟏𝟏 − (−𝟓𝟓)

=
𝟏𝟏
𝟏𝟏

= 𝟏𝟏

−𝟓𝟓 − (−𝟒𝟒)
−𝟓𝟓 − (−𝟒𝟒)

=
𝟏𝟏
𝟏𝟏

= 𝟏𝟏

−𝟒𝟒 − (−𝟐𝟐)
−𝟒𝟒 − (−𝟐𝟐)

=
𝟐𝟐
𝟐𝟐

= 𝟏𝟏

Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is
equal to 𝟏𝟏. Since the rate of change is constant so far, it could be a linear function.

b. What equation describes the function?

Clearly the equation 𝒚𝒚 = 𝟑𝟑 fits the data. It is a linear function.

c. What will the graph of the function look like? Explain.

The graph of the function will be four distinct points in a row. These four points lie on the line given by the
equation 𝒚𝒚 = 𝟑𝟑.

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Lesson 6: Graphs of Linear Functions and Rate of Change

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 6

Multi-Step Equations I

Set 1:

3𝑥𝑥 + 2 = 5𝑥𝑥 + 6

4(5𝑥𝑥 + 6) = 4(3𝑥𝑥 + 2)

3𝑥𝑥 + 2
6

=
5𝑥𝑥 + 6

Answer for each problem in this set is 𝟑𝟑 = −𝟐𝟐.

Set 2:
6 − 4𝑥𝑥 = 10𝑥𝑥 + 9

−2(−4𝑥𝑥 + 6) = −2(10𝑥𝑥 + 9)

10𝑥𝑥 + 9
5

=
6 − 4𝑥𝑥

Answer for each problem in this set is 𝟑𝟑 = − 𝟑𝟑
𝟏𝟏𝟒𝟒.

Set 3:

5𝑥𝑥 + 2 = 9𝑥𝑥 − 18

8𝑥𝑥 + 2 − 3𝑥𝑥 = 7𝑥𝑥 − 18 + 2𝑥𝑥

2 + 5𝑥𝑥
3

=
7𝑥𝑥 − 18 + 2𝑥𝑥

Answer for each problem in this set is 𝟑𝟑 = 𝟓𝟓.

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Lesson 7: Comparing Linear Functions and Graphs

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7

Lesson 7: Comparing Linear Functions and Graphs

Student Outcomes

 Students compare the properties of two functions that are represented in different ways via tables, graphs,
equations, or written descriptions.

 Students use rate of change to compare linear functions.

Lesson Notes
The Fluency Exercise included in this lesson takes approximately 10 minutes and should be assigned either at the
beginning or at the end of the lesson.

Classwork

Exploratory Challenge/Exercises 1–4 (20 minutes)

Students work in small groups to complete Exercises 1–4. Groups can select a method of their choice to answer the
questions and their methods will be a topic of discussion once the Exploratory Challenge is completed. Encourage
students to discuss the various methods (e.g., graphing, comparing rates of change, using algebra) as a group before
they begin solving.

Exploratory Challenge/Exercises 1–4

Each of Exercises 1–4 provides information about two functions. Use that information given to help you compare the two
functions and answer the questions about them.

1. Alan and Margot each drive from City A to City B, a distance of 𝟏𝟏𝟏𝟏𝟏𝟏 miles. They take the same route and drive at
constant speeds. Alan begins driving at 1:40 p.m. and arrives at City B at 4:15 p.m. Margot’s trip from City A to City
B can be described with the equation 𝒚𝒚 = 𝟔𝟔𝟏𝟏𝟔𝟔, where 𝒚𝒚 is the distance traveled in miles and 𝟔𝟔 is the time in minutes
spent traveling. Who gets from City A to City B faster?

Student solutions will vary. Sample solution is provided.

It takes Alan 𝟏𝟏𝟏𝟏𝟏𝟏 minutes to travel the 𝟏𝟏𝟏𝟏𝟏𝟏 miles. Therefore, his constant rate is
𝟏𝟏𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏

miles

per minute.

Margot drives 𝟔𝟔𝟏𝟏 miles per hour (𝟔𝟔𝟔𝟔 minutes). Therefore, her constant rate is
𝟔𝟔𝟏𝟏
𝟔𝟔𝟔𝟔

miles per

minute.

To determine who gets from City A to City B faster, we just need to compare their rates in
miles per minute.

𝟏𝟏𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏

< 𝟔𝟔𝟏𝟏 𝟔𝟔𝟔𝟔 Since Margot’s rate is faster, she will get to City B faster than Alan. Scaffolding: Providing example language for students to reference will be useful. This might consist of sentence starters, sentence frames, or a word wall. MP.1 © 2015 Great Minds eureka-math.org G8-M5-TE-1.3.0-10.2015 http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US Lesson 7: Comparing Linear Functions and Graphs 92 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7 2. You have recently begun researching phone billing plans. Phone Company A charges a flat rate of $𝟏𝟏𝟏𝟏 a month. A flat rate means that your bill will be $𝟏𝟏𝟏𝟏 each month with no additional costs. The billing plan for Phone Company B is a linear function of the number of texts that you send that month. That is, the total cost of the bill changes each month depending on how many texts you send. The table below represents some inputs and the corresponding outputs that the function assigns. Input (number of texts) Output (cost of bill in dollars) 𝟏𝟏𝟔𝟔 𝟏𝟏𝟔𝟔 𝟏𝟏𝟏𝟏𝟔𝟔 𝟔𝟔𝟔𝟔 𝟐𝟐𝟔𝟔𝟔𝟔 𝟔𝟔𝟏𝟏 𝟏𝟏𝟔𝟔𝟔𝟔 𝟗𝟗𝟏𝟏 At what number of texts would the bill from each phone plan be the same? At what number of texts is Phone Company A the better choice? At what number of texts is Phone Company B the better choice? Student solutions will vary. Sample solution is provided. The equation that represents the function for Phone Company A is 𝒚𝒚 = 𝟏𝟏𝟏𝟏. To determine the equation that represents the function for Phone Company B, we need the rate of change. (We are told it is constant.) 𝟔𝟔𝟔𝟔 − 𝟏𝟏𝟔𝟔 𝟏𝟏𝟏𝟏𝟔𝟔 − 𝟏𝟏𝟔𝟔 = 𝟏𝟏𝟔𝟔 𝟏𝟏𝟔𝟔𝟔𝟔 = 𝟔𝟔.𝟏𝟏 The equation for Phone Company B is shown below. Using the assignment of 𝟏𝟏𝟔𝟔 to 𝟏𝟏𝟔𝟔, 𝟏𝟏𝟔𝟔 = 𝟔𝟔.𝟏𝟏(𝟏𝟏𝟔𝟔) + 𝒃𝒃 𝟏𝟏𝟔𝟔 = 𝟏𝟏 + 𝒃𝒃 𝟏𝟏𝟏𝟏 = 𝒃𝒃. The equation that represents the function for Phone Company B is 𝒚𝒚 = 𝟔𝟔.𝟏𝟏𝟔𝟔 + 𝟏𝟏𝟏𝟏. We can determine at what point the phone companies charge the same amount by solving the system: �𝒚𝒚 = 𝟏𝟏𝟏𝟏 𝒚𝒚 = 𝟔𝟔.𝟏𝟏𝟔𝟔 + 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 = 𝟔𝟔.𝟏𝟏𝟔𝟔 + 𝟏𝟏𝟏𝟏 𝟑𝟑𝟔𝟔 = 𝟔𝟔.𝟏𝟏𝟔𝟔 𝟑𝟑𝟔𝟔𝟔𝟔 = 𝟔𝟔 After 𝟑𝟑𝟔𝟔𝟔𝟔 texts are sent, both companies would charge the same amount, $𝟏𝟏𝟏𝟏. More than 𝟑𝟑𝟔𝟔𝟔𝟔 texts means that the bill from Phone Company B will be higher than Phone Company A. Less than 𝟑𝟑𝟔𝟔𝟔𝟔 texts means the bill from Phone Company A will be higher. © 2015 Great Minds eureka-math.org G8-M5-TE-1.3.0-10.2015 http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US Lesson 7: Comparing Linear Functions and Graphs 93 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7 3. The function that gives the volume of water, 𝒚𝒚, that flows from Faucet A in gallons during 𝟔𝟔 minutes is a linear function with the graph shown. Faucet B’s water flow can be described by the equation 𝒚𝒚 = 𝟏𝟏 𝟔𝟔𝟔𝟔, where 𝒚𝒚 is the volume of water in gallons that flows from the faucet during 𝟔𝟔 minutes. Assume the flow of water from each faucet is constant. Which faucet has a faster rate of flow of water? Each faucet is being used to fill a tub with a volume of 𝟏𝟏𝟔𝟔 gallons. How long will it take each faucet to fill its tub? How do you know? Suppose the tub being filled by Faucet A already had 𝟏𝟏𝟏𝟏 gallons of water in it, and the tub being filled by Faucet B started empty. If now both faucets are turned on at the same time, which faucet will fill its tub fastest? Student solutions will vary. Sample solution is provided. The slope of the graph of the line is 𝟏𝟏 𝟏𝟏 because (𝟏𝟏,𝟏𝟏) is a point on the line that represents 𝟏𝟏 gallons of water that flows in 𝟏𝟏 minutes. Therefore, the rate of water flow for Faucet A is 𝟏𝟏 𝟏𝟏 . To determine which faucet has a faster flow of water, we can compare their rates. 𝟏𝟏 𝟏𝟏 < 𝟏𝟏 𝟔𝟔 Therefore, Faucet B has a faster rate of water flow. Faucet A 𝒚𝒚 = 𝟏𝟏 𝟏𝟏 𝟔𝟔 𝟏𝟏𝟔𝟔 = 𝟏𝟏 𝟏𝟏 𝟔𝟔 𝟏𝟏𝟔𝟔� 𝟏𝟏 𝟏𝟏 � = 𝟔𝟔 𝟑𝟑𝟏𝟏𝟔𝟔 𝟏𝟏 = 𝟔𝟔 𝟖𝟖𝟏𝟏.𝟏𝟏 = 𝟔𝟔 It will take 𝟖𝟖𝟏𝟏.𝟏𝟏 minutes to fill a tub of 𝟏𝟏𝟔𝟔 gallons. Faucet B 𝒚𝒚 = 𝟏𝟏 𝟔𝟔 𝟔𝟔 𝟏𝟏𝟔𝟔 = 𝟏𝟏 𝟔𝟔 𝟔𝟔 𝟏𝟏𝟔𝟔� 𝟔𝟔 𝟏𝟏 � = 𝟔𝟔 𝟔𝟔𝟔𝟔 = 𝟔𝟔 It will take 𝟔𝟔𝟔𝟔 minutes to fill a tub of 𝟏𝟏𝟔𝟔 gallons. The tub filled by Faucet A that already has 𝟏𝟏𝟏𝟏 gallons in it 𝟏𝟏𝟔𝟔 = 𝟏𝟏 𝟏𝟏 𝟔𝟔 + 𝟏𝟏𝟏𝟏 𝟑𝟑𝟏𝟏 = 𝟏𝟏 𝟏𝟏 𝟔𝟔 𝟑𝟑𝟏𝟏� 𝟏𝟏 𝟏𝟏 � = 𝟔𝟔 𝟔𝟔𝟏𝟏.𝟐𝟐𝟏𝟏 = 𝟔𝟔 Faucet B will fill the tub first because it will take Faucet A 𝟔𝟔𝟏𝟏.𝟐𝟐𝟏𝟏 minutes to fill the tub, even though it already has 𝟏𝟏𝟏𝟏 gallons in it. © 2015 Great Minds eureka-math.org G8-M5-TE-1.3.0-10.2015 http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US Lesson 7: Comparing Linear Functions and Graphs 94 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7 4. Two people, Adam and Bianca, are competing to see who can save the most money in one month. Use the table and the graph below to determine who will save the most money at the end of the month. State how much money each person had at the start of the competition. (Assume each is following a linear function in his or her saving habit.) Adam’s Savings: Bianca’s Savings: The slope of the line that represents Adam’s savings is 𝟑𝟑; therefore, the rate at which Adam is saving money is $𝟑𝟑 per day. According to the table of values for Bianca, she is also saving money at a rate of $𝟑𝟑 per day: 𝟐𝟐𝟔𝟔 − 𝟏𝟏𝟏𝟏 𝟖𝟖 − 𝟏𝟏 = 𝟗𝟗 𝟑𝟑 = 𝟑𝟑 𝟑𝟑𝟖𝟖 − 𝟐𝟐𝟔𝟔 𝟏𝟏𝟐𝟐 − 𝟖𝟖 = 𝟏𝟏𝟐𝟐 𝟏𝟏 = 𝟑𝟑 𝟔𝟔𝟐𝟐 − 𝟐𝟐𝟔𝟔 𝟐𝟐𝟔𝟔 − 𝟖𝟖 = 𝟑𝟑𝟔𝟔 𝟏𝟏𝟐𝟐 = 𝟑𝟑 Therefore, at the end of the month, Adam and Bianca will both have saved the same amount of money. According to the graph for Adam, the equation 𝒚𝒚 = 𝟑𝟑𝟔𝟔 + 𝟑𝟑 represents the function of money saved each day. On day zero, he had $𝟑𝟑. The equation that represents the function of money saved each day for Bianca is 𝒚𝒚 = 𝟑𝟑𝟔𝟔 + 𝟐𝟐 because, using the assignment of 𝟏𝟏𝟏𝟏 to 𝟏𝟏 𝟏𝟏𝟏𝟏 = 𝟑𝟑(𝟏𝟏) + 𝒃𝒃 𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏+ 𝒃𝒃 𝟐𝟐 = 𝒃𝒃. The amount of money Bianca had on day zero was $𝟐𝟐. Input (Number of Days) Output (Total amount of money in dollars) 𝟏𝟏 𝟏𝟏𝟏𝟏 𝟖𝟖 𝟐𝟐𝟔𝟔 𝟏𝟏𝟐𝟐 𝟑𝟑𝟖𝟖 𝟐𝟐𝟔𝟔 𝟔𝟔𝟐𝟐 © 2015 Great Minds eureka-math.org G8-M5-TE-1.3.0-10.2015 http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US Lesson 7: Comparing Linear Functions and Graphs 95 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7 Discussion (5 minutes) To encourage students to compare different means of presenting linear functions, have students detail the different ways linear functions were described throughout these exercises. Use the following questions to guide the discussion.  Was one style of presentation easier to work with over the others? Does everyone agree?  Was it easier to read off certain pieces of information about a linear function (its initial value, its constant rate of change, for instance) from one presentation of that function over another? Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson:  We know that functions can be expressed as equations, graphs, tables, and using verbal descriptions. Regardless of the way that a function is expressed, we can compare it with other functions. Exit Ticket (5 minutes) Fluency Exercise (10 minutes): Multi-Step Equations II Rapid White Board Exchange (RWBE): During this exercise, students solve nine multi-step equations. Each equation should be solved in about a minute. Consider having students work on personal white boards, showing their solutions after each problem is assigned. The nine equations and their answers are at the end of the lesson. Refer to the Rapid White Board Exchanges section in the Module Overview for directions to administer a RWBE. MP.1 © 2015 Great Minds eureka-math.org G8-M5-TE-1.3.0-10.2015 http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US Lesson 7: Comparing Linear Functions and Graphs 96 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7 Name Date Lesson 7: Comparing Linear Functions and Graphs Exit Ticket Brothers Paul and Pete walk 2 miles to school from home. Paul can walk to school in 24 minutes. Pete has slept in again and needs to run to school. Paul walks at a constant rate, and Pete runs at a constant rate. The graph of the function that represents Pete’s run is shown below. a. Which brother is moving at a greater rate? Explain how you know. b. If Pete leaves 5 minutes after Paul, will he catch up to Paul before they get to school? © 2015 Great Minds eureka-math.org G8-M5-TE-1.3.0-10.2015 http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US Lesson 7: Comparing Linear Functions and Graphs 97 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7 Exit Ticket Sample Solutions Brothers Paul and Pete walk 𝟐𝟐 miles to school from home. Paul can walk to school in 𝟐𝟐𝟏𝟏 minutes. Pete has slept in again and needs to run to school. Paul walks at a constant rate, and Pete runs at a constant rate. The graph of the function that represents Pete’s run is shown below. a. Which brother is moving at a greater rate? Explain how you know. Paul takes 𝟐𝟐𝟏𝟏 minutes to walk 𝟐𝟐 miles; therefore, his rate is 𝟏𝟏 𝟏𝟏𝟐𝟐 miles per minute. Pete can run 𝟖𝟖 miles in 𝟔𝟔𝟔𝟔 minutes; therefore, his rate is 𝟖𝟖 𝟔𝟔𝟔𝟔 , or 𝟐𝟐 𝟏𝟏𝟏𝟏 miles per minute. Since 𝟐𝟐 𝟏𝟏𝟏𝟏 >
𝟏𝟏
𝟏𝟏𝟐𝟐

, Pete is moving at a greater rate.

b. If Pete leaves 𝟏𝟏 minutes after Paul, will he catch up to Paul
before they get to school?

Student solution methods will vary. Sample answer is shown.

Since Pete slept in, we need to account for that fact. So, Pete’s
time would be decreased. The equation that would represent
the number of miles Pete runs, 𝒚𝒚, in 𝟔𝟔 minutes, would be

𝒚𝒚 = 𝟐𝟐
𝟏𝟏𝟏𝟏 (𝟔𝟔 − 𝟏𝟏).

The equation that would represent the number of miles Paul walks, 𝒚𝒚, in 𝟔𝟔 minutes, would be 𝒚𝒚 = 𝟏𝟏
𝟏𝟏𝟐𝟐𝟔𝟔.

To find out when they meet, solve the system of equations:

�
𝒚𝒚 =

𝟐𝟐
𝟏𝟏𝟏𝟏

𝟔𝟔 −
𝟐𝟐
𝟑𝟑

𝒚𝒚 =
𝟏𝟏
𝟏𝟏𝟐𝟐

𝟔𝟔

𝟐𝟐
𝟏𝟏𝟏𝟏

𝟔𝟔 −
𝟐𝟐
𝟑𝟑

=
𝟏𝟏
𝟏𝟏𝟐𝟐

𝟔𝟔
𝟐𝟐
𝟏𝟏𝟏𝟏

𝟔𝟔 −
𝟐𝟐
𝟑𝟑
−

𝟏𝟏
𝟏𝟏𝟐𝟐

𝟔𝟔 +
𝟐𝟐
𝟑𝟑

=
𝟏𝟏
𝟏𝟏𝟐𝟐

𝟔𝟔 −
𝟏𝟏
𝟏𝟏𝟐𝟐

𝟔𝟔 +
𝟐𝟐
𝟑𝟑

𝟏𝟏
𝟐𝟐𝟔𝟔

𝟔𝟔 =
𝟐𝟐
𝟑𝟑

�
𝟐𝟐𝟔𝟔
𝟏𝟏
�
𝟏𝟏
𝟐𝟐𝟔𝟔

𝟔𝟔 =
𝟐𝟐
𝟑𝟑
�
𝟐𝟐𝟔𝟔
𝟏𝟏
�

𝟔𝟔 =
𝟏𝟏𝟔𝟔
𝟑𝟑

𝒚𝒚 =
𝟏𝟏
𝟏𝟏𝟐𝟐

�
𝟏𝟏𝟔𝟔
𝟑𝟑
� =

𝟏𝟏𝟔𝟔
𝟗𝟗

or 𝒚𝒚 =
𝟐𝟐
𝟏𝟏𝟏𝟏

�
𝟏𝟏𝟔𝟔
𝟑𝟑
� −

𝟐𝟐
𝟑𝟑

Pete would catch up to Paul in
𝟏𝟏𝟔𝟔
𝟑𝟑

minutes, which occurs
𝟏𝟏𝟔𝟔
𝟗𝟗

miles from their home. Yes, he will catch Paul

before they get to school because it is less than the total distance, two miles, to school.

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Lesson 7: Comparing Linear Functions and Graphs

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7

Problem Set Sample Solutions

1. The graph below represents the distance in miles, 𝒚𝒚, Car A travels in 𝟔𝟔 minutes. The table represents the distance in
miles, 𝒚𝒚, Car B travels in 𝟔𝟔 minutes. It is moving at a constant rate. Which car is traveling at a greater speed? How
do you know?

Car A:

Car B:

Time in minutes
(𝟔𝟔)

Distance in miles
(𝒚𝒚)

𝟏𝟏𝟏𝟏 𝟏𝟏𝟐𝟐.𝟏𝟏

𝟑𝟑𝟔𝟔 𝟐𝟐𝟏𝟏

𝟏𝟏𝟏𝟏 𝟑𝟑𝟏𝟏.𝟏𝟏

Based on the graph, Car A is traveling at a rate of 𝟐𝟐 miles every 𝟑𝟑 minutes, 𝒎𝒎 = 𝟐𝟐
𝟑𝟑. From the table, the constant rate

that Car B is traveling is

𝟐𝟐𝟏𝟏 − 𝟏𝟏𝟐𝟐.𝟏𝟏
𝟑𝟑𝟔𝟔 − 𝟏𝟏𝟏𝟏

=
𝟏𝟏𝟐𝟐.𝟏𝟏
𝟏𝟏𝟏𝟏

=
𝟐𝟐𝟏𝟏
𝟑𝟑𝟔𝟔

=
𝟏𝟏
𝟔𝟔

Since 𝟏𝟏
𝟔𝟔

>
𝟐𝟐
𝟑𝟑

, Car B is traveling at a greater speed.

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Lesson 7: Comparing Linear Functions and Graphs

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7

2. The local park needs to replace an existing fence that is 𝟔𝟔 feet high. Fence Company A charges $𝟏𝟏,𝟔𝟔𝟔𝟔𝟔𝟔 for building
materials and $𝟐𝟐𝟔𝟔𝟔𝟔 per foot for the length of the fence. Fence Company B charges are based solely on the length of
the fence. That is, the total cost of the 𝟔𝟔-foot high fence will depend on how long the fence is. The table below
represents some inputs and their corresponding outputs that the cost function for Fence Company B assigns. It is a
linear function.

Input
(length of fence in

feet)

Output
(cost of bill in dollars)

𝟏𝟏𝟔𝟔𝟔𝟔 𝟐𝟐𝟔𝟔,𝟔𝟔𝟔𝟔𝟔𝟔

𝟏𝟏𝟐𝟐𝟔𝟔 𝟑𝟑𝟏𝟏,𝟐𝟐𝟔𝟔𝟔𝟔

𝟏𝟏𝟖𝟖𝟔𝟔 𝟏𝟏𝟔𝟔,𝟖𝟖𝟔𝟔𝟔𝟔

𝟐𝟐𝟏𝟏𝟔𝟔 𝟔𝟔𝟏𝟏,𝟔𝟔𝟔𝟔𝟔𝟔

a. Which company charges a higher rate per foot of fencing? How do you know?

Let 𝟔𝟔 represent the length of the fence and 𝒚𝒚 represent the total cost.

The equation that represents the function for Fence Company A is 𝒚𝒚 = 𝟐𝟐𝟔𝟔𝟔𝟔𝟔𝟔 + 𝟏𝟏,𝟔𝟔𝟔𝟔𝟔𝟔. So, the rate is 𝟐𝟐𝟔𝟔𝟔𝟔
dollars per foot of fence.

The rate of change for Fence Company B is given by:

𝟐𝟐𝟔𝟔,𝟔𝟔𝟔𝟔𝟔𝟔 − 𝟑𝟑𝟏𝟏,𝟐𝟐𝟔𝟔𝟔𝟔
𝟏𝟏𝟔𝟔𝟔𝟔 − 𝟏𝟏𝟐𝟐𝟔𝟔

=
−𝟏𝟏,𝟐𝟐𝟔𝟔𝟔𝟔
−𝟐𝟐𝟔𝟔

= 𝟐𝟐𝟔𝟔𝟔𝟔

Fence Company B charges $𝟐𝟐𝟔𝟔𝟔𝟔 per foot of fence, which is a higher rate per foot of fence length than Fence
Company A.

b. At what number of the length of the fence would the cost from each fence company be the same? What will
the cost be when the companies charge the same amount? If the fence you need were 𝟏𝟏𝟗𝟗𝟔𝟔 feet in length,
which company would be a better choice?

Student solutions will vary. Sample solution is provided.

The equation for Fence Company B is

𝒚𝒚 = 𝟐𝟐𝟔𝟔𝟔𝟔𝟔𝟔.

We can find out at what point the fence companies charge the same amount by solving the system

�𝒚𝒚 = 𝟐𝟐𝟔𝟔𝟔𝟔𝟔𝟔 + 𝟏𝟏𝟔𝟔𝟔𝟔𝟔𝟔
𝒚𝒚 = 𝟐𝟐𝟔𝟔𝟔𝟔𝟔𝟔

𝟐𝟐𝟔𝟔𝟔𝟔𝟔𝟔 + 𝟏𝟏,𝟔𝟔𝟔𝟔𝟔𝟔 = 𝟐𝟐𝟔𝟔𝟔𝟔𝟔𝟔
𝟏𝟏,𝟔𝟔𝟔𝟔𝟔𝟔 = 𝟔𝟔𝟔𝟔𝟔𝟔

𝟏𝟏𝟏𝟏𝟔𝟔.𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔… . . . = 𝟔𝟔
𝟏𝟏𝟏𝟏𝟔𝟔.𝟏𝟏 ≈ 𝟔𝟔

At 𝟏𝟏𝟏𝟏𝟔𝟔.𝟏𝟏 feet of fencing, both companies would charge the same amount (about $𝟑𝟑𝟔𝟔,𝟑𝟑𝟏𝟏𝟔𝟔). Less than
𝟏𝟏𝟏𝟏𝟔𝟔.𝟏𝟏 feet of fencing means that the cost from Fence Company A will be more than Fence Company B. More
than 𝟏𝟏𝟏𝟏𝟔𝟔.𝟏𝟏 feet of fencing means that the cost from Fence Company B will be more than Fence Company A.
So, for 𝟏𝟏𝟗𝟗𝟔𝟔 feet of fencing, Fence Company A is the better choice.

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Lesson 7: Comparing Linear Functions and Graphs

100

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7

3. The equation 𝒚𝒚 = 𝟏𝟏𝟐𝟐𝟑𝟑𝟔𝟔 describes the function for the number of toys, 𝒚𝒚, produced at Toys Plus in 𝟔𝟔 minutes of
production time. Another company, #1 Toys, has a similar function, also linear, that assigns the values shown in the
table below. Which company produces toys at a slower rate? Explain.

Time in minutes
(𝟔𝟔)

Toys Produced
(𝒚𝒚)

𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔

𝟏𝟏𝟏𝟏 𝟏𝟏,𝟑𝟑𝟐𝟐𝟔𝟔

𝟏𝟏𝟑𝟑 𝟏𝟏,𝟏𝟏𝟔𝟔𝟔𝟔

We are told that #1 Toys produces toys at a constant rate. That rate is:

𝟏𝟏,𝟑𝟑𝟐𝟐𝟔𝟔 − 𝟔𝟔𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏 − 𝟏𝟏

=
𝟏𝟏𝟐𝟐𝟔𝟔
𝟔𝟔

= 𝟏𝟏𝟐𝟐𝟔𝟔

The rate of production for #1 Toys is 𝟏𝟏𝟐𝟐𝟔𝟔 toys per minute. The rate of production for Toys Plus is 𝟏𝟏𝟐𝟐𝟑𝟑 toys per
minute. Since 𝟏𝟏𝟐𝟐𝟔𝟔 is less than 𝟏𝟏𝟐𝟐𝟑𝟑, #1 Toys produces toys at a slower rate.

4. A train is traveling from City A to City B, a distance of 𝟑𝟑𝟐𝟐𝟔𝟔 miles. The graph below shows the number of miles, 𝒚𝒚,
the train travels as a function of the number of hours, 𝟔𝟔, that have passed on its journey. The train travels at a
constant speed for the first four hours of its journey and then slows down to a constant speed of 48 miles per hour
for the remainder of its journey.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7

a. How long will it take the train to reach its destination?

Student solutions will vary. Sample solution is provided.

We see from the graph that the train travels 220 miles during its first four hours of travel. It has 100 miles
remaining to travel, which it shall do at a constant speed of 48 miles per hour. We see that it will take about
2 hours more to finish the trip:

𝟏𝟏𝟔𝟔𝟔𝟔 = 𝟏𝟏𝟖𝟖𝟔𝟔
𝟐𝟐.𝟔𝟔𝟖𝟖𝟑𝟑𝟑𝟑𝟑𝟑… = 𝟔𝟔

𝟐𝟐.𝟏𝟏 ≈ 𝟔𝟔.

This means it will take about 𝟔𝟔.𝟏𝟏 hours (𝟏𝟏+ 𝟐𝟐.𝟏𝟏 = 𝟔𝟔.𝟏𝟏) for the train to reach its destination.

b. If the train had not slowed down after 𝟏𝟏 hours, how long would it have taken to reach its destination?

𝟑𝟑𝟐𝟐𝟔𝟔 = 𝟏𝟏𝟏𝟏𝟔𝟔
𝟏𝟏.𝟖𝟖𝟏𝟏𝟖𝟖𝟏𝟏𝟖𝟖𝟏𝟏𝟖𝟖… . = 𝟔𝟔

𝟏𝟏.𝟖𝟖 ≈ 𝟔𝟔

The train would have reached its destination in about 𝟏𝟏.𝟖𝟖 hours had it not slowed down.

c. Suppose after 𝟏𝟏 hours, the train increased its constant speed. How fast would the train have to travel to
complete the destination in 𝟏𝟏.𝟏𝟏 hours?

Let 𝒎𝒎 represent the new constant speed of the train.

𝟏𝟏𝟔𝟔𝟔𝟔 = 𝒎𝒎(𝟏𝟏.𝟏𝟏)

𝟔𝟔𝟔𝟔.𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔… . = 𝟔𝟔
𝟔𝟔𝟔𝟔.𝟏𝟏 ≈ 𝟔𝟔

The train would have to increase its speed to about 𝟔𝟔𝟔𝟔.𝟏𝟏 miles per hour to arrive at its destination 𝟏𝟏.𝟏𝟏 hours
later.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7

a. A hose is used to fill up a 𝟏𝟏,𝟐𝟐𝟔𝟔𝟔𝟔 gallon water truck. Water flows from the hose at a constant rate. After 𝟏𝟏𝟔𝟔
minutes, there are 𝟔𝟔𝟏𝟏 gallons of water in the truck. After 𝟏𝟏𝟏𝟏 minutes, there are 𝟖𝟖𝟐𝟐 gallons of water in the
truck. How long will it take to fill up the water truck? Was the tank initially empty?

Student solutions will vary. Sample solution is provided.

Let 𝟔𝟔 represent the time in minutes it takes to pump 𝒚𝒚 gallons of water. Then, the rate can be found as
follows:

Time in minutes (𝟔𝟔) Amount of water pumped in gallons (𝒚𝒚)

𝟏𝟏𝟔𝟔 𝟔𝟔𝟏𝟏

𝟏𝟏𝟏𝟏 𝟖𝟖𝟐𝟐

𝟔𝟔𝟏𝟏 − 𝟖𝟖𝟐𝟐
𝟏𝟏𝟔𝟔 − 𝟏𝟏𝟏𝟏

=
−𝟏𝟏𝟏𝟏
−𝟏𝟏

=
𝟏𝟏𝟏𝟏
𝟏𝟏

Since the water is pumping at a constant rate, we can assume the equation is linear. Therefore, the equation
for the volume of water pumped from the hose is found by

𝟔𝟔𝟏𝟏 =
𝟏𝟏𝟏𝟏
𝟏𝟏

(𝟏𝟏𝟔𝟔) + 𝒃𝒃
𝟔𝟔𝟏𝟏 = 𝟑𝟑𝟏𝟏 + 𝒃𝒃
𝟑𝟑𝟏𝟏 = 𝒃𝒃

The equation is 𝒚𝒚 = 𝟏𝟏𝟏𝟏
𝟏𝟏 𝟔𝟔 + 𝟑𝟑𝟏𝟏, and we see that the tank initially had 31 gallons of water in it. The time to fill

the tank is given by

𝟏𝟏𝟐𝟐𝟔𝟔𝟔𝟔 =
𝟏𝟏𝟏𝟏
𝟏𝟏
𝟔𝟔 + 𝟑𝟑𝟏𝟏

𝟏𝟏𝟏𝟏𝟔𝟔𝟗𝟗 =
𝟏𝟏𝟏𝟏
𝟏𝟏
𝟔𝟔

𝟑𝟑𝟏𝟏𝟑𝟑.𝟖𝟖𝟐𝟐𝟑𝟑𝟏𝟏… = 𝟔𝟔
𝟑𝟑𝟏𝟏𝟑𝟑.𝟖𝟖 ≈ 𝟔𝟔

It would take about 𝟑𝟑𝟏𝟏𝟏𝟏 minutes or about 𝟏𝟏.𝟏𝟏 hours to fill up the truck.

b. The driver of the truck realizes that something is wrong with the hose he is using. After 𝟑𝟑𝟔𝟔 minutes, he shuts
off the hose and tries a different hose. The second hose flows at a constant rate of 𝟏𝟏𝟖𝟖 gallons per minute.
How long now does it take to fill up the truck?

Since the first hose has been pumping for 𝟑𝟑𝟔𝟔 minutes, there are 𝟏𝟏𝟑𝟑𝟑𝟑 gallons of water already in the truck.
That means the new hose only has to fill up 𝟏𝟏,𝟔𝟔𝟔𝟔𝟏𝟏 gallons. Since the second hose fills up the truck at a
constant rate of 𝟏𝟏𝟖𝟖 gallons per minute, the equation for the second hose is 𝒚𝒚 = 𝟏𝟏𝟖𝟖𝟔𝟔.

𝟏𝟏𝟔𝟔𝟔𝟔𝟏𝟏 = 𝟏𝟏𝟖𝟖𝟔𝟔
𝟏𝟏𝟗𝟗.𝟐𝟐𝟏𝟏 = 𝟔𝟔
𝟏𝟏𝟗𝟗.𝟑𝟑 ≈ 𝟔𝟔

It will take the second hose about 𝟏𝟏𝟗𝟗.𝟑𝟑 minutes (or a little less than an hour) to finish the job.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 7

Multi-Step Equations II

1. 2(𝑥𝑥 + 5) = 3(𝑥𝑥 + 6)

𝟔𝟔 = −𝟖𝟖

2. 3(𝑥𝑥 + 5) = 4(𝑥𝑥 + 6)

𝟔𝟔 = −𝟗𝟗

3. 4(𝑥𝑥 + 5) = 5(𝑥𝑥 + 6)

𝟔𝟔 = −𝟏𝟏𝟔𝟔

4. −(4𝑥𝑥 + 1) = 3(2𝑥𝑥 − 1)

𝟔𝟔 =
𝟏𝟏
𝟏𝟏

5. 3(4𝑥𝑥 + 1) = −(2𝑥𝑥 − 1)

𝟔𝟔 = −
𝟏𝟏
𝟏𝟏

6. −3(4𝑥𝑥 + 1) = 2𝑥𝑥 − 1

𝟔𝟔 = −
𝟏𝟏
𝟏𝟏

7. 15𝑥𝑥 − 12 = 9𝑥𝑥 − 6

𝟔𝟔 = 𝟏𝟏

8.
1
3

(15𝑥𝑥 − 12) = 9𝑥𝑥 − 6

𝟔𝟔 =
𝟏𝟏
𝟐𝟐

9.
2
3

(15𝑥𝑥 − 12) = 9𝑥𝑥 − 6

𝟔𝟔 = 𝟐𝟐

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Lesson 8: Graphs of Simple Nonlinear Functions

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 8

Lesson 8: Graphs of Simple Nonlinear Functions

Student Outcomes

 Students examine the average rate of change for nonlinear function over various intervals and verify that these
values are not constant.

Lesson Notes
In Exercises 4–10, students are given the option to sketch the graphs of given equations to verify their claims about them
being linear or nonlinear. For this reason, students may need graph paper to complete these exercises. Students need
graph paper to complete the Problem Set.

Classwork

Exploratory Challenge/Exercises 1–3 (19 minutes)

Students work independently or in pairs to complete Exercises 1–3.

Exploratory Challenge/Exercises 1–3

1. Consider the function that assigns to each number 𝒙𝒙 the value 𝒙𝒙𝟐𝟐.

a. Do you think the function is linear or nonlinear? Explain.

I think the function is nonlinear. The equation describing the function is not of the
form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃.

b. Develop a list of inputs and outputs for this function. Organize your work using the table below. Then,
answer the questions that follow.

Input (𝒙𝒙) Output (𝒙𝒙𝟐𝟐)

−𝟓𝟓 𝟐𝟐𝟓𝟓

−𝟒𝟒 𝟏𝟏𝟏𝟏

−𝟑𝟑 𝟗𝟗

−𝟐𝟐 𝟒𝟒

−𝟏𝟏 𝟏𝟏

𝟎𝟎 𝟎𝟎

𝟏𝟏 𝟏𝟏

𝟐𝟐 𝟒𝟒

𝟑𝟑 𝟗𝟗

𝟒𝟒 𝟏𝟏𝟏𝟏

𝟓𝟓 𝟐𝟐𝟓𝟓

Scaffolding:
Students may benefit from
exploring these exercises in
small groups.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 8

c. Plot the inputs and outputs as ordered pairs defining points on the coordinate plane.

d. What shape does the graph of the points appear to take?

It appears to take the shape of a curve.

e. Find the rate of change using rows 1 and 2 from the table above.

𝟐𝟐𝟓𝟓 − 𝟏𝟏𝟏𝟏
−𝟓𝟓 − (−𝟒𝟒) =

𝟗𝟗
−𝟏𝟏

= −𝟗𝟗

f. Find the rate of change using rows 2 and 𝟑𝟑 from the table above.

𝟏𝟏𝟏𝟏 − 𝟗𝟗
−𝟒𝟒 − (−𝟑𝟑)

=
𝟕𝟕
−𝟏𝟏

= −𝟕𝟕

g. Find the rate of change using any two other rows from the table above.

Student work will vary.

𝟏𝟏𝟏𝟏 − 𝟐𝟐𝟓𝟓
𝟒𝟒 − 𝟓𝟓

=
−𝟗𝟗
−𝟏𝟏

= 𝟗𝟗

h. Return to your initial claim about the function. Is it linear or nonlinear? Justify your answer with as many
pieces of evidence as possible.

This is definitely a nonlinear function because the rate of change is not a constant for different intervals of
inputs. Also, we would expect the graph of a linear function to be a set of points in a line, and this graph is
not a line. As was stated before, the expression 𝒙𝒙𝟐𝟐 is nonlinear.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 8

2. Consider the function that assigns to each number 𝒙𝒙 the value 𝒙𝒙𝟑𝟑.

a. Do you think the function is linear or nonlinear? Explain.

I think the function is nonlinear. The equation describing the function is not of the form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃.

b. Develop a list of inputs and outputs for this function. Organize your work using the table below. Then,
answer the questions that follow.

Input (𝒙𝒙) Output (𝒙𝒙𝟑𝟑)

−𝟐𝟐.𝟓𝟓 −𝟏𝟏𝟓𝟓.𝟏𝟏𝟐𝟐𝟓𝟓

−𝟐𝟐 −𝟖𝟖

−𝟏𝟏.𝟓𝟓 −𝟑𝟑.𝟑𝟑𝟕𝟕𝟓𝟓

−𝟏𝟏 −𝟏𝟏

−𝟎𝟎.𝟓𝟓 −𝟎𝟎.𝟏𝟏𝟐𝟐𝟓𝟓

𝟎𝟎 𝟎𝟎

𝟎𝟎.𝟓𝟓 𝟎𝟎.𝟏𝟏𝟐𝟐𝟓𝟓

𝟏𝟏 𝟏𝟏

𝟏𝟏.𝟓𝟓 𝟑𝟑.𝟑𝟑𝟕𝟕𝟓𝟓

𝟐𝟐 𝟖𝟖

𝟐𝟐.𝟓𝟓 𝟏𝟏𝟓𝟓.𝟏𝟏𝟐𝟐𝟓𝟓

c. Plot the inputs and outputs as ordered pairs defining points on
the coordinate plane.

d. What shape does the graph of the points appear to take?

It appears to take the shape of a curve.

e. Find the rate of change using rows 2 and 3 from the table above.

−𝟖𝟖− (−𝟑𝟑.𝟑𝟑𝟕𝟕𝟓𝟓)
−𝟐𝟐− (−𝟏𝟏.𝟓𝟓)

=
−𝟒𝟒.𝟏𝟏𝟐𝟐𝟓𝟓
−𝟎𝟎.𝟓𝟓

= 𝟗𝟗.𝟐𝟐𝟓𝟓

f. Find the rate of change using rows 3 and 4 from the table above.

−𝟑𝟑.𝟑𝟑𝟕𝟕𝟓𝟓 − (−𝟏𝟏)
−𝟏𝟏.𝟓𝟓 − (−𝟏𝟏)

=
−𝟐𝟐.𝟑𝟑𝟕𝟕𝟓𝟓
−𝟎𝟎.𝟓𝟓

= 𝟒𝟒.𝟕𝟕𝟓𝟓

g. Find the rate of change using rows 8 and 9 from the table above.

𝟏𝟏 − 𝟑𝟑.𝟑𝟑𝟕𝟕𝟓𝟓
𝟏𝟏 − 𝟏𝟏.𝟓𝟓

=
−𝟐𝟐.𝟑𝟑𝟕𝟕𝟓𝟓
−𝟎𝟎.𝟓𝟓

= 𝟒𝟒.𝟕𝟕𝟓𝟓

h. Return to your initial claim about the function. Is it linear or nonlinear? Justify your answer with as many
pieces of evidence as possible.

This is definitely a nonlinear function because the rate of change is not a constant for any interval of inputs.
Also, we would expect the graph of a linear function to be a line, and this graph is not a line. As was stated
before, the expression 𝒙𝒙𝟑𝟑 is nonlinear.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 8

3. Consider the function that assigns to each positive number 𝒙𝒙 the value
𝟏𝟏
𝒙𝒙

a. Do you think the function is linear or nonlinear? Explain.

I think the function is nonlinear. The equation describing the function is not of the form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃.

b. Develop a list of inputs and outputs for this function. Organize your work using the table below. Then,
answer the questions that follow.

Input (𝒙𝒙) Output �𝟏𝟏𝒙𝒙�
𝟎𝟎.𝟏𝟏 𝟏𝟏𝟎𝟎

𝟎𝟎.𝟐𝟐 𝟓𝟓

𝟎𝟎.𝟒𝟒 𝟐𝟐.𝟓𝟓

𝟎𝟎.𝟓𝟓 𝟐𝟐

𝟎𝟎.𝟖𝟖 𝟏𝟏.𝟐𝟐𝟓𝟓

𝟏𝟏 𝟏𝟏

𝟏𝟏.𝟏𝟏 𝟎𝟎.𝟏𝟏𝟐𝟐𝟓𝟓

𝟐𝟐 𝟎𝟎.𝟓𝟓

𝟐𝟐.𝟓𝟓 𝟎𝟎.𝟒𝟒

𝟒𝟒 𝟎𝟎.𝟐𝟐𝟓𝟓

𝟓𝟓 𝟎𝟎.𝟐𝟐

c. Plot the inputs and outputs as ordered pairs defining points on the coordinate plane.

d. What shape does the graph of the points appear to take?

It appears to take the shape of a curve.

e. Find the rate of change using rows 1 and 2 from the table above.

𝟏𝟏𝟎𝟎 − 𝟓𝟓
𝟎𝟎.𝟏𝟏 − 𝟎𝟎.𝟐𝟐

=
𝟓𝟓

−𝟎𝟎.𝟏𝟏
= −𝟓𝟓𝟎𝟎

f. Find the rate of change using rows 2 and 𝟑𝟑 from the table above.

𝟓𝟓 − 𝟐𝟐.𝟓𝟓
𝟎𝟎.𝟐𝟐 − 𝟎𝟎.𝟒𝟒

=
𝟐𝟐.𝟓𝟓
−𝟎𝟎.𝟐𝟐

= −𝟏𝟏𝟐𝟐.𝟓𝟓

g. Find the rate of change using any two other rows from the table above.

Student work will vary.

𝟏𝟏 − 𝟎𝟎.𝟏𝟏𝟐𝟐𝟓𝟓
𝟏𝟏 − 𝟏𝟏.𝟏𝟏

=
𝟎𝟎.𝟑𝟑𝟕𝟕𝟓𝟓
−𝟎𝟎.𝟏𝟏

= −𝟎𝟎.𝟏𝟏𝟐𝟐𝟓𝟓

h. Return to your initial claim about the function. Is it linear or nonlinear? Justify your answer with as many
pieces of evidence as possible.

before, the expression
𝟏𝟏
𝒙𝒙

is nonlinear.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 8

Discussion (4 minutes)

 What did you notice about the rates of change in the preceding three problems?

 The rates of change were not constant in each of the three problems.
 If the rate of change for pairs of inputs and corresponding outputs were the same for each and every pair, then

what can we say about the function?

 We know the function is linear.

 If the rate of change for pairs of inputs and corresponding outputs is not the same for each pair, what can we
say about the function?

 We know the function is nonlinear.
 Recall that any linear function can be described by an equation of the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏. Any equation that

cannot be written in this form is not linear, and its corresponding function is nonlinear.

Exercises 4–10 (12 minutes)

Students work independently or in pairs to complete Exercises 4–10.

Exercises 4–10

In each of Exercises 4–10, an equation describing a rule for a function is given, and a question is asked about it. If
necessary, use a table to organize pairs of inputs and outputs, and then plot each on a coordinate plane to help answer
the question.

4. What shape do you expect the graph of the function
described by 𝒚𝒚 = 𝒙𝒙 to take? Is it a linear or nonlinear
function?

I expect the shape of the graph to be a line. This
function is a linear function described by the linear
equation 𝒚𝒚 = 𝒙𝒙. The graph of this function is a line.

5. What shape do you expect the graph of the function described
by 𝒚𝒚 = 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝒙𝒙 to take? Is it a linear or nonlinear function?

I expect the shape of the graph to be something other than a
line. This function is nonlinear because its graph is not a line.
Also the equation describing the function is not of the form
𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃. It is not linear.

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6. What shape do you expect the graph of the function
described by 𝟑𝟑𝒙𝒙 + 𝟕𝟕𝒚𝒚 = 𝟖𝟖 to take? Is it a linear or
nonlinear function?

I expect the shape of the graph to be a line. This function is
a linear function described by the linear equation
𝟑𝟑𝒙𝒙 + 𝟕𝟕𝒚𝒚 = 𝟖𝟖. The graph of this function is a line. (We have

𝒚𝒚 = −𝟑𝟑
𝟕𝟕𝒙𝒙 + 𝟖𝟖

𝟕𝟕.)

7. What shape do you expect the graph of the function described by
𝒚𝒚 = 𝟒𝟒𝒙𝒙𝟑𝟑 to take? Is it a linear or nonlinear function?

I expect the shape of the graph to be something other than a line.
This function is nonlinear because its graph is not a line. Also the
equation describing the function is not of the form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃. It is
not linear.

8. What shape do you expect the graph of the function described

by 𝟑𝟑𝒙𝒙 = 𝒚𝒚 to take? Is it a linear or nonlinear function? (Assume
that an input of 𝒙𝒙 = 𝟎𝟎 is disallowed.)

9. What shape do you expect the graph of the function described

by 𝟒𝟒
𝒙𝒙𝟐𝟐

= 𝒚𝒚 to take? Is it a linear or nonlinear function?

(Assume that an input of 𝒙𝒙 = 𝟎𝟎 is disallowed.)

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Lesson Summary

One way to determine if a function is linear or nonlinear is to inspect average rates of change using a table of values. If
these average rates of change are not constant, then the function is not linear.

Another way is to examine the graph of the function. If all the points on the graph do not lie on a common line, then the
function is not linear.

If a function is described by an equation different from one equivalent to 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃 for some fixed values 𝒎𝒎 and 𝒃𝒃,
then the function is not linear.

10. What shape do you expect the graph of the equation 𝒙𝒙𝟐𝟐 + 𝒚𝒚𝟐𝟐 = 𝟑𝟑𝟏𝟏 to take? Is it a linear or nonlinear function? Is
it a function? Explain.

I expect the shape of the graph to be something other than a
line. It is nonlinear because its graph is not a line. It is not a
function because there is more than one output for any given
value of 𝒙𝒙 in the interval (−𝟏𝟏,𝟏𝟏). For example, at 𝒙𝒙 = 𝟎𝟎 the
𝒚𝒚-value is both 𝟏𝟏 and −𝟏𝟏. This does not fit the definition of
function because functions assign to each input exactly one
output. Since there is at least one instance where an input has
two outputs, it is not a function.

Closing (5 minutes)

Summarize, or ask students to summarize, the main points from the lesson.

 Students understand that, unlike linear functions, nonlinear functions do not have a constant rate of change.

 Students expect the graph of nonlinear functions to be some sort of curve.

Exit Ticket (5 minutes)

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Name Date

Lesson 8: Graphs of Simple Nonlinear Functions

Exit Ticket

1. The graph below is the graph of a function. Do you think the function is linear or nonlinear? Briefly justify your

answer.

2. Consider the function that assigns to each number 𝑚𝑚 the value
1
2
𝑚𝑚2. Do you expect the graph of this function to be a

straight line? Briefly justify your answer.

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Exit Ticket Sample Solutions

1. The graph below is the graph of a function. Do you think the function is linear or nonlinear? Briefly justify your
answer.

Student work may vary. The plot of this graph appears to be a straight line, and so the function is linear.

2. Consider the function that assigns to each number 𝒙𝒙 the value
𝟏𝟏
𝟐𝟐
𝒙𝒙𝟐𝟐 . Do you expect the graph of this function to be a

straight line? Briefly justify your answer.

The equation is nonlinear (not of the form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃), so the function is nonlinear. Its graph will not be a straight
line.

Problem Set Sample Solutions

1. Consider the function that assigns to each number 𝒙𝒙 the value 𝒙𝒙𝟐𝟐 − 𝟒𝟒.

a. Do you think the function is linear or nonlinear? Explain.

The equation describing the function is not of the form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃. It is not linear.

b. Do you expect the graph of this function to be a straight line?

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c. Develop a list of inputs and matching outputs for this function. Use them to begin a graph of the function.

Input (𝒙𝒙) Output (𝒙𝒙𝟐𝟐 − 𝟒𝟒)

−𝟑𝟑 𝟓𝟓

−𝟐𝟐 𝟎𝟎

−𝟏𝟏 −𝟑𝟑

𝟎𝟎 −𝟒𝟒

𝟏𝟏 −𝟑𝟑

𝟐𝟐 𝟎𝟎

𝟑𝟑 𝟓𝟓

d. Was your prediction to (b) correct?

Yes, the graph appears to be taking the shape
of some type of curve.

2. Consider the function that assigns to each number 𝒙𝒙 greater than −𝟑𝟑 the value 𝟏𝟏
𝒙𝒙+𝟑𝟑.

a. Is the function linear or nonlinear? Explain.

The equation describing the function is not of the form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃. It is not linear.

b. Do you expect the graph of this function to be a straight line?

c. Develop a list of inputs and matching outputs for this function. Use them to begin a graph of the function.

Input (𝒙𝒙) Output � 𝟏𝟏
𝒙𝒙+𝟑𝟑�

−𝟐𝟐 𝟏𝟏

−𝟏𝟏 𝟎𝟎.𝟓𝟓

𝟎𝟎 𝟎𝟎.𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑…

𝟏𝟏 𝟎𝟎.𝟐𝟐𝟓𝟓

𝟐𝟐 𝟎𝟎.𝟐𝟐

𝟑𝟑 𝟎𝟎.𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏…

d. Was your prediction to (b) correct?

Yes, the graph appears to be taking the shape of some type of curve.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 8

a. Is the function represented by this graph linear or nonlinear? Briefly justify your answer.

The graph is clearly not a straight line, so the function is not linear.

b. What is the average rate of change for this function from an input of 𝒙𝒙 = −𝟐𝟐 to an input of 𝒙𝒙 = −𝟏𝟏?

−𝟐𝟐− 𝟏𝟏
−𝟐𝟐 − (−𝟏𝟏)

=
−𝟑𝟑
−𝟏𝟏

= 𝟑𝟑

c. What is the average rate of change for this function from an input of 𝒙𝒙 = −𝟏𝟏 to an input of 𝒙𝒙 = 𝟎𝟎?

𝟏𝟏 − 𝟐𝟐
−𝟏𝟏− 𝟎𝟎

=
−𝟏𝟏
−𝟏𝟏

= 𝟏𝟏

As expected, the average rate of change of this function is not constant.

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Topic B: Volume

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GRADE 8 • MODULE 5

8
G R A D E

New York State Common Core

Mathematics Curriculum

Topic B

Volume

8.G.C.9

Focus Standard: 8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use
them to solve real-world and mathematical problems.

Instructional Days: 3

Lesson 9: Examples of Functions from Geometry (E)1

Lesson 10: Volumes of Familiar Solids—Cones and Cylinders (S)

Lesson 11: Volume of a Sphere (P)

In Lesson 9, students work with functions from geometry. For example, students write the rules that
represent the perimeters of various regular shapes and areas of common shapes. Along those same lines,
students write functions that represent the area of more complex shapes (e.g., the border of a picture frame).
In Lesson 10, students learn the volume formulas for cylinders and cones. Building upon their knowledge of
area of circles and the concept of congruence, students see a cylinder as a stack of circular congruent disks
and consider the total area of the disks in three dimensions as the volume of a cylinder. A physical
demonstration shows students that it takes exactly three cones of the same dimensions as a cylinder to equal
the volume of the cylinder. The demonstration leads students to the formula for the volume of cones in
general. Students apply the formulas to answer questions such as, “If a cone is filled with water to half its
height, what is the ratio of the volume of water to the container itself?” Students then use what they know
about the volume of the cylinder to derive the formula for the volume of a sphere. In Lesson 11, students
learn that the volume of a sphere is equal to two-thirds the volume of a cylinder that fits tightly around the
sphere and touches only at points. Finally, students apply what they have learned about volume to solve real-
world problems where they will need to make decisions about which formulas to apply to a given situation.

1Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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Lesson 9: Examples of Functions from Geometry

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

Lesson 9: Examples of Functions from Geometry

Student Outcomes

 Students write rules to express functions related to geometry.
 Students review what they know about volume with respect to rectangular prisms and further develop their

conceptual understanding of volume by comparing the liquid contained within a solid to the volume of a
standard rectangular prism (i.e., a prism with base area equal to one).

Classwork

Exploratory Challenge 1/Exercises 1–4 (10 minutes)

Students work independently or in pairs to complete Exercises 1–4. Once students are finished, debrief the activity. Ask
students to think about real-life situations that might require using the function they developed in Exercise 4. Some
sample responses may include area of wood needed to make a 1-inch frame for a picture, area required to make a
sidewalk border (likely larger than 1-inch) around a park or playground, or the area of a planter around a tree.

Exploratory Challenge 1/Exercises 1–4

As you complete Exercises 1–4, record the information in the table below.

Side length in inches
(𝒔𝒔)

Area in square inches
(𝑨𝑨)

Expression that
describes area of

border

Exercise 1
𝟔𝟔 𝟑𝟑𝟔𝟔

𝟔𝟔𝟔𝟔 − 𝟑𝟑𝟔𝟔
𝟖𝟖 𝟔𝟔𝟔𝟔

Exercise 2

𝟗𝟗 𝟖𝟖𝟖𝟖
𝟖𝟖𝟏𝟏𝟖𝟖 − 𝟖𝟖𝟖𝟖

𝟖𝟖𝟖𝟖 𝟖𝟖𝟏𝟏𝟖𝟖

Exercise 3
𝟖𝟖𝟑𝟑 𝟖𝟖𝟔𝟔𝟗𝟗

𝟏𝟏𝟏𝟏𝟐𝟐 − 𝟖𝟖𝟔𝟔𝟗𝟗
𝟖𝟖𝟐𝟐 𝟏𝟏𝟏𝟏𝟐𝟐

Exercise 4
𝒔𝒔 𝒔𝒔𝟏𝟏

(𝒔𝒔 + 𝟏𝟏)𝟏𝟏 − 𝒔𝒔𝟏𝟏

𝒔𝒔 + 𝟏𝟏 (𝒔𝒔 + 𝟏𝟏)𝟏𝟏

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Lesson 9: Examples of Functions from Geometry

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

1. Use the figure below to answer parts (a)–(f).

a. What is the length of one side of the smaller, inner square?

The length of one side of the smaller square is 𝟔𝟔 𝐢𝐢𝐢𝐢.

b. What is the area of the smaller, inner square?

𝟔𝟔𝟏𝟏 = 𝟑𝟑𝟔𝟔

The area of the smaller square is 𝟑𝟑𝟔𝟔 𝐢𝐢𝐢𝐢𝟏𝟏.

c. What is the length of one side of the larger, outer square?

The length of one side of the larger square is 𝟖𝟖 𝐢𝐢𝐢𝐢.

d. What is the area of the larger, outer square?

𝟖𝟖𝟏𝟏 = 𝟔𝟔𝟔𝟔

The area of the larger square is 𝟔𝟔𝟔𝟔 𝐢𝐢𝐢𝐢𝟏𝟏.

e. Use your answers in parts (b) and (d) to determine the area of the 𝟖𝟖-inch white border of the figure.

𝟔𝟔𝟔𝟔 − 𝟑𝟑𝟔𝟔 = 𝟏𝟏𝟖𝟖

The area of the 𝟖𝟖-inch white border is 𝟏𝟏𝟖𝟖 𝐢𝐢𝐢𝐢𝟏𝟏.

f. Explain your strategy for finding the area of the white border.

First, I had to determine the length of one side of the larger, outer square. Since the inner square is 𝟔𝟔 𝐢𝐢𝐢𝐢. and
the border is 𝟖𝟖 𝐢𝐢𝐢𝐢. on all sides, then the length of one side of the larger square is (𝟔𝟔+ 𝟏𝟏) 𝐢𝐢𝐢𝐢 = 𝟖𝟖 𝐢𝐢𝐢𝐢. Then, the
area of the larger square is 𝟔𝟔𝟔𝟔 𝐢𝐢𝐢𝐢𝟏𝟏. Next, I found the area of the smaller, inner square. Since one side length
is 𝟔𝟔 𝐢𝐢𝐢𝐢., the area is 𝟑𝟑𝟔𝟔 𝐢𝐢𝐢𝐢𝟏𝟏. To find the area of the white border, I needed to subtract the area of the inner
square from the area of the outer square.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

2. Use the figure below to answer parts (a)–(f).

a. What is the length of one side of the smaller, inner square?

The length of one side of the smaller square is 𝟗𝟗 𝐢𝐢𝐢𝐢.

b. What is the area of the smaller, inner square?

𝟗𝟗𝟏𝟏 = 𝟖𝟖𝟖𝟖

The area of the smaller square is 𝟖𝟖𝟖𝟖 𝐢𝐢𝐢𝐢𝟏𝟏.

c. What is the length of one side of the larger, outer square?

The length of one side of the larger square is 𝟖𝟖𝟖𝟖 𝐢𝐢𝐢𝐢.

d. What is the area of the larger, outer square?

𝟖𝟖𝟖𝟖𝟏𝟏 = 𝟖𝟖𝟏𝟏𝟖𝟖

The area of the larger square is 𝟖𝟖𝟏𝟏𝟖𝟖 𝐢𝐢𝐢𝐢𝟏𝟏.

e. Use your answers in parts (b) and (d) to determine the area of the 𝟖𝟖-inch white border of the figure.

𝟖𝟖𝟏𝟏𝟖𝟖 − 𝟖𝟖𝟖𝟖 = 𝟔𝟔𝟒𝟒

The area of the 𝟖𝟖-inch white border is 𝟔𝟔𝟒𝟒 𝐢𝐢𝐢𝐢𝟏𝟏.

f. Explain your strategy for finding the area of the white border.

First, I had to determine the length of one side of the larger, outer square. Since the inner square is 𝟗𝟗 𝐢𝐢𝐢𝐢. and
the border is 𝟖𝟖 𝐢𝐢𝐢𝐢. on all sides, the length of one side of the larger square is (𝟗𝟗+ 𝟏𝟏) 𝐢𝐢𝐢𝐢 = 𝟖𝟖𝟖𝟖 𝐢𝐢𝐢𝐢. Therefore,
the area of the larger square is 𝟖𝟖𝟏𝟏𝟖𝟖 𝐢𝐢𝐢𝐢𝟏𝟏. Then, I found the area of the smaller, inner square. Since one side
length is 𝟗𝟗 𝐢𝐢𝐢𝐢., the area is 𝟖𝟖𝟖𝟖 𝐢𝐢𝐢𝐢𝟏𝟏. To find the area of the white border, I needed to subtract the area of the
inner square from the area of the outer square.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

3. Use the figure below to answer parts (a)–(f).

a. What is the length of one side of the smaller, inner square?

The length of one side of the smaller square is 𝟖𝟖𝟑𝟑 𝐢𝐢𝐢𝐢.

b. What is the area of the smaller, inner square?

𝟖𝟖𝟑𝟑𝟏𝟏 = 𝟖𝟖𝟔𝟔𝟗𝟗

The area of the smaller square is 𝟖𝟖𝟔𝟔𝟗𝟗 𝐢𝐢𝐢𝐢𝟏𝟏.

c. What is the length of one side of the larger, outer square?

The length of one side of the larger square is 𝟖𝟖𝟐𝟐 𝐢𝐢𝐢𝐢.

d. What is the area of the larger, outer square?

𝟖𝟖𝟐𝟐𝟏𝟏 = 𝟏𝟏𝟏𝟏𝟐𝟐

The area of the larger square is 𝟏𝟏𝟏𝟏𝟐𝟐 𝐢𝐢𝐢𝐢𝟏𝟏.

e. Use your answers in parts (b) and (d) to determine the area of the 𝟖𝟖-inch white border of the figure.

𝟏𝟏𝟏𝟏𝟐𝟐 − 𝟖𝟖𝟔𝟔𝟗𝟗 = 𝟐𝟐𝟔𝟔

The area of the 𝟖𝟖-inch white border is 𝟐𝟐𝟔𝟔 𝐢𝐢𝐢𝐢𝟏𝟏.

f. Explain your strategy for finding the area of the white border.

First, I had to determine the length of one side of the larger, outer square. Since the inner square is 𝟖𝟖𝟑𝟑 𝐢𝐢𝐢𝐢.
and the border is 𝟖𝟖 𝐢𝐢𝐢𝐢. on all sides, the length of one side of the larger square is (𝟖𝟖𝟑𝟑 + 𝟏𝟏) 𝐢𝐢𝐢𝐢 = 𝟖𝟖𝟐𝟐 𝐢𝐢𝐢𝐢.
Therefore, the area of the larger square is 𝟏𝟏𝟏𝟏𝟐𝟐 𝐢𝐢𝐢𝐢𝟏𝟏. Then, I found the area of the smaller, inner square. Since
one side length is 𝟖𝟖𝟑𝟑 𝐢𝐢𝐢𝐢., the area is 𝟖𝟖𝟔𝟔𝟗𝟗 𝐢𝐢𝐢𝐢𝟏𝟏. To find the area of the white border, I needed to subtract the
area of the inner square from the area of the outer square.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

MP.8

4. Write a function that would allow you to calculate the area of a 𝟖𝟖-inch white border for any sized square picture
measured in inches.

a. Write an expression that represents the side length of the smaller, inner square.

Symbols used will vary. Expect students to use 𝒔𝒔 or 𝒙𝒙 to represent one side of the smaller, inner square.
Answers that follow will use 𝒔𝒔 as the symbol to represent one side of the smaller, inner square.

b. Write an expression that represents the area of the smaller, inner square.

𝒔𝒔𝟏𝟏

c. Write an expression that represents the side lengths of the larger, outer square.

𝒔𝒔 + 𝟏𝟏

d. Write an expression that represents the area of the larger, outer square.

(𝒔𝒔 + 𝟏𝟏)𝟏𝟏

e. Use your expressions in parts (b) and (d) to write a function for the area 𝑨𝑨 of the 𝟖𝟖-inch white border for any
sized square picture measured in inches.

𝑨𝑨 = (𝒔𝒔 + 𝟏𝟏)𝟏𝟏 − 𝒔𝒔𝟏𝟏

Discussion (6 minutes)

This discussion prepares students for the volume problems that they will work in the next two lessons. The goal is to
remind students of the concept of volume using a rectangular prism and then have them describe the volume in terms of
a function.

 Recall the concept of volume. How do you describe the volume of a three-dimensional figure? Give an
example, if necessary.

 Volume is the space that a three-dimensional figure can occupy. The volume of a glass is the amount of
liquid it can hold.

 In Grade 6 you learned the formula to determine the volume of a rectangular prism. The volume 𝑉𝑉 of a
rectangular prism is a function of the edge lengths, 𝑙𝑙, 𝑤𝑤, and ℎ. That is, the function that allows us to
determine the volume of a rectangular prism can be described by the following rule:

𝑉𝑉 = 𝑙𝑙𝑤𝑤ℎ.

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 Generally, we interpret volume in the following way:

 Fill the shell of the solid with water, and pour water into a three-dimensional figure, in this case a standard
rectangular prism (i.e., a prism with base side lengths of one), as shown.

 Then, the volume of the shell of the solid is the height 𝑣𝑣 of the water in the standard rectangular prism. Why is

the volume, 𝑣𝑣, the height of the water?

 The volume is equal to the height of the water because the area of the base is 1 square unit. Thus,
whatever the height, 𝑣𝑣, is, multiplied by 1, will be equal to 𝑣𝑣.

 If the height of water in the standard rectangular prism is 16.7 ft., what is the volume of the shell of the solid?
Explain.

 The volume of the shell of the solid would be 16.7 ft3 because the height, 16.7 ft., multiplied by the
area of the base, 1 ft2, is 16.7 ft3.

 There are a few basic assumptions that we make when we discuss volume. Have students paraphrase each
assumption after you state it to make sure they understand the concept.

(a) The volume of a solid is always a number greater than or equal to 0.
(b) The volume of a unit cube (i.e., a rectangular prism whose edges all have length 1) is by definition 1 cubic

unit.

(d) If two solids have (at most) their boundaries in common, then their total volume can be calculated by
adding the individual volumes together. (These figures are sometimes referred to as composite solids.)

Scaffolding:
 Concrete and hands-on

experiences with volume
would be useful.

 Students may know the
formulas for volume but
with different letters to
represent the values
(linked to their first
language).

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

Exercises 5–6 (5 minutes)

Exercises 5–6

5. The volume of the prism shown below is 𝟔𝟔𝟖𝟖.𝟔𝟔 𝐢𝐢𝐢𝐢𝟑𝟑. What is the height of the prism?

Let 𝒙𝒙 represent the height of the prism.

𝟔𝟔𝟖𝟖.𝟔𝟔 = 𝟖𝟖(𝟏𝟏.𝟏𝟏)𝒙𝒙
𝟔𝟔𝟖𝟖.𝟔𝟔 = 𝟖𝟖𝟏𝟏.𝟔𝟔𝒙𝒙
𝟑𝟑.𝟐𝟐 = 𝒙𝒙

The height of the prism is 𝟑𝟑.𝟐𝟐 𝐢𝐢𝐢𝐢.

6. Find the value of the ratio that compares the volume of the larger prism to the smaller prism.

Volume of larger prism:

𝑽𝑽 = 𝟏𝟏(𝟗𝟗)(𝟐𝟐)
= 𝟑𝟑𝟖𝟖𝟐𝟐

The volume of the larger prism is 𝟑𝟑𝟖𝟖𝟐𝟐 𝐜𝐜𝐦𝐦𝟑𝟑.

Volume of smaller prism:

𝑽𝑽 = 𝟏𝟏(𝟔𝟔.𝟐𝟐)(𝟑𝟑)
= 𝟏𝟏𝟏𝟏

The volume of the smaller prism is 𝟏𝟏𝟏𝟏 𝐜𝐜𝐦𝐦𝟑𝟑.

The ratio that compares the volume of the larger prism to the smaller prism is 𝟑𝟑𝟖𝟖𝟐𝟐:𝟏𝟏𝟏𝟏. The value of the ratio is
𝟑𝟑𝟖𝟖𝟐𝟐
𝟏𝟏𝟏𝟏

=
𝟑𝟑𝟐𝟐
𝟑𝟑

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

Exploratory Challenge 2/Exercises 7–10 (14 minutes)

Students work independently or in pairs to complete Exercises 7–10. Ensure that students know that when base is
referenced, it means the bottom of the prism.

Exploratory Challenge 2/Exercises 7–10

As you complete Exercises 7–10, record the information in the table below. Note that base refers to the bottom of the
prism.

Area of base in

square centimeters
(𝑩𝑩)

Height in centimeters
(𝒉𝒉)

Volume in cubic
centimeters

Exercise 7 𝟑𝟑𝟔𝟔 𝟑𝟑 𝟖𝟖𝟒𝟒𝟖𝟖

Exercise 8 𝟑𝟑𝟔𝟔 𝟖𝟖 𝟏𝟏𝟖𝟖𝟖𝟖

Exercise 9 𝟑𝟑𝟔𝟔 𝟖𝟖𝟐𝟐 𝟐𝟐𝟔𝟔𝟒𝟒

Exercise 10 𝟑𝟑𝟔𝟔 𝒙𝒙 𝟑𝟑𝟔𝟔𝒙𝒙

7. Use the figure to the right to answer parts (a)–(c).

a. What is the area of the base?

The area of the base is 𝟑𝟑𝟔𝟔 𝐜𝐜𝐦𝐦𝟏𝟏.

b. What is the height of the figure?

The height is 𝟑𝟑 𝐜𝐜𝐦𝐦.

c. What is the volume of the figure?

The volume of the rectangular prism is 𝟖𝟖𝟒𝟒𝟖𝟖 𝐜𝐜𝐦𝐦𝟑𝟑.

8. Use the figure to the right to answer parts (a)–(c).

a. What is the area of the base?

The area of the base is 𝟑𝟑𝟔𝟔 𝐜𝐜𝐦𝐦𝟏𝟏.

b. What is the height of the figure?

The height is 𝟖𝟖 𝐜𝐜𝐦𝐦.

c. What is the volume of the figure?

The volume of the rectangular prism is 𝟏𝟏𝟖𝟖𝟖𝟖 𝐜𝐜𝐦𝐦𝟑𝟑.

9. Use the figure to the right to answer parts (a)–(c).

a. What is the area of the base?

The area of the base is 𝟑𝟑𝟔𝟔 𝐜𝐜𝐦𝐦𝟏𝟏.

b. What is the height of the figure?

The height is 𝟖𝟖𝟐𝟐 𝐜𝐜𝐦𝐦.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

c. What is the volume of the figure?

The volume of the rectangular prism is 𝟐𝟐𝟔𝟔𝟒𝟒 𝐜𝐜𝐦𝐦𝟑𝟑.

10. Use the figure to the right to answer parts (a)–(c).

a. What is the area of the base?

The area of the base is 𝟑𝟑𝟔𝟔 𝐜𝐜𝐦𝐦𝟏𝟏.

b. What is the height of the figure?

The height is 𝒙𝒙 𝐜𝐜𝐦𝐦.

c. Write and describe a function that will allow you to
determine the volume of any rectangular prism that has a
base area of
𝟑𝟑𝟔𝟔 𝐜𝐜𝐦𝐦𝟏𝟏.

The rule that describes the function is 𝑽𝑽 = 𝟑𝟑𝟔𝟔𝒙𝒙, where 𝑽𝑽 is the volume and 𝒙𝒙 is the height of the rectangular
prism. The volume of a rectangular prism with a base area of 𝟑𝟑𝟔𝟔 𝐜𝐜𝐦𝐦𝟏𝟏 is a function of its height.

Closing (5 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 We know how to write functions to determine area or volume of a figure.

 We know that we can add volumes together as long as they only touch at a boundary.
 We know that identical solids will be equal in volume.

 We were reminded of the volume formula for a rectangular prism, and we used the formula to determine the
volume of rectangular prisms.

Exit Ticket (5 minutes)

MP.8

Lesson Summary

There are a few basic assumptions that are made when working with volume:

(a) The volume of a solid is always a number greater than or equal to 𝟒𝟒.

(b) The volume of a unit cube (i.e., a rectangular prism whose edges all have a length of 𝟖𝟖) is by definition
𝟖𝟖 cubic unit.

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Lesson 9: Examples of Functions from Geometry

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 9

Name Date

Lesson 9: Examples of Functions from Geometry

Exit Ticket

1. Write a function that would allow you to calculate the area in square inches, 𝐴𝐴, of a 2-inch white border for any

sized square figure with sides of length 𝑠𝑠 measured in inches.

2. The volume of the rectangular prism is 295.68 in3. What is its width?

11 in.

6.4 in.

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Exit Ticket Sample Solutions

1. Write a function that would allow you to calculate the area in square inches, 𝑨𝑨, of a 𝟏𝟏-inch white border for any
sized square figure with sides of length 𝒔𝒔 measured in inches.

Let 𝒔𝒔 represent the side length of the inner square in
inches. Then, the area of the inner square is 𝒔𝒔𝟏𝟏 square
inches. The side length of the larger square, in inches, is
𝒔𝒔 + 𝟔𝟔, and the area in square inches is (𝒔𝒔 + 𝟔𝟔)𝟏𝟏. If 𝑨𝑨 is the
area of the 𝟏𝟏-inch border, then the function that describes
𝑨𝑨 in square inches is

𝑨𝑨 = (𝒔𝒔 + 𝟔𝟔)𝟏𝟏 − 𝒔𝒔𝟏𝟏.

2. The volume of the rectangular prism is 𝟏𝟏𝟗𝟗𝟐𝟐.𝟔𝟔𝟖𝟖 𝐢𝐢𝐢𝐢𝟑𝟑. What is its width?

Let 𝒙𝒙 represent the width of the prism.

𝟏𝟏𝟗𝟗𝟐𝟐.𝟔𝟔𝟖𝟖 = 𝟖𝟖𝟖𝟖(𝟔𝟔.𝟔𝟔)𝒙𝒙
𝟏𝟏𝟗𝟗𝟐𝟐.𝟔𝟔𝟖𝟖 = 𝟏𝟏𝟒𝟒.𝟔𝟔𝒙𝒙

𝟔𝟔.𝟏𝟏 = 𝒙𝒙

The width of the prism is 𝟔𝟔.𝟏𝟏 𝐢𝐢𝐢𝐢.

Problem Set Sample Solutions

1. Calculate the area of the 𝟑𝟑-inch white border of the square figure below.

𝟖𝟖𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟖𝟖𝟗𝟗
𝟖𝟖𝟖𝟖𝟏𝟏 = 𝟖𝟖𝟏𝟏𝟖𝟖

The area of the 𝟑𝟑-inch white border is 𝟖𝟖𝟔𝟔𝟖𝟖 𝐢𝐢𝐢𝐢𝟏𝟏.

𝟖𝟖𝟖𝟖 𝐢𝐢𝐢𝐢.

𝟔𝟔.𝟔𝟔 𝐢𝐢𝐢𝐢.

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2. Write a function that would allow you to calculate the area, 𝑨𝑨, of a 𝟑𝟑-inch white border for any sized square picture
measured in inches.

Let 𝒔𝒔 represent the side length of the inner square in
inches. Then, the area of the inner square is 𝒔𝒔𝟏𝟏 square
inches. The side length of the outer square, in inches, is
𝒔𝒔 + 𝟔𝟔, which means that the area of the outer square, in
square inches, is (𝒔𝒔 + 𝟔𝟔)𝟏𝟏. The function that describes
the area, 𝑨𝑨, of the 𝟑𝟑-inch border is in square inches

𝑨𝑨 = (𝒔𝒔 + 𝟔𝟔)𝟏𝟏 − 𝒔𝒔𝟏𝟏.

3. Dartboards typically have an outer ring of numbers that represent the number of points a player can score for
getting a dart in that section. A simplified dartboard is shown below. The center of the circle is point 𝑨𝑨. Calculate
the area of the outer ring. Write an exact answer that uses 𝝅𝝅 (do not approximate your answer by using 𝟑𝟑.𝟖𝟖𝟔𝟔 for
𝝅𝝅).

Inner ring area: 𝝅𝝅𝒓𝒓𝟏𝟏 = 𝝅𝝅(𝟔𝟔𝟏𝟏) = 𝟑𝟑𝟔𝟔 𝝅𝝅

Outer ring: 𝝅𝝅𝒓𝒓𝟏𝟏 = 𝝅𝝅(𝟔𝟔+ 𝟏𝟏)𝟏𝟏 = 𝝅𝝅(𝟖𝟖𝟏𝟏) = 𝟔𝟔𝟔𝟔 𝝅𝝅

Difference in areas: 𝟔𝟔𝟔𝟔 𝝅𝝅− 𝟑𝟑𝟔𝟔 𝝅𝝅 = (𝟔𝟔𝟔𝟔 − 𝟑𝟑𝟔𝟔)𝝅𝝅 = 𝟏𝟏𝟖𝟖 𝝅𝝅

The inner ring has an area of 𝟑𝟑𝟔𝟔𝝅𝝅 𝐢𝐢𝐢𝐢𝟏𝟏. The area of the
inner ring including the border is 𝟔𝟔𝟔𝟔𝝅𝝅 𝐢𝐢𝐢𝐢𝟏𝟏. The
difference is the area of the border, 𝟏𝟏𝟖𝟖𝝅𝝅 𝐢𝐢𝐢𝐢𝟏𝟏.

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4. Write a function that would allow you to calculate the area, 𝑨𝑨, of the outer ring for any sized dartboard with radius
𝒓𝒓. Write an exact answer that uses 𝝅𝝅 (do not approximate your answer by using 𝟑𝟑.𝟖𝟖𝟔𝟔 for 𝝅𝝅).

Inner ring area: 𝝅𝝅𝒓𝒓𝟏𝟏

Outer ring: 𝝅𝝅𝒓𝒓𝟏𝟏 = 𝝅𝝅(𝒓𝒓+ 𝟏𝟏)𝟏𝟏

Difference in areas: Inner ring area: 𝝅𝝅(𝒓𝒓+ 𝟏𝟏)𝟏𝟏 − 𝝅𝝅𝒓𝒓𝟏𝟏

The inner ring has an area of 𝝅𝝅𝒓𝒓𝟏𝟏 𝐢𝐢𝐢𝐢𝟏𝟏. The area of the
inner ring including the border is 𝝅𝝅(𝒓𝒓+ 𝟏𝟏)𝟏𝟏 𝐢𝐢𝐢𝐢𝟏𝟏. Let 𝑨𝑨
be the area of the outer ring. Then, the function that
would describe that area in square inches is
𝑨𝑨 = 𝝅𝝅(𝒓𝒓+ 𝟏𝟏)𝟏𝟏 − 𝝅𝝅𝒓𝒓𝟏𝟏.

5. The shell of the solid shown was filled with water and then poured into the standard rectangular prism, as shown.
The height that the volume reaches is 𝟖𝟖𝟔𝟔.𝟏𝟏 𝐢𝐢𝐢𝐢. What is the volume of the shell of the solid?

𝑽𝑽 = 𝑩𝑩𝒉𝒉
= 𝟖𝟖(𝟖𝟖𝟔𝟔.𝟏𝟏)
= 𝟖𝟖𝟔𝟔.𝟏𝟏

The volume of the shell of the solid is
𝟖𝟖𝟔𝟔.𝟏𝟏 𝐢𝐢𝐢𝐢𝟑𝟑.

6. Determine the volume of the rectangular prism shown below.

𝟔𝟔.𝟔𝟔× 𝟐𝟐.𝟖𝟖 × 𝟖𝟖𝟒𝟒.𝟏𝟏 = 𝟑𝟑𝟑𝟑𝟏𝟏.𝟗𝟗𝟏𝟏𝟖𝟖

The volume of the prism is 𝟑𝟑𝟑𝟑𝟏𝟏.𝟗𝟗𝟏𝟏𝟖𝟖 𝐢𝐢𝐢𝐢𝟑𝟑.

𝟔𝟔.𝟔𝟔 𝐢𝐢𝐢𝐢.

𝟐𝟐.𝟖𝟖 𝐢𝐢𝐢𝐢.

𝟖𝟖𝟒𝟒.𝟏𝟏 𝐢𝐢𝐢𝐢.

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7. The volume of the prism shown below is 𝟗𝟗𝟏𝟏𝟏𝟏 𝐜𝐜𝐦𝐦𝟑𝟑. What is its length?

Let 𝒙𝒙 represent the length of the prism.

𝟗𝟗𝟏𝟏𝟏𝟏 = 𝟖𝟖.𝟖𝟖(𝟐𝟐)𝒙𝒙
𝟗𝟗𝟏𝟏𝟏𝟏 = 𝟔𝟔𝟒𝟒.𝟐𝟐𝒙𝒙
𝟏𝟏𝟔𝟔 = 𝒙𝒙

The length of the prism is 𝟏𝟏𝟔𝟔 𝐜𝐜𝐦𝐦.

8. The volume of the prism shown below is 𝟑𝟑𝟏𝟏.𝟏𝟏𝟑𝟑𝟏𝟏𝟐𝟐 ft3. What is its width?

Let 𝒙𝒙 represent the width.

𝟑𝟑𝟏𝟏.𝟏𝟏𝟑𝟑𝟏𝟏𝟐𝟐 = (𝟒𝟒.𝟏𝟏𝟐𝟐)(𝟔𝟔.𝟐𝟐)𝒙𝒙
𝟑𝟑𝟏𝟏.𝟏𝟏𝟑𝟑𝟏𝟏𝟐𝟐 = 𝟑𝟑.𝟑𝟑𝟏𝟏𝟐𝟐𝒙𝒙

𝟗𝟗.𝟏𝟏 = 𝒙𝒙

The width of the prism is 𝟗𝟗.𝟏𝟏 𝐟𝐟𝐟𝐟.

9. Determine the volume of the three-dimensional figure below. Explain how you got your answer.

𝟏𝟏 × 𝟏𝟏.𝟐𝟐 × 𝟖𝟖.𝟐𝟐 = 𝟏𝟏.𝟐𝟐
𝟏𝟏× 𝟖𝟖 × 𝟖𝟖 = 𝟏𝟏

The volume of the top rectangular prism is 𝟏𝟏.𝟐𝟐 𝐮𝐮𝐢𝐢𝐢𝐢𝐟𝐟𝐬𝐬𝟑𝟑.
The volume of the bottom rectangular prism is 𝟏𝟏 𝐮𝐮𝐢𝐢𝐢𝐢𝐟𝐟𝐬𝐬𝟑𝟑.
The figure is made of two rectangular prisms, and since the
rectangular prisms only touch at their boundaries, we can
add their volumes together to obtain the volume of the
figure. The total volume of the three-dimensional figure is
𝟗𝟗.𝟐𝟐 𝐮𝐮𝐢𝐢𝐢𝐢𝐟𝐟𝐬𝐬𝟑𝟑.

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Lesson 10: Volumes of Familiar Solids―Cones and Cylinders

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Lesson 10: Volumes of Familiar Solids―Cones and Cylinders

Student Outcomes

 Students know the volume formulas for cones and cylinders.
 Students apply the formulas for volume to real-world and mathematical problems.

Lesson Notes
For the demonstrations in this lesson, the following items are needed: a stack of same-sized note cards, a stack of same-
sized round disks, a cylinder and cone of the same dimensions, and something with which to fill the cone (e.g., rice, sand,
or water). Demonstrate to students that the volume of a rectangular prism is like finding the sum of the areas of
congruent rectangles, stacked one on top of the next. A similar demonstration is useful for the volume of a cylinder. To
demonstrate that the volume of a cone is one-third that of the volume of a cylinder with the same dimension, fill a cone
with rice, sand, or water, and show students that it takes exactly three cones to equal the volume of the cylinder.

Classwork

Opening Exercise (3 minutes)

Students complete the Opening Exercise independently. Revisit the Opening Exercise once the discussion below is
finished.

Opening Exercise

i. Write an equation to determine the volume of the rectangular prism shown below.

𝑽𝑽 = 𝟖𝟖(𝟔𝟔)(𝒉𝒉)
= 𝟒𝟒𝟖𝟖𝒉𝒉

The volume is 𝟒𝟒𝟖𝟖𝒉𝒉 𝐦𝐦𝐦𝐦𝟑𝟑.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

ii. Write an equation to determine the volume of the rectangular prism shown below.

𝑽𝑽 = 𝟏𝟏𝟏𝟏(𝟖𝟖)(𝒉𝒉)
= 𝟖𝟖𝟏𝟏𝒉𝒉

The volume is 𝟖𝟖𝟏𝟏𝒉𝒉 𝐢𝐢𝐧𝐧𝟑𝟑.

iii. Write an equation to determine the volume of the rectangular prism shown below.

𝑽𝑽 = 𝟔𝟔(𝟒𝟒)(𝒉𝒉)
= 𝟐𝟐𝟒𝟒𝒉𝒉

The volume is 𝟐𝟐𝟒𝟒𝒉𝒉 𝐜𝐜𝐦𝐦𝟑𝟑.

iv. Write an equation for volume, 𝑽𝑽, in terms of the area of the base, 𝑩𝑩.

𝑽𝑽 = 𝑩𝑩𝒉𝒉

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

b. Using what you learned in part (a), write an equation to determine the volume of the cylinder shown below.

𝑽𝑽 = 𝑩𝑩𝒉𝒉
= 𝟒𝟒𝟐𝟐𝝅𝝅𝒉𝒉
= 𝟏𝟏𝟔𝟔𝝅𝝅𝒉𝒉

The volume is 𝟏𝟏𝟔𝟔𝝅𝝅𝒉𝒉 𝐜𝐜𝐦𝐦𝟑𝟑.

Students may not know the formula to determine the volume of a cylinder, so some may not be able to respond to this
exercise until after the discussion below. This is an exercise for students to make sense of problems and persevere in
solving them.

Discussion (10 minutes)

 We will continue with an intuitive discussion of volume. The volume formula from the last lesson says that if
the dimensions of a rectangular prism are 𝑙𝑙, 𝑤𝑤, ℎ, then the volume of the rectangular prism is 𝑉𝑉 = 𝑙𝑙 ∙ 𝑤𝑤 ∙ ℎ.

 Referring to the picture, we call the blue rectangle at the bottom of the rectangular prism the base, and the

length of any one of the edges perpendicular to the base the height of the rectangular prism. Then, the
formula says

𝑉𝑉 = (area of base) ∙ height.

Scaffolding:
Demonstrate the volume of a
rectangular prism using a stack
of note cards. The volume of
the rectangular prism increases
as the height of the stack
increases. Note that the
rectangles (note cards) are
congruent.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

 Examine the volume of a cylinder with base 𝐵𝐵 and height ℎ. Is the solid (i.e., the totality of all the line
segments) of length ℎ lying above the plane so that each segment is perpendicular to the plane, and is its
lower endpoint lying on the base 𝐵𝐵 (as shown)?

 Do you know a name for the shape of the base?

 No, it is some curvy shape.
 Let’s examine another cylinder.

 Do we know the name of the shape of the base?

 It appears to be a circle.

 What do you notice about the line segments intersecting the base?

 The line segments appear to be perpendicular to the base.
 What angle does the line segment appear to make with the base?

 The angle appears to be a right angle.

 When the base of a diagram is the shape of a circle and the line segments on the base are perpendicular to the
base, then the shape of the diagram is called a right circular cylinder.

We want to use the general formula for volume of a prism to apply to this shape of a right
circular cylinder.

 What is the general formula for finding the volume of a prism?
 𝑉𝑉 = (area of base) ∙ height

 What is the area for the base of the right circular cylinder?

 The area of a circle is 𝐴𝐴 = 𝜋𝜋𝑟𝑟2.
 What information do we need to find the area of a circle?

 We need to know the radius of the circle.

 What would be the volume of a right circular cylinder?

 𝑉𝑉 = (𝜋𝜋𝑟𝑟2)ℎ
 What information is needed to find the volume of a right circular cylinder?

 We would need to know the radius of the base and the height of the cylinder.

Scaffolding:
Demonstrate the volume of a
cylinder using a stack of round
disks. The volume of the
cylinder increases as the height
of the stack increases, just like
the rectangular prism. Note
that the disks are congruent.

Scaffolding:
Clearly stating the meanings of
symbols may present
challenges for English language
learners, and as such, students
may benefit from a menu of
phrases to support their
statements. They will require
detailed instruction and
support in learning the non-
negotiable vocabulary terms
and phrases.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

Exercises 1–3 (8 minutes)

Students work independently or in pairs to complete Exercises 1–3.

Exercises 1–3

1. Use the diagram to the right to answer the questions.

a. What is the area of the base?

The area of the base is (𝟒𝟒.𝟓𝟓)(𝟖𝟖.𝟐𝟐) 𝐢𝐢𝐧𝐧𝟐𝟐or 𝟑𝟑𝟔𝟔.𝟗𝟗 𝐢𝐢𝐧𝐧𝟐𝟐.

b. What is the height?

The height of the rectangular prism is 𝟏𝟏𝟏𝟏.𝟕𝟕 𝐢𝐢𝐧𝐧.

c. What is the volume of the rectangular prism?

The volume of the rectangular prism is 𝟒𝟒𝟑𝟑𝟏𝟏.𝟕𝟕𝟑𝟑 𝐢𝐢𝐧𝐧𝟑𝟑.

2. Use the diagram to the right to answer the questions.

a. What is the area of the base?

𝑨𝑨 = 𝝅𝝅𝟐𝟐𝟐𝟐
𝑨𝑨 = 𝟒𝟒𝝅𝝅

The area of the base is 𝟒𝟒𝟒𝟒 𝐜𝐜𝐦𝐦𝟐𝟐.

b. What is the height?

The height of the right circular cylinder is 𝟓𝟓.𝟑𝟑 𝐜𝐜𝐦𝐦.

c. What is the volume of the right circular cylinder?

𝑽𝑽 = (𝝅𝝅𝒓𝒓𝟐𝟐)𝒉𝒉
𝑽𝑽 = (𝟒𝟒𝝅𝝅)𝟓𝟓.𝟑𝟑
𝑽𝑽 = 𝟐𝟐𝟏𝟏.𝟐𝟐𝝅𝝅

The volume of the right circular cylinder is 𝟐𝟐𝟏𝟏.𝟐𝟐𝝅𝝅 𝐜𝐜𝐦𝐦𝟑𝟑.

3. Use the diagram to the right to answer the questions.

a. What is the area of the base?

𝐀𝐀 = 𝟒𝟒𝟔𝟔𝟐𝟐
𝐀𝐀 = 𝟑𝟑𝟔𝟔𝟒𝟒

The area of the base is 𝟑𝟑𝟔𝟔𝟒𝟒 𝐢𝐢𝐧𝐧𝟐𝟐.

b. What is the height?

The height of the right circular cylinder is 𝟐𝟐𝟓𝟓 𝐢𝐢𝐧𝐧.

c. What is the volume of the right circular cylinder?

𝐕𝐕 = (𝟑𝟑𝟔𝟔𝟒𝟒)𝟐𝟐𝟓𝟓
𝐕𝐕 = 𝟗𝟗𝟏𝟏𝟏𝟏𝟒𝟒

The volume of the right circular cylinder is 𝟗𝟗𝟏𝟏𝟏𝟏𝝅𝝅 𝐢𝐢𝐧𝐧𝟑𝟑.

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Discussion (10 minutes)

 Next, we introduce the concept of a cone. We start with the general concept of a cylinder. Let 𝑃𝑃 be a point in
the plane that contains the top of a cylinder or height, ℎ. Then, the totality of all the segments joining 𝑃𝑃 to a
point on the base 𝐵𝐵 is a solid, called a cone, with base 𝐵𝐵 and height ℎ. The point 𝑃𝑃 is the top vertex of the
cone. Here are two examples of such cones.

 Let’s examine the diagram on the right more closely. What is the shape of the base?

 It appears to be the shape of a circle.

 Where does the line segment from the vertex to the base appear to intersect the base?
 It appears to intersect at the center of the circle.

 What type of angle do the line segment and base appear to make?

 It appears to be a right angle.

 If the vertex of a circular cone happens to lie on the line perpendicular to the circular base at its center, then
the cone is called a right circular cone.

 We want to develop a general formula for volume of right circular cones from our general formula for
cylinders.

 If we were to fill a cone of height ℎ and radius 𝑟𝑟 with rice (or sand or water), how many cones do you think it
would take to fill up a cylinder of the same height, ℎ, and radius, 𝑟𝑟?

Show students a cone filled with rice (or sand or water). Show students a cylinder of the same height and radius. Give
students time to make a conjecture about how many cones it will take to fill the cylinder. Ask students to share their
guesses and their reasoning to justify their claims. Consider having the class vote on the correct answer before the
demonstration or showing the video. Demonstrate that it would take the volume of three cones to fill up the cylinder, or
show the following short, one-minute video http://youtu.be/0ZACAU4SGyM.

 What would be the general formula for the volume of a right cone? Explain.

Provide students time to work in pairs to develop the formula for the volume of a cone.

 Since it took three cones to fill up a cylinder with the same dimensions, then the volume of the cone is
one-third that of the cylinder. We know the volume for a cylinder already, so the cone’s volume will be
1
3

of the volume of a cylinder with the same base and same height. Therefore, the formula will be

𝑉𝑉 = 1
3 (𝜋𝜋𝑟𝑟2)ℎ.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

Exercises 4–6 (5 minutes)

Students work independently or in pairs to complete Exercises 4–6 using the general formula for the volume of a cone.
Exercise 6 is a challenge problem.

Exercises 4–6

4. Use the diagram to find the volume of the right circular cone.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑

(𝝅𝝅𝒓𝒓𝟐𝟐)𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑

(𝝅𝝅𝟒𝟒𝟐𝟐)𝟗𝟗

𝑽𝑽 = 𝟒𝟒𝟖𝟖𝝅𝝅

The volume of the right circular cone is 𝟒𝟒𝟖𝟖𝝅𝝅 𝐦𝐦𝐦𝐦𝟑𝟑.

5. Use the diagram to find the volume of the right circular cone.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑

(𝝅𝝅𝒓𝒓𝟐𝟐)𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑

(𝝅𝝅𝟐𝟐.𝟑𝟑𝟐𝟐)𝟏𝟏𝟓𝟓

𝑽𝑽 = 𝟐𝟐𝟔𝟔.𝟒𝟒𝟓𝟓𝝅𝝅

The volume of the right circular cone is 𝟐𝟐𝟔𝟔.𝟒𝟒𝟓𝟓𝝅𝝅 𝐦𝐦𝟑𝟑.

6. Challenge: A container in the shape of a right circular cone has height 𝒉𝒉, and base of radius 𝒓𝒓, as shown. It is filled
with water (in its upright position) to half the height. Assume that the surface of the water is parallel to the base of
the inverted cone. Use the diagram to answer the following questions:

a. What do we know about the lengths of 𝑨𝑨𝑩𝑩 and 𝑨𝑨𝑨𝑨?

Then we know that |𝑨𝑨𝑩𝑩| = 𝒓𝒓, and |𝑨𝑨𝑨𝑨| = 𝒉𝒉.

b. What do we know about the measure of ∠𝑨𝑨𝑨𝑨𝑩𝑩 and
∠𝑨𝑨𝑶𝑶𝑶𝑶?

∠𝑨𝑨𝑨𝑨𝑩𝑩 and ∠𝑨𝑨𝑶𝑶𝑶𝑶 are both right angles.

c. What can you say about △𝑨𝑨𝑨𝑨𝑩𝑩 and △𝑨𝑨𝑶𝑶𝑶𝑶?

We have two similar triangles, △𝑨𝑨𝑨𝑨𝑩𝑩 and △𝑨𝑨𝑶𝑶𝑶𝑶 by AA
criterion.

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d. What is the ratio of the volume of water to the volume of the container itself?

Since
|𝑨𝑨𝑩𝑩|
|𝑶𝑶𝑶𝑶| =

|𝑨𝑨𝑩𝑩|
|𝑶𝑶𝑶𝑶| =

𝟐𝟐|𝑨𝑨𝑶𝑶|
|𝑶𝑶𝑨𝑨| .

Then |𝑨𝑨𝑩𝑩| = 𝟐𝟐|𝑶𝑶𝑶𝑶|.

Using the volume formula to determine the volume of the container, we have 𝑽𝑽 = 𝟏𝟏
𝟑𝟑𝝅𝝅|𝑨𝑨𝑩𝑩|𝟐𝟐|𝑨𝑨𝑨𝑨|.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟐𝟐|𝑶𝑶𝑶𝑶|)𝟐𝟐(𝟐𝟐|𝑨𝑨𝑶𝑶|)

with water.

Therefore, the volume of the water to the volume of the container is 𝟏𝟏:𝟖𝟖.

Closing (4 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 Students know the volume formulas for right circular cylinders.

 Students know the volume formula for right circular cones with relation to right circular cylinders.

 Students can apply the formulas for volume of right circular cylinders and cones.

Exit Ticket (5 minutes)

Lesson Summary

The formula to find the volume, 𝑽𝑽, of a right circular cylinder is 𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉 = 𝑩𝑩𝒉𝒉, where 𝑩𝑩 is the area of the base.

The formula to find the volume of a cone is directly related to that of the cylinder. Given a right circular cylinder
with radius 𝒓𝒓 and height 𝒉𝒉, the volume of a cone with those same dimensions is one-third of the cylinder. The

formula for the volume, 𝑽𝑽, of a circular cone is 𝑽𝑽 = 𝟏𝟏
𝟑𝟑𝝅𝝅𝒓𝒓

𝟐𝟐𝒉𝒉. More generally, the volume formula for a general cone

is 𝑽𝑽 = 𝟏𝟏
𝟑𝟑𝑩𝑩𝒉𝒉, where 𝑩𝑩 is the area of the base.

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Name Date

Lesson 10: Volumes of Familiar Solids—Cones and Cylinders

Exit Ticket

1. Use the diagram to find the total volume of the three cones shown below.

2. Use the diagram below to determine which has the greater volume, the cone or the cylinder.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

Exit Ticket Sample Solutions

1. Use the diagram to find the total volume of the three cones shown below.

Since all three cones have the same base and height, the volume of the three cones will be the same as finding the
volume of a cylinder with the same base radius and same height.

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅(𝟐𝟐)𝟐𝟐𝟑𝟑
𝑽𝑽 = 𝟏𝟏𝟐𝟐𝝅𝝅

The volume of all three cones is 𝟏𝟏𝟐𝟐𝝅𝝅 𝐟𝐟𝐭𝐭𝟑𝟑.

2. Use the diagram below to determine which has the greater volume, the cone or the cylinder.

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅𝟒𝟒𝟐𝟐(𝟔𝟔)
𝑽𝑽 = 𝟗𝟗𝟔𝟔𝝅𝝅

The volume of the cylinder is 𝟗𝟗𝟔𝟔𝝅𝝅 𝐜𝐜𝐦𝐦𝟑𝟑.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝟔𝟔𝟐𝟐(𝟖𝟖)

𝑽𝑽 = 𝟗𝟗𝟔𝟔𝝅𝝅

The volume of the cone is 𝟗𝟗𝟔𝟔𝝅𝝅 𝐜𝐜𝐦𝐦𝟑𝟑.

The volume of the cylinder and the volume of the cone are the same, 𝟗𝟗𝟔𝟔𝝅𝝅 𝐜𝐜𝐦𝐦𝟑𝟑.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

Problem Set Sample Solutions

1. Use the diagram to help you find the volume of the right circular cylinder.

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅(𝟏𝟏)𝟐𝟐(𝟏𝟏)
𝑽𝑽 = 𝝅𝝅

The volume of the right circular cylinder is 𝝅𝝅 𝐟𝐟𝐭𝐭𝟑𝟑.

2. Use the diagram to help you find the volume of the right circular cone.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟐𝟐.𝟖𝟖)𝟐𝟐(𝟒𝟒.𝟑𝟑)

𝑽𝑽 = 𝟏𝟏𝟏𝟏.𝟐𝟐𝟑𝟑𝟕𝟕𝟑𝟑𝟑𝟑𝟑𝟑…𝝅𝝅

The volume of the right circular cone is about 𝟏𝟏𝟏𝟏.𝟐𝟐𝝅𝝅 𝐜𝐜𝐦𝐦𝟑𝟑.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

3. Use the diagram to help you find the volume of the right circular cylinder.

If the diameter is 𝟏𝟏𝟐𝟐 𝐦𝐦𝐦𝐦, then the radius is 𝟔𝟔 𝐦𝐦𝐦𝐦.

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅(𝟔𝟔)𝟐𝟐(𝟏𝟏𝟕𝟕)
𝑽𝑽 = 𝟔𝟔𝟏𝟏𝟐𝟐𝝅𝝅

The volume of the right circular cylinder is 𝟔𝟔𝟏𝟏𝟐𝟐𝝅𝝅 𝐦𝐦𝐦𝐦𝟑𝟑.

4. Use the diagram to help you find the volume of the right circular cone.

If the diameter is 𝟏𝟏𝟒𝟒 𝐢𝐢𝐧𝐧., then the radius is 𝟕𝟕 𝐢𝐢𝐧𝐧.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟕𝟕)𝟐𝟐(𝟏𝟏𝟖𝟖.𝟐𝟐)

𝑽𝑽 = 𝟐𝟐𝟗𝟗𝟕𝟕.𝟐𝟐𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔…𝝅𝝅
𝑽𝑽 ≈ 𝟐𝟐𝟗𝟗𝟕𝟕.𝟑𝟑𝝅𝝅

The volume of the right cone is about 𝟐𝟐𝟗𝟗𝟕𝟕.𝟑𝟑𝝅𝝅 𝐢𝐢𝐧𝐧𝟑𝟑.

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Lesson 10: Volumes of Familiar Solids―Cones and Cylinders

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 10

5. Oscar wants to fill with water a bucket that is the shape of a right circular cylinder. It has a 𝟔𝟔-inch radius and
𝟏𝟏𝟐𝟐-inch height. He uses a shovel that has the shape of a right circular cone with a 𝟑𝟑-inch radius and 𝟒𝟒-inch height.
How many shovelfuls will it take Oscar to fill the bucket up level with the top?

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅(𝟔𝟔)𝟐𝟐(𝟏𝟏𝟐𝟐)
𝑽𝑽 = 𝟒𝟒𝟑𝟑𝟐𝟐𝝅𝝅

The volume of the bucket is 𝟒𝟒𝟑𝟑𝟐𝟐𝝅𝝅 𝐢𝐢𝐧𝐧𝟑𝟑.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟑𝟑)𝟐𝟐(𝟒𝟒)

𝑽𝑽 = 𝟏𝟏𝟐𝟐𝝅𝝅

The volume of shovel is 𝟏𝟏𝟐𝟐𝝅𝝅 𝐢𝐢𝐧𝐧𝟑𝟑.

𝟒𝟒𝟑𝟑𝟐𝟐𝝅𝝅
𝟏𝟏𝟐𝟐𝝅𝝅

= 𝟑𝟑𝟔𝟔

It would take 𝟑𝟑𝟔𝟔 shovelfuls of water to fill up the bucket.

6. A cylindrical tank (with dimensions shown below) contains water that is 𝟏𝟏-foot deep. If water is poured into the

tank at a constant rate of 𝟐𝟐𝟏𝟏 𝐟𝐟𝐭𝐭
𝟑𝟑

𝐦𝐦𝐢𝐢𝐧𝐧 for 𝟐𝟐𝟏𝟏 𝐦𝐦𝐢𝐢𝐧𝐧., will the tank overflow? Use 𝟑𝟑.𝟏𝟏𝟒𝟒 to estimate 𝝅𝝅.

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅(𝟑𝟑)𝟐𝟐(𝟏𝟏𝟐𝟐)
𝑽𝑽 = 𝟏𝟏𝟏𝟏𝟖𝟖𝝅𝝅

The volume of the tank is about 𝟑𝟑𝟑𝟑𝟗𝟗.𝟏𝟏𝟐𝟐 𝐟𝐟𝐭𝐭3.

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅(𝟑𝟑)𝟐𝟐(𝟏𝟏)
𝑽𝑽 = 𝟗𝟗𝝅𝝅

There is about 𝟐𝟐𝟖𝟖.𝟐𝟐𝟔𝟔 𝐟𝐟𝐭𝐭𝟑𝟑 of water already in the tank. There is about 𝟑𝟑𝟏𝟏𝟏𝟏.𝟖𝟖𝟔𝟔 𝐟𝐟𝐭𝐭𝟑𝟑 of space left in the tank. If the
water is poured at a constant rate for 𝟐𝟐𝟏𝟏 𝐦𝐦𝐢𝐢𝐧𝐧., 𝟒𝟒𝟏𝟏𝟏𝟏 𝐟𝐟𝐭𝐭𝟑𝟑 will be poured into the tank, and the tank will overflow.

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Lesson 11: Volume of a Sphere

Student Outcomes

 Students know the volume formula for a sphere as it relates to a right circular cylinder with the same diameter
and height.

 Students apply the formula for the volume of a sphere to real-world and mathematical problems.

Lesson Notes
The demonstrations in this lesson require a sphere (preferably one that can be filled with water, sand, or rice) and a right
circular cylinder with the same diameter and height as the diameter of the sphere. Demonstrate to students that the
volume of a sphere is two-thirds the volume of the circumscribing cylinder. If this demonstration is impossible, a video
link is included to show a demonstration.

Classwork

Discussion (10 minutes)

Show students pictures of the spheres shown below (or use real objects). Ask the class to come up with a mathematical
definition on their own.

 Finally, we come to the volume of a sphere of radius 𝑟𝑟. First recall that a sphere
of radius 𝑟𝑟 is the set of all the points in three-dimensional space of distance 𝑟𝑟
from a fixed point, called the center of the sphere. So a sphere is, by definition,
a surface, or a two-dimensional object. When we talk about the volume of a
sphere, we mean the volume of the solid inside this surface.

 The discovery of this formula was a major event in ancient mathematics. The first person to discover the
formula was Archimedes (287–212 B.C.E.), but it was also independently discovered in China by Zu Chongshi
(429–501 C.E.) and his son Zu Geng (circa 450–520 C.E.) by essentially the same method. This method has
come to be known as Cavalieri’s Principle because he announced this method at a time when he had an
audience. Cavalieri (1598–1647) was one of the forerunners of calculus.

Scaffolding:
Consider using a small bit of
clay to represent the center
and toothpicks to represent
the radius of a sphere.

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Show students a cylinder. Convince them that the diameter of the sphere is the same as the diameter and the height of
the cylinder. Give students time to make a conjecture about how much of the volume of the cylinder is taken up by the
sphere. Ask students to share their guesses and their reasoning. Consider having the class vote on the correct answer
before proceeding with the discussion.

 The derivation of this formula and its understanding requires advanced mathematics, so we will not derive it at
this time.

If possible, do a physical demonstration showing that the volume of a sphere is exactly
2
3

the volume of a cylinder with

the same diameter and height. Also consider showing the following 1:17-minute video:
http://www.youtube.com/watch?v=aLyQddyY8ik.

 Based on the demonstration (or video), we can say that

Volume(sphere) = 2
3 volume(cylinder with same diameter and height of the sphere).

Exercises 1–3 (5 minutes)

Students work independently or in pairs using the general formula for the volume of a sphere. Verify that students are
able to compute the formula for the volume of a sphere.

Exercises 1–3

1. What is the volume of a cylinder?

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

2. What is the height of the cylinder?

The height of the cylinder is the same as the diameter of the sphere. The diameter is 𝟐𝟐𝒓𝒓.

3. If 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯(𝐬𝐬𝐬𝐬𝐬𝐬𝐯𝐯𝐬𝐬𝐯𝐯) = 𝟐𝟐
𝟑𝟑𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯(𝐜𝐜𝐜𝐜𝐯𝐯𝐜𝐜𝐜𝐜𝐜𝐜𝐯𝐯𝐬𝐬 𝐰𝐰𝐜𝐜𝐰𝐰𝐬𝐬 𝐬𝐬𝐬𝐬𝐯𝐯𝐯𝐯 𝐜𝐜𝐜𝐜𝐬𝐬𝐯𝐯𝐯𝐯𝐰𝐰𝐯𝐯𝐬𝐬 𝐬𝐬𝐜𝐜𝐜𝐜 𝐬𝐬𝐯𝐯𝐜𝐜𝐡𝐡𝐬𝐬𝐰𝐰), what is the formula for the

volume of a sphere?

𝐕𝐕𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯(𝐬𝐬𝐬𝐬𝐬𝐬𝐯𝐯𝐬𝐬𝐯𝐯) =
𝟐𝟐
𝟑𝟑

(𝛑𝛑𝐬𝐬𝟐𝟐𝐬𝐬)

𝐕𝐕𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯(𝐬𝐬𝐬𝐬𝐬𝐬𝐯𝐯𝐬𝐬𝐯𝐯) =
𝟐𝟐
𝟑𝟑

(𝛑𝛑𝐬𝐬𝟐𝟐𝟐𝟐𝐬𝐬)

𝐕𝐕𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯(𝐬𝐬𝐬𝐬𝐬𝐬𝐯𝐯𝐬𝐬𝐯𝐯) =
𝟒𝟒
𝟑𝟑

(𝛑𝛑𝐬𝐬𝟑𝟑)

MP.2

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Example 1 (4 minutes)

 When working with circular two- and three-dimensional figures, we can express our answers in two ways. One
is exact and will contain the symbol for pi, 𝜋𝜋. The other is an approximation, which usually uses 3.14 for 𝜋𝜋.
Unless noted otherwise, we will have exact answers that contain the pi symbol.

 For Examples 1 and 2, use the formula from Exercise 3 to compute the exact volume for the sphere shown
below.

Example 1

Compute the exact volume for the sphere shown below.

Provide students time to work; then, have them share their solutions.

 Sample student work:

𝑉𝑉 =
4
3
𝜋𝜋𝑟𝑟3

=
4
3
𝜋𝜋(43)

=
4
3
𝜋𝜋(64)

=
256

3
𝜋𝜋

= 85
1
3
𝜋𝜋

The volume of the sphere is 85 1
3𝜋𝜋 cm3.

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Example 2 (6 minutes)

Example 2

A cylinder has a diameter of 𝟏𝟏𝟏𝟏 inches and a height of 𝟏𝟏𝟒𝟒 inches. What is the volume of the largest sphere that will fit
into the cylinder?

 What is the radius of the base of the cylinder?
 The radius of the base of the cylinder is 8 inches.

 Could the sphere have a radius of 8 inches? Explain.

 No. If the sphere had a radius of 8 inches, then it would not fit into the cylinder because the height is
only 14 inches. With a radius of 8 inches, the sphere would have a height of 2𝑟𝑟, or 16 inches. Since the
cylinder is only 14 inches high, the radius of the sphere cannot be 8 inches.

 What size radius for the sphere would fit into the cylinder? Explain.

 A radius of 7 inches would fit into the cylinder because 2𝑟𝑟 is 14, which means the sphere would touch
the top and bottom of the cylinder. A radius of 7 means the radius of the sphere would not touch the
sides of the cylinder, but would fit into it.

 Now that we know the radius of the largest sphere is 7 inches, what is the volume of the sphere?
 Sample student work:

𝑉𝑉 =
4
3
𝜋𝜋𝑟𝑟3

=
4
3
𝜋𝜋(73)

=
4
3
𝜋𝜋(343)

=
1372

3
𝜋𝜋

= 457
1
3
𝜋𝜋

The volume of the sphere is 457 1
3𝜋𝜋 cm3.

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Exercises 4–8 (10 minutes)

Students work independently or in pairs to use the general formula for the volume of a sphere.

Exercises 4–8

4. Use the diagram and the general formula to find the volume of the sphere.

𝐕𝐕 =
𝟒𝟒
𝟑𝟑
𝛑𝛑𝐬𝐬𝟑𝟑

𝐕𝐕 =
𝟒𝟒
𝟑𝟑
𝛑𝛑(𝟏𝟏𝟑𝟑)

𝐕𝐕 ≈ 𝟐𝟐𝟐𝟐𝟐𝟐𝛑𝛑

The volume of the sphere is about 𝟐𝟐𝟐𝟐𝟐𝟐𝛑𝛑 𝐜𝐜𝐜𝐜𝟑𝟑.

5. The average basketball has a diameter of 𝟗𝟗.𝟓𝟓 inches. What is the volume of an average basketball? Round your
answer to the tenths place.

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟒𝟒.𝟕𝟕𝟓𝟓𝟑𝟑)

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟏𝟏𝟏𝟏𝟕𝟕.𝟏𝟏𝟕𝟕)

𝑽𝑽 ≈ 𝟏𝟏𝟒𝟒𝟐𝟐.𝟗𝟗𝝅𝝅

The volume of an average basketball is about 𝟏𝟏𝟒𝟒𝟐𝟐.𝟗𝟗𝛑𝛑 𝐜𝐜𝐜𝐜𝟑𝟑.

6. A spherical fish tank has a radius of 𝟐𝟐 inches. Assuming the entire tank could be filled with water, what would the
volume of the tank be? Round your answer to the tenths place.

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟐𝟐𝟑𝟑)

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟓𝟓𝟏𝟏𝟐𝟐)

𝑽𝑽 ≈ 𝟏𝟏𝟐𝟐𝟐𝟐.𝟕𝟕𝝅𝝅

The volume of the fish tank is about 𝟏𝟏𝟐𝟐𝟐𝟐.𝟕𝟕𝛑𝛑 𝐜𝐜𝐜𝐜𝟑𝟑.

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

7. Use the diagram to answer the questions.

a. Predict which of the figures shown above has the greater volume. Explain.

Student answers will vary. Students will probably say the cone has more volume because it looks larger.

b. Use the diagram to find the volume of each, and determine which has the greater volume.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟐𝟐.𝟓𝟓𝟐𝟐)(𝟏𝟏𝟐𝟐.𝟏𝟏)

𝑽𝑽 = 𝟐𝟐𝟏𝟏.𝟐𝟐𝟓𝟓𝝅𝝅

The volume of the cone is 𝟐𝟐𝟏𝟏.𝟐𝟐𝟓𝟓𝛑𝛑 𝐯𝐯𝐯𝐯𝟑𝟑.

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟐𝟐.𝟐𝟐𝟑𝟑)

𝑽𝑽 = 𝟐𝟐𝟗𝟗.𝟐𝟐𝟏𝟏𝟗𝟗𝟑𝟑𝟑𝟑𝟑𝟑…𝝅𝝅

The volume of the sphere is about 𝟐𝟐𝟗𝟗.𝟐𝟐𝟕𝟕𝛑𝛑 𝐯𝐯𝐯𝐯𝟑𝟑. The volume of the sphere is greater than the volume of the
cone.

8. One of two half spheres formed by a plane through the sphere’s center is called a hemisphere. What is the formula
for the volume of a hemisphere?

Since a hemisphere is half a sphere, the 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯(𝐬𝐬𝐯𝐯𝐯𝐯𝐜𝐜𝐬𝐬𝐬𝐬𝐬𝐬𝐯𝐯𝐬𝐬𝐯𝐯) = 𝟏𝟏
𝟐𝟐 (𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 𝐯𝐯𝐨𝐨 𝐬𝐬𝐬𝐬𝐬𝐬𝐯𝐯𝐬𝐬𝐯𝐯).

𝑽𝑽 =
𝟏𝟏
𝟐𝟐
�
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑�

𝑽𝑽 =
𝟐𝟐
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Closing (5 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

 Students know the volume formula for a sphere with relation to a right circular cylinder.
 Students know the volume formula for a hemisphere.

 Students can apply the volume of a sphere to solve mathematical problems.

Exit Ticket (5 minutes)

Lesson Summary

The formula to find the volume of a sphere is directly related to that of the right circular cylinder. Given a right
circular cylinder with radius 𝒓𝒓 and height 𝒉𝒉, which is equal to 𝟐𝟐𝒓𝒓, a sphere with the same radius 𝒓𝒓 has a volume that
is exactly two-thirds of the cylinder.

Therefore, the volume of a sphere with radius 𝒓𝒓 has a volume given by the formula 𝑽𝑽 = 𝟒𝟒
𝟑𝟑𝝅𝝅𝒓𝒓

𝟑𝟑.

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Lesson 11: Volume of a Sphere

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Name Date

Lesson 11: Volume of a Sphere

Exit Ticket

1. What is the volume of the sphere shown below?

2. Which of the two figures below has the greater volume?

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Exit Ticket Sample Solutions

1. What is the volume of the sphere shown below?

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

=
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟑𝟑𝟑𝟑)

=
𝟏𝟏𝟏𝟏𝟐𝟐
𝟑𝟑

𝝅𝝅

= 𝟑𝟑𝟏𝟏𝝅𝝅

The volume of the sphere is 𝟑𝟑𝟏𝟏𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑.

2. Which of the two figures below has the greater volume?

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

=
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟒𝟒𝟑𝟑)

=
𝟐𝟐𝟓𝟓𝟏𝟏
𝟑𝟑

𝝅𝝅

= 𝟐𝟐𝟓𝟓
𝟏𝟏
𝟑𝟑
𝝅𝝅

The volume of the sphere is 𝟐𝟐𝟓𝟓𝟏𝟏𝟑𝟑𝝅𝝅 𝐯𝐯𝐯𝐯𝟑𝟑.

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

=
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟑𝟑𝟐𝟐)(𝟏𝟏.𝟓𝟓)

=
𝟓𝟓𝟐𝟐.𝟓𝟓
𝟑𝟑

𝝅𝝅

= 𝟏𝟏𝟗𝟗.𝟓𝟓𝝅𝝅

The volume of the cone is 𝟏𝟏𝟗𝟗.𝟓𝟓𝝅𝝅 𝐯𝐯𝐯𝐯𝟑𝟑. The sphere has the greater volume.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

Problem Set Sample Solutions

1. Use the diagram to find the volume of the sphere.

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟗𝟗𝟑𝟑)

𝑽𝑽 = 𝟗𝟗𝟕𝟕𝟐𝟐𝝅𝝅

The volume of the sphere is 𝟗𝟗𝟕𝟕𝟐𝟐𝛑𝛑 𝐜𝐜𝐯𝐯𝟑𝟑.

2. Determine the volume of a sphere with diameter 𝟗𝟗 𝐯𝐯𝐯𝐯, shown below.

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

=
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟒𝟒.𝟓𝟓𝟑𝟑)

=
𝟑𝟑𝟏𝟏𝟒𝟒.𝟓𝟓
𝟑𝟑

𝝅𝝅

= 𝟏𝟏𝟐𝟐𝟏𝟏.𝟓𝟓𝝅𝝅

The volume of the sphere is 𝟏𝟏𝟐𝟐𝟏𝟏.𝟓𝟓𝝅𝝅 𝐯𝐯𝐯𝐯𝟑𝟑.

3. Determine the volume of a sphere with diameter 𝟐𝟐𝟐𝟐 𝐜𝐜𝐜𝐜., shown below.

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

=
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟏𝟏𝟏𝟏𝟑𝟑)

=
𝟓𝟓𝟑𝟑𝟐𝟐𝟒𝟒
𝟑𝟑

𝝅𝝅

= 𝟏𝟏𝟕𝟕𝟕𝟕𝟒𝟒
𝟐𝟐
𝟑𝟑
𝝅𝝅

The volume of the sphere is 𝟏𝟏𝟕𝟕𝟕𝟕𝟒𝟒 𝟐𝟐𝟑𝟑𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

4. Which of the two figures below has the lesser volume?

The volume of the cone:

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

=
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟏𝟏𝟏𝟏)(𝟕𝟕)

=
𝟏𝟏𝟏𝟏𝟐𝟐
𝟑𝟑

𝝅𝝅

= 𝟑𝟑𝟕𝟕
𝟏𝟏
𝟑𝟑
𝝅𝝅

The volume of the sphere:

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

=
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟐𝟐𝟑𝟑)

=
𝟑𝟑𝟐𝟐
𝟑𝟑
𝝅𝝅

= 𝟏𝟏𝟏𝟏
𝟐𝟐
𝟑𝟑

𝝅𝝅

The cone has volume 𝟑𝟑𝟕𝟕𝟏𝟏𝟑𝟑𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑 and the sphere has volume 𝟏𝟏𝟏𝟏𝟐𝟐𝟑𝟑 𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑. The sphere has the lesser volume.

5. Which of the two figures below has the greater volume?

The volume of the cylinder:

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉

= 𝝅𝝅(𝟑𝟑𝟐𝟐)(𝟏𝟏.𝟐𝟐)

= 𝟓𝟓𝟓𝟓.𝟐𝟐𝝅𝝅

The volume of the sphere:

𝑽𝑽 =
𝟒𝟒
𝟑𝟑
𝝅𝝅𝒓𝒓𝟑𝟑

=
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟓𝟓𝟑𝟑)

=
𝟓𝟓𝟏𝟏𝟏𝟏
𝟑𝟑

𝝅𝝅

= 𝟏𝟏𝟏𝟏𝟏𝟏
𝟐𝟐
𝟑𝟑
𝝅𝝅

The cylinder has volume 𝟓𝟓𝟓𝟓.𝟐𝟐𝝅𝝅 𝐯𝐯𝐯𝐯𝟑𝟑 and the sphere has volume 𝟏𝟏𝟏𝟏𝟏𝟏𝟐𝟐𝟑𝟑𝝅𝝅 𝐯𝐯𝐯𝐯𝟑𝟑. The sphere has the greater volume.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•5 Lesson 11

6. Bridget wants to determine which ice cream option is the best choice. The chart below gives the description and
prices for her options. Use the space below each item to record your findings.

$𝟐𝟐.𝟏𝟏𝟏𝟏 $𝟑𝟑.𝟏𝟏𝟏𝟏 $𝟒𝟒.𝟏𝟏𝟏𝟏

One scoop in a cup Two scoops in a cup Three scoops in a cup

𝑽𝑽 ≈ 𝟒𝟒.𝟏𝟏𝟗𝟗 𝐜𝐜𝐜𝐜𝟑𝟑 𝑽𝑽 ≈ 𝟐𝟐.𝟑𝟑𝟕𝟕 𝐜𝐜𝐜𝐜𝟑𝟑 𝑽𝑽 ≈ 𝟏𝟏𝟐𝟐.𝟓𝟓𝟏𝟏 𝐜𝐜𝐜𝐜𝟑𝟑
Half a scoop on a cone

filled with ice cream
A cup filled with ice cream
(level to the top of the cup)

𝑽𝑽 ≈ 𝟏𝟏.𝟐𝟐 𝐜𝐜𝐜𝐜𝟑𝟑 𝑽𝑽 ≈ 𝟏𝟏𝟒𝟒.𝟏𝟏𝟑𝟑 𝐜𝐜𝐜𝐜𝟑𝟑

A scoop of ice cream is considered a perfect sphere and has a 𝟐𝟐-inch diameter. A cone has a 𝟐𝟐-inch diameter and a
height of 𝟒𝟒.𝟓𝟓 inches. A cup, considered a right circular cylinder, has a 𝟑𝟑-inch diameter and a height of 𝟐𝟐 inches.

a. Determine the volume of each choice. Use 𝟑𝟑.𝟏𝟏𝟒𝟒 to approximate 𝝅𝝅.

First, find the volume of one scoop of ice cream.

𝐕𝐕𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 𝐯𝐯𝐨𝐨 𝐯𝐯𝐜𝐜𝐯𝐯 𝐬𝐬𝐜𝐜𝐯𝐯𝐯𝐯𝐬𝐬 =
𝟒𝟒
𝟑𝟑
𝝅𝝅(𝟏𝟏𝟑𝟑)

The volume of one scoop of ice cream is
𝟒𝟒
𝟑𝟑
𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑, or approximately 𝟒𝟒.𝟏𝟏𝟗𝟗 𝐜𝐜𝐜𝐜𝟑𝟑.

The volume of two scoops of ice cream is
𝟐𝟐
𝟑𝟑
𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑, or approximately 𝟐𝟐.𝟑𝟑𝟕𝟕 𝐜𝐜𝐜𝐜𝟑𝟑.

The volume of three scoops of ice cream is
𝟏𝟏𝟐𝟐
𝟑𝟑
𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑, or approximately 𝟏𝟏𝟐𝟐.𝟓𝟓𝟏𝟏 𝐜𝐜𝐜𝐜𝟑𝟑.

𝐕𝐕𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 𝐯𝐯𝐨𝐨 𝐬𝐬𝐬𝐬𝐯𝐯𝐨𝐨 𝐬𝐬𝐜𝐜𝐯𝐯𝐯𝐯𝐬𝐬 =
𝟐𝟐
𝟑𝟑
𝛑𝛑(𝟏𝟏𝟑𝟑)

The volume of half a scoop of ice cream is
𝟐𝟐
𝟑𝟑
𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑, or approximately 𝟐𝟐.𝟏𝟏𝟗𝟗 𝐜𝐜𝐜𝐜𝟑𝟑.

𝐕𝐕𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 𝐯𝐯𝐨𝐨 𝐜𝐜𝐯𝐯𝐜𝐜𝐯𝐯 =
𝟏𝟏
𝟑𝟑

(𝝅𝝅𝒓𝒓𝟐𝟐)𝒉𝒉

𝑽𝑽 =
𝟏𝟏
𝟑𝟑
𝝅𝝅(𝟏𝟏𝟐𝟐)𝟒𝟒.𝟓𝟓

𝑽𝑽 = 𝟏𝟏.𝟓𝟓𝝅𝝅

The volume of the cone is 𝟏𝟏.𝟓𝟓𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑, or approximately 𝟒𝟒.𝟕𝟕𝟏𝟏 𝐜𝐜𝐜𝐜𝟑𝟑. Then, the cone with half a scoop of ice
cream on top is approximately 𝟏𝟏.𝟐𝟐 𝐜𝐜𝐜𝐜𝟑𝟑.

𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐𝒉𝒉
𝑽𝑽 = 𝝅𝝅𝟏𝟏.𝟓𝟓𝟐𝟐(𝟐𝟐)
𝑽𝑽 = 𝟒𝟒.𝟓𝟓𝝅𝝅

The volume of the cup is 𝟒𝟒.𝟓𝟓𝝅𝝅 𝐜𝐜𝐜𝐜𝟑𝟑, or approximately 𝟏𝟏𝟒𝟒.𝟏𝟏𝟑𝟑 𝐜𝐜𝐜𝐜𝟑𝟑.

b. Determine which choice is the best value for her money. Explain your reasoning.

Student answers may vary.

Checking the cost for every 𝐜𝐜𝐜𝐜𝟑𝟑 of each choice:
𝟐𝟐

𝟒𝟒.𝟏𝟏𝟗𝟗
≈ 𝟏𝟏.𝟒𝟒𝟕𝟕𝟕𝟕𝟐𝟐𝟑𝟑…

𝟐𝟐
𝟏𝟏.𝟐𝟐

≈ 𝟏𝟏.𝟐𝟐𝟗𝟗𝟒𝟒𝟏𝟏𝟏𝟏…

𝟑𝟑
𝟐𝟐.𝟑𝟑𝟕𝟕

≈ 𝟏𝟏.𝟑𝟑𝟓𝟓𝟐𝟐𝟒𝟒𝟐𝟐…

𝟒𝟒
𝟏𝟏𝟐𝟐.𝟓𝟓𝟏𝟏

≈ 𝟏𝟏.𝟑𝟑𝟏𝟏𝟐𝟐𝟒𝟒𝟕𝟕…

𝟒𝟒
𝟏𝟏𝟒𝟒.𝟏𝟏𝟑𝟑

≈ 𝟏𝟏.𝟐𝟐𝟐𝟐𝟑𝟑𝟏𝟏𝟐𝟐…

The best value for her money is the cup filled with ice cream since it costs about 𝟐𝟐𝟐𝟐 cents for every 𝐜𝐜𝐜𝐜𝟑𝟑.

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8•5 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Name Date

a. We define 𝑥𝑥 as a year between 2008 and 2013 and 𝑦𝑦 as the total number of smartphones sold that
year, in millions. The table shows values of 𝑥𝑥 and corresponding 𝑦𝑦 values.

i. How many smartphones were sold in 2009?

ii. In which year were 90 million smartphones sold?

iii. Is 𝑦𝑦 a function of 𝑥𝑥? Explain why or why not.

b. Randy began completing the table below to represent a particular linear function. Write an equation
to represent the function he was using and complete the table for him.

Year
(𝒙𝒙) 2008 2009 2010 2011 2012 2013

Number of
smartphones

in millions
(𝒚𝒚)

3.7 17.3 42.4 90 125 153.2

Input
(𝒙𝒙) −3 −1 0

1
2

1 2 3

Output
(𝒚𝒚)

−5 4 13

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8•5 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

c. Create the graph of the function in part (b).

d. At NYU in 2013, the cost of the weekly meal plan options could be described as a function of the

number of meals. Is the cost of the meal plan a linear or nonlinear function? Explain.

8 meals: $125/week
10 meals: $135/week
12 meals: $155/week
21 meals: $220/week

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8•5 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

2. The cost to enter and go on rides at a local water park, Wally’s Water World, is shown in the graph below.

A new water park, Tony’s Tidal Takeover, just opened. You have not heard anything specific about how
much it costs to go to this park, but some of your friends have told you what they spent. The information
is organized in the table below.

Number of rides 0 2 4 6
Dollars spent 12.00 13.50 15.00 16.50

Each park charges a different admission fee and a different fee per ride, but the cost of each ride remains
the same.

a. If you only have $14 to spend, which park would you attend (assume the rides are the same

quality)? Explain.

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8•5 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

b. Another water park, Splash, opens, and they charge an admission fee of $30 with no additional fee
for rides. At what number of rides does it become more expensive to go to Wally’s Water World
than Splash? At what number of rides does it become more expensive to go to Tony’s Tidal
Takeover than Splash?

c. For all three water parks, the cost is a function of the number of rides. Compare the functions for all
three water parks in terms of their rate of change. Describe the impact it has on the total cost of
attending each park.

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8•5 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

3. For each part below, leave your answers in terms of 𝜋𝜋.

a. Determine the volume for each three-dimensional figure shown below.

b. You want to fill the cylinder shown below with water. All you have is a container shaped like a cone
with a radius of 3 inches and a height of 5 inches; you can use this cone-shaped container to take
water from a faucet and fill the cylinder. How many cones will it take to fill the cylinder?

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8•5 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

c. You have a cylinder with a diameter of 15 inches and height of 12 inches. What is the volume of the
largest sphere that will fit inside of it?

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8•5 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

A Progression Toward Mastery

Assessment
Task Item

STEP 1
Missing or
incorrect answer
and little evidence
of reasoning or
application of
mathematics to
solve the problem.

STEP 2
Missing or incorrect
answer but
evidence of some
reasoning or
application of
mathematics to
solve the problem.

STEP 3
A correct answer
with some evidence
of reasoning or
application of
mathematics to
solve the problem,
or an incorrect
answer with
substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem.

STEP 4
A correct answer
supported by
substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem.

8.F.A.1

Student makes little or
no attempt to solve the
problem.

Student answers at least
one of the three
questions correctly as
17.3 million, 2011, or
yes. Student does not
provide an explanation
as to why 𝑦𝑦 is a function
of 𝑥𝑥.

Student answers all
three questions correctly
as 17.3 million, 2011,
and yes. Student
provides an explanation
as to why 𝑦𝑦 is a function
of 𝑥𝑥. Student may not
have used vocabulary
related to functions.

Student answers all
three questions correctly
as 17.3 million, 2011,
and yes. Student
provides a compelling
explanation as to why 𝑦𝑦
is a function of 𝑥𝑥 and
uses appropriate
vocabulary related to
functions (e.g.,
assignment, input, and
output).

8.F.A.1

Student makes little or
no attempt to solve the
problem.
Student does not write
a function or equation.
The outputs may or
may not be calculated
correctly.

Student does not
correctly write the
equation to describe the
function.
The outputs may be
correct for the function
described by the
student.
The outputs may or may
not be calculated
correctly.
Student may have made
calculation errors.
Two or more of the
outputs are calculated
correctly.

Student correctly writes
the equation to describe
the function as
𝑦𝑦 = 3𝑥𝑥 + 4.
Three or more of the
outputs are calculated
correctly.
Student may have made
calculation errors.

Student correctly writes
the equation to describe
the function as
𝑦𝑦 = 3𝑥𝑥 + 4.
All four of the outputs
are calculated correctly
as when 𝑥𝑥 = −1, 𝑦𝑦 = 1;

when 𝑥𝑥 = 1
2, 𝑦𝑦 = 11

2 ;
when 𝑥𝑥 = 1, 𝑦𝑦 = 7; and
when 𝑥𝑥 = 2, 𝑦𝑦 = 10.

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8.F.A.1

Student makes little or
no attempt to solve the
problem.
Student may have
graphed some or all of
the input/outputs
given.

Student graphs the
input/outputs incorrectly
(e.g., (4,0) instead of
(0,4)).
The input/outputs do
not appear to be linear.

Student may or may not
have graphed the
input/outputs correctly
(e.g., (4,0) instead of
(0,4)).
The input/outputs
appear to be linear.

Student graphs the
input/outputs correctly
as (0,4).
The input/outputs
appear to be linear.

8.F.A.3

Student makes little or
no attempt to solve the
problem.
Student may or may
not have made a
choice.
Student does not give
an explanation.

Student incorrectly
determines that the
meal plan is linear or
correctly determines
that it is nonlinear.
Student does not give an
explanation, or the
explanation does not
include any
mathematical reasoning.

Student correctly
determines that the
meal plan is nonlinear.
Explanation includes
some mathematical
reasoning.
Explanation may or may
not include reference to
the graph.

Student correctly
determines that the
meal plan is nonlinear.
Explanation includes
substantial mathematical
reasoning.
Explanation includes
reference to the graph.

8.F.A.2

Student makes little or
no attempt to solve the
problem.
Student may or may
not have made a
choice.
Student does not give
an explanation.

Student identifies either
choice.
Student makes
significant calculation
errors.
Student gives little or no
explanation.

Student identifies either
choice.
Student may have made
calculation errors.
Explanation may or may
not have included the
calculation errors.

Student identifies
Wally’s Water World as
the better choice.
Student references that
for $14 he can ride three
rides at Wally’s Water
World but only two rides
at Tony’s Tidal Takeover.

8.F.A.2

Student makes little or
no attempt to solve the
problem.
Student does not give
an explanation.

Student identifies the
number of rides at both
parks incorrectly.
Student may or may not
identify functions to
solve the problem. For
example, student uses
the table or counting
method.
Student makes some
attempt to find the
function for one or both
of the parks.
The functions used are
incorrect.

Student identifies the
number of rides at one
of the parks correctly.
Student makes some
attempt to identify the
function for one or both
of the parks.
Student may or may not
identify functions to
solve the problem. For
example, student uses
the table or counting
method.
One function used is
correct.

Student identifies that
the 25th ride at Tony’s
Tidal Takeover makes it
more expensive than
Splash. Student may
have stated that he
could ride 24 rides for
$30 at Tony’s. Student
identifies that the 12th
ride at Wally’s Water
World makes it more
expensive than Splash.
Student may have stated
that he could ride 11
rides for $30 at Wally’s.
Student identifies
functions to solve the
problem (e.g., if 𝑥𝑥 is the
number of rides,
𝑤𝑤 = 2𝑥𝑥 + 8 for the cost
of Wally’s, and
𝑡𝑡 = 0.75𝑥𝑥 + 12 for the
cost of Tony’s).

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8.F.A.2

Student makes little or
no attempt to solve the
problem.

Student may have
identified the rate of
change for each park but
does so incorrectly.
Student may not have
compared the rate of
change for each park.
Student may have
described the impact of
the rate of change on
total cost for one or two
of the parks but draws
incorrect conclusions.

Student correctly
identifies the rate of
change for each park.
Student may or may not
have compared the rate
of change for each park.
Student may have
described the impact of
the rate of change on
total cost for all parks
but makes minor
mistakes in the
description.

Student correctly
identifies the rate of
change for each park:
Wally’s is 2, Tony’s is
0.75, and Splash is 0.
Student compares the
rate of change for each
park and identifies which
park has the greatest
rate of change (or least
rate of change) as part of
the comparison.
Student describes the
impact of the rate of
change on the total cost
for each park.

8.G.C.9

Student makes little or
no attempt to solve the
problem.
Student finds none or
one of the volumes
correctly.
Student may or may
not have included
correct units.
Student may have
omitted 𝜋𝜋 from one or
more of the volumes
(i.e., the volume of the
cone is 48).

Student finds two out of
three volumes correctly.
Student may or may not
have included correct
units.
Student may have
omitted 𝜋𝜋 from one or
more of the volumes
(i.e., the volume of the
cone is 48).

Student finds all three of
the volumes correctly.
Student does not include
the correct units.
Student may have
omitted 𝜋𝜋 from one or
more of the volumes
(i.e., the volume of the
cone is 48).

Student finds all three of
the volumes correctly,
that is, the volume of the
cone is 48𝜋𝜋 mm3, the
volume of the cylinder is
21.2𝜋𝜋 cm3, and the
volume of the sphere is
36𝜋𝜋 in3.
Student includes the
correct units.

8.G.C.9

Student makes little or
no attempt to solve the
problem.

Student does not
correctly calculate the
number of cones.
Student makes
significant calculation
errors.
Student may have used
the wrong formula for
volume of the cylinder or
the cone.
Student may not have
answered in a complete
sentence.

Student may have
correctly calculated the
number of cones, but
does not correctly
calculate the volume of
the cylinder or cone
(e.g., volume of the cone
is 192, omitting the 𝜋𝜋).
Student correctly
calculates the volume of
the cone at 15𝜋𝜋 in3 or
the volume of the
cylinder at 192𝜋𝜋 in3 but
not both.
Student may have used
incorrect units.
Student may have made
minor calculation errors.
Student may not answer
in a complete sentence.

Student correctly
calculates that it will
take 12.8 cones to fill
the cylinder.
Student correctly
calculates the volume of
the cone at 15𝜋𝜋 in3 and
the volume of the
cylinder at 192𝜋𝜋 in3.
Student answers in a
complete sentence.

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8.G.C.9

Student makes little or
no attempt to solve the
problem.

Student does not
correctly calculate the
volume.
Student may have used
the diameter instead of
the radius for
calculations.
Student may have made
calculation errors.
Student may or may not
have omitted 𝜋𝜋.
Student may or may not
have included the units.

Student correctly
calculates the volume
but does not include the
units or includes
incorrect units (e.g., in2).
Student uses the radius
of 6 to calculate the
volume.
Student may have
calculated the volume as
288 (𝜋𝜋 is omitted).

Student correctly
calculates the volume as
288𝜋𝜋 in3.
Student uses the radius
of 6 to calculate the
volume.
Student includes correct
units.

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Name Date

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8
G R A D E

New York State Common Core

Mathematics Curriculum
GRADE 8 • MODULE 6

Module 6: Linear Functions

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Table of Contents1

Linear Functions
Module Overview ……………………………………………………………………………………………………………………………….. 2

Topic A: Linear Functions (8.F.B.4, 8.F.B.5) …………………………………………………………………………………………….. 7

Lesson 1: Modeling Linear Relationships ……………………………………………………………………………………. 8

Lesson 2: Interpreting Rate of Change and Initial Value ……………………………………………………………… 20

Lesson 3: Representations of a Line …………………………………………………………………………………………. 28

Lessons 4–5: Increasing and Decreasing Functions …………………………………………………………………….. 39

Topic B: Bivariate Numerical Data (8.SP.A.1, 8.SP.A.2) ………………………………………………………………………….. 67

Lesson 6: Scatter Plots ……………………………………………………………………………………………………………. 68

Lesson 7: Patterns in Scatter Plots …………………………………………………………………………………………… 79

Lesson 8: Informally Fitting a Line ……………………………………………………………………………………………. 95

Lesson 9: Determining the Equation of a Line Fit to Data ………………………………………………………….. 107

Mid-Module Assessment and Rubric …………………………………………………………………………………………………. 119
Topics A through B (assessment 1 day, return 1 day, remediation or further applications 1 day)

Topic C: Linear and Nonlinear Models (8.SP.A.1, 8.SP.A.2, 8.SP.A.3) ……………………………………………………… 129

Lesson 10: Linear Models ……………………………………………………………………………………………………… 130

Lesson 11: Using Linear Models in a Data Context ……………………………………………………………………. 142

Lesson 12: Nonlinear Models in a Data Context (Optional) ……………………………………………………….. 153

Topic D: Bivariate Categorical Data (8.SP.A.4) …………………………………………………………………………………….. 167

Lesson 13: Summarizing Bivariate Categorical Data in a Two-Way Table …………………………………….. 168

Lesson 14: Association Between Categorical Variables ……………………………………………………………… 180

End-of-Module Assessment and Rubric ……………………………………………………………………………………………… 190
Topics A through D (assessment 1 day, return 1 day, remediation or further applications 1 day)

1Each lesson is ONE day, and ONE day is considered a 45-minute period.

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8•6 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

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Grade 8 • Module 6

Linear Functions

OVERVIEW
In Grades 6 and 7, students worked with data involving a single variable. This module introduces students to
bivariate data. Students are introduced to a function as a rule that assigns exactly one value to each input. In
this module, students use their understanding of functions to model the relationships of bivariate data. This
module is important in setting a foundation for students’ work in Algebra I.

Topic A examines the relationship between two variables using linear functions (8.F.B.4). Linear functions are
connected to a context using the initial value and slope as a rate of change to interpret the context. Students
represent linear functions by using tables and graphs and by specifying rate of change and initial value. Slope
is also interpreted as an indication of whether the function is increasing or decreasing and as an indication of
the steepness of the graph of the linear function (8.F.B.5). Nonlinear functions are explored by examining
nonlinear graphs and verbal descriptions of nonlinear behavior.

In Topic B, students use linear functions to model the relationship between two quantitative variables as
students move to the domain of statistics and probability. Students make scatter plots based on data. They
also examine the patterns of their scatter plots or given scatter plots. Students assess the fit of a linear model
by judging the closeness of the data points to the line (8.SP.A.1, 8.SP.A.2).

In Topic C, students use linear and nonlinear models to answer questions in context (8.SP.A.1, 8.SP.A.2).
They interpret the rate of change and the initial value in context (8.SP.A.3). They use the equation of a linear
function and its graph to make predictions. Students also examine graphs of nonlinear functions and use
nonlinear functions to model relationships that are nonlinear. Students gain experience with the
mathematical practice of “modeling with mathematics” (MP.4).

In Topic D, students examine bivariate categorical data by using two-way tables to determine relative
frequencies. They use the relative frequencies calculated from tables to informally assess possible
associations between two categorical variables (8.SP.A.4).

Focus Standards
Use functions to model relationships between quantities.

8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the
rate of change and initial value of the function from a description of a relationship or from
two (𝑥𝑥,𝑦𝑦) values, including reading these from a table or from a graph. Interpret the rate of
change and initial value of a linear function in terms of the situation it models, and in terms of
its graph or a table of values.

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8•6 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

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8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph
(e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that
exhibits the qualitative features of a function that has been described verbally.

Investigate patterns of association in bivariate data.2

8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns
of association between two quantities. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.

8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative
variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the data points to the line.

8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement
data, interpreting the slope and intercept. For example, in a linear model for a biology
experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight
each day is associated with an additional 1.5 cm in mature plant height.

8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way table. Construct and interpret a
two-way table summarizing data on two categorical variables collected from the same
subjects. Use relative frequencies calculated for rows or columns to describe possible
association between the two variables. For example, collect data from students in your class
on whether or not they have a curfew on school nights and whether or not they have assigned
chores at home. Is there evidence that those who have a curfew also tend to have chores?

Foundational Standards
Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.

7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct
simple equations and inequalities to solve problems by reasoning about the quantities.

Define, evaluate, and compare functions.

8.F.B.1 Understand that a function is a rule that assigns to each input exactly one output. The graph
of a function is the set of ordered pairs consisting of an input and the corresponding output.3

8.F.B.2 Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a linear
function represented by a table of values and a linear function represented by an algebraic
expression, determine which function has the greater rate of change.

28.SP standards are used as applications to the work done with 8.F standards.
3Function notation is not required in Grade 8.

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8•6 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

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8.F.B.3 Interpret the equation 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 as defining a linear function, whose graph is a straight
line; give examples of functions that are not linear. For example, the function 𝐴𝐴 = 𝑠𝑠2 giving
the area of a square as a function of its side length is not linear because its graph contains the
points (1,1), (2,4) and (3,9), which are not on a straight line.

Focus Standards for Mathematical Practice
MP.2 Reason abstractly and quantitatively. Students reason quantitatively by symbolically

representing the verbal description of a relationship between two bivariate variables. They
attend to the meaning of data based on the context of problems and the possible linear or
nonlinear functions that explain the relationships of the variables.

MP.4 Model with mathematics. Students model relationships between variables using linear and
nonlinear functions. They interpret models in the context of the data and reflect on whether
or not the models make sense based on slopes, initial values, or the fit to the data.

MP.6 Attend to precision. Students evaluate functions to model a relationship between numerical
variables. They evaluate the function by assessing the closeness of the data points to the line.
They use care in interpreting the slope and the 𝑦𝑦-intercept in linear functions.

MP.7 Look for and make use of structure. Students identify pattern or structure in scatter plots.
They fit lines to data displayed in a scatter plot and determine the equations of lines based on
points or the slope and initial value.

Terminology
New or Recently Introduced Terms

 Association (description) (An association is a relationship between the two variables of a bivariate
data set.
The relationship is often expressed in terms of relative frequencies (described using two-way tables
of the two domains of variables of the data set) or numerical relationships that can be modeled by
functions (most often as linear relationships between the two domains of the two variables for the
data set).)

 Bivariate Data Set (description) (A bivariate data set is an ordered list of ordered pairs of data values
(called data points).
Data sets and bivariate data sets are both called data sets. Data values can be either numerical or
categorical. If both are numerical, then the data set is called a numerical bivariate data set.)

 Column Relative Frequency (In a two-way table, a column relative frequency is a cell frequency
divided by the column total for that cell.)

 Row Relative Frequency (In a two-way table, a row relative frequency is the number given by
dividing the cell frequency by the row total for that cell.)

 Scatter Plot (A scatter plot is a graph of a numerical bivariate data set.)

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Module 6: Linear Functions

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 Two-Way Frequency Table (description) (A two-way frequency table is a rectangular table used to
summarize data on two categorical variables of a bivariate data set. The rows of the table
correspond to the possible categories for one of the variables, and the columns correspond to the
possible categories for the other variable. Entries in the cells of the table indicate the number of
times that a particular category combination occurs in the data set; the value is the frequency for
that combination.)

 Variable (description) (A variable is a symbol (such as a letter) that is a placeholder for a data value
from a specified set of data values. The specified set of data values is called the domain of the
variable.)

Familiar Terms and Symbols4

 Categorical variable
 Intercept or initial value
 Numerical variable
 Slope

Suggested Tools and Representations
 Graphing calculator
 Scatter Plot
 Two-way frequency tables

Scatter Plot

Curfew No Curfew Total

Assigned
Chores

25 10 35

Not
Assigned
Chores

8 7 15

Total 33 17 50

Two-Way Frequency Table

4These are terms and symbols students have seen previously.

Mare Weight (kg)

Fo
al

W
ei

gh
t

(k
g)

5905805705605505405305205105000

130

120

110

100

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Module 6: Linear Functions

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Assessment Summary
Assessment Type Administered Format Standards Addressed

Mid-Module
Assessment Task After Topic B Constructed response with rubric 8.F.B.4, 8.F.B.5,

8.SP.A.1, 8.SP.A.2

End-of-Module
Assessment Task After Topic D Constructed response with rubric

8.F.B.4, 8.F.B.5,
8.SP.A.1, 8.SP.A.2,
8.SP.A.3, 8.SP.A.4

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8
G R A D E

New York State Common Core

Mathematics Curriculum
GRADE 8 • MODULE 6

Topic A: Linear Functions

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Topic A

Linear Functions

8.F.B.4, 8.F.B.5

Focus Standards: 8.F.B.4 Construct a function to model a linear relationship between two quantities.
Determine the rate of change and initial value of the function from a
description of a relationship or from two (𝑥𝑥,𝑦𝑦) values, including reading these
from a table or from a graph. Interpret the rate of change and initial value of a
linear function in terms of the situation it models, and in terms of its graph or a
table of values.

8.F.B.5 Describe qualitatively the functional relationship between two quantities by
analyzing a graph (e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a function
that has been described verbally.

Instructional Days: 5

Lesson 1: Modeling Linear Relationships (P)1

Lesson 2: Interpreting Rate of Change and Initial Value (P)

Lesson 3: Representations of a Line (P)

Lessons 4–5: Increasing and Decreasing Functions (P, P)

In Topic A, students build on their study of functions by recognizing a linear relationship between two
variables (8.F.B.4). Students use the context of a problem to construct a function to model a linear
relationship (8.F.B.4). In Lesson 1, students are given a verbal description of a linear relationship between
two variables and then must describe a linear model. Students graph linear functions using a table of values
and by plotting points. They recognize a linear function given in terms of the slope and initial value, or 𝑦𝑦-
intercept. In Lesson 2, students interpret the rate of change and the 𝑦𝑦-intercept, or initial value, in the
context of the problem. They interpret the sign of the rate of change as indicating that a linear function is
increasing or decreasing (8.F.B.5) and as indicating the steepness of a line. In Lesson 3, students graph the
line of a given linear function. They express the equation of a linear function as 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏, or an equivalent
form, when given the initial value and slope. In Lessons 4 and 5, students describe and interpret a linear
function given two points or its graph.

1Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 1

Lesson 1: Modeling Linear Relationships

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Lesson 1: Modeling Linear Relationships

Student Outcomes

 Students determine a linear function given a verbal description of a linear relationship between two quantities.
 Students interpret linear functions based on the context of a problem.

 Students sketch the graph of a linear function by constructing a table of values, plotting points, and connecting
points by a line.

Lesson Notes
In this first lesson, students construct linear functions based on verbal descriptions of bivariate data. They graph the
linear functions by creating a table of values, plotting points, and drawing the line. Throughout this lesson, provide
students with the opportunity to explain the functions in terms of the equation of the line and the relationship between
the two variables. Emphasize the context with students as they explain the rates of change and the initial values.

Classwork

Example 1 (2–3 minutes): Logging On

Read through the example as a class. Convey to students that the information presented in the example can be
organized into ordered pairs or points. Minutes can be represented by 𝑥𝑥 and cost by 𝑦𝑦.

Example 1: Logging On

Lenore has just purchased a tablet computer, and she is considering purchasing an Internet access plan so that she can
connect to the Internet wirelessly from virtually anywhere in the world. One company offers an Internet access plan so
that when a person connects to the company’s wireless network, the person is charged a fixed access fee for connecting
plus an amount for the number of minutes connected based upon a constant usage rate in dollars per minute.

Lenore is considering this company’s plan, but the company’s advertisement does not state how much the fixed access
fee for connecting is, nor does it state the usage rate. However, the company’s website says that a 𝟏𝟏𝟏𝟏-minute session
costs $𝟏𝟏.𝟒𝟒𝟏𝟏, a 𝟐𝟐𝟏𝟏-minute session costs $𝟏𝟏.𝟕𝟕𝟏𝟏, and a 𝟑𝟑𝟏𝟏-minute session costs $𝟏𝟏.𝟏𝟏𝟏𝟏. Lenore decides to use these pieces
of information to determine both the fixed access fee for connecting and the usage rate.

Exercises 1–6 (10 minutes)

This exercise set introduces students to constant rate of change and initial value and how those values are used to
construct a function to model a situation. Pose each exercise to the class, one at a time, using the following questions to
encourage discussion.

After Exercise 1, discuss as a class the need to graph this real-world problem only in the first quadrant. Begin by asking
students the following:

 If we used the entire coordinate plane to graph this line, what would the negative 𝑥𝑥 values represent?

 The 𝑥𝑥-axis represents minutes. So, time would be negative, which does not make sense in the context
of the problem.

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Lesson 1: Modeling Linear Relationships

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For Exercise 2, use the table to demonstrate constant rate of change to students.

 How could we use the table to determine the constant rate of change?

Exercises 1–6

1. Lenore makes a table of this information and a graph where number of minutes is represented by the horizontal axis
and total session cost is represented by the vertical axis. Plot the three given points on the graph. These three
points appear to lie on a line. What information about the access plan suggests that the correct model is indeed a
linear relationship?

The amount charged for the minutes connected is based upon a constant usage rate in dollars per minute.

2. The rate of change describes how the total cost changes with respect to time.

a. When the number of minutes increases by 𝟏𝟏𝟏𝟏 (e.g., from 𝟏𝟏𝟏𝟏 minutes to 𝟐𝟐𝟏𝟏 minutes or from 𝟐𝟐𝟏𝟏 minutes to
𝟑𝟑𝟏𝟏 minutes), how much does the charge increase?

When the number of minutes increases by 𝟏𝟏𝟏𝟏 (e.g., from 𝟏𝟏𝟏𝟏 minutes to 𝟐𝟐𝟏𝟏 minutes or from 𝟐𝟐𝟏𝟏 minutes to 𝟑𝟑𝟏𝟏
minutes), the cost increases by $𝟏𝟏.𝟑𝟑𝟏𝟏 (𝟑𝟑𝟏𝟏 cents).

b. Another way to say this would be the usage charge per 𝟏𝟏𝟏𝟏 minutes of use. Use that information to determine
the increase in cost based on only 𝟏𝟏 minute of additional usage. In other words, find the usage charge per
minute of use.

If $𝟏𝟏.𝟑𝟑𝟏𝟏 is the usage charge per 𝟏𝟏𝟏𝟏 minutes of use, then $𝟏𝟏.𝟏𝟏𝟑𝟑 is the usage charge per 𝟏𝟏 minute of use (i.e.,
the usage rate). Since the usage rate is constant, students should use what they have learned in Module 4.

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Lesson 1: Modeling Linear Relationships

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3. The company’s pricing plan states that the usage rate is constant for any number of minutes connected to the
Internet. In other words, the increase in cost for 𝟏𝟏𝟏𝟏 more minutes of use (the value that you calculated in Exercise
2) is the same whether you increase from 𝟐𝟐𝟏𝟏 to 𝟑𝟑𝟏𝟏 minutes, 𝟑𝟑𝟏𝟏 to 𝟒𝟒𝟏𝟏 minutes, etc. Using this information,
determine the total cost for 𝟒𝟒𝟏𝟏 minutes, 𝟓𝟓𝟏𝟏 minutes, and 𝟔𝟔𝟏𝟏 minutes of use. Record those values in the table, and
plot the corresponding points on the graph in Exercise 1.

Consider the following table and graphs.

4. Using the table and the graph in Exercise 1, compute the hypothetical cost for 𝟏𝟏 minutes of use. What does that
value represent in the context of the values that Lenore is trying to figure out?

Since there is a $𝟏𝟏.𝟑𝟑𝟏𝟏 decrease in cost for each decrease of 𝟏𝟏𝟏𝟏 minutes of use, one could subtract $𝟏𝟏.𝟑𝟑𝟏𝟏 from the
cost value for 𝟏𝟏𝟏𝟏 minutes and arrive at the hypothetical cost value for 𝟏𝟏 minutes. That cost would be $𝟏𝟏.𝟏𝟏𝟏𝟏.
Students may notice that such a value follows the regular pattern in the table and would represent the fixed access
fee for connecting. (This value could also be found from the graph after completing Exercise 6.)

Convey to students that this is known as the initial value.

 Why is this a hypothetical cost?

 Because it is impossible to connect for 0 minutes; the connection is always for some interval of time.

5. On the graph in Exercise 1, draw a line through the points representing 𝟏𝟏 to 𝟔𝟔𝟏𝟏 minutes of use under this company’s
plan. The slope of this line is equal to the constant rate of change, which in this case is the usage rate.

6. Using 𝒙𝒙 for the number of minutes and 𝒚𝒚 for the total cost in dollars, write a function to model the linear
relationship between minutes of use and total cost.

𝒚𝒚 = 𝟏𝟏.𝟏𝟏𝟑𝟑𝒙𝒙 + 𝟏𝟏.𝟏𝟏𝟏𝟏

Number of
Minutes

Total Session
Cost (in
dollars)

𝟏𝟏

𝟏𝟏𝟏𝟏 𝟏𝟏.𝟒𝟒𝟏𝟏

𝟐𝟐𝟏𝟏 𝟏𝟏.𝟕𝟕𝟏𝟏

𝟑𝟑𝟏𝟏 𝟏𝟏.𝟏𝟏𝟏𝟏

𝟒𝟒𝟏𝟏 𝟏𝟏.𝟑𝟑𝟏𝟏

𝟓𝟓𝟏𝟏 𝟏𝟏.𝟔𝟔𝟏𝟏

𝟔𝟔𝟏𝟏 𝟏𝟏.𝟗𝟗𝟏𝟏

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MP.2

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Example 2 (2–3 minutes): Another Rate Plan

Provide students time to read the example. As a whole group, summarize this alternative rate plan.

Example 2: Another Rate Plan

A second wireless access company has a similar method for computing its costs. Unlike the first company that Lenore was
considering, this second company explicitly states its access fee is $𝟏𝟏.𝟏𝟏𝟓𝟓, and its usage rate is $𝟏𝟏.𝟏𝟏𝟒𝟒 per minute.

Total Session Cost = $𝟏𝟏.𝟏𝟏𝟓𝟓+ $𝟏𝟏.𝟏𝟏𝟒𝟒 (number of minutes)

 How is this plan presented differently?
 In this case, we are given the access fee and usage rate with an equation. In the first example, just data

points were given.

 Based on the work with the first set of problems, how do you think the two plans are different?

 The values for the access fee and usage charge per minute are different, or the initial value and the rate
of change are different.

Exercises 7–9 (7 minutes)

Allow students to work independently on these exercises. After most students have completed the problems, discuss
them as a whole group.

Exercises 7–16

7. Let 𝒙𝒙 represent the number of minutes used and 𝒚𝒚 represent the total session cost in dollars. Construct a linear
function that models the total session cost based on the number of minutes used.

𝒚𝒚 = 𝟏𝟏.𝟏𝟏𝟒𝟒𝒙𝒙 + 𝟏𝟏.𝟏𝟏𝟓𝟓

8. Using the linear function constructed in Exercise 7, determine the total session cost for sessions of 𝟏𝟏, 𝟏𝟏𝟏𝟏, 𝟐𝟐𝟏𝟏, 𝟑𝟑𝟏𝟏,
𝟒𝟒𝟏𝟏, 𝟓𝟓𝟏𝟏, and 𝟔𝟔𝟏𝟏 minutes, and fill in these values in the table below.

Number of
Minutes

Total Session
Cost (in
dollars)

𝟏𝟏 𝟏𝟏.𝟏𝟏𝟓𝟓

𝟏𝟏𝟏𝟏 𝟏𝟏.𝟓𝟓𝟓𝟓

𝟐𝟐𝟏𝟏 𝟏𝟏.𝟗𝟗𝟓𝟓

𝟑𝟑𝟏𝟏 𝟏𝟏.𝟑𝟑𝟓𝟓

𝟒𝟒𝟏𝟏 𝟏𝟏.𝟕𝟕𝟓𝟓

𝟓𝟓𝟏𝟏 𝟐𝟐.𝟏𝟏𝟓𝟓

𝟔𝟔𝟏𝟏 𝟐𝟐.𝟓𝟓𝟓𝟓

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9. Plot these points on the original graph in Exercise 1, and draw a line through these points. In what ways does the
line that represents this second company’s access plan differ from the line that represents the first company’s access
plan?

The second company’s plan line begins at a greater initial value. The same plan also increases in total cost more
quickly over time; in other words, the slope of the line for the second company’s plan is steeper.

Exercises 10–12 (7 minutes)

MP3 download sites are a popular forum for selling music. Different sites offer pricing that depends on whether or not
you want to purchase an entire album or individual songs à la carte. One site offers MP3 downloads of individual songs
with the following price structure: a $𝟑𝟑 fixed fee for a monthly subscription plus a charge of $𝟏𝟏.𝟐𝟐𝟓𝟓 per song.

10. Using 𝒙𝒙 for the number of songs downloaded and 𝒚𝒚 for the total monthly cost in dollars, construct a linear function
to model the relationship between the number of songs downloaded and the total monthly cost.

Since $𝟑𝟑 is the initial cost and there is a 𝟐𝟐𝟓𝟓 cent increase per song, the function would be

𝒚𝒚 = 𝟑𝟑+ 𝟏𝟏.𝟐𝟐𝟓𝟓𝒙𝒙 or 𝒚𝒚 = 𝟏𝟏.𝟐𝟐𝟓𝟓𝒙𝒙 + 𝟑𝟑.

11. Using the linear function you wrote in Exercise 10, construct a table to record the total monthly cost (in dollars) for
MP3 downloads of 𝟏𝟏𝟏𝟏 songs, 𝟐𝟐𝟏𝟏 songs, and so on up to 𝟏𝟏𝟏𝟏𝟏𝟏 songs.

Number of Songs Total Monthly Cost
(in dollars)

𝟏𝟏𝟏𝟏 𝟓𝟓.𝟓𝟓𝟏𝟏
𝟐𝟐𝟏𝟏 𝟖𝟖.𝟏𝟏𝟏𝟏
𝟑𝟑𝟏𝟏 𝟏𝟏𝟏𝟏.𝟓𝟓𝟏𝟏
𝟒𝟒𝟏𝟏 𝟏𝟏𝟑𝟑.𝟏𝟏𝟏𝟏
𝟓𝟓𝟏𝟏 𝟏𝟏𝟓𝟓.𝟓𝟓𝟏𝟏
𝟔𝟔𝟏𝟏 𝟏𝟏𝟖𝟖.𝟏𝟏𝟏𝟏
𝟕𝟕𝟏𝟏 𝟐𝟐𝟏𝟏.𝟓𝟓𝟏𝟏
𝟖𝟖𝟏𝟏 𝟐𝟐𝟑𝟑.𝟏𝟏𝟏𝟏
𝟗𝟗𝟏𝟏 𝟐𝟐𝟓𝟓.𝟓𝟓𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏 𝟐𝟐𝟖𝟖.𝟏𝟏𝟏𝟏

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Company’s Plan
First

Company’s Plan
Second

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12. Plot the 𝟏𝟏𝟏𝟏 data points in the table on a coordinate plane. Let the 𝒙𝒙-axis represent the number of songs
downloaded and the 𝒚𝒚-axis represent the total monthly cost (in dollars) for MP3 downloads.

Exercises 13–16 (7–8 minutes)

Read through the problem as a class. The data in this exercise set are presented as two points given in context. Point
out the difference by asking students the following:

 How are the data in this problem different from the data in the MP3 problem?

 In this problem, the data can be organized as ordered pairs. In the MP3 problem, a rate of change and
initial value were given.

A band will be paid a flat fee for playing a concert. Additionally, the band will receive a fixed amount for every ticket
sold. If 𝟒𝟒𝟏𝟏 tickets are sold, the band will be paid $𝟐𝟐𝟏𝟏𝟏𝟏. If 𝟕𝟕𝟏𝟏 tickets are sold, the band will be paid $𝟐𝟐𝟔𝟔𝟏𝟏.

13. Determine the rate of change.

The points (𝟒𝟒𝟏𝟏,𝟐𝟐𝟏𝟏𝟏𝟏) and (𝟕𝟕𝟏𝟏,𝟐𝟐𝟔𝟔𝟏𝟏) have been given.

So, the rate of change is 𝟐𝟐 because
𝟐𝟐𝟔𝟔𝟏𝟏 − 𝟐𝟐𝟏𝟏𝟏𝟏
𝟕𝟕𝟏𝟏 − 𝟒𝟒𝟏𝟏

= 𝟐𝟐.

14. Let 𝒙𝒙 represent the number of tickets sold and 𝒚𝒚 represent the amount the band will be paid in dollars. Construct a
linear function to represent the relationship between the number of tickets sold and the amount the band will be
paid.

Using the rate of change and (𝟒𝟒𝟏𝟏,𝟐𝟐𝟏𝟏𝟏𝟏):

𝟐𝟐𝟏𝟏𝟏𝟏 = 𝟐𝟐(𝟒𝟒𝟏𝟏) + 𝒃𝒃
𝟐𝟐𝟏𝟏𝟏𝟏 = 𝟖𝟖𝟏𝟏 + 𝒃𝒃
𝟏𝟏𝟐𝟐𝟏𝟏 = 𝒃𝒃

Therefore, the function is 𝒚𝒚 = 𝟐𝟐𝒙𝒙 + 𝟏𝟏𝟐𝟐𝟏𝟏.

15. What flat fee will the band be paid for playing the concert regardless of the number of tickets sold?

The band will be paid a flat fee of $𝟏𝟏𝟐𝟐𝟏𝟏 for playing the concert.

16. How much will the band receive for each ticket sold?

The band receives $𝟐𝟐 per ticket.

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Closing (5 minutes)

Consider posing the following questions; allow a few student responses for each.

 In Exercise 9 when the two pricing models that Lenore was considering were both displayed on the same
graph, was there ever a point at which the second company’s model was a better, less expensive choice than
the first company’s model?

 No. The second company always had the more expensive plan; its line was always above the other
company’s line.

 When comparing the equations of the two models, which value in the second company’s model (the $0.15
access fee or $0.04 cost per minute) led you to think that it would increase at a faster rate than the first
model?
 The $0.04 cost per minute led me to believe it would increase at a faster rate. The other company’s

plan only increased at a rate of $0.03 per minute.

Exit Ticket (5 minutes)

Lesson Summary

A linear function can be used to model a linear relationship between two types of quantities. The graph of a linear
function is a straight line.

A linear function can be constructed using a rate of change and an initial value. It can be interpreted as an equation
of a line in which:

 The rate of change is the slope of the line and describes how one quantity changes with respect to
another quantity.

 The initial value is the 𝒚𝒚-intercept.

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Lesson 1: Modeling Linear Relationships

Exit Ticket

A rental car company offers a rental package for a midsize car. The cost comprises a fixed $30 administrative fee for the
cleaning and maintenance of the car plus a rental cost of $35 per day.

1. Using 𝑥𝑥 for the number of days and 𝑦𝑦 for the total cost in dollars, construct a function to model the relationship
between the number of days and the total cost of renting a midsize car.

2. The same company is advertising a deal on compact car rentals. The linear function 𝑦𝑦 = 30𝑥𝑥 + 15 can be used to

model the relationship between the number of days, 𝑥𝑥, and the total cost in dollars, 𝑦𝑦, of renting a compact car.

a. What is the fixed administrative fee?

b. What is the rental cost per day?

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Exit Ticket Sample Solutions

A rental car company offers a rental package for a midsize car. The cost comprises a fixed $𝟑𝟑𝟏𝟏 administrative fee for the
cleaning and maintenance of the car plus a rental cost of $𝟑𝟑𝟓𝟓 per day.

1. Using 𝒙𝒙 for the number of days and 𝒚𝒚 for the total cost in dollars, construct a function to model the relationship
between the number of days and the total cost of renting a midsize car.

𝒚𝒚 = 𝟑𝟑𝟓𝟓𝒙𝒙 + 𝟑𝟑𝟏𝟏

2. The same company is advertising a deal on compact car rentals. The linear function 𝒚𝒚 = 𝟑𝟑𝟏𝟏𝒙𝒙 + 𝟏𝟏𝟓𝟓 can be used to
model the relationship between the number of days, 𝒙𝒙, and the total cost in dollars, 𝒚𝒚, of renting a compact car.

a. What is the fixed administrative fee?

The administrative fee is $𝟏𝟏𝟓𝟓.

b. What is the rental cost per day?

It costs $𝟑𝟑𝟏𝟏 per day to rent the compact car.

Problem Set Sample Solutions

1. Recall that Lenore was investigating two wireless access plans. Her friend in Europe says that he uses a plan in
which he pays a monthly fee of 𝟑𝟑𝟏𝟏 euro plus 𝟏𝟏.𝟏𝟏𝟐𝟐 euro per minute of use.

a. Construct a table of values for his plan’s monthly cost based on 𝟏𝟏𝟏𝟏𝟏𝟏 minutes of use for the month, 𝟐𝟐𝟏𝟏𝟏𝟏
minutes of use, and so on up to 𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏 minutes of use. (The charge of 𝟏𝟏.𝟏𝟏𝟐𝟐 euro per minute of use is
equivalent to 𝟐𝟐 euro per 𝟏𝟏𝟏𝟏𝟏𝟏 minutes of use.)

Number of Minutes Total Monthly Cost (€)
𝟏𝟏𝟏𝟏𝟏𝟏 𝟑𝟑𝟐𝟐.𝟏𝟏𝟏𝟏
𝟐𝟐𝟏𝟏𝟏𝟏 𝟑𝟑𝟒𝟒.𝟏𝟏𝟏𝟏
𝟑𝟑𝟏𝟏𝟏𝟏 𝟑𝟑𝟔𝟔.𝟏𝟏𝟏𝟏
𝟒𝟒𝟏𝟏𝟏𝟏 𝟑𝟑𝟖𝟖.𝟏𝟏𝟏𝟏
𝟓𝟓𝟏𝟏𝟏𝟏 𝟒𝟒𝟏𝟏.𝟏𝟏𝟏𝟏
𝟔𝟔𝟏𝟏𝟏𝟏 𝟒𝟒𝟐𝟐.𝟏𝟏𝟏𝟏
𝟕𝟕𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒.𝟏𝟏𝟏𝟏
𝟖𝟖𝟏𝟏𝟏𝟏 𝟒𝟒𝟔𝟔.𝟏𝟏𝟏𝟏
𝟗𝟗𝟏𝟏𝟏𝟏 𝟒𝟒𝟖𝟖.𝟏𝟏𝟏𝟏
𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏 𝟓𝟓𝟏𝟏.𝟏𝟏𝟏𝟏

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 1

Lesson 1: Modeling Linear Relationships

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b. Plot these 𝟏𝟏𝟏𝟏 points on a carefully labeled graph, and draw the line that contains these points.

c. Let 𝒙𝒙 represent minutes of use and 𝒚𝒚 represent the total monthly cost in euro. Construct a linear function
that determines the monthly cost based on minutes of use.

𝒚𝒚 = 𝟑𝟑𝟏𝟏+ 𝟏𝟏.𝟏𝟏𝟐𝟐𝒙𝒙

d. Use the function to calculate the cost under this plan for 𝟕𝟕𝟓𝟓𝟏𝟏 minutes of use. If this point were added to the
graph, would it be above the line, below the line, or on the line?

The cost for 𝟕𝟕𝟓𝟓𝟏𝟏 minutes would be €𝟒𝟒𝟓𝟓. The point (𝟕𝟕𝟓𝟓𝟏𝟏,𝟒𝟒𝟓𝟓) would be on the line.

2. A shipping company charges a $𝟒𝟒.𝟒𝟒𝟓𝟓 handling fee in addition to $𝟏𝟏.𝟐𝟐𝟕𝟕 per pound to ship a package.

a. Using 𝒙𝒙 for the weight in pounds and 𝒚𝒚 for the cost of shipping in dollars, write a linear function that
determines the cost of shipping based on weight.

𝒚𝒚 = 𝟒𝟒.𝟒𝟒𝟓𝟓 + 𝟏𝟏.𝟐𝟐𝟕𝟕𝒙𝒙

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Lesson 1: Modeling Linear Relationships

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b. Which line (solid, dotted, or dashed) on the following graph represents the shipping company’s pricing
method? Explain.

The solid line would be the correct line. Its initial value is 𝟒𝟒.𝟒𝟒𝟓𝟓, and its slope is 𝟏𝟏.𝟐𝟐𝟕𝟕. The dashed line shows
the cost decreasing as the weight increases, so that is not correct. The dotted line starts at an initial value
that is too low.

3. Kelly wants to add new music to her MP3 player. Another subscription site offers its downloading service using the
following: 𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐌𝐌𝐓𝐓𝐌𝐌𝐓𝐓𝐌𝐌𝐓𝐓𝐌𝐌 𝐂𝐂𝐓𝐓𝐂𝐂𝐓𝐓 = 𝟓𝟓.𝟐𝟐𝟓𝟓 + 𝟏𝟏.𝟑𝟑𝟏𝟏(𝐌𝐌𝐧𝐧𝐧𝐧𝐧𝐧𝐧𝐧𝐧𝐧 𝐓𝐓𝐨𝐨 𝐂𝐂𝐓𝐓𝐌𝐌𝐬𝐬𝐂𝐂).

a. Write a sentence (all words, no math symbols) that the company could use on its website to explain how it
determines the price for MP3 downloads for the month.

“We charge a $𝟓𝟓.𝟐𝟐𝟓𝟓 subscription fee plus 𝟑𝟑𝟏𝟏 cents per song downloaded.”

b. Let 𝒙𝒙 represent the number of songs downloaded and 𝒚𝒚 represent the total monthly cost in dollars. Construct
a function to model the relationship between the number of songs downloaded and the total monthly cost.

𝒚𝒚 = 𝟓𝟓.𝟐𝟐𝟓𝟓 + 𝟏𝟏.𝟑𝟑𝟏𝟏𝒙𝒙

c. Determine the cost of downloading 𝟏𝟏𝟏𝟏 songs.

𝟓𝟓.𝟐𝟐𝟓𝟓 + 𝟏𝟏.𝟑𝟑𝟏𝟏(𝟏𝟏𝟏𝟏) = 𝟖𝟖.𝟐𝟐𝟓𝟓

The cost of downloading 𝟏𝟏𝟏𝟏 songs is $𝟖𝟖.𝟐𝟐𝟓𝟓.

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Lesson 1: Modeling Linear Relationships

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4. Li Na is saving money. Her parents gave her an amount to start, and since then she has been putting aside a fixed
amount each week. After six weeks, Li Na has a total of $𝟖𝟖𝟐𝟐 of her own savings in addition to the amount her
parents gave her. Fourteen weeks from the start of the process, Li Na has $𝟏𝟏𝟏𝟏𝟖𝟖.

a. Using 𝒙𝒙 for the number of weeks and 𝒚𝒚 for the amount in savings (in dollars), construct a linear function that
describes the relationship between the number of weeks and the amount in savings.

The points (𝟔𝟔,𝟖𝟖𝟐𝟐) and (𝟏𝟏𝟒𝟒,𝟏𝟏𝟏𝟏𝟖𝟖) have been given.

So, the rate of change is 𝟒𝟒.𝟓𝟓 because 𝟏𝟏𝟏𝟏𝟖𝟖 − 𝟖𝟖𝟐𝟐
𝟏𝟏𝟒𝟒 − 𝟔𝟔 = 𝟑𝟑𝟔𝟔

𝟖𝟖 = 𝟒𝟒.𝟓𝟓.

Using the rate of change and (𝟔𝟔,𝟖𝟖𝟐𝟐):

𝟖𝟖𝟐𝟐 = 𝟒𝟒.𝟓𝟓(𝟔𝟔) + 𝒃𝒃
𝟖𝟖𝟐𝟐 = 𝟐𝟐𝟕𝟕 + 𝒃𝒃
𝟓𝟓𝟓𝟓 = 𝒃𝒃

The function is 𝒚𝒚 = 𝟒𝟒.𝟓𝟓𝒙𝒙 + 𝟓𝟓𝟓𝟓.

b. How much did Li Na’s parents give her to start?

Li Na’s parents gave her $𝟓𝟓𝟓𝟓 to start.

c. How much does Li Na set aside each week?

Li Na is setting aside $𝟒𝟒.𝟓𝟓𝟏𝟏 every week for savings.

d. Draw the graph of the linear function below (start by plotting the points for 𝒙𝒙 = 𝟏𝟏 and 𝒙𝒙 = 𝟐𝟐𝟏𝟏).

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 2

Lesson 2: Interpreting Rate of Change and Initial Value

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Lesson 2: Interpreting Rate of Change and Initial Value

Student Outcomes

 Students interpret the constant rate of change and initial value of a line in context.
 Students interpret slope as rate of change and relate slope to the steepness of a line and the sign of the slope,

indicating that a linear function is increasing if the slope is positive and decreasing if the slope is negative.

Lesson Notes
In this lesson, students work with linear functions and the equations that define the linear functions. They specifically
interpret the slope of a linear function as a rate of change. Rate of change is used to mean constant rate of change in
the subsequent lessons. Students also explain whether the rate of change of a linear function is increasing or
decreasing. Each example in this lesson has a context. Connect students to the context of each problem by having them
summarize what they think a function indicates about the problem. For example, have students explain a slope in terms
of the units and the rate of change. Also, ask students to explain how knowing the value of one of the variables predicts
the value of the second variable.

Classwork

Linear functions are defined by the equation of a line. The graphs and the equations of the lines are important for
understanding the relationship between the two variables represented in the following example as 𝒙𝒙 and 𝒚𝒚.

Example 1 (5 minutes): Rate of Change and Initial Value

Read through the site’s pricing plan. Convey to students that the rate of change and initial value can immediately be
found when given an equation written in the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 or 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏𝑚𝑚. Pay careful attention to the
interpretation of the rate of change and initial value. Give students a moment to answer parts (a) and (b) individually.
Then, discuss as a class the interpretation of rate of change and initial value in context, and generalize the interpretation
of rate of change and initial value in contextual situations. Discuss why the sign of the rate of change affects whether or
not the linear function increases or decreases.

Example 1: Rate of Change and Initial Value

The equation of a line can be interpreted as defining a linear function. The graphs and the equations of lines are
important in understanding the relationship between two types of quantities (represented in the following examples by 𝒙𝒙
and 𝒚𝒚).

In a previous lesson, you encountered an MP3 download site that offers downloads of individual songs with the following
price structure: a $𝟑𝟑 fixed fee for a monthly subscription plus a fee of $𝟎𝟎.𝟐𝟐𝟐𝟐 per song. The linear function that models
the relationship between the number of songs downloaded and the total monthly cost of downloading songs can be
written as

𝒚𝒚 = 𝟎𝟎.𝟐𝟐𝟐𝟐𝒙𝒙 + 𝟑𝟑,

where 𝒙𝒙 represents the number of songs downloaded and 𝒚𝒚 represents the total monthly cost (in dollars) for
MP3 downloads.

MP.2

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Lesson 2: Interpreting Rate of Change and Initial Value

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a. In your own words, explain the meaning of 𝟎𝟎.𝟐𝟐𝟐𝟐 within the context of the problem.

In the example on the previous page, the value 𝟎𝟎.𝟐𝟐𝟐𝟐 means there is a cost increase of $𝟎𝟎.𝟐𝟐𝟐𝟐 for every 𝟏𝟏 song
downloaded.

b. In your own words, explain the meaning of 𝟑𝟑 within the context of the problem.

In the example on the previous page, the value of 𝟑𝟑 represents an initial cost of $𝟑𝟑 for downloading 𝟎𝟎 songs.
In other words, there is a fixed cost of $𝟑𝟑 to subscribe to the site.

The values represented in the function can be interpreted in the following way:

𝒚𝒚 = 𝟎𝟎.𝟐𝟐𝟐𝟐𝒙𝒙 + 𝟑𝟑

The coefficient of 𝒙𝒙 is referred to as the rate of
change. It can be interpreted as the change in the
values of 𝒚𝒚 for every one-unit increase in the values
of 𝒙𝒙.

When the rate of change is positive, the linear
function is increasing. In other words, increasing
indicates that as the 𝒙𝒙-value increases, so does the
𝒚𝒚-value.

When the rate of change is negative, the linear
function is decreasing. Decreasing indicates that as
the 𝒙𝒙-value increases, the 𝒚𝒚-value decreases.

The constant value is referred to as the initial value
or 𝒚𝒚-intercept and can be interpreted as the value of
𝒚𝒚 when 𝒙𝒙 = 𝟎𝟎.

Exercises 1–6 (15 minutes): Is It a Better Deal?

Discuss the other site’s pricing plan. This second plan results in a different linear function to determine pricing. Let
students work independently, and then discuss as a class the linear function and compare it to the first company that is
summarized in the lesson.

Exercises 1–6: Is It a Better Deal?

Another site offers MP3 downloads with a different price structure: a $𝟐𝟐 fixed fee for a monthly subscription plus a fee of
$𝟎𝟎.𝟒𝟒𝟎𝟎 per song.

1. Write a linear function to model the relationship between the number of songs downloaded and the total monthly
cost. As before, let 𝒙𝒙 represent the number of songs downloaded and 𝒚𝒚 represent the total monthly cost (in dollars)
of downloading songs.

𝒚𝒚 = 𝟎𝟎.𝟒𝟒𝒙𝒙 + 𝟐𝟐

2. Determine the cost of downloading 𝟎𝟎 songs and 𝟏𝟏𝟎𝟎 songs from this site.

𝒚𝒚 = 𝟎𝟎.𝟒𝟒(𝟎𝟎) + 𝟐𝟐 = 𝟐𝟐.𝟎𝟎𝟎𝟎. For 𝟎𝟎 songs, the cost is $𝟐𝟐.𝟎𝟎𝟎𝟎.

𝒚𝒚 = 𝟎𝟎.𝟒𝟒(𝟏𝟏𝟎𝟎) + 𝟐𝟐 = 𝟔𝟔.𝟎𝟎𝟎𝟎. For 𝟏𝟏𝟎𝟎 songs, the cost is $𝟔𝟔.𝟎𝟎𝟎𝟎.

rate of
change

initial value

MP.2

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Lesson 2: Interpreting Rate of Change and Initial Value

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3. The graph below already shows the linear model for the first subscription site (Company 1): 𝒚𝒚 = 𝟎𝟎.𝟐𝟐𝟐𝟐𝒙𝒙 + 𝟑𝟑. Graph
the equation of the line for the second subscription site (Company 2) by marking the two points from your work in
Exercise 2 (for 𝟎𝟎 songs and 𝟏𝟏𝟎𝟎 songs) and drawing a line through those two points.

4. Which line has a steeper slope? Which company’s model has the more expensive cost per song?

The line modeled by the second subscription site (Company 2) is steeper. It has the larger slope value and the
greater cost per song.

5. Which function has the greater initial value?

The first subscription site (Company 1) has the greater initial value. Its monthly subscription fee is $𝟑𝟑 compared to
only $𝟐𝟐 for the second site.

6. Which subscription site would you choose if you only wanted to download 𝟐𝟐 songs per month? Which company
would you choose if you wanted to download 𝟏𝟏𝟎𝟎 songs? Explain your reasoning.

For 𝟐𝟐 songs: Company 1’s cost is $𝟒𝟒.𝟐𝟐𝟐𝟐 (𝒚𝒚 = 𝟐𝟐𝟐𝟐(𝟐𝟐) + 𝟑𝟑); Company 2’s cost is $𝟒𝟒.𝟎𝟎𝟎𝟎 (𝒚𝒚 = 𝟎𝟎.𝟒𝟒(𝟐𝟐) + 𝟐𝟐). So,
Company 2 would be the better choice. Graphically, Company 2’s model also has the smaller 𝒚𝒚-value when 𝒙𝒙 = 𝟐𝟐.

For 𝟏𝟏𝟎𝟎 songs: Company 1’s cost is $𝟐𝟐.𝟐𝟐𝟎𝟎 (𝒚𝒚 = 𝟎𝟎.𝟐𝟐𝟐𝟐(𝟏𝟏𝟎𝟎) + 𝟑𝟑); Company 2’s cost is $𝟔𝟔.𝟎𝟎𝟎𝟎 (𝒚𝒚 = 𝟎𝟎.𝟒𝟒(𝟏𝟏𝟎𝟎) + 𝟐𝟐). So,
Company 1 would be the better choice. Graphically, Company 1’s model also has the smaller 𝒚𝒚-value at 𝒙𝒙 = 𝟏𝟏𝟎𝟎.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 2

Lesson 2: Interpreting Rate of Change and Initial Value

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Exercises 7–9 (10–15 minutes): Aging Autos

Let students work independently, and then discuss the answers as a class. Note the linear equation provided in Exercise
8 is written in the form 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏𝑚𝑚. Students may mix up the values for rate of change and initial value. If class time is
running short, choose two of the exercises for students to work on, and assign the other exercise of the Problem Set for
homework.

Exercises 7–9: Aging Autos

7. When someone purchases a new car and begins to drive it, the mileage (meaning the number of miles the car has
traveled) immediately increases. Let 𝒙𝒙 represent the number of years since the car was purchased and 𝒚𝒚 represent
the total miles traveled. The linear function that models the relationship between the number of years since
purchase and the total miles traveled is 𝒚𝒚 = 𝟏𝟏𝟐𝟐𝟎𝟎𝟎𝟎𝟎𝟎𝒙𝒙.

a. Identify and interpret the rate of change.

The rate of change is 𝟏𝟏𝟐𝟐,𝟎𝟎𝟎𝟎𝟎𝟎. It means that the mileage is increasing by 𝟏𝟏𝟐𝟐,𝟎𝟎𝟎𝟎𝟎𝟎 miles per year.

b. Identify and interpret the initial value.

The initial value is 𝟎𝟎. This means that there were no miles on the car when it was purchased.

c. Is the mileage increasing or decreasing each year according to the model? Explain your reasoning.

Since the rate of change is positive, it means the mileage is increasing each year.

8. When someone purchases a new car and begins to drive it, generally speaking, the resale value of the car (in dollars)
goes down each year. Let 𝒙𝒙 represent the number of years since purchase and 𝒚𝒚 represent the resale value of the
car (in dollars). The linear function that models the resale value based on the number of years since purchase is
𝒚𝒚 = 𝟐𝟐𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 − 𝟏𝟏𝟐𝟐𝟎𝟎𝟎𝟎𝒙𝒙.

a. Identify and interpret the rate of change.

The rate of change is −𝟏𝟏,𝟐𝟐𝟎𝟎𝟎𝟎. The resale value of the car is decreasing $𝟏𝟏,𝟐𝟐𝟎𝟎𝟎𝟎 every year since purchase.

b. Identify and interpret the initial value.

The initial value is $𝟐𝟐𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎. The car’s value at the time of purchase was $𝟐𝟐𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎.

c. Is the resale value increasing or decreasing each year according to the model? Explain.

The slope is negative. This means that the resale value decreases each year.

9. Suppose you are given the linear function 𝒚𝒚 = 𝟐𝟐.𝟐𝟐𝒙𝒙 + 𝟏𝟏𝟎𝟎.

a. Write a story that can be modeled by the given linear function.

Answers will vary. I am ordering cupcakes for a birthday party. The bakery is going to charge $𝟐𝟐.𝟐𝟐𝟎𝟎 per
cupcake in addition to a $𝟏𝟏𝟎𝟎 decorating fee.

b. What is the rate of change? Explain its meaning with respect to your story.

The rate of change is 𝟐𝟐.𝟐𝟐, which means that the cost increases $𝟐𝟐.𝟐𝟐𝟎𝟎 for every additional cupcake ordered.

c. What is the initial value? Explain its meaning with respect to your story.

The initial value is 𝟏𝟏𝟎𝟎, which in this story means that there is a flat fee of $𝟏𝟏𝟎𝟎 to decorate the cupcakes.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 2

Lesson 2: Interpreting Rate of Change and Initial Value

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Closing (5 minutes)

Consider posing the following questions. Allow a few student responses for each.

 In Exercise 3, for what number of songs would the total monthly cost be the same regardless of the company
selected? What visual attribute of the graph supports this answer?

 7 songs; point of intersection

It may be necessary to discuss why the answer is 7 and not 6 2
3
(the solution you would get if you solved the system

algebraically).
 Just by looking at the graph for Exercise 3, which company would you select if you had 12 songs to download?

Explain why this is the better choice.

 Company 1 has the lower cost for more than 7 songs since its linear model is below the Company 2
linear model after 7 songs.

Exit Ticket (10 minutes)

Lesson Summary

When a linear function is given by the equation of a line of the form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃, the rate of change is 𝒎𝒎, and the
initial value is 𝒃𝒃. Both are easy to identify.

The rate of change of a linear function is the slope of the line it represents. It is the change in the values of 𝒚𝒚 per a
one-unit increase in the values of 𝒙𝒙.

 A positive rate of change indicates that a linear function is increasing.

 A negative rate of change indicates that a linear function is decreasing.

 Given two lines each with positive slope, the function represented by the steeper line has a greater rate
of change.

The initial value of a linear function is the value of the 𝒚𝒚-variable when the 𝒙𝒙-value is zero.

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Lesson 2: Interpreting Rate of Change and Initial Value

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Lesson 2: Interpreting Rate of Change and Initial Value

Exit Ticket

In 2008, a collector of sports memorabilia purchased 5 specific baseball cards as an investment. Let 𝑦𝑦 represent each
card’s resale value (in dollars) and 𝑚𝑚 represent the number of years since purchase. Each card’s resale value after 0, 1, 2,
3, and 4 years could be modeled by linear equations as follows:

Card A: 𝑦𝑦 = 5 − 0.7𝑚𝑚

Card B: 𝑦𝑦 = 4 + 2.6𝑚𝑚

Card C: 𝑦𝑦 = 10 + 0.9𝑚𝑚

Card D: 𝑦𝑦 = 10 − 1.1𝑚𝑚

Card E: 𝑦𝑦 = 8 + 0.25𝑚𝑚

1. Which card(s) are decreasing in value each year? How can you tell?

2. Which card(s) had the greatest initial value at purchase (at 0 years)?

3. Which card(s) is increasing in value the fastest from year to year? How can you tell?

4. If you were to graph the equations of the resale values of Card B and Card C, which card’s graph line would be
steeper? Explain.

5. Write a sentence explaining the 0.9 value in Card C’s equation.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 2

Lesson 2: Interpreting Rate of Change and Initial Value

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Exit Ticket Sample Solutions

In 2008, a collector of sports memorabilia purchased 𝟐𝟐 specific baseball cards as an investment. Let 𝒚𝒚 represent each
card’s resale value (in dollars) and 𝒙𝒙 represent the number of years since purchase. Each card’s resale value after 𝟎𝟎, 𝟏𝟏, 𝟐𝟐,
𝟑𝟑, and 𝟒𝟒 years could be modeled by linear equations as follows:

Card A: 𝒚𝒚 = 𝟐𝟐 − 𝟎𝟎.𝟕𝟕𝒙𝒙

Card B: 𝒚𝒚 = 𝟒𝟒+ 𝟐𝟐.𝟔𝟔𝒙𝒙

Card C: 𝒚𝒚 = 𝟏𝟏𝟎𝟎+ 𝟎𝟎.𝟗𝟗𝒙𝒙

Card D: 𝒚𝒚 = 𝟏𝟏𝟎𝟎 − 𝟏𝟏.𝟏𝟏𝒙𝒙

Card E: 𝒚𝒚 = 𝟖𝟖+ 𝟎𝟎.𝟐𝟐𝟐𝟐𝒙𝒙

1. Which card(s) are decreasing in value each year? How can you tell?

Cards A and D are decreasing in value, as shown by the negative values for rate of change in each equation.

2. Which card(s) had the greatest initial value at purchase (at 𝟎𝟎 years)?

Since all of the models are in slope-intercept form, Cards C and D have the greatest initial values at $𝟏𝟏𝟎𝟎 each.

3. Which card(s) is increasing in value the fastest from year to year? How can you tell?

Card B is increasing in value the fastest from year to year. Its model has the greatest rate of change.

4. If you were to graph the equations of the resale values of Card B and Card C, which card’s graph line would be
steeper? Explain.

The Card B line would be steeper because the function for Card B has the greatest rate of change; the card’s value is
increasing at a faster rate than the other values of other cards.

5. Write a sentence explaining the 𝟎𝟎.𝟗𝟗 value in Card C’s equation.

The 𝟎𝟎.𝟗𝟗 value means that Card C’s value increases by 𝟗𝟗𝟎𝟎 cents per year.

Problem Set Sample Solutions

1. A rental car company offers the following two pricing methods for its customers to choose from for a
one-month rental:

Method 1: Pay $𝟒𝟒𝟎𝟎𝟎𝟎 for the month, or

Method 2: Pay $𝟎𝟎.𝟑𝟑𝟎𝟎 per mile plus a standard maintenance fee of $𝟑𝟑𝟐𝟐.

a. Construct a linear function that models the relationship between the miles driven and the total rental cost for
Method 2. Let 𝒙𝒙 represent the number of miles driven and 𝒚𝒚 represent the rental cost (in dollars).

𝒚𝒚 = 𝟑𝟑𝟐𝟐+ 𝟎𝟎.𝟑𝟑𝟎𝟎𝒙𝒙

b. If you plan to drive 𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 miles for the month, which method would you choose? Explain your reasoning.

Method 1 has a flat rate of $𝟒𝟒𝟎𝟎𝟎𝟎 regardless of miles. Using Method 2, the cost would be $𝟑𝟑𝟔𝟔𝟐𝟐
(𝒚𝒚 = 𝟑𝟑𝟐𝟐 + 𝟎𝟎.𝟑𝟑(𝟏𝟏𝟏𝟏𝟎𝟎𝟎𝟎)). So, Method 2 would be preferred.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 2

Lesson 2: Interpreting Rate of Change and Initial Value

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2. Recall from a previous lesson that Kelly wants to add new music to her MP3 player. She was interested in a monthly
subscription site that offered its MP3 downloading service for a monthly subscription fee plus a fee per song. The
linear function that modeled the total monthly cost in dollars (𝒚𝒚) based on the number of songs downloaded (𝒙𝒙) is
𝒚𝒚 = 𝟐𝟐.𝟐𝟐𝟐𝟐 + 𝟎𝟎.𝟑𝟑𝟎𝟎𝒙𝒙.

The site has suddenly changed its monthly price structure. The linear function that models the new total monthly
cost in dollars (𝒚𝒚) based on the number of songs downloaded (𝒙𝒙) is 𝒚𝒚 = 𝟎𝟎.𝟑𝟑𝟐𝟐𝒙𝒙 + 𝟒𝟒.𝟐𝟐𝟎𝟎.

a. Explain the meaning of the value 𝟒𝟒.𝟐𝟐𝟎𝟎 in the new equation. Is this a better situation for Kelly than before?

The initial value is 𝟒𝟒.𝟐𝟐𝟎𝟎 and means that the monthly subscription cost is now $𝟒𝟒.𝟐𝟐𝟎𝟎. This is lower than
before, which is good for Kelly.

b. Explain the meaning of the value 𝟎𝟎.𝟑𝟑𝟐𝟐 in the new equation. Is this a better situation for Kelly than before?

The rate of change is 𝟎𝟎.𝟑𝟑𝟐𝟐. This means that the cost is increasing by $𝟎𝟎.𝟑𝟑𝟐𝟐 for every song downloaded. This
is more than the download cost for the original plan.

c. If you were to graph the two equations (old versus new), which line would have the steeper slope? What
does this mean in the context of the problem?

The slope of the new line is steeper because the new linear function has a greater rate of change. It means
that the total monthly cost of the new plan is increasing at a faster rate per song compared to the cost of the
old plan.

d. Which subscription plan provides the better value if Kelly downloads fewer than 𝟏𝟏𝟐𝟐 songs per month?

If Kelly were to download 𝟏𝟏𝟐𝟐 songs, both plans will cost the same ($𝟗𝟗.𝟕𝟕𝟐𝟐). Therefore, the new plan is
cheaper if Kelly downloads fewer than 𝟏𝟏𝟐𝟐 songs.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 3

Lesson 3: Representations of a Line

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Lesson 3: Representations of a Line

Student Outcomes

 Students graph a line specified by a linear function.
 Students graph a line specified by an initial value and a rate of change of a function and construct the linear

function by interpreting the graph.

 Students graph a line specified by two points of a linear relationship and provide the linear function.

Lesson Notes
Linear functions are defined by the equations of a line. This lesson reviews students’ work with the representation of a
line and, in particular, the determination of the rate of change and the initial value of a linear function from two points
on the graph or from the equation of the line defined by the function in the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 or an equivalent form.
Students interpret the rate of change and the initial value based on the graph of the equation of the line in addition to
the context of the variables.

Classwork

Example 1 (10 minutes): Rate of Change and Initial Value Given in the Context of the Problem

Here, verbal information giving an initial value and a rate of change is translated into a function and its graph. Work
through the example as a class.

In part (b), explain why the value 0.5 given in the question is the rate of change.

• It is a good idea to show this on the graph, demonstrating that each increase of 1 unit for 𝑚𝑚 (miles) results in an
increase of 0.5 for 𝐶𝐶 (cost in dollars). An increase of 1,000 for 𝑚𝑚 results in an increase of 500 units for 𝐶𝐶.

• Point out that if the question stated that each mile driven reduced the cost by $0.50, then the line would have a
negative slope.

It is important for students to understand that when the scales on the two axes are different, the rate of change cannot
be used to plot points by simply counting the squares. Encourage students to use the rate of change by holding on to
the idea of increasing the variable shown on the horizontal axis and showing the resulting increase in the variable shown
on the vertical axis (as explained in the previous paragraph).

Given the rate of change and initial value, the linear function can be written in slope-intercept form (𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏) or an
equivalent form such as 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏𝑚𝑚. Students should pay careful attention to variables presented in the problem; 𝑚𝑚
and 𝐶𝐶 are used in place of 𝑚𝑚 and 𝑦𝑦.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 3

Lesson 3: Representations of a Line

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Example 1: Rate of Change and Initial Value Given in the Context of the Problem

A truck rental company charges a $𝟏𝟏𝟏𝟏𝟏𝟏 rental fee in addition to a charge of $𝟏𝟏.𝟏𝟏𝟏𝟏 per mile driven. Graph the linear
function relating the total cost of the rental in dollars, 𝑪𝑪, to the number of miles driven, 𝒎𝒎, on the axes below.

a. If the truck is driven 𝟏𝟏 miles, what is the cost to the customer? How is this shown on the graph?

$𝟏𝟏𝟏𝟏𝟏𝟏, shown as the point (𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏). This is the initial value. Some students might say “𝒃𝒃.” Help them to use
the term initial value.

b. What is the rate of change that relates cost to number of miles driven? Explain what it means within the
context of the problem.

The rate of change is 𝟏𝟏.𝟏𝟏. It means that the cost increases by $𝟏𝟏.𝟏𝟏𝟏𝟏 for every mile driven.

c. On the axes given, sketch the graph of the linear function that relates 𝑪𝑪 to 𝒎𝒎.

Students can plot the initial value (𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏) and then use the rate of change to identify additional points as
needed. A 𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏-unit increase in 𝒎𝒎 results in a 𝟏𝟏𝟏𝟏𝟏𝟏-unit increase for 𝑪𝑪, so another point on the line is
(𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏,𝟔𝟔𝟏𝟏𝟏𝟏).

d. Write the equation of the linear function that models the relationship between number of miles driven and
total rental cost.

𝑪𝑪 = 𝟏𝟏.𝟏𝟏𝒎𝒎 + 𝟏𝟏𝟏𝟏𝟏𝟏

Exercises 1–5 (10 minutes)

Here, students have an opportunity to practice the ideas to which they have just been introduced. Let students work
independently on these exercises. Then, discuss and confirm answers as a class.

Exercise 3, part (c), provides an excellent opportunity for discussion about the model and whether or not it continues to
make sense over time.

 In Exercise 3, you found that the price of the car in year seven was less than $600. Does this make sense in
general?

 Not really
 Under what conditions might the car be worth less than $600 after seven years?

 The car may have been in an accident or damaged.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 3

Lesson 3: Representations of a Line

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Exercises

Jenna bought a used car for $𝟏𝟏𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏. She has been told that the value of the car is likely to decrease by $𝟐𝟐,𝟏𝟏𝟏𝟏𝟏𝟏 for each
year that she owns the car. Let the value of the car in dollars be 𝑽𝑽 and the number of years Jenna has owned the car be 𝒕𝒕.

1. What is the value of the car when 𝒕𝒕 = 𝟏𝟏? Show this point on the graph.

$𝟏𝟏𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏. Shown by the point (𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏)

2. What is the rate of change that relates 𝑽𝑽 to 𝒕𝒕? (Hint: Is it positive or negative? How can you tell?)

−𝟐𝟐,𝟏𝟏𝟏𝟏𝟏𝟏. The rate of change is negative because the value of the car is decreasing.

3. Find the value of the car when:

a. 𝒕𝒕 = 𝟏𝟏

$𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − $𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏 = $𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

b. 𝒕𝒕 = 𝟐𝟐

$𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟐𝟐($𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏) = $𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

c. 𝒕𝒕 = 𝟕𝟕

$𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟕𝟕($𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏) = $𝟏𝟏𝟏𝟏𝟏𝟏

4. Plot the points for the values you found in Exercise 3, and draw the line (using a straightedge) that passes through
those points.

See the graph above.

5. Write the linear function that models the relationship between the number of years Jenna has owned the car and
the value of the car.

𝑽𝑽 = 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏𝒕𝒕 or 𝑽𝑽 = −𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏𝒕𝒕+ 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

Exercises 6–10 (10 minutes)

Here, in the context of the pricing of a book, students are given two points on the graph of a linear equation and are
expected to draw the graph, determine the rate of change, and answer questions by referring to the graph.

MP.7

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 3

Lesson 3: Representations of a Line

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Point out that the horizontal axis does not start at 0. Ask students the following question:

 Why do you think the first value is at 15?

 The online bookseller may not sell the book for less than $15.

In Exercise 8, students are asked to find the rate of change; it might be worthwhile to check that they are using the
scales on the axes, not purely counting squares.

For Exercises 9 and 10, encourage students to show their work by drawing vertical and horizontal lines on the graph, as
shown in the sample student answers below.

Let students work with a partner. Then, discuss and confirm answers as a class.

An online bookseller has a new book in print. The company estimates that if the book is priced at $𝟏𝟏𝟏𝟏, then 𝟏𝟏𝟏𝟏𝟏𝟏 copies
of the book will be sold per day, and if the book is priced at $𝟐𝟐𝟏𝟏, then 𝟏𝟏𝟏𝟏𝟏𝟏 copies of the book will be sold per day.

6. Identify the ordered pairs given in the problem. Then, plot both on the graph.

The ordered pairs are (𝟏𝟏𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏) and (𝟐𝟐𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏). See the graph above.

7. Assume that the relationship between the number of books sold and the price is linear. (In other words, assume
that the graph is a straight line.) Using a straightedge, draw the line that passes through the two points.

See the graph above.

8. What is the rate of change relating number of copies sold to price?

Between the points (𝟏𝟏𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏) and (𝟐𝟐𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏), the run is 𝟏𝟏, and the rise is −(𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟏𝟏𝟏𝟏𝟏𝟏) = −𝟐𝟐𝟏𝟏𝟏𝟏. So, the rate of

change is −𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏 = −𝟏𝟏𝟏𝟏.

9. Based on the graph, if the company prices the book at $𝟏𝟏𝟏𝟏, about how many copies of the book can they expect to
sell per day?

𝟔𝟔𝟏𝟏𝟏𝟏

10. Based on the graph, approximately what price should the company charge in order to sell 𝟕𝟕𝟏𝟏𝟏𝟏 copies of the book
per day?

$𝟏𝟏𝟕𝟕

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Lesson 3: Representations of a Line

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Closing (5 minutes)

If time allows, consider posing the following questions:

 How would you interpret the meaning of the rate of change (−50) from Exercise 8?
 Answers will vary; pay careful attention to wording. The number of copies sold would decrease by 50

units as the price increased by $1, or for every dollar increase in the price, the number of copies sold
would decrease by 50 units.

 Does it seem reasonable that the number of copies sold would decrease with respect to an increase in price?

 Yes, if the book is really expensive, someone may not want to buy it. If the cost remains low, it seems
reasonable that more people would want to buy it.

 How is the information given in the truck rental problem different from the information given in the book-
pricing problem?

 In the book pricing problem, the information is given as ordered pairs. In the truck rental problem, the
information is given in the form of a slope and an initial value.

Exit Ticket (10 minutes)

Lesson Summary

When the rate of change, 𝒃𝒃, and an initial value, 𝒂𝒂, are given in the context of a problem, the linear function that
models the situation is given by the equation 𝒚𝒚 = 𝒂𝒂 + 𝒃𝒃𝒃𝒃.

The rate of change and initial value can also be used to sketch the graph of the linear function that models the
situation.

When two or more ordered pairs are given in the context of a problem that involves a linear relationship, the graph
of the linear function is the line that passes through those points. The linear function can be represented by the
equation of that line.

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Lesson 3: Representations of a Line

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Name ___________________________________________________ Date____________________

Lesson 3: Representations of a Line

Exit Ticket

1. A car starts a journey with 18 gallons of fuel. Assuming a constant rate, the car consumes 0.04 gallon for every mile

driven. Let 𝐴𝐴 represent the amount of gas in the tank (in gallons) and 𝑚𝑚 represent the number of miles driven.

a. How much gas is in the tank if 0 miles have been driven? How would this be represented on the axes above?

b. What is the rate of change that relates the amount of gas in the tank to the number of miles driven? Explain
what it means within the context of the problem.

c. On the axes above, draw the line that represents the graph of the linear function that relates 𝐴𝐴 to 𝑚𝑚.

d. Write the linear function that models the relationship between the number of miles driven and the amount of
gas in the tank.

Number of Miles

Am
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o

f G
as

in
G

al
lo

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2. Andrew works in a restaurant. The graph below shows the relationship between the amount Andrew earns in
dollars and the number of hours he works.

a. If Andrew works for 7 hours, approximately how much does he earn in dollars?

b. Estimate how long Andrew has to work in order to earn $64.

c. What is the rate of change of the function given by the graph? Interpret the value within the context of the
problem.

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Exit Ticket Sample Solutions

1. A car starts a journey with 𝟏𝟏𝟏𝟏 gallons of fuel. Assuming a constant rate, the car consumes 𝟏𝟏.𝟏𝟏𝟎𝟎 gallon for every
mile driven. Let 𝑨𝑨 represent the amount of gas in the tank (in gallons) and 𝒎𝒎 represent the number of miles driven.

a. How much gas is in the tank if 𝟏𝟏 miles have been driven? How would this be represented on the axes above?

There are 𝟏𝟏𝟏𝟏 gallons in the tank. This would be represented as (𝟏𝟏,𝟏𝟏𝟏𝟏), the initial value, on the graph above.

b. What is the rate of change that relates the amount of gas in the tank to the number of miles driven? Explain
what it means within the context of the problem.

−𝟏𝟏.𝟏𝟏𝟎𝟎; the car consumes 𝟏𝟏.𝟏𝟏𝟎𝟎 gallon for every mile driven. It relates the amount of fuel to the miles driven.

c. On the axes above, draw the line that represents the graph of the linear function that relates 𝑨𝑨 to 𝒎𝒎.

See the graph above. Students can plot the initial value (𝟏𝟏,𝟏𝟏𝟏𝟏) and then use the rate of change to identify
additional points as needed. A 𝟏𝟏𝟏𝟏-unit increase in 𝒎𝒎 results in a 𝟐𝟐-unit decrease for 𝑨𝑨, so another point on
the line is (𝟏𝟏𝟏𝟏,𝟏𝟏𝟔𝟔).

d. Write the linear function that models the relationship between the number of miles driven and the amount
of gas in the tank.

𝑨𝑨 = 𝟏𝟏𝟏𝟏 − 𝟏𝟏.𝟏𝟏𝟎𝟎𝒎𝒎 or 𝑨𝑨 = −𝟏𝟏.𝟏𝟏𝟎𝟎𝒎𝒎 + 𝟏𝟏𝟏𝟏

2. Andrew works in a restaurant. The graph below shows the relationship between the amount Andrew earns in
dollars and the number of hours he works.

Am
ou

nt
o

f G
as

in
G

al
lo

Number of Miles

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 3

Lesson 3: Representations of a Line

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a. If Andrew works for 𝟕𝟕 hours, approximately how much does he earn in dollars?

$𝟗𝟗𝟔𝟔

b. Estimate how long Andrew has to work in order to earn $𝟔𝟔𝟎𝟎.

𝟏𝟏 hours

c. What is the rate of change of the function given by the graph? Interpret the value within the context of the
problem.

Using the ordered pairs (𝟕𝟕,𝟗𝟗𝟔𝟔) and (𝟏𝟏,𝟔𝟔𝟎𝟎), the slope is 𝟏𝟏. It means that the amount Andrew earns increases
by $𝟏𝟏 for every hour worked.

Problem Set Sample Solutions

1. A plumbing company charges a service fee of $𝟏𝟏𝟐𝟐𝟏𝟏, plus $𝟎𝟎𝟏𝟏 for each hour worked. Sketch the graph of the linear
function relating the cost to the customer (in dollars), 𝑪𝑪, to the time worked by the plumber (in hours), 𝒕𝒕, on the
axes below.

a. If the plumber works for 𝟏𝟏 hours, what is the cost to the customer? How is this shown on the graph?

$𝟏𝟏𝟐𝟐𝟏𝟏 This is shown on the graph by the point (𝟏𝟏,𝟏𝟏𝟐𝟐𝟏𝟏).

b. What is the rate of change that relates cost to time?

𝟎𝟎𝟏𝟏

c. Write a linear function that models the relationship between the hours worked and the cost to the customer.

𝑪𝑪 = 𝟎𝟎𝟏𝟏𝒕𝒕+ 𝟏𝟏𝟐𝟐𝟏𝟏

d. Find the cost to the customer if the plumber works for each of the following number of hours.

i. 𝟏𝟏 hour

$𝟏𝟏𝟔𝟔𝟏𝟏

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Lesson 3: Representations of a Line

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ii. 𝟐𝟐 hours

$𝟐𝟐𝟏𝟏𝟏𝟏

iii. 𝟔𝟔 hours

$𝟏𝟏𝟔𝟔𝟏𝟏

e. Plot the points for these times on the coordinate plane, and use a straightedge to draw the line through the
points.

See the graph on the previous page.

2. An author has been paid a writer’s fee of $𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏 plus $𝟏𝟏.𝟏𝟏𝟏𝟏 for every copy of the book that is sold.

a. Sketch the graph of the linear function that relates the total amount of money earned in dollars, 𝑨𝑨, to the
number of books sold, 𝒏𝒏, on the axes below.

b. What is the rate of change that relates the total amount of money earned to the number of books sold?

𝟏𝟏.𝟏𝟏

c. What is the initial value of the linear function based on the graph?

𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏

d. Let the number of books sold be 𝒏𝒏 and the total amount earned be 𝑨𝑨. Construct a linear function that models
the relationship between the number of books sold and the total amount earned.

𝑨𝑨 = 𝟏𝟏.𝟏𝟏𝒏𝒏 + 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

Number of Books Sold

To
ta

l A
m

ou
nt

o
f M

on
ey

E
ar

ne
d

(in
d

ol
la

rs
)

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 3

Lesson 3: Representations of a Line

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3. Suppose that the price of gasoline has been falling. At the beginning of last month (𝒕𝒕 = 𝟏𝟏), the price was $𝟎𝟎.𝟔𝟔𝟏𝟏 per
gallon. Twenty days later (𝒕𝒕 = 𝟐𝟐𝟏𝟏), the price was $𝟎𝟎.𝟐𝟐𝟏𝟏 per gallon. Assume that the price per gallon, 𝑷𝑷, fell at a
constant rate over the twenty days.

a. Identify the ordered pairs given in the problem. Plot both points on the coordinate plane above.

(𝟏𝟏,𝟎𝟎.𝟔𝟔𝟏𝟏) and (𝟐𝟐𝟏𝟏,𝟎𝟎.𝟐𝟐𝟏𝟏); see the graph above.

b. Using a straightedge, draw the line that contains the two points.

See the graph above.

c. What is the rate of change? What does it mean within the context of the problem?

Using points (𝟏𝟏,𝟎𝟎.𝟔𝟔𝟏𝟏) and (𝟐𝟐𝟏𝟏,𝟎𝟎.𝟐𝟐𝟏𝟏), the rate of change is −𝟏𝟏.𝟏𝟏𝟐𝟐 because
𝟎𝟎.𝟐𝟐𝟏𝟏 − 𝟎𝟎.𝟔𝟔𝟏𝟏
𝟐𝟐𝟏𝟏 − 𝟏𝟏

=
−𝟏𝟏.𝟎𝟎
𝟐𝟐𝟏𝟏

= −𝟏𝟏. 𝟏𝟏𝟐𝟐.

The price of gas is decreasing $𝟏𝟏.𝟏𝟏𝟐𝟐 each day.

d. What is the function that models the relationship between the number of days and the price per gallon?

𝑷𝑷 = −𝟏𝟏.𝟏𝟏𝟐𝟐𝒕𝒕+ 𝟎𝟎.𝟔𝟔

e. What was the price of gasoline after 𝟗𝟗 days?

$𝟎𝟎.𝟎𝟎𝟐𝟐; see the graph above.

f. After how many days was the price $𝟎𝟎.𝟏𝟏𝟐𝟐?

𝟏𝟏𝟎𝟎 days; see the graph above.

Time in Days

Pr
ic

e
pe

r G
al

lo
n

in
D

ol
la

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 4

Lesson 4: Increasing and Decreasing Functions

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Lesson 4: Increasing and Decreasing Functions

Student Outcomes

 Students describe qualitatively the functional relationship between two types of quantities by analyzing a
graph.

 Students sketch a graph that exhibits the qualitative features of a function based on a verbal description.

Lesson Notes
This lesson focuses on graphs and the role they play in analyzing functional relationships between quantities. Students
have been introduced to increasing and decreasing functions in a prior lesson in Grade 8. This lesson references a
constant function, one in which the graph of the function is a line with zero slope. Piecewise functions are also used
throughout the lesson to demonstrate how the functional relationship can increase or decrease between different
intervals. Rate of change should be discussed among the intervals, but the term piecewise function does not need to be
defined. This lesson also focuses on linear relationships. Nonlinear examples are presented in the next lesson.

Classwork

Opening

Graphs are useful tools in terms of representing data. They provide a visual story, highlighting important facts that
surround the relationship between quantities.

The graph of a linear function is a line. The slope of the line can provide useful information about the functional
relationship between the two types of quantities:

 A linear function whose graph has a positive slope is said to be an increasing function.

 A linear function whose graph has a negative slope is said to be a decreasing function.

 A linear function whose graph has a zero slope is said to be a constant function.

Exercise 1 (7–9 minutes)

Read through the opening text with students. Remind students that knowing the slope of the line that represents the
function tells them if the function is increasing or decreasing. Introduce the term constant function. Present examples
of functions that are constant; for example, your cell phone bill is $79 every month for unlimited calls and data. Let
students work independently on Exercise 1; then, discuss and confirm answers as a class.

Exercises

1. Read through each of the scenarios, and choose the graph of the function that best matches the situation. Explain
the reason behind each choice.

a. A bathtub is filled at a constant rate of 𝟏𝟏.𝟕𝟕𝟕𝟕 gallons per minute.

b. A bathtub is drained at a constant rate of 𝟐𝟐.𝟕𝟕 gallons per minute.

c. A bathtub contains 𝟐𝟐.𝟕𝟕 gallons of water.

d. A bathtub is filled at a constant rate of 𝟐𝟐.𝟕𝟕 gallons per minute.

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Scenario: c

Explanation: The amount of water
in the tub does not change over
time; it remains constant at 𝟐𝟐.𝟕𝟕
gallons.

Scenario: b

Explanation: The bathtub is being
drained at a constant rate of 𝟐𝟐.𝟕𝟕
gallons per minute. So, the amount
of water is decreasing, which means
that the slope of the line is negative.
The graph of the function also shows
that there are initially 𝟐𝟐𝟐𝟐 gallons of
water in the tub.

Scenario: d

Explanation: The tub is being filled
at a constant rate of 𝟐𝟐.𝟕𝟕 gallons per
minute, which implies that the
amount of water in the tub is
increasing, so the line has a positive
slope. Based on the graph, the
amount of water is also increasing
at a faster rate than in choice (a).

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Scenario: a

Explanation: The bathtub is being
filled at a constant rate of 𝟏𝟏.𝟕𝟕𝟕𝟕
gallons per minute, so the amount
of water is increasing. This implies
that the slope should be positive;
however, the rate at which the tub is
being filled is less than the rate for
choice (d).

Exercise 2 (8–10 minutes)

In this exercise, students sketch a graph of a functional relationship given a verbal description. Allow students to work
with a partner, and then confirm answers as a class. Refer to the functions as increasing or decreasing when discussing
answers.

Students may misinterpret the meaning of flat rate in part (a). Discuss the meaning as a class. Tell students that it could
also be called a flat fee.

After students have graphed the scenario presented in part (b), consider generating another graph where “meters under
water” is represented using negative numbers. This provides an opportunity for students to see a real-world scenario
with a negative slope graphed in the second quadrant. Ask students if both graphs model the same situation.

2. Read through each of the scenarios, and sketch a graph of a function that models the situation.

a. A messenger service charges a flat rate of $𝟒𝟒.𝟗𝟗𝟕𝟕 to deliver a package regardless of the distance to the
destination.

The delivery charge remains constant regardless of the distance to the destination.

MP.2

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b. At sea level, the air that surrounds us presses down on our bodies at 𝟏𝟏𝟒𝟒.𝟕𝟕 pounds per square inch (𝐩𝐩𝐩𝐩𝐩𝐩). For
every 𝟏𝟏𝟐𝟐 meters that you dive under water, the pressure increases by 𝟏𝟏𝟒𝟒.𝟕𝟕 𝐩𝐩𝐩𝐩𝐩𝐩.

The initial value is 𝟏𝟏𝟒𝟒.𝟕𝟕 𝐩𝐩𝐩𝐩𝐩𝐩. The function increases at a rate of 𝟏𝟏𝟒𝟒.𝟕𝟕 𝐩𝐩𝐩𝐩𝐩𝐩 for every 𝟏𝟏𝟐𝟐 meters, or 𝟏𝟏.𝟒𝟒𝟕𝟕 𝐩𝐩𝐩𝐩𝐩𝐩
per meter.

c. The range (driving distance per charge) of an electric car varies based on the average speed the car is driven.
The initial range of the electric car after a full charge is 𝟒𝟒𝟐𝟐𝟐𝟐 miles. However, the range is reduced by 𝟐𝟐𝟐𝟐 miles
for every 𝟏𝟏𝟐𝟐 𝐦𝐦𝐩𝐩𝐦𝐦 increase in average speed the car is driven.

The initial value of the function is 𝟒𝟒𝟐𝟐𝟐𝟐. The function is decreasing by 𝟐𝟐𝟐𝟐 miles for every 𝟏𝟏𝟐𝟐 𝐦𝐦𝐩𝐩𝐦𝐦 increase in
speed. In other words, the function decreases by 𝟐𝟐 miles for every 𝟏𝟏 𝐦𝐦𝐩𝐩𝐦𝐦 increase in speed.

Exercise 3 (7–9 minutes)

Graphs of piecewise functions are introduced in this exercise. Students match verbal descriptions to a given graph.
Let students work with a partner. Then, discuss and confirm answers as a class.

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3. The graph below represents the total number of smartphones that are shipped to a retail store over the course of
𝟕𝟕𝟐𝟐 days.

Match each part of the graph (A, B, and C) to its verbal description. Explain the reasoning behind your choice.

i. Half of the factory workers went on strike, and not enough smartphones were produced for normal
shipments.

C; if half of the workers went on strike, then the number of smartphones produced would be less than
normal. The rate of change for C is less than the rate of change for A.

ii. The production schedule was normal, and smartphones were shipped to the retail store at a constant
rate.

A; if the production schedule is normal, the rate of change of interval A is greater than the rate of
change of interval C.

iii. A defective electronic chip was found, and the factory had to shut down, so no smartphones were
shipped.

B; if no smartphones are shipped to the store, the total number remains constant during that time.

Exercise 4 (10–12 minutes)

Let students work in small groups to create a story around the function represented by the graph. Then, compare
stories as a class. Consider asking the following questions to connect the graph of the function to real-world experiences
before groups begin writing their stories.

 What reason might explain why the account balance increases between Days 6 and 9 and then decreases
between Days 9 and 14?

 Answers will vary. Maybe the person holding the account earned $15 each day mowing lawns and
deposited the money each day to his account. Then, the same person needed to debit his account $6
each day to pay for lunch.

 What reason might explain why the account balance does not change during the first few days?
 Answers will vary. Jameson is sick and cannot work to earn money to deposit into his account.

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4. The relationship between Jameson’s account balance and time is modeled by the graph below.

a. Write a story that models the situation represented by the graph.

Answers will vary.

Jameson was sick and did not work for almost a whole week. Then, he mowed several lawns over the next
few days and deposited the money into his account after each job. It rained several days, so instead of
working, Jameson withdrew money from his account each day to go to the movies and out to lunch with
friends.

b. When is the function represented by the graph increasing? How does this relate to your story?

It is increasing between 𝟔𝟔 and 𝟗𝟗 days. Jameson earned money mowing lawns and made a deposit to his
account each day. The money earned for each day was constant for these days. This is represented by a
straight line.

c. When is the function represented by the graph decreasing? How does this relate to your story?

It is decreasing between 𝟗𝟗 and 𝟏𝟏𝟒𝟒 days. Since Days 𝟗𝟗–𝟏𝟏𝟒𝟒 are represented by a straight line, this means that
Jameson spent the money constantly over these days. Jameson cannot work because it is raining. Perhaps he
withdraws money from his account to spend on different activities each day because he cannot work.

Closing (3–4 minutes)

Review the Lesson Summary with the class.

 Refer back to Exercise 1. In parts (a) and (d), the bathtub was being filled at a constant rate. Is it reasonable
within the context of the problem for the function in the graph to continue increasing?
 No. At some point, the tub will be full, and the amount of water cannot continue to increase.

 Refer back to Exercise 2, part (b). The amount of pressure that an underwater diver experiences continues to
increase as the diver continues to descend. Is it reasonable within the context of the problem for the function
in the graph to continue increasing?

 No. At some point, the pressure will be too great, and the diver will not be able to descend any farther.
 Is there a scenario that would require a function that modeled the situation to increase indefinitely? Explain.

 Yes. Students may use the example of money left in a savings account.

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Lesson 4: Increasing and Decreasing Functions

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It may need to be pointed out that this scenario is not necessarily linear, but if no money is withdrawn, the total will
continue to increase.

Exit Ticket (8 minutes)

Lesson Summary

The graph of a function can be used to help describe the relationship between two types of quantities.

The slope of the line can provide useful information about the functional relationship between the quantities
represented by the line:

 A function whose graph has a positive slope is said to be an increasing function.

 A function whose graph has a negative slope is said to be a decreasing function.

 A function whose graph has a zero slope is said to be a constant function.

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Lesson 4: Increasing and Decreasing Functions

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Name ___________________________________________________ Date____________________

Lesson 4: Increasing and Decreasing Functions

Exit Ticket

1. The graph below shows the relationship between a car’s value and time.

Match each part of the graph (A, B, and C) to its verbal description. Explain the reasoning behind your choice.

i. The value of the car holds steady due to a positive consumer report on the same model.

ii. There is a shortage of used cars on the market, and the value of the car rises at a constant rate.

iii. The value of the car depreciates at a constant rate.

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2. Henry and Roxy both drive electric cars that need to be recharged before use. Henry uses a standard charger at his
home to recharge his car. The graph below represents the relationship between the battery charge and the amount
of time it has been connected to the power source for Henry’s car.

a. Describe how Henry’s car battery is being recharged with respect to time.

b. Roxy has a supercharger at her home that can charge about half of the battery in 20 minutes. There is no

remaining charge left when she begins recharging the battery. Sketch a graph that represents the relationship
between the battery charge and the amount of time on the axes above. Assume the relationship is linear.

c. Which person’s car will be recharged to full capacity first? Explain.

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Exit Ticket Sample Solutions

1. The graph below shows the relationship between a car’s value and time.

Match each part of the graph (A, B, and C) to its verbal description. Explain the reasoning behind your choice.

i. The value of the car holds steady due to a positive consumer report on the same model.

B; if the value is holding steady, there is no change in the car’s value between years.

ii. There is a shortage of used cars on the market, and the value of the car rises at a constant rate.

C; if the value of the car is rising, it represents an increasing function.

iii. The value of the car depreciates at a constant rate.

A; if the value depreciates, it represents a decreasing function.

a. Describe how Henry’s car battery is being recharged with respect to time.

The battery charge is increasing at a constant rate of 𝟏𝟏𝟐𝟐% every 𝟏𝟏𝟐𝟐 minutes.

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b. Roxy has a supercharger at her home that can charge about half of the battery in 𝟐𝟐𝟐𝟐 minutes. There is no
remaining charge left when she begins recharging the battery. Sketch a graph that represents the
relationship between the battery charge and the amount of time on the axes above. Assume the relationship
is linear.

See the graph on the previous page.

c. Which person’s car will be recharged to full capacity first? Explain.

Roxy’s car will be completely recharged first. Her supercharger has a greater rate of change compared to
Henry’s charger.

Problem Set Sample Solutions

1. Read through each of the scenarios, and choose the graph of the function that best matches the situation. Explain
the reason behind each choice.

a. The tire pressure on Regina’s car remains at 𝟑𝟑𝟐𝟐 𝐩𝐩𝐩𝐩𝐩𝐩.
b. Carlita inflates her tire at a constant rate for 𝟒𝟒 minutes.

c. Air is leaking from Courtney’s tire at a constant rate.

Scenario: c

Explanation: The tire
pressure decreases each
minute at a constant rate.

Scenario: a

Explanation: The tire
pressure remains at 𝟑𝟑𝟐𝟐 𝐩𝐩𝐩𝐩𝐩𝐩.

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Lesson 4: Increasing and Decreasing Functions

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Scenario: b

Explanation: The tire
pressure is increasing each
minute at a constant rate for
𝟒𝟒 minutes.

2. A home was purchased for $𝟐𝟐𝟕𝟕𝟕𝟕,𝟐𝟐𝟐𝟐𝟐𝟐. Due to a recession, the value of the home fell at a constant rate over the
next 𝟕𝟕 years.

a. Sketch a graph of a function that models the situation.

Graphs will vary; a sample graph is provided.

b. Based on your graph, how is the home value changing with respect to time?

Answers will vary; a sample answer is provided.

The value is decreasing by $𝟐𝟐𝟕𝟕,𝟐𝟐𝟐𝟐𝟐𝟐 over 𝟕𝟕 years or at a constant rate of $𝟕𝟕,𝟐𝟐𝟐𝟐𝟐𝟐 per year.

3. The graph below displays the first hour of Sam’s bike ride.

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Match each part of the graph (A, B, and C) to its verbal description. Explain the reasoning behind your choice.

i. Sam rides his bike to his friend’s house at a constant rate.

A; the distance from home should be increasing as Sam is riding toward his friend’s house.

ii. Sam and his friend bike together to an ice cream shop that is between their houses.

C; Sam was at his friend’s house, but as they start biking to the ice cream shop, the distance from Sam’s
home begins to decrease.

iii. Sam plays at his friend’s house.

B; Sam remains at the same distance from home while he is at his friend’s house.

4. Using the axes below, create a story about the relationship between two quantities.

a. Write a story about the relationship between two quantities. Any quantities can be used (e.g., distance and
time, money and hours, age and growth). Be creative. Include keywords in your story such as increase and
decrease to describe the relationship.

Answers will vary. Give students the freedom to write a basic linear story or a piecewise story.

A rock climber begins her descent from a height of 𝟕𝟕𝟐𝟐 feet. She slowly descends at a constant rate for 𝟒𝟒
minutes. She takes a break for 𝟏𝟏 minute; she then realizes she left some of her gear on top of the rock and
climbs more quickly back to the top at a constant rate.

b. Label each axis with the quantities of your choice, and sketch a graph of the function that models the
relationship described in the story.

Answers will vary based on the story from part (a).

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 5

Lesson 5: Increasing and Decreasing Functions

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Lesson 5: Increasing and Decreasing Functions

Student Outcomes

 Students qualitatively describe the functional relationship between two types of quantities by analyzing a
graph.

 Students sketch a graph that exhibits the qualitative features of linear and nonlinear functions based on a
verbal description.

Lesson Notes
This lesson extends the concepts introduced in Lesson 4 and focuses on graphs and the role they play in analyzing
functional relationships between quantities. Students begin the lesson by comparing and contrasting linear and
nonlinear functions. Encourage students to distinguish a linear function from a nonlinear function by analyzing a graph
using the rate of change for an interval instead of just stating that “it looks like a straight line.” Students sketch
nonlinear functions given a contextual situation but do not construct the functions.

Classwork

Example 1 (3–5 minutes): Nonlinear Functions in the Real World

Read through the scenarios as a class. A linear function is used to model the first scenario, and a nonlinear function is
used to model the second scenario.

Example 1: Nonlinear Functions in the Real World

Not all real-world situations can be modeled by a linear function. There are times when a nonlinear function is needed to
describe the relationship between two types of quantities. Compare the two scenarios:

a. Aleph is running at a constant rate on a flat, paved road. The graph below represents the total distance he
covers with respect to time.

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b. Shannon is running on a flat, rocky trail that eventually rises up a steep mountain. The graph below
represents the total distance she covers with respect to time.

Exercises 1–2 (5–7 minutes)

Students look at the rate of change for different intervals for the scenarios presented in Example 1. Let students work
with a partner. Then, discuss answers as a class. Remind students of increasing, decreasing, and constant linear
functions from the previous lesson.

 Why might the distance that Shannon runs during each 15-minute interval decrease?

 Shannon is running up a mountain. Maybe the mountain is getting steeper, which is causing her to run
slower.

 Are these increasing or decreasing functions?

 They are both increasing functions because the total distance is increasing with respect to time. The
function that models Aleph’s total distance is an increasing linear function, and Shannon’s total
distance is an increasing nonlinear function.

Exercises 1–2

1. In your own words, describe what is happening as Aleph is running during the following intervals of time.

a. 𝟎𝟎 to 𝟏𝟏𝟏𝟏 minutes

Aleph runs 𝟐𝟐 miles in 𝟏𝟏𝟏𝟏 minutes.

b. 𝟏𝟏𝟏𝟏 to 𝟑𝟑𝟎𝟎 minutes

Aleph runs another 𝟐𝟐 miles in 𝟏𝟏𝟏𝟏 minutes for a total of 𝟒𝟒 miles.

c. 𝟑𝟑𝟎𝟎 to 𝟒𝟒𝟏𝟏 minutes

Aleph runs another 𝟐𝟐 miles in 𝟏𝟏𝟏𝟏 minutes for a total of 𝟔𝟔 miles.

d. 𝟒𝟒𝟏𝟏 to 𝟔𝟔𝟎𝟎 minutes

Aleph runs another 𝟐𝟐 miles in 𝟏𝟏𝟏𝟏 minutes for a total of 𝟖𝟖 miles.

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2. In your own words, describe what is happening as Shannon is running during the following intervals of time.

a. 𝟎𝟎 to 𝟏𝟏𝟏𝟏 minutes

Shannon runs 𝟏𝟏.𝟏𝟏 miles in 𝟏𝟏𝟏𝟏 minutes.

b. 𝟏𝟏𝟏𝟏 to 𝟑𝟑𝟎𝟎 minutes

Shannon runs another 𝟎𝟎.𝟔𝟔 mile in 𝟏𝟏𝟏𝟏 minutes for a total of 𝟐𝟐.𝟏𝟏 miles.

c. 𝟑𝟑𝟎𝟎 to 𝟒𝟒𝟏𝟏 minutes

Shannon runs another 𝟎𝟎.𝟏𝟏 mile in 𝟏𝟏𝟏𝟏 minutes for a total of 𝟐𝟐.𝟔𝟔 miles.

d. 𝟒𝟒𝟏𝟏 to 𝟔𝟔𝟎𝟎 minutes

Shannon runs another 𝟎𝟎.𝟒𝟒 mile in 𝟏𝟏𝟏𝟏 minutes for a total of 𝟑𝟑.𝟎𝟎 miles.

Example 2 (5 minutes): Increasing and Decreasing Functions

Convey to students that linear functions have a constant rate of change while nonlinear functions do not have a constant
rate of change. Consider using a table of values for additional clarification using the information from Exercises 1 and 2.

 How would you describe the rate of change of the function modeling Shannon’s total distance? Explain.

 The function is increasing but at a decreasing rate of change. The rate of change is decreasing for
every 15-minute interval.

Example 2: Increasing and Decreasing Functions

The rate of change of a function can provide useful information about the relationship between two quantities.
A linear function has a constant rate of change. A nonlinear function has a variable rate of change.

Linear Functions Nonlinear Functions

Linear function increasing at a constant rate

Nonlinear function increasing at a variable rate

Linear function decreasing at a constant rate

Nonlinear function decreasing at a variable rate

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Linear function with a constant rate

𝒙𝒙 𝒚𝒚
𝟎𝟎 𝟕𝟕
𝟏𝟏 𝟏𝟏𝟎𝟎
𝟐𝟐 𝟏𝟏𝟑𝟑
𝟑𝟑 𝟏𝟏𝟔𝟔
𝟒𝟒 𝟏𝟏𝟏𝟏

Nonlinear function with a variable rate

𝒙𝒙 𝒚𝒚
𝟎𝟎 𝟎𝟎
𝟏𝟏 𝟐𝟐
𝟐𝟐 𝟒𝟒
𝟑𝟑 𝟖𝟖
𝟒𝟒 𝟏𝟏𝟔𝟔

Exercises 3–5 (15 minutes)

Students sketch graphs of functions based on a verbal description. Note that the graph should just be a rough sketch
that matches the verbal description. Allow students to work with a partner or in a small group. Discuss and compare
answers as a class.

Exercises 3–5

3. Different breeds of dogs have different growth rates. A large breed dog typically experiences a rapid growth rate
from birth to age 𝟔𝟔 months. At that point, the growth rate begins to slow down until the dog reaches full growth
around 𝟐𝟐 years of age.

a. Sketch a graph that represents the weight of a large breed dog from birth to 𝟐𝟐 years of age.

Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.

The function is nonlinear because the growth rate is not constant.

c. Is the function represented by the graph increasing or decreasing? Explain.

The function is increasing but at a decreasing rate. There is rapid growth during the first 𝟔𝟔 months, and then
the growth rate decreases.

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4. Nikka took her laptop to school and drained the battery while typing a research paper. When she returned home,
Nikka connected her laptop to a power source, and the battery recharged at a constant rate.

a. Sketch a graph that represents the battery charge with respect to time.

Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.

The function is linear because the battery is recharging at a constant rate.

c. Is the function represented by the graph increasing or decreasing? Explain.

The function is increasing because the battery is being recharged.

5. The long jump is a track-and-field event where an athlete attempts to leap as far as possible from a given point.
Mike Powell of the United States set the long jump world record of 𝟖𝟖.𝟏𝟏𝟏𝟏 meters (𝟐𝟐𝟏𝟏.𝟒𝟒 feet) during the 1991 World
Championships in Tokyo, Japan.

a. Sketch a graph that represents the path of a high school athlete attempting the long jump.

Answers will vary.

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Note: If students have trouble visualizing the path of a jump, use the following table for students to begin their sketches.
Remind students to draw a curve and not to connect points with a straight line.

𝒙𝒙 𝒚𝒚
0 0
1 0.75
2 1.2
3 1.35
4 1.2
5 0.75
6 0

b. Is the function represented by the graph linear or nonlinear? Explain.

The function is nonlinear. The rate of change is not constant.

c. Is the function represented by the graph increasing or decreasing? Explain.

The function both increases and decreases over different intervals. The function increases as the athlete
begins the jump and reaches a maximum height. The function decreases after the athlete reaches maximum
height and begins descending back toward the ground.

Example 3 (5–7 minutes): Ferris Wheel

This example presents students with a graph of a nonlinear function that both increases and decreases over different
intervals of time. Students may have a difficult time connecting the graph to the scenario. Remind students that the
graph is relating time to a rider’s distance above the ground. Consider doing a rough sketch of the Ferris wheel scenario
on a personal white board for further clarification using a similar object such as a hamster wheel or a K’NEX construction
toy. There are also videos that can be found online that relate this type of motion to nonlinear curves. The website
www.graphingstories.com has a great video that relates the motion of a playground merry-go-round to the distance of a
camera that produces a graph similar to the Ferris wheel example.

Example 3: Ferris Wheel

Lamar and his sister are riding a Ferris wheel at a state fair. Using their watches, they find that it takes 𝟖𝟖 seconds for the
Ferris wheel to make a complete revolution. The graph below represents Lamar and his sister’s distance above the
ground with respect to time.

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Exercises 6–9 (5–7 minutes)

Allow students to work with a partner or in a small group to complete the following exercises.
Confirm answers as a class.

Exercises 6–9

6. Use the graph from Example 3 to answer the following questions.

a. Is the function represented by the graph linear or nonlinear?

The function is nonlinear. The rate of change is not constant.

b. Where is the function increasing? What does this mean within the context of the problem?

The function is increasing during the following intervals of time: 𝟎𝟎 to 𝟒𝟒 seconds, 𝟖𝟖 to 𝟏𝟏𝟐𝟐 seconds, 𝟏𝟏𝟔𝟔 to 𝟐𝟐𝟎𝟎
seconds, 𝟐𝟐𝟒𝟒 to 𝟐𝟐𝟖𝟖 seconds, and 𝟑𝟑𝟐𝟐 to 𝟑𝟑𝟔𝟔 seconds. It means that Lamar and his sister are rising in the air.

c. Where is the function decreasing? What does this mean within the context of the problem?

The function is decreasing during the following intervals of time: 𝟒𝟒 to 𝟖𝟖 seconds, 𝟏𝟏𝟐𝟐 to 𝟏𝟏𝟔𝟔 seconds, 𝟐𝟐𝟎𝟎 to 𝟐𝟐𝟒𝟒
seconds, 𝟐𝟐𝟖𝟖 to 𝟑𝟑𝟐𝟐 seconds, and 𝟑𝟑𝟔𝟔 to 𝟒𝟒𝟎𝟎 seconds. Lamar and his sister are traveling back down toward the
ground.

7. How high above the ground is the platform for passengers to get on the Ferris wheel? Explain your reasoning.

The lowest point on the graph, which is at 𝟏𝟏 feet, can represent the platform where the riders get on the
Ferris wheel.

8. Based on the graph, how many revolutions does the Ferris wheel complete during the 𝟒𝟒𝟎𝟎-second time interval?
Explain your reasoning.

The Ferris wheel completes 𝟏𝟏 revolutions. The lowest points on the graph can represent Lamar and his sister at the
beginning of a revolution or at the entrance platform of the Ferris wheel. So, one revolution occurs between 𝟎𝟎 and
𝟖𝟖 seconds, 𝟖𝟖 and 𝟏𝟏𝟔𝟔 seconds, 𝟏𝟏𝟔𝟔 and 𝟐𝟐𝟒𝟒 seconds, 𝟐𝟐𝟒𝟒 and 𝟑𝟑𝟐𝟐 seconds, and 𝟑𝟑𝟐𝟐 and 𝟒𝟒𝟎𝟎 seconds.

9. What is the diameter of the Ferris wheel? Explain your reasoning.

The diameter of the Ferris wheel is 𝟒𝟒𝟎𝟎 feet. The lowest point on the graph represents the base of the Ferris wheel,
and the highest point on the graph represents the top of the Ferris wheel. The difference between the two values is
𝟒𝟒𝟎𝟎 feet, which is the diameter of the wheel.

Closing (2 minutes)

Review the Lesson Summary with students.

 Refer back to Exercises 3 and 4 (dog growth rate and laptop battery recharge problems). Note that both
functions were increasing. Is it possible for those functions to continue to increase within the context of the
problem? Explain.

 No. Both functions cannot continue to increase.

 The dog’s weight will increase until it reaches full growth. At that point, the weight would remain
constant or may fluctuate based on diet and exercise.

 The laptop battery capacity can only reach 100%. At that point, it is fully charged. The function could
not continue to increase.

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Exit Ticket (5 minutes)

Lesson Summary

The graph of a function can be used to help describe the relationship between the quantities it represents.

A linear function has a constant rate of change. A nonlinear function does not have a constant rate of change.

 A function whose graph has a positive rate of change is an increasing function.

 A function whose graph has a negative rate of change is a decreasing function.

 Some functions may increase and decrease over different intervals.

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Name Date

Lesson 5: Increasing and Decreasing Functions

Exit Ticket

Lamar and his sister continue to ride the Ferris wheel. The graph below represents Lamar and his sister’s distance above
the ground with respect to time during the next 40 seconds of their ride.

a. Name one interval where the function is increasing.

b. Name one interval where the function is decreasing.

c. Is the function linear or nonlinear? Explain.

d. What could be happening during the interval of time from 60 to 64 seconds?

e. Based on the graph, how many complete revolutions are made during this 40-second interval?

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Exit Ticket Sample Solutions

Lamar and his sister continue to ride the Ferris wheel. The graph below represents Lamar and his sister’s distance above
the ground with respect to time during the next 𝟒𝟒𝟎𝟎 seconds of their ride.

a. Name one interval where the function is increasing.

The function is increasing during the following intervals of time: 𝟒𝟒𝟎𝟎 to 𝟒𝟒𝟒𝟒 seconds, 𝟒𝟒𝟖𝟖 to 𝟏𝟏𝟐𝟐 seconds, 𝟏𝟏𝟔𝟔 to
𝟔𝟔𝟎𝟎 seconds, and 𝟕𝟕𝟐𝟐 to 𝟕𝟕𝟔𝟔 seconds.

b. Name one interval where the function is decreasing.

The function is decreasing during the following intervals of time: 𝟒𝟒𝟒𝟒 to 𝟒𝟒𝟖𝟖 seconds, 𝟏𝟏𝟐𝟐 to 𝟏𝟏𝟔𝟔 seconds, 𝟔𝟔𝟒𝟒 to
𝟔𝟔𝟔𝟔 seconds, 𝟕𝟕𝟎𝟎 to 𝟕𝟕𝟐𝟐 seconds, and 𝟕𝟕𝟔𝟔 to 𝟖𝟖𝟎𝟎 seconds.

c. Is the function linear or nonlinear? Explain.

The function is both linear and nonlinear during different intervals of time. It is linear from 𝟔𝟔𝟎𝟎 to 𝟔𝟔𝟒𝟒 seconds
and from 𝟔𝟔𝟔𝟔 to 𝟕𝟕𝟎𝟎 seconds. It is nonlinear from 𝟒𝟒𝟎𝟎 to 𝟔𝟔𝟎𝟎 seconds and from 𝟕𝟕𝟎𝟎 to 𝟖𝟖𝟎𝟎 seconds.

d. What could be happening during the interval of time from 𝟔𝟔𝟎𝟎 to 𝟔𝟔𝟒𝟒 seconds?

The Ferris wheel is not moving during that time, so riders may be getting off or getting on.

e. Based on the graph, how many complete revolutions are made during this 𝟒𝟒𝟎𝟎-second interval?

Four revolutions are made during this time period.

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Problem Set Sample Solutions

1. Read through the following scenarios, and match each to its graph. Explain the reasoning behind your choice.

a. This shows the change in a smartphone battery charge as a person uses the phone more frequently.

b. A child takes a ride on a swing.

c. A savings account earns simple interest at a constant rate.

d. A baseball has been hit at a youth baseball game.

Scenario: c

The savings account is earning interest at a constant
rate, which means that the function is linear.

Scenario: d

The baseball is hit into the air, reaches a maximum
height, and falls back to the ground at a variable rate.

Scenario: b

The distance from the ground increases as the child
swings up into the air and then decreases as the child
swings back down toward the ground.

Scenario: a

The battery charge is decreasing as a person uses the
phone more frequently.

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2. The graph below shows the volume of water for a given creek bed during a 𝟐𝟐𝟒𝟒-hour period. On this particular day,
there was wet weather with a period of heavy rain.

Describe how each part (A, B, and C) of the graph relates to the scenario.

A: The rain begins, and the volume of water flowing in the creek bed begins to increase.

B: A period of heavy rain occurs, causing the volume of water to increase.

C: The heavy rain begins to subside, and the volume of water continues to increase.

3. Half-life is the time required for a quantity to fall to half of its value measured at the beginning of the time period. If
there are 𝟏𝟏𝟎𝟎𝟎𝟎 grams of a radioactive element to begin with, there will be 𝟏𝟏𝟎𝟎 grams after the first half-life, 𝟐𝟐𝟏𝟏 grams
after the second half-life, and so on.

a. Sketch a graph that represents the amount of the radioactive element left with respect to the number of half-
lives that have passed.

Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.

The function is nonlinear. The rate of change is not constant with respect to time.

c. Is the function represented by the graph increasing or decreasing?

The function is decreasing.

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4. Lanae parked her car in a no-parking zone. Consequently, her car was towed to an impound lot. In order to release
her car, she needs to pay the impound lot charges. There is an initial charge on the day the car is brought to the lot.
However, 𝟏𝟏𝟎𝟎% of the previous day’s charges will be added to the total charge for every day the car remains in the
lot.

a. Sketch a graph that represents the total charges with respect to the number of days a car remains in the
impound lot.

Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.

The function is nonlinear. The function is increasing.

c. Is the function represented by the graph increasing or decreasing? Explain.

The function is increasing. The total charge is increasing as the number of days the car is left in the
lot increases.

5. Kern won a $𝟏𝟏𝟎𝟎 gift card to his favorite coffee shop. Every time he visits the shop, he purchases the same coffee
drink.

a. Sketch a graph of a function that can be used to represent the amount of money that remains on the gift card
with respect to the number of drinks purchased.

Answers will vary.

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b. Is the function represented by the graph linear or nonlinear? Explain.

The function is linear. Since Kern purchases the same drink every visit, the balance is decreasing by the same
amount or, in other words, at a constant rate of change.

c. Is the function represented by the graph increasing or decreasing? Explain.

The function is decreasing. With each drink purchased, the amount of money on the card decreases.

6. Jay and Brooke are racing on bikes to a park 𝟖𝟖 miles away. The tables below display the total distance each person
biked with respect to time.

Jay

Time
(minutes)

Distance
(miles)

𝟎𝟎 𝟎𝟎
𝟏𝟏 𝟎𝟎.𝟖𝟖𝟒𝟒
𝟏𝟏𝟎𝟎 𝟏𝟏.𝟖𝟖𝟔𝟔
𝟏𝟏𝟏𝟏 𝟑𝟑.𝟎𝟎𝟎𝟎
𝟐𝟐𝟎𝟎 𝟒𝟒.𝟐𝟐𝟕𝟕
𝟐𝟐𝟏𝟏 𝟏𝟏.𝟔𝟔𝟕𝟕

Brooke

Time
(minutes)

Distance
(miles)

𝟎𝟎 𝟎𝟎
𝟏𝟏 𝟏𝟏.𝟐𝟐
𝟏𝟏𝟎𝟎 𝟐𝟐.𝟒𝟒
𝟏𝟏𝟏𝟏 𝟑𝟑.𝟔𝟔
𝟐𝟐𝟎𝟎 𝟒𝟒.𝟖𝟖
𝟐𝟐𝟏𝟏 𝟔𝟔.𝟎𝟎

a. Which person’s biking distance could be modeled by a nonlinear function? Explain.

The distance that Jay biked could be modeled by a nonlinear function because the rate of change is not
constant. The distance that Brooke biked could be modeled by a linear function because the rate of change is
constant.

b. Who would you expect to win the race? Explain.

Jay will win the race. The distance he bikes during each five-minute interval is increasing, while Brooke’s
biking distance remains constant. If the trend remains the same, it is estimated that both Jay and Brooke will
travel about 𝟕𝟕.𝟐𝟐 miles in 𝟑𝟑𝟎𝟎 minutes. So, Jay will overtake Brooke during the last 𝟏𝟏 minutes to win the race.

7. Using the axes in Problem 7(b), create a story about the relationship between two quantities.

a. Write a story about the relationship between two quantities. Any quantities can be used (e.g., distance and
time, money and hours, age and growth). Be creative! Include keywords in your story such as increase and
decrease to describe the relationship.

Answers will vary.

A person in a car is at a red stoplight. The light turns green, and the person presses down on the accelerator
with increasing pressure. The car begins to move and accelerate. The rate at which the car accelerates is
not constant.

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Lesson 5: Increasing and Decreasing Functions

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b. Label each axis with the quantities of your choice, and sketch a graph of the function that models the
relationship described in the story.

Answers will vary based on the story from part (a).

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Topic B: Bivariate Numerical Data

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8
G R A D E

New York State Common Core

Mathematics Curriculum
GRADE 8 • MODULE 6

Topic B

Bivariate Numerical Data

8.SP.A.1, 8.SP.A.2

Focus Standards: 8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to
investigate patterns of association between two quantities. Describe patterns
such as clustering, outliers, positive or negative association, linear association,
and nonlinear association.

8.SP.A.2 Know that straight lines are widely used to model relationships between two
quantitative variables. For scatter plots that suggest a linear association,
informally fit a straight line, and informally assess the model fit by judging the
closeness of the data points to the line.

Instructional Days: 4

Lesson 6: Scatter Plots (P)1

Lesson 7: Patterns in Scatter Plots (P)

Lesson 8: Informally Fitting a Line (P)

Lesson 9: Determining the Equation of a Line Fit to Data (P)

In Topic B, students connect their study of linear functions to applications involving bivariate data. A key tool
in developing this connection is a scatter plot. In Lesson 6, students construct scatter plots and focus on
identifying linear versus nonlinear patterns (8.SP.A.1). They distinguish positive linear association and
negative linear association based on the scatter plot. Students describe trends in the scatter plot along with
clusters and outliers (points that do not fit the pattern). In Lesson 8, students informally fit a straight line to
data displayed in a scatter plot (8.SP.A.2) by judging the closeness of the data points to the line. In Lesson 9,
students interpret and determine the equation of the line they fit to the data and use the equation to make
predictions and to evaluate possible association of the variables. Based on these predictions, students
address the need for a best-fit line, which is formally introduced in Algebra I.

1Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 6

Lesson 6: Scatter Plots

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Lesson 6: Scatter Plots

Student Outcomes

 Students construct scatter plots.
 Students use scatter plots to investigate relationships.

 Students understand that a trend in a scatter plot does not establish cause-and-effect.

Lesson Notes
This lesson is the first in a set of lessons dealing with relationships between numerical variables. In this lesson, students
learn how to construct a scatter plot and look for patterns that suggest a statistical relationship between two numerical
variables. Note that in this, and subsequent lessons, there is notation on the graphs indicating that not all of the
intervals are represented on the axes. Students may need explanation that connects the “zigzag” to the idea that there
are numbers on the axes that are just not shown in the graph.

Classwork

Example 1 (5 minutes)

Spend a few minutes introducing the context of this example. Make sure that students
understand that in this context, an observation can be thought of as an ordered pair
consisting of the value for each of two variables.

Example 1

A bivariate data set consists of observations on two variables. For example, you might collect
data on 𝟏𝟏𝟏𝟏 different car models. Each observation in the data set would consist of an (𝒙𝒙,𝒚𝒚) pair.

𝒙𝒙: weight (in pounds, rounded to the nearest 𝟓𝟓𝟓𝟓 pounds)

and

𝒚𝒚: fuel efficiency (in miles per gallon, 𝐦𝐦𝐦𝐦𝐦𝐦)

The table below shows the weight and fuel efficiency for 𝟏𝟏𝟏𝟏 car models with automatic
transmissions manufactured in 2009 by Chevrolet.

Model Weight (pounds) Fuel Efficiency (𝐦𝐦𝐦𝐦𝐦𝐦)
𝟏𝟏 𝟏𝟏,𝟐𝟐𝟓𝟓𝟓𝟓 𝟐𝟐𝟏𝟏
𝟐𝟐 𝟐𝟐,𝟓𝟓𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐
𝟏𝟏 𝟒𝟒,𝟓𝟓𝟓𝟓𝟓𝟓 𝟏𝟏𝟏𝟏
𝟒𝟒 𝟒𝟒,𝟓𝟓𝟓𝟓𝟓𝟓 𝟐𝟐𝟓𝟓
𝟓𝟓 𝟏𝟏,𝟕𝟕𝟓𝟓𝟓𝟓 𝟐𝟐𝟓𝟓
𝟔𝟔 𝟏𝟏,𝟓𝟓𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐
𝟕𝟕 𝟏𝟏,𝟓𝟓𝟓𝟓𝟓𝟓 𝟏𝟏𝟏𝟏
𝟐𝟐 𝟏𝟏,𝟓𝟓𝟓𝟓𝟓𝟓 𝟐𝟐𝟓𝟓
𝟏𝟏 𝟒𝟒,𝟔𝟔𝟓𝟓𝟓𝟓 𝟏𝟏𝟔𝟔
𝟏𝟏𝟓𝟓 𝟓𝟓,𝟐𝟐𝟓𝟓𝟓𝟓 𝟏𝟏𝟐𝟐
𝟏𝟏𝟏𝟏 𝟓𝟓,𝟔𝟔𝟓𝟓𝟓𝟓 𝟏𝟏𝟔𝟔
𝟏𝟏𝟐𝟐 𝟒𝟒,𝟓𝟓𝟓𝟓𝟓𝟓 𝟏𝟏𝟔𝟔
𝟏𝟏𝟏𝟏 𝟒𝟒,𝟐𝟐𝟓𝟓𝟓𝟓 𝟏𝟏𝟓𝟓

Scaffolding:
 Point out to students that

the word bivariate is
composed of the prefix bi-
and the stem variate.

 Bi- means two.
 Variate indicates a

variable.
 The focus in this lesson is

on two numerical
variables.

Scaffolding:
 English language learners

new to the curriculum may
be familiar with the metric
system (kilometers,
kilograms, and liters) but
unfamiliar with the English
system (miles, pounds,
and gallons).

 It may be helpful to
provide conversions:
1 kg ≈ 2.2 lb.
1 lb.≈ 0.45 kg
1 km ≈ 0.62 mi.
1 mi.≈ 1.61 km

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Lesson 6: Scatter Plots

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Exercises 1–3 (10–12 minutes)

After students have had a chance to think about Exercise 1, make sure that everyone understands what an observation
(an ordered pair) represents in the context of this example. Relate plotting the point that corresponds to the first
observation to students’ previous work with plotting points in a rectangular coordinate system. As a way of encouraging
the need to look at a graph of the data, consider asking students to try to determine if there is a relationship between
weight and fuel efficiency by just looking at the table. Allow students time to complete the scatter plot and complete
Exercise 3. Have students share their answers to Exercise 3.

Exercises 1–8

1. In the Example 1 table, the observation corresponding to Model 1 is (𝟏𝟏𝟐𝟐𝟓𝟓𝟓𝟓,𝟐𝟐𝟏𝟏). What is the fuel efficiency of this
car? What is the weight of this car?

The fuel efficiency is 𝟐𝟐𝟏𝟏 miles per gallon, and the weight is 𝟏𝟏,𝟐𝟐𝟓𝟓𝟓𝟓 pounds.

One question of interest is whether there is a relationship between the car weight and fuel efficiency. The best way to
begin to investigate is to construct a graph of the data. A scatter plot is a graph of the (𝑥𝑥,𝑦𝑦) pairs in the data set. Each
(𝑥𝑥,𝑦𝑦) pair is plotted as a point in a rectangular coordinate system.

For example, the observation (3200, 23) would be plotted as a point located above 3,200 on the 𝑥𝑥-axis and across from
23 on the 𝑦𝑦-axis, as shown below.

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2. Add the points corresponding to the other 𝟏𝟏𝟐𝟐 observations to the scatter plot.

3. Do you notice a pattern in the scatter plot? What does this imply about the relationship between weight (𝒙𝒙) and
fuel efficiency (𝒚𝒚)?

There does seem to be a pattern in the plot. Higher weights tend to be paired with lesser fuel efficiencies, so it looks
like heavier cars generally have lower fuel efficiency.

Exercises 4–8 (6–8 minutes)

These exercises give students additional practice creating a scatter plot and identifying a pattern in the plot. Students
should work individually on these exercises and then discuss their answers to Exercises 7 and 8 with a partner. However,
some English language learners may benefit from paired or small group work, particularly if their English literacy is not
strong.

Is there a relationship between price and the quality of athletic shoes? The data in the table below are from the
Consumer Reports website.

𝒙𝒙: price (in dollars)

and

𝒚𝒚: Consumer Reports quality rating

The quality rating is on a scale of 𝟓𝟓 to 𝟏𝟏𝟓𝟓𝟓𝟓, with 𝟏𝟏𝟓𝟓𝟓𝟓 being the highest quality.

Shoe Price (dollars) Quality Rating
𝟏𝟏 𝟔𝟔𝟓𝟓 𝟕𝟕𝟏𝟏
𝟐𝟐 𝟒𝟒𝟓𝟓 𝟕𝟕𝟓𝟓
𝟏𝟏 𝟒𝟒𝟓𝟓 𝟔𝟔𝟐𝟐
𝟒𝟒 𝟐𝟐𝟓𝟓 𝟓𝟓𝟏𝟏
𝟓𝟓 𝟏𝟏𝟏𝟏𝟓𝟓 𝟓𝟓𝟐𝟐
𝟔𝟔 𝟏𝟏𝟏𝟏𝟓𝟓 𝟓𝟓𝟕𝟕
𝟕𝟕 𝟏𝟏𝟓𝟓 𝟓𝟓𝟔𝟔
𝟐𝟐 𝟐𝟐𝟓𝟓 𝟓𝟓𝟐𝟐
𝟏𝟏 𝟏𝟏𝟏𝟏𝟓𝟓 𝟓𝟓𝟏𝟏
𝟏𝟏𝟓𝟓 𝟕𝟕𝟓𝟓 𝟓𝟓𝟏𝟏

MP.7

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4. One observation in the data set is (𝟏𝟏𝟏𝟏𝟓𝟓,𝟓𝟓𝟕𝟕). What does this ordered pair represent in terms of cost and quality?

The pair represents a shoe that costs $𝟏𝟏𝟏𝟏𝟓𝟓 with a quality rating of 𝟓𝟓𝟕𝟕.

5. To construct a scatter plot of these data, you need to start by thinking about appropriate scales for the axes of the
scatter plot. The prices in the data set range from $𝟏𝟏𝟓𝟓 to $𝟏𝟏𝟏𝟏𝟓𝟓, so one reasonable choice for the scale of the 𝒙𝒙-axis
would range from $𝟐𝟐𝟓𝟓 to $𝟏𝟏𝟐𝟐𝟓𝟓, as shown below. What would be a reasonable choice for a scale for the 𝒚𝒚-axis?

Sample response: The smallest 𝒚𝒚-value is 𝟓𝟓𝟏𝟏, and the largest 𝒚𝒚-value is 𝟕𝟕𝟏𝟏. So, the 𝒚𝒚-axis could be scaled from
𝟓𝟓𝟓𝟓 to 𝟕𝟕𝟓𝟓.

6. Add a scale to the 𝒚𝒚-axis. Then, use these axes to construct a scatter plot of the data.

7. Do you see any pattern in the scatter plot indicating that there is a relationship between
price and quality rating for athletic shoes?

Answers will vary. Students may say that they do not see a pattern, or they may say that
they see a slight downward trend.

8. Some people think that if shoes have a high price, they must be of high quality. How would
you respond?

Answers will vary. The data do not support this. Students will either respond that there
does not appear to be a relationship between price and quality, or if they saw a downward
trend in the scatter plot, they might even indicate that the higher-priced shoes tend to have
lower quality. Look for consistency between the answer to this question and how students
answered the previous question.

Example 2 (5–10 minutes): Statistical Relationships

This example makes a very important point. While discussing this example with the class, make sure students
understand the distinction between a statistical relationship and a cause-and-effect relationship. After discussing the
example, ask students if they can think of other examples of numerical variables that might have a statistical relationship
but which probably do not have a cause-and-effect relationship.

Scaffolding:
For more complicated and
reflective answers, consider
allowing English language
learners to use one or more of
the following options:
collaborate with a same-
language peer, illustrate their
responses, or provide a first-
language narration or
response.

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Lesson 6: Scatter Plots

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Example 2: Statistical Relationships

A pattern in a scatter plot indicates that the values of one variable tend to vary in a predictable way as the values of the
other variable change. This is called a statistical relationship. In the fuel efficiency and car weight example, fuel
efficiency tended to decrease as car weight increased.

This is useful information, but be careful not to jump to the conclusion that increasing the weight of a car causes the fuel
efficiency to go down. There may be some other explanation for this. For example, heavier cars may also have bigger
engines, and bigger engines may be less efficient. You cannot conclude that changes to one variable cause changes in the
other variable just because there is a statistical relationship in a scatter plot.

Exercises 9–10 (5 minutes)

Students can work individually or with a partner on these exercises. Then, confirm answers as a class.

Exercises 9–10

9. Data were collected on

𝒙𝒙: shoe size

and

𝒚𝒚: score on a reading ability test

for 𝟐𝟐𝟏𝟏 elementary school students. The scatter plot of these data is shown below. Does there appear to be a
statistical relationship between shoe size and score on the reading test?

Possible response: The pattern in the scatter plot appears to follow a line. As shoe sizes increase, the reading scores
also seem to increase. There does appear to be a statistical relationship because there is a pattern in the
scatter plot.

10. Explain why it is not reasonable to conclude that having big feet causes a high reading score. Can you think of a
different explanation for why you might see a pattern like this?

Possible response: You cannot conclude that just because there is a statistical relationship between shoe size and
reading score that one causes the other. These data were for students completing a reading test for younger
elementary school children. Older children, who would have bigger feet than younger children, would probably tend
to score higher on a reading test for younger students.

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Lesson 6: Scatter Plots

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Closing (3 minutes)

Consider posing the following questions; allow a few student responses for each.

 Why is it helpful to make a scatter plot when you have data on two numerical variables?
 A scatter plot makes it easier to see patterns in the data and to see if there is a statistical relationship

between the two variables.

 Can you think of an example of two variables that would have a statistical relationship but not a cause-and-
effect relationship?

 One famous example is the number of people who must be rescued by lifeguards at the beach and the
number of ice cream sales. Both of these variables have higher values when the temperature is high
and lower values when the temperature is low. So, there is a statistical relationship between them—
they tend to vary in a predictable way. However, it would be silly to say that an increase in ice cream
sales causes more beach rescues.

Exit Ticket (5 minutes)

Lesson Summary
 A scatter plot is a graph of numerical data on two variables.

 A pattern in a scatter plot suggests that there may be a relationship between the two variables used to
construct the scatter plot.

 If two variables tend to vary together in a predictable way, we can say that there is a statistical
relationship between the two variables.

 A statistical relationship between two variables does not imply that a change in one variable causes a
change in the other variable (a cause-and-effect relationship).

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Name ___________________________________________________ Date____________________

Lesson 6: Scatter Plots

Exit Ticket

Energy is measured in kilowatt-hours. The table below shows the cost of building a facility to produce energy and the
ongoing cost of operating the facility for five different types of energy.

Type of Energy
Cost to Operate

(cents per kilowatt-hour)
Cost to Build

(dollars per kilowatt-hour)
Hydroelectric 0.4 2,200

Wind 1.0 1,900
Nuclear 2.0 3,500

Coal 2.2 2,500
Natural Gas 4.8 1,000

1. Construct a scatter plot of the cost to build the facility in dollars per kilowatt-hour (𝑥𝑥) and the cost to operate the
facility in cents per kilowatt-hour (𝑦𝑦). Use the grid below, and be sure to add an appropriate scale to the axes.

2. Do you think that there is a statistical relationship between building cost and operating cost? If so, describe the
nature of the relationship.

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3. Based on the scatter plot, can you conclude that decreased building cost is the cause of increased operating cost?
Explain.

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Exit Ticket Sample Solutions

Energy is measured in kilowatt-hours. The table below shows the cost of building a facility to produce energy and the
ongoing cost of operating the facility for five different types of energy.

Type of Energy Cost to Operate
(cents per kilowatt-hour)

Cost to Build
(dollars per kilowatt-hour)

Hydroelectric 𝟓𝟓.𝟒𝟒 𝟐𝟐,𝟐𝟐𝟓𝟓𝟓𝟓
Wind 𝟏𝟏.𝟓𝟓 𝟏𝟏,𝟏𝟏𝟓𝟓𝟓𝟓

Nuclear 𝟐𝟐.𝟓𝟓 𝟏𝟏,𝟓𝟓𝟓𝟓𝟓𝟓
Coal 𝟐𝟐.𝟐𝟐 𝟐𝟐,𝟓𝟓𝟓𝟓𝟓𝟓

Natural Gas 𝟒𝟒.𝟐𝟐 𝟏𝟏,𝟓𝟓𝟓𝟓𝟓𝟓

1. Construct a scatter plot of the cost to build the facility in dollars per kilowatt-hour (𝒙𝒙) and the cost to operate the
facility in cents per kilowatt-hour (𝒚𝒚). Use the grid below, and be sure to add an appropriate scale to the axes.

2. Do you think that there is a statistical relationship between building cost and operating cost? If so, describe the
nature of the relationship.

Answers may vary. Sample response: Yes, because it looks like there is a downward pattern in the scatter plot. It
appears that the types of energy that have facilities that are more expensive to build are less expensive to operate.

3. Based on the scatter plot, can you conclude that decreased building cost is the cause of increased operating cost?
Explain.

Sample response: No. Just because there may be a statistical relationship between cost to build and cost to operate
does not mean that there is a cause-and-effect relationship.

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Lesson 6: Scatter Plots

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Problem Set Sample Solutions
The Problem Set is intended to reinforce material from the lesson and have students think about the meaning of points
in a scatter plot, clusters, positive and negative linear trends, and trends that are not linear.

1. The table below shows the price and overall quality rating for 𝟏𝟏𝟓𝟓 different brands of bike helmets.

Data source: www.consumerreports.org

Helmet Price (dollars) Quality Rating
A 𝟏𝟏𝟓𝟓 𝟔𝟔𝟓𝟓
B 𝟐𝟐𝟓𝟓 𝟔𝟔𝟏𝟏
C 𝟏𝟏𝟓𝟓 𝟔𝟔𝟓𝟓
D 𝟒𝟒𝟓𝟓 𝟓𝟓𝟓𝟓
E 𝟓𝟓𝟓𝟓 𝟓𝟓𝟒𝟒
F 𝟐𝟐𝟏𝟏 𝟒𝟒𝟕𝟕
G 𝟏𝟏𝟓𝟓 𝟒𝟒𝟕𝟕
H 𝟏𝟏𝟐𝟐 𝟒𝟒𝟏𝟏
I 𝟒𝟒𝟓𝟓 𝟒𝟒𝟐𝟐
J 𝟐𝟐𝟐𝟐 𝟒𝟒𝟏𝟏
K 𝟐𝟐𝟓𝟓 𝟒𝟒𝟓𝟓
L 𝟐𝟐𝟓𝟓 𝟏𝟏𝟐𝟐
M 𝟏𝟏𝟓𝟓 𝟔𝟔𝟏𝟏
N 𝟏𝟏𝟓𝟓 𝟔𝟔𝟏𝟏
O 𝟒𝟒𝟓𝟓 𝟓𝟓𝟏𝟏

Construct a scatter plot of price (𝒙𝒙) and quality rating (𝒚𝒚). Use the grid below.

2. Do you think that there is a statistical relationship between price and quality rating? If so, describe the nature of
the relationship.

Sample response: No. There is no pattern visible in the scatter plot. There does not appear to be a relationship
between price and the quality rating for bike helmets.

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Lesson 6: Scatter Plots

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3. Scientists are interested in finding out how different species adapt to finding food sources. One group studied
crocodilian species to find out how their bite force was related to body mass and diet. The table below displays the
information they collected on body mass (in pounds) and bite force (in pounds).

Species Body Mass (pounds) Bite Force (pounds)
Dwarf crocodile 𝟏𝟏𝟓𝟓 𝟒𝟒𝟓𝟓𝟓𝟓

Crocodile F 𝟒𝟒𝟓𝟓 𝟐𝟐𝟔𝟔𝟓𝟓
Alligator A 𝟏𝟏𝟓𝟓 𝟐𝟐𝟓𝟓𝟓𝟓
Caiman A 𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏𝟓𝟓
Caiman B 𝟏𝟏𝟕𝟕 𝟐𝟐𝟒𝟒𝟓𝟓
Caiman C 𝟒𝟒𝟓𝟓 𝟐𝟐𝟓𝟓𝟓𝟓

Crocodile A 𝟏𝟏𝟏𝟏𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓
Nile crocodile 𝟐𝟐𝟕𝟕𝟓𝟓 𝟔𝟔𝟓𝟓𝟓𝟓

Crocodile B 𝟏𝟏𝟏𝟏𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓
Crocodile C 𝟏𝟏𝟏𝟏𝟓𝟓 𝟔𝟔𝟓𝟓𝟓𝟓
Crocodile D 𝟏𝟏𝟏𝟏𝟓𝟓 𝟕𝟕𝟓𝟓𝟓𝟓
Caiman D 𝟏𝟏𝟐𝟐𝟓𝟓 𝟓𝟓𝟓𝟓𝟓𝟓

Indian Gharial crocodile 𝟐𝟐𝟐𝟐𝟓𝟓 𝟒𝟒𝟓𝟓𝟓𝟓
Crocodile G 𝟐𝟐𝟐𝟐𝟓𝟓 𝟏𝟏,𝟓𝟓𝟓𝟓𝟓𝟓

American crocodile 𝟐𝟐𝟕𝟕𝟓𝟓 𝟏𝟏𝟓𝟓𝟓𝟓
Crocodile E 𝟐𝟐𝟐𝟐𝟓𝟓 𝟕𝟕𝟓𝟓𝟓𝟓
Crocodile F 𝟒𝟒𝟐𝟐𝟓𝟓 𝟏𝟏,𝟔𝟔𝟓𝟓𝟓𝟓

American alligator 𝟏𝟏𝟓𝟓𝟓𝟓 𝟏𝟏,𝟏𝟏𝟓𝟓𝟓𝟓
Alligator B 𝟏𝟏𝟐𝟐𝟓𝟓 𝟏𝟏,𝟐𝟐𝟓𝟓𝟓𝟓
Alligator C 𝟏𝟏𝟔𝟔𝟓𝟓 𝟏𝟏,𝟒𝟒𝟓𝟓𝟓𝟓

Data Source: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0031781#pone-0031781-t001

(Note: Body mass and bite force have been converted to pounds from kilograms and newtons, respectively.)

Construct a scatter plot of body mass (𝒙𝒙) and bite force (𝒚𝒚). Use the grid below, and be sure to add an appropriate
scale to the axes.

4. Do you think that there is a statistical relationship between body mass and bite force? If so, describe the nature of

the relationship.

Sample response: Yes, because it looks like there is an upward pattern in the scatter plot. It appears that alligators
with larger body mass also tend to have greater bite force.

5. Based on the scatter plot, can you conclude that increased body mass causes increased bite force? Explain.

Sample response: No. Just because there is a statistical relationship between body mass and bite force does not
mean that there is a cause-and-effect relationship.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 7

Lesson 7: Patterns in Scatter Plots

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Lesson 7: Patterns in Scatter Plots

Student Outcomes

 Students distinguish linear patterns from nonlinear patterns based on scatter plots.
 Students describe positive and negative trends in a scatter plot.

 Students identify and describe unusual features in scatter plots, such as clusters and outliers.

Lesson Notes
This lesson asks students to look for and describe patterns in scatter plots. It
provides a foundation for later lessons in which students use a line to describe
the relationship between two numerical variables when the pattern in the
scatter plot is linear. Students distinguish between linear and nonlinear
relationships as well as positive and negative linear relationships. The terms
clusters and outliers are also introduced, and students look for these features in
scatter plots and investigate what clusters and outliers reveal about the data.

Classwork

Example 1 (3–5 minutes)

Spend a few minutes going over the three questions posed as a way to help students structure their thinking about data
displayed in a scatter plot. Students should see that looking for patterns in a scatter plot is a logical extension of their
work in the previous lesson where they learned to make a scatter plot. Make sure that students understand the
distinction between a positive linear relationship and a negative linear relationship before moving on to Exercises 1–5.
Students have a chance to practice answering these questions in the exercises that follow. To highlight MP.7, consider
asking students to examine the five scatter plots and describe their similarities and differences before telling students
what to look for.

Example 1

In the previous lesson, you learned that scatter plots show trends in bivariate data.

When you look at a scatter plot, you should ask yourself the following questions:

a. Does it look like there is a relationship between the two variables used to make the
scatter plot?

b. If there is a relationship, does it appear to be linear?

c. If the relationship appears to be linear, is the relationship a positive linear
relationship or a negative linear relationship?

To answer the first question, look for patterns in the scatter plot. Does there appear to be a general pattern to the points
in the scatter plot, or do the points look as if they are scattered at random? If you see a pattern, you can answer the
second question by thinking about whether the pattern would be well described by a line. Answering the third question
requires you to distinguish between a positive linear relationship and a negative linear relationship. A positive linear
relationship is one that is described by a line with a positive slope. A negative linear relationship is one that is described
by a line with a negative slope.

Scaffolding:
 Point out to students that in this

lesson, the meaning of the word
relationship is not the same as the
use of the word describing a
familial connection, such as a sister
or cousin.

 In this lesson, a relationship
indicates that two numerical
variables have a connection that
can be described either verbally or
with mathematical symbols.

Scaffolding:
For English language learners,
teachers may need to read
aloud the information in
Example 1, highlighting each
key point with a visual example
as students record it in a
graphic organizer for reference.

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Lesson 7: Patterns in Scatter Plots

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Exercises 1–5 (8–10 minutes)

Consider answering Exercise 1 as part of a whole-class discussion, and then allow students to work individually or in pairs
on Exercises 2–5. Have students share answers to these exercises, and discuss any of the exercises where there is
disagreement on the answers. Additionally, point out to students that scatter plots that more closely resemble a linear
pattern are sometimes called strong. Scatter plots that are linear but not as close to a line are sometimes known as
weak. A linear relationship may sometimes be referred to as strong positive, weak positive, strong negative, or weak
negative. Consider using these terms with students while discussing their scatter plots.

Exercises 1–9

Take a look at the following five scatter plots. Answer the three questions in Example 1 for each scatter plot.

1. Scatter Plot 1

Is there a relationship?

Yes

If there is a relationship, does it appear to be linear?

Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Negative

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2. Scatter Plot 2

Is there a relationship?

Yes

If there is a relationship, does it appear to be linear?

Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Positive

3. Scatter Plot 3

Is there a relationship?

If there is a relationship, does it appear to be linear?

Not applicable

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Not applicable

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Lesson 7: Patterns in Scatter Plots

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4. Scatter Plot 4

Is there a relationship?

Yes

If there is a relationship, does it appear to be linear?

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Not applicable

5. Scatter Plot 5

Is there a relationship?

Yes

If there is a relationship, does it appear to be linear?

Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?

Negative

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Lesson 7: Patterns in Scatter Plots

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Exercises 6–9 (10 minutes)

Let students work in pairs on Exercises 6–9. Encourage students to use terms such as
linear and nonlinear and positive and negative in their descriptions. Also, remind students
that their descriptions should be written making use of the context of the problem. Point
out that a good description would provide answers to the three questions they answered
in the previous exercises.

6. Below is a scatter plot of data on weight in pounds (𝒙𝒙) and fuel efficiency in miles per gallon
(𝒚𝒚) for 𝟏𝟏𝟏𝟏 cars. Using the questions at the beginning of this lesson as a guide, write a few
sentences describing any possible relationship between 𝒙𝒙 and 𝒚𝒚.

Possible response: There appears to be a negative linear relationship between fuel efficiency and weight. Students
may note that this is a fairly strong negative relationship. The cars with greater weight tend to have lesser
fuel efficiency.

7. Below is a scatter plot of data on price in dollars (𝒙𝒙) and quality rating (𝒚𝒚) for 𝟏𝟏𝟏𝟏 bike helmets. Using the questions
at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between
𝒙𝒙 and 𝒚𝒚.

Possible response: There does not appear to be a relationship between quality rating and price. The points in the
scatter plot appear to be scattered at random, and there is no apparent pattern in the scatter plot.

Scaffolding:
 It may be helpful to

provide sentence frames
on the classroom board to
help students articulate
their observations.

 For example, “I see a
negative or positive linear
relationship between
and . The higher or
lower the , the higher
or lower the .”

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Lesson 7: Patterns in Scatter Plots

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8. Below is a scatter plot of data on shell length in millimeters (𝒙𝒙) and age in years (𝒚𝒚) for 𝟐𝟐𝟐𝟐 lobsters of known age.
Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible
relationship between 𝒙𝒙 and 𝒚𝒚.

Possible response: There appears to be a relationship between shell length and age, but the pattern in the scatter
plot is curved rather than linear. Age appears to increase as shell length increases, but the increase is not at a
constant rate.

9. Below is a scatter plot of data from crocodiles on body mass in pounds (𝒙𝒙) and bite force in pounds (𝒚𝒚). Using the
questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship
between 𝒙𝒙 and 𝒚𝒚.

Data Source: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0031781#pone-0031781-t001

(Note: Body mass and bite force have been converted to pounds from kilograms and newtons, respectively.)

Possible response: There appears to be a positive linear relationship between bite force and body mass. For
crocodiles, the greater the body mass, the greater the bite force tends to be. Students may notice that this is a
positive relationship but not quite as strong as the relationship noted in Exercise 6.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 7

Lesson 7: Patterns in Scatter Plots

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Example 2 (5 minutes): Clusters and Outliers

Spend a few minutes introducing the meaning of the terms clusters and outliers in the
context of scatter plots. Consider asking students to sketch a scatter plot that has an
outlier and a scatter plot that has two clusters as a way of checking their understanding of
these terms before moving on to the exercises that follow.

Example 2: Clusters and Outliers

In addition to looking for a general pattern in a scatter plot, you should also look for other
interesting features that might help you understand the relationship between two variables. Two
things to watch for are as follows:

 CLUSTERS: Usually, the points in a scatter plot form a single cloud of points, but
sometimes the points may form two or more distinct clouds of points. These clouds
are called clusters. Investigating these clusters may tell you something useful about
the data.

 OUTLIERS: An outlier is an unusual point in a scatter plot that does not seem to fit the
general pattern or that is far away from the other points in the scatter plot.

The scatter plot below was constructed using data from a study of Rocky Mountain elk
(“Estimating Elk Weight from Chest Girth,” Wildlife Society Bulletin, 1996). The variables studied
were chest girth in centimeters (𝒙𝒙) and weight in kilograms (𝒚𝒚).

Exercises 10–12 (8 minutes)

Have students work individually or in pairs on Exercises 10–12. Then, have students share answers to these exercises,
and discuss any of the exercises where there is disagreement on the answers.

Exercises 10–12

10. Do you notice any point in the scatter plot of elk weight versus chest girth that might be described as an outlier? If
so, which one?

Possible response: The point in the lower left-hand corner of the plot corresponding to an elk with a chest girth of
about 𝟗𝟗𝟗𝟗 𝐜𝐜𝐜𝐜 and a weight of about 𝟏𝟏𝟏𝟏𝟏𝟏 𝐤𝐤𝐤𝐤 could be described as an outlier. There are no other points in the
scatter plot that are near this one.

Scaffolding:
English language learners need
the chance to practice using
the terms clusters and outliers
in both oral and written
contexts. Sentence frames
may be useful for students to
communicate initial ideas.

Scaffolding:
 The terms elk and girth

may not be familiar to
English language learners.

 An elk is a large mammal,
similar to a deer.

 Girth refers to the
measurement around
something. For this
problem, girth refers to
the measurement around
the elk from behind the
front legs and under the
belly. A visual aid of an elk
(found on several
websites) would help
explain an elk’s chest girth.

 Consider providing
students with sentence
frames or word banks, and
allow students to respond
in their first language to
these exercises.

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Lesson 7: Patterns in Scatter Plots

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11. If you identified an outlier in Exercise 10, write a sentence describing how this data observation differs from the
others in the data set.

Possible response: This point corresponds to an observation for an elk that is much smaller than the other elk in the
data set, both in terms of chest girth and weight.

12. Do you notice any clusters in the scatter plot? If so, how would you distinguish between the clusters in terms of
chest girth? Can you think of a reason these clusters might have occurred?

Possible response: Other than the outlier, there appear to be three clusters of points. One cluster corresponds to elk
with chest girths between about 𝟏𝟏𝟏𝟏𝟏𝟏 𝐜𝐜𝐜𝐜 and 𝟏𝟏𝟏𝟏𝟏𝟏 𝐜𝐜𝐜𝐜. A second cluster includes elk with chest girths between
about 𝟏𝟏𝟐𝟐𝟏𝟏 𝐜𝐜𝐜𝐜 and 𝟏𝟏𝟏𝟏𝟏𝟏 𝐜𝐜𝐜𝐜. The third cluster includes elk with chest girths above 𝟏𝟏𝟏𝟏𝟏𝟏 𝐜𝐜𝐜𝐜. It may be that age and
sex play a role. Maybe the cluster with the smaller chest girths includes young elk. The two other clusters might
correspond to females and males if there is a difference in size for the two sexes for Rocky Mountain elk. If we had
data on age and sex, we could investigate this further.

Closing (3–5 minutes)

Consider posing the following questions; allow a few student responses for each.

 Why do you think it is a good idea to look at a scatter plot when you have data
on two numerical variables?

 Possible response: Looking at a scatter plot makes it easier to see if
there is a relationship between the two variables. It is hard to determine
if there is a relationship when you just have the data in a table or a list.

 What should you look for when you are looking at a scatter plot?

 Possible response: First, you should look for any general patterns. If there are patterns, you then want
to consider whether the pattern is linear or nonlinear, and if it is linear, whether the relationship is
positive or negative. Finally, it is also a good idea to look for any other interesting features such as
outliers or clusters. The closer the points are to a line, the “stronger” the linear relationship.

Exit Ticket (5 minutes)

Lesson Summary

 A scatter plot might show a linear relationship, a nonlinear relationship, or no relationship.

 A positive linear relationship is one that would be modeled using a line with a positive slope. A
negative linear relationship is one that would be modeled by a line with a negative slope.

 Outliers in a scatter plot are unusual points that do not seem to fit the general pattern in the plot or
that are far away from the other points in the scatter plot.

 Clusters occur when the points in the scatter plot appear to form two or more distinct clouds of
points.

Scaffolding:
Allowing English language
learners to brainstorm with a
partner first may elicit a
greater response in the whole-
group setting.

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Lesson 7: Patterns in Scatter Plots

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Lesson 7: Patterns in Scatter Plots

Exit Ticket

1. Which of the following scatter plots shows a negative linear relationship? Explain how you know.

Scatter Plot 1

Scatter Plot 2

Scatter Plot 3

Scatter Plot 4

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2. The scatter plot below was constructed using data from eighth-grade students on number of hours playing video
games per week (𝑥𝑥) and number of hours of sleep per night (𝑦𝑦). Write a few sentences describing the relationship
between sleep time and time spent playing video games for these students. Are there any noticeable clusters or
outliers?

3. In a scatter plot, if the values of 𝑦𝑦 tend to increase as the value of 𝑥𝑥 increases, would you say that there is a positive
relationship or a negative relationship between 𝑥𝑥 and 𝑦𝑦? Explain your answer.

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Exit Ticket Sample Solutions

1. Which of the following scatter plots shows a negative linear relationship? Explain how you know.

Scatter Plot 1

Scatter Plot 2

Scatter Plot 3

Scatter Plot 4

Only Scatter Plot 3 shows a negative linear relationship because the 𝒚𝒚-values tend to decrease as the value of
𝒙𝒙 increases.

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Lesson 7: Patterns in Scatter Plots

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2. The scatter plot below was constructed using data from eighth-grade students on number of hours playing video
games per week (𝒙𝒙) and number of hours of sleep per night (𝒚𝒚). Write a few sentences describing the relationship
between sleep time and time spent playing video games for these students. Are there any noticeable clusters
or outliers?

Answers will vary. Sample response: There appears to be a negative linear relationship between the number of
hours per week a student plays video games and the number of hours per night the student sleeps. As video game
time increases, the number of hours of sleep tends to decrease. There is one observation that might be considered
an outlier—the point corresponding to a student who plays video games 𝟏𝟏𝟐𝟐 hours per week. Other than the outlier,
there are two clusters—one corresponding to students who spend very little time playing video games and a second
corresponding to students who play video games between about 𝟏𝟏𝟏𝟏 and 𝟐𝟐𝟏𝟏 hours per week.

3. In a scatter plot, if the value of 𝒚𝒚 tends to increase as the value of 𝒙𝒙 increases, would you say that there is a positive
relationship or a negative relationship between 𝒙𝒙 and 𝒚𝒚? Explain your answer.

There is a positive relationship. If the value of 𝒚𝒚 increases as the value of 𝒙𝒙 increases, the points go up on the scatter
plot from left to right.

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Lesson 7: Patterns in Scatter Plots

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1. Suppose data was collected on size in square feet (𝒙𝒙) of several houses and price in dollars (𝒚𝒚). The data was then
used to construct the scatterplot below. Write a few sentences describing the relationship between price and size
for these houses. Are there any noticeable clusters or outliers?

Answers will vary. Possible response: There appears to be a positive linear relationship between size and price.
Price tends to increase as size increases. There appear to be two clusters of houses—one that includes houses that
are less than 𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏 square feet in size and another that includes houses that are more than 𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏 square feet
in size.

2. The scatter plot below was constructed using data on length in inches (𝒙𝒙) of several alligators and weight in pounds
(𝒚𝒚). Write a few sentences describing the relationship between weight and length for these alligators. Are there
any noticeable clusters or outliers?

Data Source: Exploring Data, Quantitative Literacy Series, James Landwehr and Ann Watkins, 1987.

Answers will vary. Possible response: There appears to be a positive relationship between length and weight, but
the relationship is not linear. Weight tends to increase as length increases. There are three observations that stand
out as outliers. These correspond to alligators that are much bigger in terms of both length and weight than the
other alligators in the sample. Without these three alligators, the relationship between length and weight would
look linear. It might be possible to use a line to model the relationship between weight and length for alligators that
have lengths of fewer than 𝟏𝟏𝟏𝟏𝟏𝟏 inches.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 7

Lesson 7: Patterns in Scatter Plots

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3. Suppose the scatter plot below was constructed using data on age in years (𝒙𝒙) of several Honda Civics and price in
dollars (𝒚𝒚). Write a few sentences describing the relationship between price and age for these cars. Are there any
noticeable clusters or outliers?

Answers will vary. Possible response: There appears to be a negative linear relationship between price and age.
Price tends to decrease as age increases. There is one car that looks like an outlier—the car that is 𝟏𝟏𝟏𝟏 years old.
This car has a price that is lower than expected based on the pattern of the other points in the scatter plot.

4. Samples of students in each of the U.S. states periodically take part in a large-scale assessment called the National
Assessment of Educational Progress (NAEP). The table below shows the percent of students in the northeastern
states (as defined by the U.S. Census Bureau) who answered Problems 7 and 15 correctly on the 2011 eighth-grade
test. The scatter plot shows the percent of eighth-grade students who got Problems 7 and 15 correct on the 2011
NAEP.

State
Percent Correct

Problem 7
Percent Correct

Problem 15

Connecticut 𝟐𝟐𝟗𝟗 𝟏𝟏𝟏𝟏

New York 𝟐𝟐𝟐𝟐 𝟏𝟏𝟐𝟐

Rhode Island 𝟐𝟐𝟗𝟗 𝟏𝟏𝟐𝟐

Maine 𝟐𝟐𝟐𝟐 𝟏𝟏𝟏𝟏

Pennsylvania 𝟐𝟐𝟗𝟗 𝟏𝟏𝟐𝟐

Vermont 𝟏𝟏𝟐𝟐 𝟏𝟏𝟐𝟐

New Jersey 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏

New Hampshire 𝟐𝟐𝟗𝟗 𝟏𝟏𝟐𝟐

Massachusetts 𝟏𝟏𝟏𝟏 𝟏𝟏𝟗𝟗

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 7

Lesson 7: Patterns in Scatter Plots

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Percent Correct for Problems 7 and 15 on 2011 Eighth-Grade NAEP

a. Why does it appear that there are only eight points in the scatter plot for nine states?

Two of the states, New Hampshire and Rhode Island, had exactly the same percent correct on each of the
questions, (𝟐𝟐𝟗𝟗,𝟏𝟏𝟐𝟐).

b. What is true of the states represented by the cluster of five points in the lower left corner of the graph?

Answers will vary; those states had lower percentages correct than the other three states in the upper right.

c. Which state did the best on these two problems? Explain your reasoning.

Answers will vary; some students might argue that Massachusetts at (𝟏𝟏𝟏𝟏,𝟏𝟏𝟗𝟗) did the best. Even though
Vermont actually did a bit better on Problem 15, it was lower on Problem 7.

d. Is there a trend in the data? Explain your thinking.

Answers will vary; there seems to be a positive linear trend, as a large percent correct on one question
suggests a large percent correct on the other, and a low percent on one suggests a low percent on the other.

5. The plot below shows the mean percent of sunshine during the year and the mean amount of precipitation in inches
per year for the states in the United States.

Data source: www.currentresults.com/Weather/US/average-annual-state-sunshine.php

www.currentresults.com/Weather/US/average-annual-state-precipitation.php

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 7

Lesson 7: Patterns in Scatter Plots

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a. Where on the graph are the states that have a large amount of precipitation and a small percent of sunshine?

Those states will be in the lower right-hand corner of the graph.

b. The state of New York is the point (𝟏𝟏𝟗𝟗,𝟏𝟏𝟏𝟏.𝟐𝟐). Describe how the mean amount of precipitation and percent
of sunshine in New York compare to the rest of the United States.

New York has a little over 𝟏𝟏𝟏𝟏 inches of precipitation per year and is sunny about 𝟏𝟏𝟏𝟏% of the time. It has a
smaller percent of sunshine over the year than most states and is about in the middle of the states in terms of
the amount of precipitation, which goes from about 𝟏𝟏𝟏𝟏 to 𝟗𝟗𝟏𝟏 inches per year.

c. Write a few sentences describing the relationship between mean amount of precipitation and percent of
sunshine.

There is a negative relationship, or the more precipitation, the less percent of sun. If you took away the three
states at the top left with a large percent of sun and very little precipitation, the trend would not be as
pronounced. The relationship is not linear.

6. At a dinner party, every person shakes hands with every other person present.
a. If three people are in a room and everyone shakes hands with everyone else, how many handshakes take

place?

Three handshakes

b. Make a table for the number of handshakes in the room for one to six people. You may want to make a
diagram or list to help you count the number of handshakes.

Number People Handshakes Number People Handshakes
𝟏𝟏 𝟏𝟏 𝟏𝟏 𝟗𝟗
𝟐𝟐 𝟏𝟏 𝟏𝟏 𝟏𝟏𝟏𝟏
𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟏𝟏𝟏𝟏

c. Make a scatter plot of number of people (𝒙𝒙) and number of handshakes (𝒚𝒚). Explain your thinking.

d. Does the trend seem to be linear? Why or why not?

The trend is increasing, but it is not linear. As the number of people increases, the number of handshakes also
increases. It does not increase at a constant rate.

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Lesson 8: Informally Fitting a Line

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 8

Lesson 8: Informally Fitting a Line

Student Outcomes

 Students informally fit a straight line to data displayed in a scatter plot.
 Students make predictions based on the graph of a line that has been fit to data.

Lesson Notes
In this lesson, students investigate scatter plots of data and informally fit a line to the pattern observed in the plot.
Students then make predictions based on their lines. Students informally evaluate their predictions based on the fit of
the line to the data.

Classwork

Example 1 (2–3 minutes): Housing Costs

Introduce the data presented in the table and the scatter plot of the data. Ask students
the following:

 Examine the scatter plot. What trend do you see? How would you describe this
trend?

 It appears to be a positive linear trend. The scatter plot indicates that
the larger the size, the higher the price.

(Note: Make sure to give students an opportunity to explain why they think there is a positive linear trend between
price and size.)

Example 1: Housing Costs

Let’s look at some data from one midwestern city that indicate the sizes and sale prices of various houses sold in this city.

Size (square feet) Price (dollars) Size (square feet) Price (dollars)
𝟓𝟓,𝟐𝟐𝟐𝟐𝟐𝟐 𝟏𝟏,𝟎𝟎𝟓𝟓𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎 𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎
𝟏𝟏,𝟖𝟖𝟖𝟖𝟓𝟓 𝟏𝟏𝟖𝟖𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟏𝟏,𝟖𝟖𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎
𝟏𝟏,𝟎𝟎𝟐𝟐𝟏𝟏 𝟖𝟖𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟏𝟏𝟓𝟓𝟏𝟏 𝟓𝟓𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎
𝟏𝟏,𝟏𝟏𝟐𝟐𝟖𝟖 𝟐𝟐𝟏𝟏𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟏𝟏𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎
𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟖𝟖𝟏𝟏𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟏𝟏,𝟐𝟐𝟏𝟏𝟐𝟐 𝟏𝟏𝟓𝟓𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎
𝟐𝟐,𝟎𝟎𝟎𝟎𝟎𝟎 𝟐𝟐𝟎𝟎𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟏𝟏,𝟏𝟏𝟏𝟏𝟖𝟖 𝟏𝟏𝟓𝟓𝟏𝟏,𝟏𝟏𝟏𝟏𝟏𝟏
𝟐𝟐,𝟏𝟏𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟐𝟐,𝟏𝟏𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏,𝟎𝟎𝟎𝟎𝟎𝟎
𝟏𝟏,𝟓𝟓𝟏𝟏𝟏𝟏 𝟏𝟏𝟖𝟖𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎 𝟐𝟐,𝟏𝟏𝟐𝟐𝟏𝟏 𝟐𝟐𝟏𝟏𝟏𝟏,𝟏𝟏𝟎𝟎𝟎𝟎

Data Source: http://www.trulia.com/for sale/Milwaukee,WI/5 p, accessed in 2013

Scaffolding:
 The terms house and home

are used interchangeably
throughout the example.

 This may be confusing for
English language learners
and should be clarified.

MP.2
&

MP.7

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Lesson 8: Informally Fitting a Line

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 8

A scatter plot of the data is given below.

Exercises 1–6 (15 minutes)

In these exercises, be sure that students retain the units as they write and discuss the solutions, being mindful of the
Mathematical Practice standard of attending to precision. Students might use a transparent ruler or a piece of uncooked
spaghetti to help draw and decide where to place their lines. To avoid problems with the size of the numbers and to
have students focus on drawing their lines, the teacher should provide a worksheet for students with the points already
plotted on a grid. Students should concentrate on the general form of the scatter plot rather than worrying too much
about the exact placement of points in the scatter plot. The primary focus of the work in these exercises is to have
students think about the trend, use a line to describe the trend, and make predictions based on the line.

Work through the exercises as a class, allowing time to discuss multiple responses.

Exercises 1–6

1. What can you tell about the price of large homes compared to the price of small homes from the table?

Answers will vary. Students should make the observation that, overall, the larger homes cost more, and the smaller
homes cost less. However, it is hard to generalize because one of the smaller homes costs nearly $𝟏𝟏𝟓𝟓𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎.

2. Use the scatter plot to answer the following questions.

a. Does the scatter plot seem to support the statement that larger houses tend to cost more? Explain your
thinking.

Yes, because the trend is positive, the larger the size of the house, the more the house tends to cost.

b. What is the cost of the most expensive house, and where is that point on the scatter plot?

The house with a size of 𝟓𝟓,𝟐𝟐𝟐𝟐𝟐𝟐 square feet costs $𝟏𝟏,𝟎𝟎𝟓𝟓𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎, which is the most expensive. It is in the upper
right corner of the scatter plot.

Size (square feet)

Pr
ic

e
(d

ol
la

rs
)

6000500040003000200010000

1,200,000

1,000,000

800,000

600,000

400,000

200,000

MP.6

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Lesson 8: Informally Fitting a Line

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 8

c. Some people might consider a given amount of money and then predict what size house they could buy.
Others might consider what size house they want and then predict how much it would cost. How would you
use the scatter plot in Example 1?

Answers will vary. Since the size of the house is on the horizontal axis and the price is on the vertical axis, the
scatter plot is set up with price as the dependent variable and size as the independent variable. This is the
way you would set it up if you wanted to predict price based on size. Although various answers are
appropriate, move the discussion along using size to predict price.

d. Estimate the cost of a 𝟐𝟐,𝟎𝟎𝟎𝟎𝟎𝟎-square-foot house.

Answers will vary. Reasonable answers range between $𝟐𝟐𝟎𝟎𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎 and $𝟏𝟏𝟎𝟎𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎.

e. Do you think a line would provide a reasonable way to describe how price and size are related? How could
you use a line to predict the price of a house if you are given its size?

Answers will vary; however, use this question to develop the idea that a line would provide a way to estimate
the cost given the size of a house. The challenge is how to make that line. Note: Students are encouraged in
the next exercise to first make a line and then evaluate whether or not it fits the data. This will provide a
reasonable estimate of the cost of a house in relation to its size.

3. Draw a line in the plot that you think would fit the trend in the data.

Answers will vary. Discuss several of the lines students have drawn by encouraging students
to share their lines with the class. At this point, do not evaluate the lines as good or bad.
Students may want to know a precise procedure or process to draw their lines. If that
question comes up, indicate to students that a procedure will be developed in their future
work (Algebra I) with statistics. For now, the goal is to simply draw a line that can be used
to describe the relationship between the size of a home and its cost. Indicate that strategies
for drawing a line will be explored in Exercise 5. Use the lines provided by students to
evaluate the predictions in the following exercise. These predictions are used to develop a
strategy for drawing a line. Use the line drawn by students to highlight their understanding
of the data.

4. Use your line to answer the following questions:

a. What is your prediction of the price of a 𝟐𝟐,𝟎𝟎𝟎𝟎𝟎𝟎-square-foot house?

Answers will vary. A reasonable prediction is around $𝟓𝟓𝟎𝟎𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎.

b. What is the prediction of the price of a 𝟏𝟏,𝟓𝟓𝟎𝟎𝟎𝟎-square-foot house?

Answers will vary. A reasonable prediction is around $𝟐𝟐𝟎𝟎𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎.

Display various predictions students found for these two examples. Consider using a chart
similar to the following to discuss the different predictions.

Student
Estimate of the Price for a
𝟐𝟐,𝟎𝟎𝟎𝟎𝟎𝟎-Square-Foot House

Estimate of the Price for a
𝟏𝟏,𝟓𝟓𝟎𝟎𝟎𝟎-Square-Foot House

Student 1 $300,000 $100,000

Student 2 $600,000 $400,000

MP.1

Scaffolding:
 Point out to students that

the word trend is not
connected to the use of
this word in describing
fashion or music. (For
example, “the trend in
music is for more use of
drums.”)

 In this lesson, trend
describes the pattern or
lack of a pattern in the
scatter plot.

 Ask students to highlight
words that they think
would describe a trend in
the scatter plots that are
examined in this lesson.

 Explain to English language
learners that scatter plot
may be referred to as just
plot.

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Lesson 8: Informally Fitting a Line

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 8

Discuss that predictions vary as a result of the different lines that students used to describe the pattern in the scatter
plot. What line makes the most sense for these data?

Before discussing answers to that question, encourage students to explain how they drew their lines and why their
predictions might have been higher (or lower) than other students’. For example, students with lines that are visibly
above most of the points may have predictions that are higher than the predictions of students with lines below several
of the points. Ask students to summarize their theories of how to draw a line as a strategy for drawing a line. After they
provide their own descriptions, provide students an opportunity to think about the following strategies that might have
been used to draw a line.

5. Consider the following general strategies students use for drawing a line. Do you think they represent a good
strategy for drawing a line that fits the data? Explain why or why not, or draw a line for the scatter plot using the
strategy that would indicate why it is or why it is not a good strategy.

a. Laure thought she might draw her line using the very first point (farthest to the left) and the very last point
(farthest to the right) in the scatter plot.

Answers will vary. This may work in some cases, but those points might not capture the trend in the data.
For example, the first point in the lower left might not be in line with the other points.

b. Phil wants to be sure that he has the same number of points above and below the line.

Answers will vary. You could draw a nearly horizontal line that has half of the points above and half below,
but that might not represent the trend in the data at all. Note: For many students just starting out, this
seems like a reasonable strategy, but it often can result in lines that clearly do not fit the data. As indicated,
drawing a nearly horizontal line is a good way to indicate that this is not a good strategy.

c. Sandie thought she might try to get a line that had the most points right on it.

Answers will vary. That might result in, perhaps, three points on the line (knowing it only takes two to make
a line), but the others could be anywhere. The line might even go in the wrong direction. Note: For students
just beginning to think of how to draw a line, this seems like a reasonable goal; however, point out that this
strategy may result in lines that are not good for predicting price.

d. Maree decided to get her line as close to as many of the points as possible.

Answers will vary. If you can figure out how to do this, Maree’s approach seems like a reasonable way to find
a line that takes all of the points into account.

6. Based on the strategies discussed in Exercise 5, would you change how you draw a line through the points? Explain
your answer.

Answers will vary based on how a student drew his original line. Summarize that the goal is to draw a line that is as
close as possible to the points in the scatter plot. More precise methods are developed in Algebra I.

Example 2 (2–3 minutes): Deep Water

Introduce students to the data in the table. Pose the questions in the text, and allow for multiple responses.

Example 2: Deep Water

Does the current in the water go faster or slower when the water is shallow? The data on the depth and velocity of the
Columbia River at various locations in Washington State listed on the next page can help you think about the answer.

MP.1

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Lesson 8: Informally Fitting a Line

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 8

Depth and Velocity in the Columbia River, Washington State

Depth (feet) Velocity (feet/second)
𝟎𝟎.𝟖𝟖 𝟏𝟏.𝟓𝟓𝟓𝟓
𝟐𝟐.𝟎𝟎 𝟏𝟏.𝟏𝟏𝟏𝟏
𝟐𝟐.𝟏𝟏 𝟏𝟏.𝟏𝟏𝟐𝟐
𝟐𝟐.𝟐𝟐 𝟏𝟏.𝟐𝟐𝟏𝟏
𝟏𝟏.𝟏𝟏 𝟏𝟏.𝟐𝟐𝟏𝟏
𝟓𝟓.𝟏𝟏 𝟏𝟏.𝟏𝟏𝟏𝟏
𝟖𝟖.𝟐𝟐 𝟎𝟎.𝟏𝟏𝟏𝟏
𝟖𝟖.𝟏𝟏 𝟎𝟎.𝟓𝟓𝟏𝟏
𝟏𝟏.𝟏𝟏 𝟎𝟎.𝟓𝟓𝟏𝟏
𝟏𝟏𝟎𝟎.𝟏𝟏 𝟎𝟎.𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏.𝟐𝟐 𝟎𝟎.𝟐𝟐𝟐𝟐

Data Source: www.seattlecentral.edu/qelp/sets/011/011.html

a. What can you tell about the relationship between the depth and velocity by looking at the numbers in the
table?

Answers will vary. According to the table, as the depth increases, the velocity appears to decrease.

b. If you were to make a scatter plot of the data, which variable would you put on the horizontal axis, and why?

Answers will vary. It might be easier to measure the depth and use that information to predict the velocity of
the water, so the depth should go on the horizontal axis.

Exercises 7–9 (12–15 minutes)

These exercises engage students in a context where the trend has a negative slope. Again, students should pay careful
attention to units and interpretation of rate of change. They evaluate the line by assessing its closeness to the data
points. Let students work with a partner. If time allows, discuss the answers as a class.

Exercises 7–9

7. A scatter plot of the Columbia River data is shown below.

a. Choose a data point in the scatter plot, and describe what it means in terms of the context.

Answers will vary. For example, (𝟏𝟏.𝟏𝟏,𝟏𝟏.𝟐𝟐𝟏𝟏) would represent a place in the river that was 𝟏𝟏.𝟏𝟏 feet deep and
had a velocity of 𝟏𝟏.𝟐𝟐𝟏𝟏 𝐟𝐟𝐟𝐟/𝐬𝐬𝐬𝐬𝐬𝐬.

Depth (feet)

Ve
lo

ci
ty

(
fe

et
/s

ec
on

121086420

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

MP.6

Scaffolding:
English language learners may
need support in recognizing the
relationship between the
words depth and deep.

Scaffolding:
 The word current has

multiple meanings that
English language learners
may be familiar with from
a social studies class
(current events) or from a
science class (electrical
current).

 In this example, current
refers to the flow or
velocity of the river.

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 8

b. Based on the scatter plot, describe the relationship between velocity and depth.

The deeper the water, the slower the current velocity tends to be.

c. How would you explain the relationship between the velocity and depth of the water?

Answers will vary. Sample response: Velocity may be a result of the volume of water. Shallow water has less
volume, and as a result, the water runs faster. Note: Students may have several explanations. For example,
they may say that depth is a result of less water runoff; therefore, water depth increases.

d. If the river is two feet deep at a certain spot, how fast do you think the current would be? Explain your
reasoning.

Answers will vary. Based on the data, it could be around 𝟏𝟏.𝟏𝟏𝟏𝟏 𝐟𝐟𝐟𝐟/𝐬𝐬𝐬𝐬𝐬𝐬, or it could be closer to 𝟏𝟏.𝟏𝟏𝟐𝟐 𝐟𝐟𝐟𝐟/𝐬𝐬𝐬𝐬𝐬𝐬,
which is more in line with the pattern for the other points.

8. Consider the following questions:

a. If you draw a line to represent the trend in the plot, would it make it easier to predict the velocity of the
water if you know the depth? Why or why not?

Answers will vary. A line will help you determine a better prediction for 𝟏𝟏.𝟓𝟓 𝐟𝐟𝐟𝐟. or 𝟓𝟓 𝐟𝐟𝐟𝐟., where the points are
a bit scattered.

b. Draw a line that you think does a reasonable job of modeling the trend on the scatter plot in Exercise 7. Use
the line to predict the velocity when the water is 𝟖𝟖 feet deep.

Answers will vary. A line is drawn in the following graph. Using this line, when the water is 𝟖𝟖 𝐟𝐟𝐟𝐟. deep, the
velocity is predicted to be 𝟎𝟎.𝟖𝟖𝟏𝟏 𝐟𝐟𝐟𝐟/𝐬𝐬𝐬𝐬𝐬𝐬.

9. Use the line to predict the velocity for a depth of 𝟖𝟖.𝟏𝟏 feet. How far off was your prediction from the actual
observed velocity for the location that had a depth of 𝟖𝟖.𝟏𝟏 feet?

Answers will vary. Sample response: The current would be moving at a velocity of 𝟎𝟎.𝟏𝟏𝟖𝟖 𝐟𝐟𝐟𝐟/𝐬𝐬𝐬𝐬𝐬𝐬. The observed
velocity was 𝟎𝟎.𝟓𝟓𝟏𝟏 𝐟𝐟𝐟𝐟/𝐬𝐬𝐬𝐬𝐬𝐬, so the line predicted a velocity that was 𝟎𝟎.𝟎𝟎𝟏𝟏 𝐟𝐟𝐟𝐟/𝐬𝐬𝐬𝐬𝐬𝐬 faster than the observed value.

Depth (feet)

Ve
lo

ci
ty

(
fe

et
/s

ec
on

121086420

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

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Closing (5 minutes)

Consider posing the following questions; allow a few student responses for each.

 How do scatter plots and tables of data differ in helping you understand the “story” when looking at bivariate
numerical data?

 The numbers in a table can give you a sense of how big or small the values are, but it is easier to see a
relationship between the variables in a scatter plot.

 What is the difference between predicting an outcome by looking at a scatter plot and predicting the outcome
using a line that models the trend?

 When you look at the plot, the points are sometimes very spread out, and for a given value of an
independent variable, some values you might be interested in may not be included in the data set.
Using a line takes all of the points into consideration, and your prediction is based on an overall pattern
rather than just one or two points.

 In a scatter plot, which variable goes on the horizontal axis, and which goes on the vertical axis?

 The independent variable (or the variable not changed by other variables) goes on the horizontal axis,
and the dependent variable (or the variable to be predicted by the independent variable) goes on the
vertical axis.

Exit Ticket (5 minutes)

Lesson Summary
 When constructing a scatter plot, the variable that you want to predict (i.e., the dependent or response

variable) goes on the vertical axis. The independent variable (i.e., the variable not related to other
variables) goes on the horizontal axis.

 When the pattern in a scatter plot is approximately linear, a line can be used to describe the
linear relationship.

 A line that describes the relationship between a dependent variable and an independent variable can
be used to make predictions of the value of the dependent variable given a value of the
independent variable.

 When informally fitting a line, you want to find a line for which the points in the scatter plot tend to
be closest.

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Name ___________________________________________________ Date____________________

Lesson 8: Informally Fitting a Line

Exit Ticket

The plot below is a scatter plot of mean temperature in July and mean inches of rain per year for a sample of
midwestern cities. A line is drawn to fit the data.

July Temperatures and Rainfall in Selected Midwestern Cities

Data Source: http://countrystudies.us/united-states/weather/

1. Choose a point in the scatter plot, and explain what it represents.

2. Use the line provided to predict the mean number of inches of rain per year for a city that has a mean temperature
of 70°F in July.

3. Do you think the line provided is a good one for this scatter plot? Explain your answer.

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Exit Ticket Sample Solutions

The plot below is a scatter plot of mean temperature in July and mean inches of rain per year for a sample of midwestern
cities. A line is drawn to fit the data.

July Temperatures and Rainfall in Selected Midwestern Cities

Data Source: http://countrystudies.us/united-states/weather/

1. Choose a point in the scatter plot, and explain what it represents.

Answers will vary. Sample response: The point at about (𝟖𝟖𝟐𝟐,𝟐𝟐𝟓𝟓) represents a Midwestern city where the mean
temperature in July is about 𝟖𝟖𝟐𝟐°𝐅𝐅 and where the rainfall per year is about 𝟐𝟐𝟓𝟓 inches.

2. Use the line provided to predict the mean number of inches of rain per year for a city that has a mean temperature
of 𝟖𝟖𝟎𝟎°𝐅𝐅 in July.

Predicted rainfall is 𝟐𝟐𝟐𝟐 inches of rain per year. (Some students will state approximately 𝟐𝟐𝟐𝟐.𝟓𝟓 inches of rain.)

3. Do you think the line provided is a good one for this scatter plot? Explain your answer.

Yes. The line follows the general pattern in the scatter plot, and it does not look like there is another area in the
scatter plot where the points would be any closer to the line.

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Problem Set Sample Solutions

1. The table below shows the mean temperature in July and the mean amount of rainfall per year for 𝟏𝟏𝟏𝟏 cities in the
Midwest.

City
Mean Temperature in July

(degrees Fahrenheit)
Mean Rainfall per Year

(inches)

Chicago, IL 𝟖𝟖𝟐𝟐.𝟐𝟐 𝟐𝟐𝟏𝟏.𝟐𝟐𝟖𝟖
Cleveland, OH 𝟖𝟖𝟏𝟏.𝟏𝟏 𝟐𝟐𝟖𝟖.𝟖𝟖𝟏𝟏
Columbus, OH 𝟖𝟖𝟓𝟓.𝟏𝟏 𝟐𝟐𝟖𝟖.𝟓𝟓𝟐𝟐
Des Moines, IA 𝟖𝟖𝟏𝟏.𝟏𝟏 𝟐𝟐𝟏𝟏.𝟖𝟖𝟐𝟐
Detroit, MI 𝟖𝟖𝟐𝟐.𝟓𝟓 𝟐𝟐𝟐𝟐.𝟖𝟖𝟏𝟏
Duluth, MN 𝟏𝟏𝟓𝟓.𝟓𝟓 𝟐𝟐𝟏𝟏.𝟎𝟎𝟎𝟎
Grand Rapids, MI 𝟖𝟖𝟏𝟏.𝟏𝟏 𝟐𝟐𝟖𝟖.𝟏𝟏𝟐𝟐
Indianapolis, IN 𝟖𝟖𝟓𝟓.𝟏𝟏 𝟏𝟏𝟎𝟎.𝟏𝟏𝟓𝟓
Marquette, MI 𝟖𝟖𝟏𝟏.𝟏𝟏 𝟐𝟐𝟐𝟐.𝟏𝟏𝟓𝟓
Milwaukee, WI 𝟖𝟖𝟐𝟐.𝟎𝟎 𝟐𝟐𝟏𝟏.𝟖𝟖𝟏𝟏
Minneapolis–St. Paul, MN 𝟖𝟖𝟐𝟐.𝟐𝟐 𝟐𝟐𝟏𝟏.𝟏𝟏𝟏𝟏
Springfield, MO 𝟖𝟖𝟏𝟏.𝟐𝟐 𝟐𝟐𝟓𝟓.𝟓𝟓𝟏𝟏
St. Louis, MO 𝟖𝟖𝟎𝟎.𝟐𝟐 𝟐𝟐𝟖𝟖.𝟖𝟖𝟓𝟓
Rapid City, SD 𝟖𝟖𝟐𝟐.𝟎𝟎 𝟐𝟐𝟐𝟐.𝟐𝟐𝟏𝟏

Data Source: http://countrystudies.us/united-states/weather/

a. What do you observe from looking at the data in the table?

Answers will vary. Many of the temperatures were in the 𝟖𝟖𝟎𝟎’s, and many of the mean inches of rain were in
the 𝟐𝟐𝟎𝟎’s. It also appears that, in general, as the rainfall increased, the mean temperature also increased.

b. Look at the scatter plot below. A line is drawn to fit the data. The plot in the Exit Ticket had the mean July
temperatures for the cities on the horizontal axis. How is this plot different, and what does it mean for the
way you think about the relationship between the two variables—temperature and rain?

July Rainfall and Temperatures in Selected Midwestern Cities

This scatter plot has the labels on the axes reversed: (mean inches of rain, mean temperature). This is the
scatter plot I would use if I wanted to predict the mean temperature in July knowing the mean amount of rain
per year.

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c. The line has been drawn to model the relationship between the amount of rain and the temperature in those
Midwestern cities. Use the line to predict the mean July temperature for a Midwestern city that has a mean
of 𝟐𝟐𝟐𝟐 inches of rain per year.

Answers will vary. For 𝟐𝟐𝟐𝟐 𝐢𝐢𝐢𝐢. of rain per year, the line indicates a mean July temperature of approximately
𝟖𝟖𝟎𝟎°𝐅𝐅.

d. For which of the cities in the sample does the line do the worst job of predicting the mean temperature? The
best? Explain your reasoning with as much detail as possible.

Answers will vary. I looked for points that were really close to the line and ones that were far away. The line
prediction for temperature would be farthest off for Minneapolis. For 𝟐𝟐𝟏𝟏.𝟏𝟏𝟏𝟏 𝐢𝐢𝐢𝐢. of rain in Minneapolis, the
line predicted approximately 𝟏𝟏𝟖𝟖°𝐅𝐅, whereas the actual mean temperature in July was 𝟖𝟖𝟐𝟐.𝟐𝟐°𝐅𝐅. The line
predicted very well for Milwaukee. For 𝟐𝟐𝟐𝟐.𝟏𝟏𝟓𝟓 𝐢𝐢𝐢𝐢. of rain in Milwaukee, the line predicted approximately
𝟖𝟖𝟐𝟐°𝐅𝐅, whereas the actual mean temperature in July was 𝟖𝟖𝟐𝟐°𝐅𝐅 and was only off by about 𝟏𝟏°𝐅𝐅. The line was
also close for Marquette. For 𝟐𝟐𝟏𝟏.𝟖𝟖𝟏𝟏 𝐢𝐢𝐢𝐢. of rain in Marquette, the line predicted approximately 𝟖𝟖𝟏𝟏°𝐅𝐅, whereas
the actual mean temperature in July was 𝟖𝟖𝟏𝟏.𝟏𝟏°𝐅𝐅 and was only off by about 𝟏𝟏°𝐅𝐅.

2. The scatter plot below shows the results of a survey of eighth-grade students who were asked to report the number
of hours per week they spend playing video games and the typical number of hours they sleep each night.

Mean Hours Sleep per Night Versus Mean Hours Playing Video Games per Week

a. What trend do you observe in the data?

The more hours that students play video games, the fewer hours they tend to sleep.

b. What was the fewest number of hours per week that students who were surveyed spent playing video
games? The most?

Two students spent 𝟎𝟎 hours, and one student spent 𝟐𝟐𝟐𝟐 hours per week playing games.

c. What was the fewest number of hours per night that students who were surveyed typically slept? The most?

The fewest hours of sleep per night was around 𝟓𝟓 hours, and the most was around 𝟏𝟏𝟎𝟎 hours.

Video Game Time (hours per week)

Sl
ee

p
Ti

m
e

(h
ou

rs
p

er
n

ig
ht

)

35302520151050

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 8

Video Game Time (hours per week)
Sl

ee
p

Ti
m

e
(h

ou
rs

p
er

n
ig

ht
)

35302520151050

d. Draw a line that seems to fit the trend in the data, and find its equation. Use the line to predict the number
of hours of sleep for a student who spends about 𝟏𝟏𝟓𝟓 hours per week playing video games.

Answers will vary. A student
who spent 𝟏𝟏𝟓𝟓 hours per week
playing games would get about
𝟖𝟖 hours of sleep per night.

3. Scientists can take very good pictures of alligators from airplanes or helicopters. Scientists in Florida are interested
in studying the relationship between the length and the weight of alligators in the waters around Florida.

a. Would it be easier to collect data on length or weight? Explain your thinking.

Answers will vary. You could measure the length from the pictures, but you would have to actually have the
alligators to weigh them.

b. Use your answer to decide which variable you would want to put on the horizontal axis and which variable
you might want to predict.

You would probably want to predict the weight of the alligator knowing the length; therefore, the length
would go on the horizontal axis and the weight on the vertical axis.

4. Scientists captured a small sample of alligators and measured both their length (in inches) and weight (in pounds).
Torre used their data to create the following scatter plot and drew a line to capture the trend in the data. She and
Steve then had a discussion about the way the line fit the data. What do you think they were discussing, and why?

Alligator Length (inches) and Weight (pounds)

Data Source: James Landwehr and Ann Watkins, Exploring Data, Quantitative Literacy Series
(Dale Seymour, 1987).

Answers will vary. Sample response: The pattern in the scatter plot is curved instead of linear. All of the data points
in the middle of the scatter plot fall below the line, and the line does not really capture the pattern in the scatter
plot. A line does not pass through the cluster of points between 𝟏𝟏𝟎𝟎 to 𝟖𝟖𝟎𝟎 𝐢𝐢𝐢𝐢. in length that fit the other points.
A model other than a line might be a better fit.

Length (inches)

W
ei

gh
t

(p
ou

nd
s)

15014013012011010090807060500

700
650
600
550
500
450
400
350
300
250
200
150
100
50
0

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 9

Lesson 9: Determining the Equation of a Line Fit to Data

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Lesson 9: Determining the Equation of a Line Fit to Data

Student Outcomes

 Students informally fit a straight line to data displayed in a scatter plot.
 Students determine the equation of a line fit to data.

 Students make predictions based on the equation of a line fit to data.

Lesson Notes
In this lesson, students informally fit a line to data by drawing a line that describes a linear pattern in a scatter plot and
then use their lines to make predictions. They determine the equation of the line and informally analyze different lines
fit to the same data. This lesson begins developing the foundation for finding an objective way to judge how well a line
fits the trend in a scatter plot and the notion of a best-fit line in Algebra I.

Classwork

Example 1 (5 minutes): Crocodiles and Alligators

Discuss the data presented in the table and scatter plot. Consider starting by asking if students are familiar with
crocodiles and alligators and how they differ. Ask students if they can imagine what a bite force of 100 pounds would
feel like. Ask them if they know what body mass indicates. If students understand that body mass is an indication of the
weight of a crocodilian and bite force is a measure of the strength of a crocodilian’s bite, the data can be investigated
even if they do not understand the technical definitions and how these variables are measured. Also, ask students if any
other aspects of the data surprised them. For example, did they realize that there are so many different species of
crocodilians? Did the wide range of body mass and bite force surprise them? If time permits, suggest that students do
further research on crocodilians.

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Lesson 9: Determining the Equation of a Line Fit to Data

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Body Mass (pounds)

Bi
te

F
or

ce
(

po
un

ds
)

4003002001000

1800

1600

1400

1200

1000

800

600

400

200

Example 1: Crocodiles and Alligators

Scientists are interested in finding out how different species adapt to finding food sources. One group studied
crocodilians to find out how their bite force was related to body mass and diet. The table below displays the information
they collected on body mass (in pounds) and bite force (in pounds).

Crocodilian Biting

Species Body Mass (pounds) Bite Force (pounds)
Dwarf crocodile 𝟑𝟑𝟑𝟑 𝟒𝟒𝟑𝟑𝟒𝟒

Crocodile F 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐𝟒𝟒
Alligator A 𝟑𝟑𝟒𝟒 𝟐𝟐𝟑𝟑𝟒𝟒
Caiman A 𝟐𝟐𝟐𝟐 𝟐𝟐𝟑𝟑𝟒𝟒
Caiman B 𝟑𝟑𝟑𝟑 𝟐𝟐𝟒𝟒𝟒𝟒
Caiman C 𝟒𝟒𝟑𝟑 𝟐𝟐𝟑𝟑𝟑𝟑

Crocodile A 𝟏𝟏𝟏𝟏𝟒𝟒 𝟑𝟑𝟑𝟑𝟒𝟒
Nile crocodile 𝟐𝟐𝟑𝟑𝟑𝟑 𝟐𝟐𝟑𝟑𝟒𝟒

Crocodile B 𝟏𝟏𝟑𝟑𝟒𝟒 𝟑𝟑𝟒𝟒𝟒𝟒
Crocodile C 𝟏𝟏𝟑𝟑𝟑𝟑 𝟐𝟐𝟒𝟒𝟒𝟒
Crocodile D 𝟏𝟏𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑𝟒𝟒
Caiman D 𝟏𝟏𝟐𝟐𝟑𝟑 𝟑𝟑𝟑𝟑𝟒𝟒

Indian gharial crocodile 𝟐𝟐𝟐𝟐𝟑𝟑 𝟒𝟒𝟒𝟒𝟒𝟒
Crocodile G 𝟐𝟐𝟐𝟐𝟒𝟒 𝟏𝟏,𝟒𝟒𝟒𝟒𝟒𝟒

American crocodile 𝟐𝟐𝟑𝟑𝟒𝟒 𝟗𝟗𝟒𝟒𝟒𝟒
Crocodile E 𝟐𝟐𝟐𝟐𝟑𝟑 𝟑𝟑𝟑𝟑𝟒𝟒
Crocodile F 𝟒𝟒𝟐𝟐𝟑𝟑 𝟏𝟏,𝟐𝟐𝟑𝟑𝟒𝟒

American alligator 𝟑𝟑𝟒𝟒𝟒𝟒 𝟏𝟏,𝟏𝟏𝟑𝟑𝟒𝟒
Alligator B 𝟑𝟑𝟐𝟐𝟑𝟑 𝟏𝟏,𝟐𝟐𝟒𝟒𝟒𝟒
Alligator C 𝟑𝟑𝟐𝟐𝟑𝟑 𝟏𝟏,𝟒𝟒𝟑𝟑𝟒𝟒

Data Source: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0031781#pone-0031781-t001

(Note: Body mass and bite force have been converted to pounds from kilograms and newtons, respectively.)

As you learned in the previous lesson, it is a good idea to begin by looking at what a scatter plot tells you about the data.
The scatter plot below displays the data on body mass and bite force for the crocodilians in the study.

Scaffolding:
 The word crocodilian

refers to any reptile of the
order Crocodylia.

 This includes crocodiles,
alligators, caimans, and
gavials. Showing students
a visual aid with pictures
of these animals may help
them understand.

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Body Mass (pounds)

Bi
te

F
or

ce
(

po
un

ds
)

400350300250200150100500

1800

1600

1400

1200

1000

800

600

400

200

Sol’s Line

Body Mass (pounds)

Bi
te

F
or

ce
(

po
un

ds
)

400350300250200150100500

1800

1600

1400

1200

1000

800

600

400

200

Marrisa’s Line

Body Mass (pounds)

Bi
te

F
or

ce
(

po
u n

ds
)

400350300250200150100500

1800

1600

1400

1200

1000

800

600

400

200

Patti’s Line

Exercises 1–5 (14 minutes)

Exercises 1 through 5 ask students to consider the fit of a line. Each student (or small group of students) draws a line
that would be a good representation of the trend in the data. Students evaluate their lines and the lines of the four
students introduced in Exercise 4.

In Exercise 2, students draw a line they think is a good representation of the trend in the data. Ask them to compare
their lines with other students’. As a group, decide who might have the best line, and ask students why they made that
choice. Have groups share their ideas. Point out that it would be helpful to agree on a standard method for judging the
fit of a line. One method is to look at how well the line predicts for the given data or how often it is over or under the
actual or observed value.

Exercises 1–6

1. Describe the relationship between body mass and bite force for the crocodilians shown in
the scatter plot.

As the body mass increases, the bite force tends to also increase.

2. Draw a line to represent the trend in the data. Comment on what you considered in drawing your line.

The line should be as close as possible to the points in the scatter plot. Students explored this idea in Lesson 8.

3. Based on your line, predict the bite force for a crocodilian that weighs 𝟐𝟐𝟐𝟐𝟒𝟒 pounds. How does this prediction
compare to the actual bite force of the 𝟐𝟐𝟐𝟐𝟒𝟒-pound crocodilian in the data set?

Answers will vary. A reasonable prediction is around 𝟐𝟐𝟑𝟑𝟒𝟒 to 𝟑𝟑𝟒𝟒𝟒𝟒 pounds. The actual bite force was 𝟏𝟏,𝟒𝟒𝟒𝟒𝟒𝟒 pounds,
so the prediction based on the line was not very close for this crocodilian.

4. Several students decided to draw lines to represent the trend in the data. Consider the lines drawn by Sol, Patti,
Marrisa, and Taylor, which are shown below.

MP.2

Scaffolding:
Point out to English language
learners that the terms body
mass and weight are used
interchangeably in this lesson.

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Lesson 9: Determining the Equation of a Line Fit to Data

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Body Mass (pounds)

Bi
te

F
or

ce
(

po
un

ds
)

400350300250200150100500

1800

1600

1400

1200

1000

800

600

400

200

Dwarf Croc

Indian Gharial Croc

For each student, indicate whether or not you think the line would be a good line to use to make predictions.
Explain your thinking.

a. Sol’s line

In general, it looks like Sol’s line overestimates the bite force for heavier crocodilians and underestimates the
bite force for crocodilians that do not weigh as much.

b. Patti’s line

Patti’s line looks like it fits the data well, so it would probably produce good predictions. The line goes
through the middle of the points in the scatter plot, and the points are fairly close to the line.

c. Marrisa’s line

It looks like Marrisa’s line overestimates the bite force because almost all of the points are below the line.

d. Taylor’s line

It looks like Taylor’s line tends to underestimate the bite force. There are many points above the line.

5. What is the equation of your line? Show the steps you used to determine your line. Based on your equation, what
is your prediction for the bite force of a crocodilian weighing 𝟐𝟐𝟒𝟒𝟒𝟒 pounds?

Answers will vary. Students have learned from previous modules how to find the equation of a line. Anticipate
students to first determine the slope based on two points on their lines. Students then use a point on the line to
obtain an equation in the form 𝒚𝒚 = 𝒎𝒎𝒎𝒎 + 𝒃𝒃 (or 𝒚𝒚 = 𝒂𝒂 + 𝒃𝒃𝒎𝒎). Students use their lines to predict a bite force for a
crocodilian that weighs 𝟐𝟐𝟒𝟒𝟒𝟒 pounds. A reasonable answer would be around 𝟐𝟐𝟒𝟒𝟒𝟒 pounds.

6. Patti drew vertical line segments from two points to the line in her scatter plot. The first point she selected was for
a dwarf crocodile. The second point she selected was for an Indian gharial crocodile.

a. Would Patti’s line have resulted in a predicted bite force that was closer to the actual bite force for the dwarf
crocodile or for the Indian gharial crocodile? What aspect of the scatter plot supports your answer?

The prediction would be closer to the actual bite force for the dwarf crocodile. That point is closer to the line
(the vertical line segment connecting it to the line is shorter) than the point for the Indian gharial crocodile.

b. Would it be preferable to describe the trend in a scatter plot using a line that makes the differences in the
actual and predicted values large or small? Explain your answer.

It would be better for the differences to be as small as possible. Small differences are closer to the line.

MP.2

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Exercise 7 (14 minutes): Used Cars

This exercise provides additional practice for students. Students use the equation of a line to make predictions and
informally assess the fit of the line.

Exercise 7: Used Cars

7. Suppose the plot below shows the age (in years) and price (in dollars) of used compact cars that were advertised in a
local newspaper.

a. Based on the scatter plot above, describe the relationship between the age and price of the used cars.

The older the car, the lower the price tends to be.

b. Nora drew a line she thought was close to many of the points and found the equation of the line. She used
the points (𝟏𝟏𝟑𝟑,𝟐𝟐𝟒𝟒𝟒𝟒𝟒𝟒) and (𝟑𝟑,𝟏𝟏𝟐𝟐𝟒𝟒𝟒𝟒𝟒𝟒) on her line to find the equation. Explain why those points made
finding the equation easy.

The points are at the intersection of the grid lines in the graph, so it is easy to determine the coordinates of
these points.

c. Find the equation of Nora’s line for predicting the price of a used car given its age. Summarize the trend
described by this equation.

Using the points, the equation is 𝒚𝒚 = −𝟏𝟏𝟒𝟒𝟒𝟒𝟒𝟒𝒎𝒎+ 𝟏𝟏𝟗𝟗𝟒𝟒𝟒𝟒𝟒𝟒, or 𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏 = −𝟏𝟏𝟒𝟒𝟒𝟒𝟒𝟒(𝐚𝐚𝐚𝐚𝐏𝐏) + 𝟏𝟏𝟗𝟗𝟒𝟒𝟒𝟒𝟒𝟒. The slope of
the line is negative, so the line indicates that the price of used cars decreases as cars get older.

Age (years)

Pr
ic

e
(d

ol
la

rs
)

1817161514131211109876543210

16000
15000
14000
13000
12000
11000
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000

Age (years)

Pr
ic

e
(d

ol
la

rs
)

1817161514131211109876543210

16000
15000
14000
13000
12000
11000
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000

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d. Based on the line, for which car in the data set would the predicted value be farthest from the actual value?
How can you tell?

It would be farthest for the car that is 𝟏𝟏𝟒𝟒 years old. It is the point in the scatter plot that is farthest from the
line.

e. What does the equation predict for the cost of a 𝟏𝟏𝟒𝟒-year-old car? How close was the prediction using the line
to the actual cost of the 𝟏𝟏𝟒𝟒-year-old car in the data set? Given the context of the data set, do you think the
difference between the predicted price and the actual price is large or small?

The line predicts a 𝟏𝟏𝟒𝟒-year-old car would cost about $𝟗𝟗,𝟒𝟒𝟒𝟒𝟒𝟒. −𝟏𝟏𝟒𝟒𝟒𝟒𝟒𝟒(𝟏𝟏𝟒𝟒) + 𝟏𝟏𝟗𝟗𝟒𝟒𝟒𝟒𝟒𝟒 = 𝟗𝟗𝟒𝟒𝟒𝟒𝟒𝟒. Compared to
$𝟒𝟒,𝟒𝟒𝟒𝟒𝟒𝟒 for the 𝟏𝟏𝟒𝟒-year-old car in the data set, the difference would be $𝟒𝟒,𝟗𝟗𝟐𝟐𝟒𝟒. The prediction is off by
about $𝟑𝟑,𝟒𝟒𝟒𝟒𝟒𝟒, which seems like a lot of money, given the prices of the cars in the data set.

f. Is $𝟑𝟑,𝟒𝟒𝟒𝟒𝟒𝟒 typical of the differences between predicted prices and actual prices for the cars in this data set?
Justify your answer.

No, most of the differences would be much smaller than $𝟑𝟑,𝟒𝟒𝟒𝟒𝟒𝟒. Most of the points are much closer to the
line, and most predictions would be within about $𝟏𝟏,𝟒𝟒𝟒𝟒𝟒𝟒 of the actual value.

Closing (2 minutes)

 When you use a line to describe a linear relationship in a data set, what are characteristics of a good fit?

 The line should be as close as possible to the points in the scatter plot. The line should go through the
“middle” of the points.

Exit Ticket (5 minutes)

Lesson Summary

 A line can be used to represent the trend in a scatter plot.

 Evaluating the equation of the line for a value of the independent variable determines a value predicted
by the line.

 A good line for prediction is one that goes through the middle of the points in a scatter plot and for
which the points tend to fall close to the line.

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Name ___________________________________________________ Date____________________

Lesson 9: Determining the Equation of a Line Fit to Data

Exit Ticket

1. The scatter plot below shows the height and speed of some of the world’s fastest roller coasters. Draw a line that

you think is a good fit for the data.

Data Source: http://rcdb.com/rhr.htm

2. Find the equation of your line. Show your steps.

3. For the two roller coasters identified in the scatter plot, use the line to find the approximate difference between the

observed speeds and the predicted speeds.

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Exit Ticket Sample Solutions

1. The scatter plot below shows the height and speed of some of the world’s fastest roller coasters. Draw a line that
you think is a good fit for the data.

Students would draw a line based on the goal of a best fit for the given scatter plot. A possible line is drawn below.

Data Source: http://rcdb.com/rhr.htm

2. Find the equation of your line. Show your steps.

Answers will vary based on the line drawn. Let 𝑺𝑺 equal the speed of the roller coaster and 𝑯𝑯 equal the maximum
height of the roller coaster.

𝒎𝒎 =
𝟏𝟏𝟏𝟏𝟑𝟑 − 𝟐𝟐𝟑𝟑
𝟑𝟑𝟒𝟒𝟒𝟒 − 𝟐𝟐𝟐𝟐𝟑𝟑

≈ 𝟒𝟒.𝟏𝟏𝟏𝟏

𝑺𝑺 = 𝟒𝟒.𝟏𝟏𝟏𝟏𝑯𝑯 + 𝒃𝒃

𝟐𝟐𝟑𝟑 = 𝟒𝟒.𝟏𝟏𝟏𝟏(𝟐𝟐𝟐𝟐𝟑𝟑) + 𝒃𝒃
𝒃𝒃 ≈ 𝟐𝟐𝟒𝟒

Therefore, the equation of the line drawn in Problem 1 is 𝑺𝑺 = 𝟒𝟒.𝟏𝟏𝟏𝟏𝑯𝑯 + 𝟐𝟐𝟒𝟒.

3. For the two roller coasters identified in the scatter plot, use the line to find the approximate difference between the
observed speeds and the predicted speeds.

Answers will vary depending on the line drawn by a student or the equation of the line. For the Top Thrill, the
maximum height is about 𝟒𝟒𝟏𝟏𝟑𝟑 feet and the speed is about 𝟏𝟏𝟒𝟒𝟒𝟒 miles per hour. The line indicated in Problem 2
predicts a speed of 𝟏𝟏𝟒𝟒𝟐𝟐 miles per hour, so the difference is about 𝟐𝟐 miles per hour over the actual speed. For the
Kinga Ka, the maximum height is about 𝟒𝟒𝟐𝟐𝟒𝟒 feet with a speed of 𝟏𝟏𝟐𝟐𝟒𝟒 miles per hour. The line predicts a speed of
about 𝟏𝟏𝟒𝟒𝟑𝟑 miles per hour, for a difference of 𝟏𝟏𝟑𝟑 miles per hour under the actual speed. (Students can use the graph
or the equation to find the predicted speed.)

Maximum Height (feet)

Sp
ee

d
(m

ph
)

5004504003503002502000

130

120

110

100

Kinga Ka

Top Thrill

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Problem Set Sample Solutions

1. The Monopoly board game is popular in many countries. The scatter plot below shows the distance from “Go” to a
property (in number of spaces moving from “Go” in a clockwise direction) and the price of the properties on the
Monopoly board. The equation of the line is 𝑷𝑷 = 𝟐𝟐𝒎𝒎 + 𝟒𝟒𝟒𝟒, where 𝑷𝑷 represents the price (in Monopoly dollars) and
𝒎𝒎 represents the distance (in number of spaces).

Distance from “Go”
(number of spaces)

Price of Property
(Monopoly dollars)

Distance from “Go”
(number of spaces)

Price of Property
(Monopoly dollars)

𝟏𝟏 𝟐𝟐𝟒𝟒 𝟐𝟐𝟏𝟏 𝟐𝟐𝟐𝟐𝟒𝟒
𝟑𝟑 𝟐𝟐𝟒𝟒 𝟐𝟐𝟑𝟑 𝟐𝟐𝟐𝟐𝟒𝟒
𝟑𝟑 𝟐𝟐𝟒𝟒𝟒𝟒 𝟐𝟐𝟒𝟒 𝟐𝟐𝟒𝟒𝟒𝟒
𝟐𝟐 𝟏𝟏𝟒𝟒𝟒𝟒 𝟐𝟐𝟑𝟑 𝟐𝟐𝟒𝟒𝟒𝟒
𝟐𝟐 𝟏𝟏𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐𝟒𝟒
𝟗𝟗 𝟏𝟏𝟐𝟐𝟒𝟒 𝟐𝟐𝟑𝟑 𝟐𝟐𝟐𝟐𝟒𝟒
𝟏𝟏𝟏𝟏 𝟏𝟏𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟏𝟏𝟑𝟑𝟒𝟒
𝟏𝟏𝟐𝟐 𝟏𝟏𝟑𝟑𝟒𝟒 𝟐𝟐𝟗𝟗 𝟐𝟐𝟐𝟐𝟒𝟒
𝟏𝟏𝟑𝟑 𝟏𝟏𝟒𝟒𝟒𝟒 𝟑𝟑𝟏𝟏 𝟑𝟑𝟒𝟒𝟒𝟒
𝟏𝟏𝟒𝟒 𝟏𝟏𝟐𝟐𝟒𝟒 𝟑𝟑𝟐𝟐 𝟑𝟑𝟒𝟒𝟒𝟒
𝟏𝟏𝟑𝟑 𝟐𝟐𝟒𝟒𝟒𝟒 𝟑𝟑𝟒𝟒 𝟑𝟑𝟐𝟐𝟒𝟒
𝟏𝟏𝟐𝟐 𝟏𝟏𝟐𝟐𝟒𝟒 𝟑𝟑𝟑𝟑 𝟐𝟐𝟒𝟒𝟒𝟒
𝟏𝟏𝟐𝟐 𝟏𝟏𝟐𝟐𝟒𝟒 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑𝟒𝟒
𝟏𝟏𝟗𝟗 𝟐𝟐𝟒𝟒𝟒𝟒 𝟑𝟑𝟗𝟗 𝟒𝟒𝟒𝟒𝟒𝟒

Price of Property Versus Distance from “Go” in Monopoly

a. Use the equation to find the difference (observed value−predicted value) for the most expensive property
and for the property that is 𝟑𝟑𝟑𝟑 spaces from “Go.”

The most expensive property is 𝟑𝟑𝟗𝟗 spaces from “Go” and costs $𝟒𝟒𝟒𝟒𝟒𝟒. The price predicted by the line would be
𝟐𝟐(𝟑𝟑𝟗𝟗) + 𝟒𝟒𝟒𝟒, or $𝟑𝟑𝟑𝟑𝟐𝟐. Observed price − predicted price would be $𝟒𝟒𝟒𝟒𝟒𝟒 − $𝟑𝟑𝟑𝟑𝟐𝟐 = $𝟒𝟒𝟐𝟐. The price predicted
for 𝟑𝟑𝟑𝟑 spaces from “Go” would be 𝟐𝟐(𝟑𝟑𝟑𝟑) + 𝟒𝟒𝟒𝟒, or $𝟑𝟑𝟐𝟐𝟒𝟒. Observed price − predicted price would be $𝟐𝟐𝟒𝟒𝟒𝟒 −
$𝟑𝟑𝟐𝟐𝟒𝟒 = −$𝟏𝟏𝟐𝟐𝟒𝟒.

b. Five of the points seem to lie in a horizontal line. What do these points have in common? What is the
equation of the line containing those five points?

These points all have the same price. The equation of the horizontal line through those points would be
𝑷𝑷 = 𝟐𝟐𝟒𝟒𝟒𝟒.

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c. Four of the five points described in part (b) are the railroads. If you were fitting a line to predict price with
distance from “Go,” would you use those four points? Why or why not?

Answers will vary. Because the four points are not part of the overall trend in the price of the properties, I
would not use them to determine a line that describes the relationship. I can show this by finding the total
error to measure the fit of the line.

2. The table below gives the coordinates of the five points shown in the scatter plots that follow. The scatter plots
show two different lines.

Data Point Independent Variable Response Variable
A 𝟐𝟐𝟒𝟒 𝟐𝟐𝟑𝟑
B 𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏
C 𝟐𝟐𝟑𝟑 𝟐𝟐𝟒𝟒
D 𝟑𝟑𝟏𝟏 𝟏𝟏𝟐𝟐
E 𝟒𝟒𝟒𝟒 𝟏𝟏𝟐𝟐

Line 1 Line 2

a. Find the predicted response values for each of the two lines.

Independent Observed Response Response Predicted
by Line 1

Response Predicted
by Line 2

𝟐𝟐𝟒𝟒 𝟐𝟐𝟑𝟑 𝟐𝟐𝟑𝟑 𝟐𝟐𝟐𝟐
𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏 𝟐𝟐𝟑𝟑.𝟐𝟐 𝟐𝟐𝟒𝟒.𝟐𝟐
𝟐𝟐𝟑𝟑 𝟐𝟐𝟒𝟒 𝟐𝟐𝟐𝟐.𝟑𝟑 𝟐𝟐𝟐𝟐.𝟑𝟑
𝟑𝟑𝟏𝟏 𝟏𝟏𝟐𝟐 𝟏𝟏𝟑𝟑.𝟏𝟏 𝟏𝟏𝟐𝟐.𝟑𝟑
𝟒𝟒𝟒𝟒 𝟏𝟏𝟐𝟐 𝟗𝟗 𝟏𝟏𝟐𝟐

b. For which data points is the prediction based on Line 1 closer to the actual value than the prediction based on
Line 2?

It is only for data point A. For data point C, both lines are off by the same amount.

c. Which line (Line 1 or Line 2) would you select as a better fit? Explain.

Line 2 is a better fit because it is closer to more of the data points.

Independent Variable

Re
sp

on
se

V
ar

ia
bl

40353025200

Independent Variable

Re
sp

on
se

V
ar

ia
bl

40353025200

𝒚𝒚 = −𝟒𝟒.𝟗𝟗𝒎𝒎 + 𝟒𝟒𝟑𝟑 𝒚𝒚 = −𝟒𝟒.𝟑𝟑𝒎𝒎 + 𝟒𝟒𝟒𝟒

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Body Mass (pounds)

Bi
te

F
or

ce
(

po
u n

d s
)

400350300250200150100500

1800

1600

1400

1200

1000

800

600

400

200

Line 1

Body Mass (pounds)

Bi
te

F
or

ce
(

po
un

ds
)

400350300250200150100500

1800

1600

1400

1200

1000

800

600

400

200

Line 2

3. The scatter plots below show different lines that students used to model the relationship between body mass (in
pounds) and bite force (in pounds) for crocodilians.

a. Match each graph to one of the equations below, and explain your reasoning. Let 𝑩𝑩 represent bite force (in
pounds) and 𝑾𝑾 represent body mass (in pounds).

Equation 1

Equation 2 Equation 3

𝑩𝑩 = 𝟑𝟑.𝟐𝟐𝟐𝟐𝑾𝑾 + 𝟏𝟏𝟐𝟐𝟐𝟐 𝑩𝑩 = 𝟑𝟑.𝟒𝟒𝟒𝟒𝑾𝑾 + 𝟑𝟑𝟑𝟑𝟏𝟏 𝑩𝑩 = 𝟐𝟐.𝟏𝟏𝟐𝟐𝑾𝑾 + 𝟐𝟐𝟐𝟐𝟑𝟑

Equation: 3

The intercept of 𝟐𝟐𝟐𝟐𝟑𝟑 appears to match the
graph, which has the second largest
intercept.

Equation: 2

The intercept of Equation 2 is larger, so it
matches Line 2, which has a 𝒚𝒚-intercept
closer to 𝟒𝟒𝟒𝟒𝟒𝟒.

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Equation: 1

The intercept of Equation 1 is the smallest,
which seems to match the graph.

b. Which of the lines would best fit the trend in the data? Explain your thinking.

Answers will vary. Line 3 would be better than the other two lines. Line 1 is not a good fit for larger weights,
and Line 2 is above nearly all of the points and pretty far away from most of them. It looks like Line 3 would
be closer to most of the points.

4. Comment on the following statements:

a. A line modeling a trend in a scatter plot always goes through the origin.

Some trend lines go through the origin, but others may not. Often, the value (𝟒𝟒,𝟒𝟒) does not make sense for
the data.

b. If the response variable increases as the independent variable decreases, the slope of a line modeling the
trend is negative.

If the trend is from the upper left to the lower right, the slope for the line is negative because for each unit
increase in the independent variable, the response decreases.

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8•6 Mid-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

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Name Date

1. Many computers come with a Solitaire card game. The player moves cards in certain ways to complete

specific patterns. The goal is to finish the game in the shortest number of moves possible, and a player’s
score is determined by the number of moves. A statistics teacher played the game 16 times and recorded
the number of moves and the final score after each game. The line represents the linear function that is
used to determine the score from the number of moves.

a. Was this person’s average score closer to 1130 or 1110? Explain how you decided.

b. The first two games she played took 169 moves (1131 points) and 153 moves (1147 points). Based
on this information, determine the equation of the linear function used by the computer to calculate
the score from the number of moves. Explain your work.

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c. Based on the linear function, each time the player makes a move, how many points does she lose?

d. Based on the linear function, how many points does the player start with in this game? Explain your
reasoning.

2. To save money, drivers often try to increase their mileage, which is measured in miles per gallon (mpg).
One theory is that speed traveled impacts mileage. Suppose the following data are recorded for five
different 300-mile tests, with the car traveling at different speeds in miles per hour (mph) for each test.

Speed (mph) Mileage
50 32
60 29
70 24
80 20
90 17

a. For the data in this table, is the association positive or negative? Explain how you decided.

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b. Construct a scatter plot of these data using the following coordinate grid. The vertical axis
represents the mileage, and the horizontal axis represents the speed in miles per hour (mph).

c. Draw a line on your scatter plot that you think is a reasonable model for predicting the mileage from
the car speed.

d. Estimate and interpret the slope of the line you found in part (c).

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Suppose additional data were measured for three more tests. These results have been added to the
previous tests, and the combined data are shown in the table below.

Speed (mph) Mileage
20 25
30 27
40 30
50 32
60 29
70 24
80 20
90 17

e. Does the association for these data appear to be linear? Why or why not?

f. If your only concern was mileage and you had no traffic constraints, what speed would you
recommend traveling based on these data? Explain your choice.

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A Progression Toward Mastery

Assessment
Task Item

STEP 1
Missing or
incorrect answer
and little
evidence of
reasoning or
application of
mathematics to
solve the problem

STEP 2
Missing or incorrect
answer but evidence
of some reasoning or
application of
mathematics to
solve the problem

STEP 3
A correct answer with
some evidence of
reasoning or
application of
mathematics to solve
the problem,
OR an incorrect
answer with
substantial evidence
of solid reasoning or
application of
mathematics to solve
the problem

STEP 4
A correct answer
supported by
substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem

8.SP.A.1

Student makes no use
of the given data.

Student chooses 1110
based solely on it being
the midpoint of the 𝑦𝑦-axis
values.

Student chooses 1130, but
reasoning is incomplete or
missing.

Student chooses 1130
based on the higher
concentration of red
dots around those 𝑦𝑦-
values.

8.F.B.4

Student cannot obtain
a line.

Student attempts to
estimate a line from the
graph.

Student uses a reasonable
approach but does not
obtain the correct line (e.g.,
interchanges slope and
intercept in the equation,
sets up an inverse of the
slope equation, or shows
insufficient work).

Student finds the
correct equation (or
with minor errors)
from slope =
(1131−1147)
169−153

= −1, and
intercept from 1131 =
𝑎𝑎 − 169, so 𝑎𝑎 = 1300.
Equation:
𝑦𝑦 = 1300− 𝑥𝑥, where
𝑦𝑦 represents points
and 𝑥𝑥 represents
number of moves.

8.F.B.4

Student makes no use
of the given data.

Student does not
recognize this as a
question about slope.

Student estimates the
slope from the graph.

Student reports the
slope (−1) found in
part (b).

8.F.B.4

Student makes no use
of the given data.

Student does not
recognize this as a
question about intercept.

Student estimates the
intercept from the graph or
solves the equation with
𝑥𝑥 = 0 without recognizing
a connection to the
equation.

Student reports the
intercept (1300) found
in part (b).

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8.F.B.4

Student makes no use
of the given data.

Student bases the answer
solely on the content
(e.g., faster cars are less
fuel efficient).

Student refers to the
scatter plot in part (b) or
makes a minor error (e.g.,
misspeaks and describes a
negative association but
appears to unintentionally
call it a positive
association).

Student notes that
mileage values are
decreasing while
speeds (mph) are
increasing and states
that this is a negative
association.
OR
Student solves for the
slope and notes the
sign of the slope.

8.SP.A.1

Student makes no use
of the given data.

Student does not
construct a scatter plot
with the correct number
of dots.

Student constructs a
scatter plot but reverses
the roles of speed and
mileage.

Student constructs a
scatter plot that has
five dots in the correct
locations.

8.SP.A.2

Student does not
answer the question.

Student does not draw a
line but rather connects
the dots.

Student draws a line that
does not reasonably
describe the behavior of
the plotted data.

Student draws a line
that reasonably
describes the behavior
of the plotted data.

8.F.B.4

Student makes no use
of the given data.

Student uses the correct
approach but makes
major calculation errors
(e.g., using only values
from the table or failing to
interpret the slope).

Student uses the correct
approach but makes minor
errors in calculation or in
interpretation.

Student estimates the
coordinates for two
locations and
determines the change
in 𝑦𝑦-values divided by
the change in 𝑥𝑥-values,
for example, (50, 33)
and (80, 20), which
yields �− 13

30
� ≈

−0.433� , and interprets
this as the decrease in
mileage per additional
mph in speed.

8.F.B.5

Student does not
comment on the
increasing or
decreasing pattern in
the values.

Student attempts to
sketch a graph of the data
and comments on the
overall pattern but does
not comment on the
change in the direction of
the association.

Student comments only on
how the change in the
mileage is not constant
without commenting on
the change in the sign of
the differences.

Student comments on
the increasing and
then decreasing
behavior of the
mileage column as the
mileage column
steadily increases.

8.F.B.4

Student does not
answer the question.

Student recommends
55 mph based only on
anecdote and does not
provide any reasoning.

Student recommends a
reasonable speed but does
not fully justify the choice.

Student recommends
and gives justification
for a speed between
40 and 50 mph, or at
50 mph, based on the
association “peaking”
at 50 mph.

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Name Date

1. Many computers come with a Solitaire card game. The player moves cards in certain ways to complete

a. Was this person’s average score closer to 1130 or 1110? Explain how you decided.

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c. Based on the linear function, each time the player makes a move, how many points does she lose?

d. Based on the linear function, how many points does the player start with in this game? Explain your
reasoning.

Speed (mph) Mileage
50 32
60 29
70 24
80 20
90 17

a. For the data in this table, is the association positive or negative? Explain how you decided.

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b. Construct a scatter plot of these data using the following coordinate grid. The vertical axis represents
the mileage and the horizontal axis represents the speed in miles per hour (mph).

c. Draw a line on your scatter plot that you think is a reasonable model for predicting the mileage from
the car speed.

d. Estimate and interpret the slope of the line you found in part (c).

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Suppose additional data were measured for three more tests. These results have been added to the
previous tests, and the combined data are shown in the table below.

Speed (mph) Mileage
20 25
30 27
40 30
50 32
60 29
70 24
80 20
90 17

e. Does the association for these data appear to be linear? Why or why not?

f. If your only concern was mileage and you had no traffic constraints, what speed would you
recommend traveling based on these data? Explain your choice.

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Topic C: Linear and Nonlinear Models

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129

8
G R A D E

New York State Common Core

Mathematics Curriculum
GRADE 8 • MODULE 6

Topic C

Linear and Nonlinear Models

8.SP.A.1, 8.SP.A.2, 8.SP.A.3

Focus Standards: 8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate
patterns of association between two quantities. Describe patterns such as clustering,
outliers, positive or negative association, linear association, and nonlinear association.

8.SP.A.2 Know that straight lines are widely used to model relationships between two
quantitative variables. For scatter plots that suggest a linear association, informally fit a
straight line, and informally assess the model fit by judging the closeness of the data
points to the line.

8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate
measurement data, interpreting the slope and intercept. For example, in a linear model
for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional
hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

Instructional Days: 3

Lesson 10: Linear Models (P)1

Lesson 11: Using Linear Models in a Data Context (P)

Lesson 12: Nonlinear Models in a Data Context (Optional) (P)

In Topic C, students interpret and use linear models. They provide verbal descriptions based on how one
variable changes as the other variable changes (8.SP.A.3). Students identify and describe how one variable
changes as the other variable changes for linear and nonlinear associations. They describe patterns of
positive and negative associations using scatter plots (8.SP.A.1, 8.SP.A.2). In Lesson 10, students identify
applications in which a linear function models the relationship between two numerical variables. In
Lesson 11, students use a linear model to answer questions about the relationship between two numerical
variables by interpreting the context of a data set (8.SP.A.1). Students use graphs and the patterns of linear
association to answer questions about the relationship of the data. In Lesson 12, students also examine
patterns and graphs that describe nonlinear associations of data (8.SP.A.1).

1Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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Lesson 10: Linear Models

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Lesson 10: Linear Models

Student Outcomes

 Students identify situations where it is reasonable to use a linear function to model the relationship between
two numerical variables.

 Students interpret slope and the initial value in a data context.

Lesson Notes
In previous lessons, students were given a set of bivariate data on variables that were linearly related. Students
constructed a scatter plot of the data, informally fit a line to the data, and found the equation of their prediction line.
The lessons also discussed criteria students could use to determine what might be considered the best-fitting prediction
line for a given set of data. A more formal discussion of this topic occurs in Algebra I.

This lesson introduces a formal statistical terminology for the two variables that define a bivariate data set. In a
prediction context, the 𝑥𝑥-variable is referred to as the independent variable, explanatory variable, or predictor variable.
The 𝑦𝑦-variable is referred to as the dependent variable, response variable, or predicted variable. Students should
become equally comfortable with using the pairings (independent, dependent), (explanatory, response), and (predictor,
predicted). Statistics builds on data, and in this lesson, students investigate bivariate data that are linearly related.
Students examine how the dependent variable relates to the independent variable or how the predicted variable relates
to the predictor variable. Students also need to connect the linear function in words to a symbolic form that represents
a linear function. In most cases, the independent variable is denoted by 𝑥𝑥 and the dependent variable by 𝑦𝑦.

Similar to lessons at the beginning of this module, this lesson works with exact linear relationships. This is done to build
conceptual understanding of how structural elements of the modeling equation are explained in context. Students apply
this thinking to more authentic data contexts in the next lesson.

Classwork

In previous lessons, you used data that follow a linear trend either in the positive direction or the
negative direction and informally fit a line through the data. You determined the equation of an
informal fitted line and used it to make predictions.

In this lesson, you use a function to model a linear relationship between two numerical variables
and interpret the slope and intercept of the linear model in the context of the data. Recall that a
function is a rule that relates a dependent variable to an independent variable.

In statistics, a dependent variable is also called a response variable or a predicted variable.
An independent variable is also called an explanatory variable or a predictor variable.

Scaffolding:
 A dependent variable is

also called a response or
predicted variable.

 An independent variable is
also called an explanatory
or a predictor variable.

 It is important to make the
interchangeability of these
terms clear to English
language learners.

 For each of the pairings,
students should have the
chance to read, write,
speak, and hear them on
multiple occasions.

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Example 1 (5 minutes)

This lesson begins by challenging students’ understanding of the terminology. Read through the opening text, and
explain the difference between dependent and independent variables. Pose the question to the class at the end of the
example, and allow for multiple responses.

 What are some other possible numerical independent variables that could relate to how well you are going to
do on the quiz?

 How many hours of sleep I got the night before

Example 1

Predicting the value of a numerical dependent (response) variable based on the value of a given numerical independent
variable has many applications in statistics. The first step in the process is to identify the dependent (predicted) variable
and the independent (predictor) variable.

There may be several independent variables that might be used to predict a given dependent variable. For example,
suppose you want to predict how well you are going to do on an upcoming statistics quiz. One possible independent
variable is how much time you spent studying for the quiz. What are some other possible numerical independent
variables that could relate to how well you are going to do on the quiz?

Exercise 1 (5 minutes)

Exercise 1 requires students to write two possible explanatory variables that might be used for each of several given
response variables. Give students a moment to think about each response variable, and then discuss the answers as a
class. Allow for multiple student responses.

Exercises 1–2

1. For each of the following dependent (response) variables, identify two possible numerical independent
(explanatory) variables that might be used to predict the value of the dependent variable.

Answers will vary. Here again, make sure that students are defining their explanatory variables (predictors)
correctly and that they are numerical.

Response Variable Possible Explanatory Variables

Height of a son
1. Height of the boy’s father
2. Height of the boy’s mother

Number of points scored in a game
by a basketball player

1. Number of shots taken in the game
2. Number of minutes played in the game

Number of hamburgers to make
for a family picnic

1. Number of people in the family
2. Price of hamburger meat

Time it takes a person to run a mile
1. Height above sea level of the track field
2. Number of practice days

Amount of money won by a contestant
on Jeopardy!TM (television game show)

1. IQ of the contestant
2. Number of questions correctly answered

Fuel efficiency (in miles per gallon) for a car
1. Weight of the car
2. Size of the car’s engine

Number of honey bees in a beehive
at a particular time

1. Size of a queen bee
2. Amount of honey harvested from the hive

Number of blooms on a dahlia plant
1. Amount of fertilizer applied to the plant
2. Amount of water applied to the plant

Number of forest fires in a state during a particular year
1. Number of acres of forest in the state
2. Amount of rain in the state that year

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Exercise 2 (5 minutes)

This exercise reverses the format and asks students to provide a response variable for each of several given explanatory
variables. Again, give students a moment to consider each independent variable. Then, discuss the dependent variables
as a class. Allow for multiple student responses.

2. Now, reverse your thinking. For each of the following numerical independent variables, write a possible numerical
dependent variable.

Dependent Variable Possible Independent Variables

Time it takes a student to run a mile Age of a student

Distance a golfer drives a ball from a tee Height of a golfer

Time it takes pain to disappear Amount of a pain reliever taken

Amount of money a person makes in a lifetime Number of years of education

Number of tomatoes harvested in a season Amount of fertilizer used on a garden

Price of a diamond ring Size of a diamond in a ring

A baseball team’s batting average Total salary for all of a team’s players

Example 2 (3–5 minutes)

This example begins the study of an exact linear relationship between two numerical variables. Example 2 and Exercises
3–9 address bivariate data that have an exact functional form, namely, linear. Students become familiar with an
equation of the form 𝑦𝑦 = intercept + (slope)𝑥𝑥. They connect this representation to the equation of a linear function
(𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏 or 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏𝑥𝑥) developed in previous modules. Make sure students clearly identify the slope and the
𝑦𝑦-intercept as they describe a linear function. Students interpret slope as the change in the dependent variable
(the 𝑦𝑦-variable) for an increase of one unit in the independent variable (the 𝑥𝑥-variable).

For example, if exam score = 57 + 8 (study time), or equivalently, 𝑦𝑦 = 57 + 8𝑥𝑥, where 𝑦𝑦 represents the exam score and
𝑥𝑥 represents the study time in hours, then an increase of one hour in study time produces an increase of 8 points in the
predicted exam score. Encourage students to interpret slope in the context of the problem. Their interpretation of
slope as simply “rise over run” is not sufficient in a statistical setting.

Students should become comfortable writing linear models using descriptive words (such as exam score and study time)
or using symbols, such as 𝑥𝑥 and 𝑦𝑦, to represent variables. Using descriptive words when writing model equations can
help students keep the context in mind, which is important in statistics.

Note that bivariate numerical data that do not have an exact linear functional form but do have a linear trend are
covered in the next lesson. Starting with Example 2, this lesson covers only contexts in which the linear relationship is
exact.

Give students a moment to read through Example 2. For English language learners, consider reading the example aloud.

Example 2

A cell phone company offers the following basic cell phone plan to its customers: A customer pays a monthly fee of
$𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒. In addition, the customer pays $𝟒𝟒.𝟏𝟏𝟏𝟏 per text message sent from the cell phone. There is no limit to the
number of text messages per month that could be sent, and there is no charge for receiving text messages.

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Exercises 3–9 (10–15 minutes)

These exercises build on earlier lessons in Module 6. Provide time for students to develop answers to the exercises.
Then, confirm their answers as a class.

Exercises 3–11

3. Determine the following:

a. Justin never sends a text message. What would be his total monthly
cost?

Justin’s monthly cost would be $𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒.

b. During a typical month, Abbey sends 𝟐𝟐𝟏𝟏 text messages. What is her
total cost for a typical month?

Abbey’s monthly cost would be $𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒 + $𝟒𝟒.𝟏𝟏𝟏𝟏(𝟐𝟐𝟏𝟏), or $𝟒𝟒𝟒𝟒.𝟕𝟕𝟏𝟏.

c. Robert sends at least 𝟐𝟐𝟏𝟏𝟒𝟒 text messages a month. What would be an
estimate of the least his total monthly cost is likely to be?

Robert’s monthly cost would be $𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒 + $𝟒𝟒.𝟏𝟏𝟏𝟏(𝟐𝟐𝟏𝟏𝟒𝟒), or $𝟕𝟕𝟕𝟕.𝟏𝟏𝟒𝟒.

4. Use descriptive words to write a linear model describing the relationship between the number of text messages sent
and the total monthly cost.

𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦 𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓 = $𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒+ (𝐦𝐦𝐧𝐧𝐦𝐦𝐧𝐧𝐧𝐧𝐧𝐧 𝐓𝐓𝐨𝐨 𝐓𝐓𝐧𝐧𝐭𝐭𝐓𝐓 𝐦𝐦𝐧𝐧𝐬𝐬𝐬𝐬𝐓𝐓𝐦𝐦𝐧𝐧𝐬𝐬) ⋅ $𝟒𝟒.𝟏𝟏𝟏𝟏

5. Is the relationship between the number of text messages sent and the total monthly cost linear? Explain your
answer.

Yes. For each text message, the total monthly cost goes up by $𝟒𝟒.𝟏𝟏𝟏𝟏. From our previous work with linear functions,
this would indicate a linear relationship.

6. Let 𝒙𝒙 represent the independent variable and 𝒚𝒚 represent the dependent variable. Use the variables 𝒙𝒙 and 𝒚𝒚 to
write the function representing the relationship you indicated in Exercise 4.

Students show the process in developing a model of the relationship between the two variables.
𝒚𝒚 = 𝟒𝟒.𝟏𝟏𝟏𝟏𝒙𝒙 + 𝟒𝟒𝟒𝟒 or 𝒚𝒚 = 𝟒𝟒𝟒𝟒 + 𝟒𝟒.𝟏𝟏𝟏𝟏𝒙𝒙

7. Explain what $𝟒𝟒.𝟏𝟏𝟏𝟏 represents in this relationship.

$𝟒𝟒.𝟏𝟏𝟏𝟏 represents the slope of the linear relationship, or the change in the total monthly cost is $𝟒𝟒.𝟏𝟏𝟏𝟏 for an increase
of one text message. (Students need to clearly explain that slope is the change in the dependent variable for a
𝟏𝟏-unit increase in the independent variable.)

8. Explain what $𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒 represents in this relationship.

$𝟒𝟒𝟒𝟒.𝟒𝟒𝟒𝟒 represents the fixed monthly fee or the 𝒚𝒚-intercept of this relationship. This is the value of the total monthly
cost when the number of text messages is 𝟒𝟒.

Scaffolding:
Using a table may help students better
understand the relationship between the
number of text messages and total monthly
cost.

MP.4

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9. Sketch a graph of this relationship on the following coordinate grid. Clearly label the axes, and include units in the
labels.

Anticipated response: Students label the 𝒙𝒙-axis as the number of text messages. They label the 𝒚𝒚-axis as the total
monthly cost. Students use any two points they derived in Exercise 3. The following graph uses the point of (𝟒𝟒,𝟒𝟒𝟒𝟒)
for Justin and the point (𝟐𝟐𝟏𝟏𝟒𝟒,𝟕𝟕𝟕𝟕.𝟏𝟏) for Robert. Highlight the intercept of (𝟒𝟒,𝟒𝟒𝟒𝟒), along with the slope of the line
they sketched. Also, point out that the line students draw should be a dotted line (and not a solid line). The number
of text messages can only be whole numbers, and as a result, the line representing this relationship should indicate
that values in between the whole numbers representing the text messages are not part of the data.

Exercise 10 (5 minutes)

If time is running short, teachers may want to choose either Exercise 10 or 11 to develop in class and assign the other to
the Problem Set. Let students continue to work with a partner, and confirm answers as a class.

10. LaMoyne needs four more pieces of lumber for his Scout project. The pieces can be cut from one large piece of
lumber according to the following pattern.

The lumberyard will make the cuts for LaMoyne at a fixed cost of $𝟐𝟐.𝟐𝟐𝟏𝟏 plus an additional cost of 𝟐𝟐𝟏𝟏 cents per cut.
One cut is free.

a. What is the functional relationship between the total cost of cutting a piece of lumber and the number of cuts
required? What is the equation of this function? Be sure to define the variables in the context of this
problem.

As students uncover the information in this problem, they should realize that the functional relationship
between the total cost and number of cuts is linear. Noting that one cut is free, the equation could be written
in one of the following ways:

𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓 𝐨𝐨𝐓𝐓𝐧𝐧 𝐜𝐜𝐧𝐧𝐓𝐓𝐓𝐓𝐜𝐜𝐦𝐦𝐦𝐦 = 𝟐𝟐.𝟐𝟐𝟏𝟏+ (𝟒𝟒.𝟐𝟐𝟏𝟏)(𝐦𝐦𝐧𝐧𝐦𝐦𝐧𝐧𝐧𝐧𝐧𝐧 𝐓𝐓𝐨𝐨 𝐜𝐜𝐧𝐧𝐓𝐓𝐬𝐬 − 𝟏𝟏)
𝒚𝒚 = 𝟐𝟐.𝟐𝟐𝟏𝟏 + (𝟒𝟒.𝟐𝟐𝟏𝟏)(𝒙𝒙 − 𝟏𝟏), where 𝒙𝒙 is the number of cuts and 𝒚𝒚 is the total cost for cutting

𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓 𝐨𝐨𝐓𝐓𝐧𝐧 𝐜𝐜𝐧𝐧𝐓𝐓𝐓𝐓𝐜𝐜𝐦𝐦𝐦𝐦 = 𝟐𝟐 + (𝟒𝟒.𝟐𝟐𝟏𝟏)(𝐦𝐦𝐧𝐧𝐦𝐦𝐧𝐧𝐧𝐧𝐧𝐧 𝐓𝐓𝐨𝐨 𝐜𝐜𝐧𝐧𝐓𝐓𝐬𝐬)
𝒚𝒚 = 𝟐𝟐+ 𝟒𝟒.𝟐𝟐𝟏𝟏𝒙𝒙, where 𝒙𝒙 is the number of cuts and 𝒚𝒚 is the total cost for cutting

𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓 𝐨𝐨𝐓𝐓𝐧𝐧 𝐜𝐜𝐧𝐧𝐓𝐓𝐓𝐓𝐜𝐜𝐦𝐦𝐦𝐦 = 𝟐𝟐.𝟐𝟐𝟏𝟏+ (𝟒𝟒.𝟐𝟐𝟏𝟏)(𝐦𝐦𝐧𝐧𝐦𝐦𝐧𝐧𝐧𝐧𝐧𝐧 𝐓𝐓𝐨𝐨 𝐩𝐩𝐓𝐓𝐜𝐜𝐩𝐩 𝐜𝐜𝐧𝐧𝐓𝐓𝐬𝐬)
𝒚𝒚 = 𝟐𝟐.𝟐𝟐𝟏𝟏 + 𝟒𝟒.𝟐𝟐𝟏𝟏𝒙𝒙, where 𝒙𝒙 is the number of paid cuts and 𝒚𝒚 is the total cost for cutting

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b. Use the equation to determine LaMoyne’s total cost for cutting.

LaMoyne requires three cuts, one of which is free. Using any of the three forms given in part (a) yields a total
cost for cutting of $𝟐𝟐.𝟕𝟕𝟏𝟏.

c. In the context of this problem, interpret the slope of the equation in words.

Using any of the three forms, each additional cut beyond the free one adds $𝟒𝟒.𝟐𝟐𝟏𝟏 to the total cost for cutting.

d. Interpret the 𝒚𝒚-intercept of your equation in words in the context of this problem. Does interpreting the
intercept make sense in this problem? Explain.

If no cuts are required, then there is no fixed cost for cutting. So, it does not make sense to interpret the
intercept in the context of this problem.

Exercise 11 (5–7 minutes)

Let students work with a partner. Then, confirm answers as a class.

11. Omar and Olivia were curious about the size of coins. They measured the diameter and circumference of several
coins and found the following data.

U.S. Coin
Diameter

(millimeters)
Circumference
(millimeters)

Penny 𝟏𝟏𝟏𝟏.𝟒𝟒 𝟏𝟏𝟏𝟏.𝟕𝟕
Nickel 𝟐𝟐𝟏𝟏.𝟐𝟐 𝟔𝟔𝟔𝟔.𝟔𝟔
Dime 𝟏𝟏𝟕𝟕.𝟏𝟏 𝟏𝟏𝟔𝟔.𝟐𝟐

Quarter 𝟐𝟐𝟒𝟒.𝟒𝟒 𝟕𝟕𝟔𝟔.𝟒𝟒
Half Dollar 𝟒𝟒𝟒𝟒.𝟔𝟔 𝟏𝟏𝟔𝟔.𝟏𝟏

a. Wondering if there was any relationship between diameter and circumference, they thought about drawing a
picture. Draw a scatter plot that displays circumference in terms of diameter.

Students may need some help in deciding which is the independent variable and which is the dependent
variable. Hopefully, they have seen from previous problems that whenever one variable, say variable A, is to
be expressed in terms of some variable B, then variable A is the dependent variable, and variable B is the
independent variable. So, circumference is being taken as the dependent variable in this problem, and
diameter is being taken as the independent variable.

Diameter (mm)

Ci
rc

um
fe

re
nc

e
(m

m
)

32302826242220180

100

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b. Do you think that circumference and diameter are related? Explain.

It may be necessary to point out to students that because the data are rounded to one decimal place, the
points on the scatter plot may not fall exactly on a line; however, they should. Circumference and diameter
are linearly related.

c. Find the equation of the function relating circumference to the diameter of a coin.

Again, because of a rounding error, equations that students find may be slightly different depending on which
points they choose to do their calculations. Hopefully, they all arrive at something close to a circumference
equal to 𝟒𝟒.𝟏𝟏𝟒𝟒, or 𝝅𝝅, multiplied by diameter.

For example, the slope of the line containing points (𝟏𝟏𝟏𝟏,𝟏𝟏𝟏𝟏.𝟕𝟕) and (𝟒𝟒𝟒𝟒.𝟔𝟔,𝟏𝟏𝟔𝟔.𝟏𝟏) is
𝟏𝟏𝟔𝟔.𝟏𝟏 − 𝟏𝟏𝟏𝟏.𝟕𝟕
𝟒𝟒𝟒𝟒.𝟔𝟔 − 𝟏𝟏𝟏𝟏

= 𝟒𝟒. 𝟏𝟏𝟒𝟒𝟕𝟕𝟏𝟏,

which rounds to 𝟒𝟒.𝟏𝟏𝟒𝟒.

The intercept may be found using 𝟏𝟏𝟏𝟏.𝟕𝟕 = 𝒂𝒂 + (𝟒𝟒.𝟏𝟏𝟒𝟒)(𝟏𝟏𝟏𝟏.𝟒𝟒), which yields 𝒂𝒂 = 𝟒𝟒.𝟒𝟒𝟒𝟒, which rounds to 𝟒𝟒.

Therefore, 𝑪𝑪 = 𝟒𝟒.𝟏𝟏𝟒𝟒𝟏𝟏+ 𝟒𝟒 = 𝟒𝟒.𝟏𝟏𝟒𝟒𝟏𝟏.

d. The value of the slope is approximately equal to the value of 𝝅𝝅. Explain why this makes sense.

The slope is identified as being approximately equal to 𝝅𝝅. (Note: Most students have previously studied the
relationship between circumference and diameter of a circle. However, if students have not yet seen this
result, discuss the interesting result that if the circumference of a circle is divided by its diameter, the result is
a constant, namely, 𝟒𝟒.𝟏𝟏𝟒𝟒 rounded to two decimal places, no matter what circle is being considered.)

e. What is the value of the 𝒚𝒚-intercept? Explain why this makes sense.

If the diameter of a circle is 𝟒𝟒 (a point), then according to the equation, its circumference is 𝟒𝟒. That is true, so
interpreting the intercept of 𝟒𝟒 makes sense in this problem.

Closing (2–3 minutes)

 Think back to Exercise 10. If the equation that models LaMoyne’s total cost of cutting is given by
𝑦𝑦 = 2.25 + 0.25𝑥𝑥, what are the dependent and independent variables?

 The independent variable is the number of paid cuts. The dependent variable is the total cost for
cutting.

 What are the meanings of the 𝑦𝑦-intercept and slope in context?

 The 𝑦𝑦-intercept is the fee for the first cut; however, if no cuts are required, then there is no fixed cost for
cutting. The slope is the cost per cut after the first.

 How are these examples different from the data we have been studying before this lesson?

 These examples are exact linear relationships.

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Exit Ticket (5–7 minutes)

Lesson Summary

 A linear functional relationship between a dependent and an independent numerical variable has the
form 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃 or 𝒚𝒚 = 𝒂𝒂 + 𝒃𝒃𝒙𝒙.

 In statistics, a dependent variable is one that is predicted, and an independent variable is the one that
is used to make the prediction.

 The graph of a linear function describing the relationship between two variables is a line.

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Name Date

Lesson 10: Linear Models

Exit Ticket

Suppose that a cell phone monthly rate plan costs the user 5 cents per minute beyond a fixed monthly fee of $20.
This implies that the relationship between monthly cost and monthly number of minutes is linear.

1. Write an equation in words that relates total monthly cost to monthly minutes used. Explain how you found your
answer.

2. Write an equation in symbols that relates the total monthly cost in dollars (𝑦𝑦) to monthly minutes used (𝑥𝑥).

3. What is the cost for a month in which 182 minutes are used? Express your answer in words in the context of this
problem.

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Exit Ticket Sample Solutions

Suppose that a cell phone monthly rate plan costs the user 𝟏𝟏 cents per minute beyond a fixed monthly fee of $𝟐𝟐𝟒𝟒.
This implies that the relationship between monthly cost and monthly number of minutes is linear.

1. Write an equation in words that relates total monthly cost to monthly minutes used. Explain how you found your
answer.

The equation is given by 𝒕𝒕𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦 𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓 = 𝟐𝟐𝟒𝟒 + 𝟒𝟒.𝟒𝟒𝟏𝟏 (𝐦𝐦𝐧𝐧𝐦𝐦𝐧𝐧𝐧𝐧𝐧𝐧 𝐓𝐓𝐨𝐨 𝐦𝐦𝐜𝐜𝐦𝐦𝐧𝐧𝐓𝐓𝐧𝐧𝐬𝐬 𝐧𝐧𝐬𝐬𝐧𝐧𝐩𝐩 𝐨𝐨𝐓𝐓𝐧𝐧 𝐓𝐓 𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦).

The 𝒚𝒚-intercept in the equation is the fixed monthly cost, $𝟐𝟐𝟒𝟒.

The slope is the amount paid per minute of cell phone usage, or $𝟒𝟒.𝟒𝟒𝟏𝟏 per minute.

The linear form is 𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦 𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓 = 𝐨𝐨𝐜𝐜𝐭𝐭𝐧𝐧𝐩𝐩 𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓+
𝐜𝐜𝐓𝐓𝐬𝐬𝐓𝐓 𝐩𝐩𝐧𝐧𝐧𝐧 𝐦𝐦𝐜𝐜𝐦𝐦𝐧𝐧𝐓𝐓𝐧𝐧 (𝐦𝐦𝐧𝐧𝐦𝐦𝐧𝐧𝐧𝐧𝐧𝐧 𝐓𝐓𝐨𝐨 𝐦𝐦𝐜𝐜𝐦𝐦𝐧𝐧𝐓𝐓𝐧𝐧𝐬𝐬 𝐧𝐧𝐬𝐬𝐧𝐧𝐩𝐩 𝐨𝐨𝐓𝐓𝐧𝐧 𝐓𝐓 𝐦𝐦𝐓𝐓𝐦𝐦𝐓𝐓𝐦𝐦).

2. Write an equation in symbols that relates the total monthly cost in dollars (𝒚𝒚) to monthly minutes used (𝒙𝒙).

The equation is 𝒚𝒚 = 𝟐𝟐𝟒𝟒+ 𝟒𝟒 .𝟒𝟒𝟏𝟏𝒙𝒙, where 𝒚𝒚 is the total cost for a month in dollars and 𝒙𝒙 is cell phone usage for the
month in minutes.

3. What is the cost for a month in which 𝟏𝟏𝟏𝟏𝟐𝟐 minutes are used? Express your answer in words in the context of this
problem.

𝟐𝟐𝟒𝟒 + (𝟒𝟒.𝟒𝟒𝟏𝟏)(𝟏𝟏𝟏𝟏𝟐𝟐) = 𝟐𝟐𝟏𝟏.𝟏𝟏𝟒𝟒

The total monthly cost in a month using 𝟏𝟏𝟏𝟏𝟐𝟐 minutes would be $𝟐𝟐𝟏𝟏.𝟏𝟏𝟒𝟒.

Problem Set Sample Solutions

1. The Mathematics Club at your school is having a meeting. The advisor decides to bring bagels and his award-
winning strawberry cream cheese. To determine his cost, from past experience he figures 𝟏𝟏.𝟏𝟏 bagels per student.
A bagel costs 𝟔𝟔𝟏𝟏 cents, and the special cream cheese costs $𝟒𝟒.𝟏𝟏𝟏𝟏 and will be able to serve all of the anticipated
students attending the meeting.

a. Find an equation that relates his total cost to the number of students he thinks will attend the meeting.

Encourage students to write a problem in words in its context. For example, the advisor’s total cost = cream
cheese fixed cost + cost of bagels. The cost of bagels depends on the unit cost of a bagel times the number of
bagels per student times the number of students. So, with symbols, if 𝒄𝒄 denotes the total cost in dollars and 𝒏𝒏
denotes the number of students, then 𝒄𝒄 = 𝟒𝟒.𝟏𝟏𝟏𝟏+ (𝟒𝟒.𝟔𝟔𝟏𝟏)(𝟏𝟏.𝟏𝟏)(𝒏𝒏), or 𝒄𝒄 = 𝟒𝟒.𝟏𝟏𝟏𝟏+ 𝟒𝟒.𝟏𝟏𝟕𝟕𝟏𝟏𝒏𝒏.

b. In the context of the problem, interpret the slope of the equation in words.

For each additional student, the cost goes up by 𝟒𝟒.𝟏𝟏𝟕𝟕𝟏𝟏 dollar, or 𝟏𝟏𝟕𝟕.𝟏𝟏 cents.

c. In the context of the problem, interpret the 𝒚𝒚-intercept of the equation in words. Does interpreting the
intercept make sense? Explain.

If there are no students, the total cost is $𝟒𝟒.𝟏𝟏𝟏𝟏. Students could interpret this by saying that the meeting was
called off before any bagels were bought, but the advisor had already made his award-winning cream cheese,
so the cost is $𝟒𝟒.𝟏𝟏𝟏𝟏. The intercept makes sense. Other students might argue otherwise.

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2. John, Dawn, and Ron agree to walk/jog for 𝟒𝟒𝟏𝟏 minutes. John has arthritic knees but manages to walk 𝟏𝟏𝟏𝟏𝟐𝟐 miles.

Dawn walks 𝟐𝟐𝟏𝟏𝟒𝟒 miles, while Ron manages to jog 𝟔𝟔 miles.
a. Draw an appropriate graph, and connect the points to show that there is a linear relationship between the

distance that each traveled based on how fast each traveled (speed). Note that the speed for a person who

travels 𝟒𝟒 miles in 𝟒𝟒𝟏𝟏 minutes, or
𝟒𝟒
𝟒𝟒

hour, is found using the expression 𝟒𝟒÷ 𝟒𝟒
𝟒𝟒, which is 𝟒𝟒 miles per hour.

John’s speed is 𝟐𝟐 miles per hour because 𝟏𝟏𝟏𝟏𝟐𝟐 ÷ 𝟒𝟒
𝟒𝟒 = 𝟐𝟐. Dawn’s speed is 𝟒𝟒 miles per hour because 𝟐𝟐𝟏𝟏𝟒𝟒 ÷ 𝟒𝟒

𝟒𝟒 = 𝟒𝟒.

Ron’s speed is 𝟏𝟏 miles per hour because 𝟔𝟔÷ 𝟒𝟒
𝟒𝟒 = 𝟏𝟏. Students may draw the scatter plot incorrectly. Note that

distance is to be expressed in terms of speed so that distance is the dependent variable on the vertical axis,
and speed is the independent variable on the horizontal axis.

b. Find an equation that expresses distance in terms of speed (how fast one goes).

The slope is
𝟔𝟔 − 𝟏𝟏.𝟏𝟏
𝟏𝟏 − 𝟐𝟐

= 𝟒𝟒. 𝟕𝟕𝟏𝟏, so the equation of the line through these points is

𝐩𝐩𝐜𝐜𝐬𝐬𝐓𝐓𝐓𝐓𝐦𝐦𝐜𝐜𝐧𝐧 = 𝒂𝒂 + (𝟒𝟒.𝟕𝟕𝟏𝟏)(𝐬𝐬𝐩𝐩𝐧𝐧𝐧𝐧𝐩𝐩).

Next, find the intercept. For example, solve 𝟔𝟔 = 𝒂𝒂 + (𝟒𝟒.𝟕𝟕𝟏𝟏)(𝟏𝟏) for 𝒂𝒂, which yields 𝒂𝒂 = 𝟒𝟒.

So, the equation is 𝐩𝐩𝐜𝐜𝐬𝐬𝐓𝐓𝐓𝐓𝐦𝐦𝐜𝐜𝐧𝐧 = 𝟒𝟒.𝟕𝟕𝟏𝟏(𝐬𝐬𝐩𝐩𝐧𝐧𝐧𝐧𝐩𝐩).

c. In the context of the problem, interpret the slope of the equation in words.

If someone increases her speed by 𝟏𝟏 mile per hour, then that person travels 𝟒𝟒.𝟕𝟕𝟏𝟏 additional mile in
𝟒𝟒𝟏𝟏 minutes.

d. In the context of the problem, interpret the 𝒚𝒚-intercept of the equation in words. Does interpreting the
intercept make sense? Explain.

The intercept of 𝟒𝟒 makes sense because if the speed is 𝟒𝟒 miles per hour, then the person is not moving.
So, the person travels no distance.

Note: Simple interest is developed in the next problem. It is an excellent example of an application of a linear function.
If students have not worked previously with finance problems of this type, it may be necessary to carefully explain
simple interest as stated in the problem. It is an important discussion to have with students if time permits. If this
discussion is not possible and students have not worked previously with any financial applications, then omit this
problem.

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3. Simple interest is money that is paid on a loan. Simple interest is calculated by taking the amount of the loan and
multiplying it by the rate of interest per year and the number of years the loan is outstanding. For college, Jodie’s
older brother has taken out a student loan for $𝟒𝟒,𝟏𝟏𝟒𝟒𝟒𝟒 at an annual interest rate of 𝟏𝟏.𝟔𝟔%, or 𝟒𝟒.𝟒𝟒𝟏𝟏𝟔𝟔. When he
graduates in four years, he has to pay back the loan amount plus interest for four years. Jodie is curious as to how
much her brother has to pay.

a. Jodie claims that her brother has to pay a total of $𝟏𝟏,𝟏𝟏𝟒𝟒𝟏𝟏. Do you agree? Explain. As an example, a $𝟏𝟏,𝟐𝟐𝟒𝟒𝟒𝟒
loan has an 𝟏𝟏% annual interest rate. The simple interest for one year is $𝟏𝟏𝟔𝟔 because (𝟒𝟒.𝟒𝟒𝟏𝟏)(𝟏𝟏,𝟐𝟐𝟒𝟒𝟒𝟒) = 𝟏𝟏𝟔𝟔.
The simple interest for two years would be $𝟏𝟏𝟏𝟏𝟐𝟐 because (𝟐𝟐)(𝟏𝟏𝟔𝟔) = 𝟏𝟏𝟏𝟏𝟐𝟐.

The total cost to repay = amount of the loan + interest on the loan.

Interest on the loan is the annual interest times the number of years the loan is outstanding.

The annual interest amount is (𝟒𝟒.𝟒𝟒𝟏𝟏𝟔𝟔)($𝟒𝟒,𝟏𝟏𝟒𝟒𝟒𝟒) = $𝟐𝟐𝟏𝟏𝟐𝟐.

For four years, the simple interest amount is 𝟒𝟒($𝟐𝟐𝟏𝟏𝟐𝟐) = $𝟏𝟏,𝟒𝟒𝟒𝟒𝟏𝟏.

So, the total cost to repay the loan is $𝟒𝟒,𝟏𝟏𝟒𝟒𝟒𝟒+ $𝟏𝟏,𝟒𝟒𝟒𝟒𝟏𝟏 = $𝟏𝟏,𝟏𝟏𝟒𝟒𝟏𝟏. Jodie is right.

b. Write an equation for the total cost to repay a loan of $𝑷𝑷 if the rate of interest for a year is 𝒓𝒓 (expressed as a
decimal) for a time span of 𝒕𝒕 years.

Note: Work with students in identifying variables to represent the values discussed in this exercise. For
example, the total cost to repay a loan is the amount of the loan plus the simple interest, or 𝑷𝑷+ 𝑰𝑰, where 𝑷𝑷
represents the amount of the loan and 𝑰𝑰 represents the simple interest over 𝒕𝒕 years.

The amount of interest per year is 𝑷𝑷 times the annual interest. Let 𝒓𝒓 represent the interest rate per year as a
decimal.

The amount of interest per year is the amount of the loan, 𝑷𝑷, multiplied by the annual interest rate as a
decimal, 𝒓𝒓 (e.g., 𝟏𝟏% is 𝟒𝟒.𝟒𝟒𝟏𝟏). The simple interest for 𝒕𝒕 years, 𝑰𝑰, is the amount of interest per year multiplied
by the number of years: 𝑰𝑰 = (𝒓𝒓𝒕𝒕)𝑷𝑷.

The total cost to repay the loan, 𝒄𝒄, is the amount of the loan plus the amount of simple interest; therefore,
𝒄𝒄 = 𝑷𝑷 + (𝒓𝒓𝒕𝒕)𝑷𝑷.

c. If 𝑷𝑷 and 𝒓𝒓 are known, is the equation a linear equation?

If 𝑷𝑷 and 𝒓𝒓 are known, then the equation should be written as 𝒄𝒄 = 𝑷𝑷 + (𝒓𝒓𝑷𝑷)𝒕𝒕, which is the linear form where
𝒄𝒄 is the dependent variable and 𝒕𝒕 is the independent variable.

d. In the context of this problem, interpret the slope of the equation in words.

For each additional year that the loan is outstanding, the total cost to repay the loan is increased by $𝒓𝒓𝑷𝑷.

As an example, consider Jodie’s brother’s equation for 𝒕𝒕 years: 𝒄𝒄 = 𝟒𝟒𝟏𝟏𝟒𝟒𝟒𝟒+ (𝟒𝟒.𝟒𝟒𝟏𝟏𝟔𝟔)(𝟒𝟒𝟏𝟏𝟒𝟒𝟒𝟒)𝒕𝒕, or
𝒄𝒄 = 𝟒𝟒𝟏𝟏𝟒𝟒𝟒𝟒 + 𝟐𝟐𝟏𝟏𝟐𝟐𝒕𝒕. For each additional year that the loan is not paid off, the total cost increases by $𝟐𝟐𝟏𝟏𝟐𝟐.

e. In the context of this problem, interpret the 𝒚𝒚-intercept of the equation in words. Does interpreting the
intercept make sense? Explain.

The 𝒚𝒚-intercept is the value where 𝒕𝒕 = 𝟒𝟒. In this problem, it is the cost of the loan at the time that the loan
was taken out. This makes sense because after 𝟒𝟒 years, the cost to repay the loan would be $𝟒𝟒,𝟏𝟏𝟒𝟒𝟒𝟒, which is
the amount of the original loan.

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Lesson 11: Using Linear Models in a Data Context

Student Outcomes

 Students recognize and justify that a linear model can be used to fit data.
 Students interpret the slope of a linear model to answer questions or to solve a problem.

Lesson Notes
In a previous lesson, students were given bivariate numerical data where there was an exact linear relationship between
two variables. Students identified which variable was the predictor variable (i.e., independent variable) and which was
the predicted variable (i.e., dependent variable). They found the equation of the line that fit the data and interpreted
the intercept and slope in words in the context of the problem. Students also calculated a prediction for a given value of
the predictor variable. This lesson introduces students to data that are not exactly linear but that have a linear trend.
Students informally fit a line and use it to make predictions and answer questions in context.

Although students may want to rely on using symbolic representations for lines, it is important to challenge them to
express their equations in words in the context of the problem. Keep emphasizing the meaning of slope in context, and
avoid the use of “rise over run.” Slope is the impact that increasing the value of the predictor variable by one unit has on
the predicted value.

Classwork

Exercise 1 (10–12 minutes)

Introduce the data in the exercise. Using a short video may help students (especially English language learners) to better
understand the context of the data. Then, work through each part of the exercise as a class. Ask students the following:

 Looking at the table, what trend appears in the data?

 There is a positive trend. As one variable increases in value, so does the other.

 Looking at the scatter plots, is there an exact linear relationship between the variables?

 No, the four points cannot be connected by a straight line.

Exercises

1. Old Faithful is a geyser in Yellowstone National Park. The following table offers some rough estimates of the length
of an eruption (in minutes) and the amount of water (in gallons) in that eruption.

Length (minutes) 𝟏𝟏.𝟓𝟓 𝟐𝟐 𝟑𝟑 𝟒𝟒.𝟓𝟓

Amount of Water (gallons) 𝟑𝟑,𝟕𝟕𝟕𝟕𝟕𝟕 𝟒𝟒,𝟏𝟏𝟕𝟕𝟕𝟕 𝟔𝟔,𝟒𝟒𝟓𝟓𝟕𝟕 𝟖𝟖,𝟒𝟒𝟕𝟕𝟕𝟕

This data is consistent with actual eruption and summary statistics that can be found at the following links:

http://geysertimes.org/geyser.php?id=OldFaithful and http://www.yellowstonepark.com/2011/07/about-old-faithful/

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http://geysertimes.org/geyser.php?id=OldFaithful

About Old Faithful, Yellowstone’s Famous Geyser

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a. Chang wants to predict the amount of water in an eruption based on the length of
the eruption. What should he use as the dependent variable? Why?

Since Chang wants to predict the amount of water in an eruption, the time length
(in minutes) is the predictor, and the amount of water is the dependent variable.

b. Which of the following two scatter plots should Chang use to build his prediction
model? Explain.

The predicted variable goes on the vertical axis with the predictor on the horizontal axis. So, the amount of
water goes on the 𝒚𝒚-axis. The plot on the graph on the right should be used.

c. Suppose that Chang believes the variables to be linearly related. Use the first and last data points in the table
to create a linear prediction model.

𝒎𝒎 =
𝟖𝟖𝟒𝟒𝟕𝟕𝟕𝟕 − 𝟑𝟑𝟕𝟕𝟕𝟕𝟕𝟕
𝟒𝟒.𝟓𝟓 − 𝟏𝟏.𝟓𝟓

≈ 𝟏𝟏,𝟓𝟓𝟔𝟔𝟔𝟔.𝟕𝟕

So, 𝒚𝒚 = 𝒂𝒂 + (𝟏𝟏,𝟓𝟓𝟔𝟔𝟔𝟔.𝟕𝟕)𝒙𝒙.

Using either (𝟏𝟏.𝟓𝟓,𝟑𝟑𝟕𝟕𝟕𝟕𝟕𝟕) or (𝟒𝟒.𝟓𝟓,𝟖𝟖𝟒𝟒𝟕𝟕𝟕𝟕) allows students to solve for the intercept. For example, solving
𝟑𝟑,𝟕𝟕𝟕𝟕𝟕𝟕 = 𝒂𝒂 + (𝟏𝟏,𝟓𝟓𝟔𝟔𝟔𝟔.𝟕𝟕)(𝟏𝟏.𝟓𝟓) for 𝒂𝒂 yields 𝒂𝒂 = 𝟏𝟏,𝟑𝟑𝟒𝟒𝟑𝟑.𝟑𝟑𝟓𝟓, or rounded to 𝟏𝟏,𝟑𝟑𝟓𝟓𝟕𝟕.𝟕𝟕 gallons. Be sure students
talk through the units in each step of the calculations.

The (informal) linear prediction model is 𝒚𝒚 = 𝟏𝟏,𝟑𝟑𝟓𝟓𝟕𝟕.𝟕𝟕+ 𝟏𝟏,𝟓𝟓𝟔𝟔𝟔𝟔.𝟕𝟕𝒙𝒙. The amount of water (𝒚𝒚) is in gallons,
and the length of the eruption (𝒙𝒙) is in minutes.

d. A friend of Chang’s told him that Old Faithful produces about 𝟑𝟑,𝟕𝟕𝟕𝟕𝟕𝟕 gallons of water for every minute that it
erupts. Does the linear model from part (c) support what Chang’s friend said? Explain.

This question requires students to interpret slope. An additional minute in eruption length results in a
prediction of an additional 𝟏𝟏,𝟓𝟓𝟔𝟔𝟔𝟔.𝟕𝟕 gallons of water produced. So, Chang’s friend who claims Old Faithful
produces 𝟑𝟑,𝟕𝟕𝟕𝟕𝟕𝟕 gallons of water a minute must be thinking of a different geyser.

e. Using the linear model from part (c), does it make sense to interpret the 𝒚𝒚-intercept in the context of this
problem? Explain.

No, it doesn’t make sense because if the length of an eruption is 𝟕𝟕, then it cannot produce 𝟏𝟏,𝟑𝟑𝟓𝟓𝟕𝟕 gallons of
water. (Convey to students that some linear models have 𝒚𝒚-intercepts that do not make sense within the
context of a problem.)

Scaffolding:
Make the interchangeability of
the terms linearly related and
linear relationship clear to
students.

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Exercise 2 (15–20 minutes)

Let students work in small groups or with a partner. Introduce the data in the table. Note that the mean times of the
three medal winners are provided for each year. Let students work on the exercise, and confirm answers to parts (c)–(f)
as a class. After answers have been confirmed, ask the class:

 What is the meaning of the 𝑦𝑦-intercept from part (c)?

 The 𝑦𝑦-intercept from part (c) is (0, 34.91). It does not make sense within the context of the problem.
In Year 0, the mean medal time was 34.91 seconds.

2. The following table gives the times of the gold, silver, and bronze medal winners for the men’s 𝟏𝟏𝟕𝟕𝟕𝟕-meter race
(in seconds) for the past 𝟏𝟏𝟕𝟕 Olympic Games.

Year 2012 2008 2004 2000 1996 1992 1988 1984 1980 1976
Gold 𝟑𝟑.𝟔𝟔𝟑𝟑 𝟑𝟑.𝟔𝟔𝟑𝟑 𝟑𝟑.𝟖𝟖𝟓𝟓 𝟑𝟑.𝟖𝟖𝟕𝟕 𝟑𝟑.𝟖𝟖𝟒𝟒 𝟑𝟑.𝟑𝟑𝟔𝟔 𝟑𝟑.𝟑𝟑𝟐𝟐 𝟑𝟑.𝟑𝟑𝟑𝟑 𝟏𝟏𝟕𝟕.𝟐𝟐𝟓𝟓 𝟏𝟏𝟕𝟕.𝟕𝟕𝟔𝟔
Silver 𝟑𝟑.𝟕𝟕𝟓𝟓 𝟑𝟑.𝟖𝟖𝟑𝟑 𝟑𝟑.𝟖𝟖𝟔𝟔 𝟑𝟑.𝟑𝟑𝟑𝟑 𝟑𝟑.𝟖𝟖𝟑𝟑 𝟏𝟏𝟕𝟕.𝟕𝟕𝟐𝟐 𝟑𝟑.𝟑𝟑𝟕𝟕 𝟏𝟏𝟕𝟕.𝟏𝟏𝟑𝟑 𝟏𝟏𝟕𝟕.𝟐𝟐𝟓𝟓 𝟏𝟏𝟕𝟕.𝟕𝟕𝟕𝟕

Bronze 𝟑𝟑.𝟕𝟕𝟑𝟑 𝟑𝟑.𝟑𝟑𝟏𝟏 𝟑𝟑.𝟖𝟖𝟕𝟕 𝟏𝟏𝟕𝟕.𝟕𝟕𝟒𝟒 𝟑𝟑.𝟑𝟑𝟕𝟕 𝟏𝟏𝟕𝟕.𝟕𝟕𝟒𝟒 𝟑𝟑.𝟑𝟑𝟑𝟑 𝟏𝟏𝟕𝟕.𝟐𝟐𝟐𝟐 𝟏𝟏𝟕𝟕.𝟑𝟑𝟑𝟑 𝟏𝟏𝟕𝟕.𝟏𝟏𝟒𝟒
Mean
Time 𝟑𝟑.𝟕𝟕𝟐𝟐 𝟑𝟑.𝟖𝟖𝟑𝟑 𝟑𝟑.𝟖𝟖𝟔𝟔 𝟑𝟑.𝟑𝟑𝟕𝟕 𝟑𝟑.𝟖𝟖𝟖𝟖 𝟏𝟏𝟕𝟕.𝟕𝟕𝟏𝟏 𝟑𝟑.𝟑𝟑𝟔𝟔 𝟏𝟏𝟕𝟕.𝟏𝟏𝟑𝟑 𝟏𝟏𝟕𝟕.𝟑𝟑𝟕𝟕 𝟏𝟏𝟕𝟕.𝟕𝟕𝟑𝟑

Data Source: https://en.wikipedia.org/wiki/100 metres at the Olympics#Men

a. If you wanted to describe how mean times change over the years, which variable would you use as the
independent variable, and which would you use as the dependent variable?

Mean medal time (dependent variable) is being predicted based on year (independent variable).

b. Draw a scatter plot to determine if the relationship between mean time and year appears to be linear.
Comment on any trend or pattern that you see in the scatter plot.

The scatter plot indicates a negative trend, meaning that, in general, the mean race times have been
decreasing over the years even though there is not a perfect linear pattern.

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c. One reasonable line goes through the 1992 and 2004 data. Find the equation of that line.

The slope of the line through (𝟏𝟏𝟑𝟑𝟑𝟑𝟐𝟐,𝟏𝟏𝟕𝟕.𝟕𝟕𝟏𝟏) and (𝟐𝟐𝟕𝟕𝟕𝟕𝟒𝟒,𝟑𝟑.𝟖𝟖𝟔𝟔) is
𝟏𝟏𝟕𝟕.𝟕𝟕𝟏𝟏 − 𝟑𝟑.𝟖𝟖𝟔𝟔
𝟏𝟏𝟑𝟑𝟑𝟑𝟐𝟐 − 𝟐𝟐𝟕𝟕𝟕𝟕𝟒𝟒

= −𝟕𝟕. 𝟕𝟕𝟏𝟏𝟐𝟐𝟓𝟓.

To find the intercept using (𝟏𝟏𝟑𝟑𝟑𝟑𝟐𝟐,𝟏𝟏𝟕𝟕.𝟕𝟕𝟏𝟏), solve 𝟏𝟏𝟕𝟕.𝟕𝟕𝟏𝟏 = 𝒂𝒂 + (−𝟕𝟕.𝟕𝟕𝟏𝟏𝟐𝟐𝟓𝟓)(𝟏𝟏𝟑𝟑𝟑𝟑𝟐𝟐) for 𝒂𝒂, which yields
𝒂𝒂 = 𝟑𝟑𝟒𝟒.𝟑𝟑𝟏𝟏.

The equation that predicts the mean medal race time for an Olympic year is 𝒚𝒚 = 𝟑𝟑𝟒𝟒.𝟑𝟑𝟏𝟏 + (−𝟕𝟕.𝟕𝟕𝟏𝟏𝟐𝟐𝟓𝟓)𝒙𝒙.
The mean medal race time (𝒚𝒚) is in seconds, and the time (𝒙𝒙) is in years.

Note to Teacher: In Algebra I, students learn a formal method called least squares for determining a “best-
fitting” line. For comparison, the least squares prediction line is 𝒚𝒚 = 𝟑𝟑𝟒𝟒.𝟑𝟑𝟓𝟓𝟔𝟔𝟐𝟐+ (−𝟕𝟕.𝟕𝟕𝟏𝟏𝟐𝟐𝟐𝟐)𝒙𝒙.

d. Before he saw these data, Chang guessed that the mean time of the three Olympic medal winners decreased
by about 𝟕𝟕.𝟕𝟕𝟓𝟓 second from one Olympic Game to the next. Does the prediction model you found in part (c)
support his guess? Explain.

The slope −𝟕𝟕.𝟕𝟕𝟏𝟏𝟐𝟐𝟓𝟓 means that from one calendar year to the next, the predicted mean race time for the top
three medals decreases by 𝟕𝟕.𝟕𝟕𝟏𝟏𝟐𝟐𝟓𝟓 second. So, between successive Olympic Games, which occur every four
years, the predicted mean race time is reduced by 𝟕𝟕.𝟕𝟕𝟓𝟓 second because 𝟒𝟒(𝟕𝟕.𝟕𝟕𝟏𝟏𝟐𝟐𝟓𝟓) = 𝟕𝟕.𝟕𝟕𝟓𝟓.

e. If the trend continues, what mean race time would you predict for the gold, silver, and bronze medal winners
in the 2016 Olympic Games? Explain how you got this prediction.

If the linear pattern were to continue, the predicted mean time for the 2016 Olympics is 𝟑𝟑.𝟕𝟕𝟏𝟏 seconds
because 𝟑𝟑𝟒𝟒.𝟑𝟑𝟏𝟏 − (𝟕𝟕.𝟕𝟕𝟏𝟏𝟐𝟐𝟓𝟓)(𝟐𝟐𝟕𝟕𝟏𝟏𝟔𝟔) = 𝟑𝟑.𝟕𝟕𝟏𝟏.

f. The data point (𝟏𝟏𝟑𝟑𝟖𝟖𝟕𝟕,𝟏𝟏𝟕𝟕.𝟑𝟑) appears to have an unusually high value for the mean race time (𝟏𝟏𝟕𝟕.𝟑𝟑). Using
your library or the Internet, see if you can find a possible explanation for why that might have happened.

The mean race time in 1980 was an unusually high 𝟏𝟏𝟕𝟕.𝟑𝟑 seconds. In their research of the 1980 Olympic
Games, students find that the United States and several other countries boycotted the games, which were
held in Moscow. Perhaps the field of runners was not the typical Olympic quality as a result. Atypical points
in a set of data are called outliers. They may influence the analysis of the data.

Following these two examples, ask students to summarize (in written or spoken form) how to make predictions from
data.

Closing (2–3 minutes)

If time allows, revisit the linear model from Exercise 2. Explain that the data can be modified to create a model in which
the 𝑦𝑦-intercept makes sense within the context of the problem.

Year 2012 2008 2004 2000 1996 1992 1988 1984 1980 1976
Number of Years (since 1976) 36 32 28 24 20 16 12 8 4 0
Gold 9.63 9.69 9.85 9.87 9.84 9.96 9.92 9.99 10.25 10.06
Silver 9.75 9.89 9.86 9.99 9.89 10.02 9.97 10.19 10.25 10.07
Bronze 9.79 9.91 9.87 10.04 9.90 10.04 9.99 10.22 10.39 10.14
Mean Time 9.72 9.83 9.86 9.97 9.88 10.01 9.96 10.13 10.30 10.09

Data Source: https://en.wikipedia.org/wiki/100 metres at the Olympics#Men

MP.7

MP.2

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Lesson Summary

In the real world, it is rare that two numerical variables are exactly linearly related. If the data are roughly linearly
related, then a line can be drawn through the data. This line can then be used to make predictions and to answer
questions. For now, the line is informally drawn, but in later grades more formal methods for determining a best-
fitting line are presented.

 Using the data points for 1992 and 2004, (16, 10.01) and (28, 9.86), the linear model is
𝑦𝑦 = 10.21 + (−0.0125)𝑥𝑥.

 Note that the slope is the same as the linear model in Exercise 2.

 The 𝑦𝑦-intercept is now (0, 10.21), which means that in 1976 (0 years since 1976), the mean medal time was
10.21 seconds.

Review the Lesson Summary with students.

Exit Ticket (8–10 minutes)

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Name Date

Lesson 11: Using Linear Models in a Data Context

Exit Ticket

According to the Bureau of Vital Statistics for the New York City Department of Health and Mental Hygiene, the life
expectancy at birth (in years) for New York City babies is as follows.

Year of Birth 2001 2002 2003 2004 2005 2006 2007 2008 2009
Life Expectancy 77.9 78.2 78.5 79.0 79.2 79.7 80.1 80.2 80.6

Data Source: http://www.nyc.gov/html/om/pdf/2012/pr465-12 charts

a. If you are interested in predicting life expectancy for babies born in a given year, which variable is the
independent variable, and which is the dependent variable?

b. Draw a scatter plot to determine if there appears to be a linear relationship between the year of birth and life

expectancy.

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c. Fit a line to the data. Show your work.

d. Based on the context of the problem, interpret in words the intercept and slope of the line you found in
part (c).

e. Use your line to predict life expectancy for babies born in New York City in 2010.

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Exit Ticket Sample Solutions

According to the Bureau of Vital Statistics for the New York City Department of Health and Mental Hygiene, the life
expectancy at birth (in years) for New York City babies is as follows.

Year of Birth 2001 2002 2003 2004 2005 2006 2007 2008 2009
Life Expectancy 𝟕𝟕𝟕𝟕.𝟑𝟑 𝟕𝟕𝟖𝟖.𝟐𝟐 𝟕𝟕𝟖𝟖.𝟓𝟓 𝟕𝟕𝟑𝟑.𝟕𝟕 𝟕𝟕𝟑𝟑.𝟐𝟐 𝟕𝟕𝟑𝟑.𝟕𝟕 𝟖𝟖𝟕𝟕.𝟏𝟏 𝟖𝟖𝟕𝟕.𝟐𝟐 𝟖𝟖𝟕𝟕.𝟔𝟔

Data Source: http://www.nyc.gov/html/om/pdf/2012/pr465-12 charts

a. If you are interested in predicting life expectancy for babies born in a given year, which variable is the
independent variable, and which is the dependent variable?

Year of birth is the independent variable, and life expectancy in years is the dependent variable.

b. Draw a scatter plot to determine if there appears to be a linear relationship between the year of birth and life
expectancy.

Life expectancy and year of birth appear to be linearly related.

c. Fit a line to the data. Show your work.

Answers will vary. For example, the line through (𝟐𝟐𝟕𝟕𝟕𝟕𝟏𝟏,𝟕𝟕𝟕𝟕.𝟑𝟑) and (𝟐𝟐𝟕𝟕𝟕𝟕𝟑𝟑,𝟖𝟖𝟕𝟕.𝟔𝟔) is 𝒚𝒚 = −𝟓𝟓𝟑𝟑𝟕𝟕.𝟒𝟒𝟑𝟑𝟖𝟖+
(𝟕𝟕.𝟑𝟑𝟑𝟑𝟕𝟕𝟓𝟓)𝒙𝒙, where life expectancy (𝒚𝒚) is in years, and the time (𝒙𝒙) is in years.

Note to Teacher: The formal least squares line (Algebra I) is 𝒚𝒚 = −𝟔𝟔𝟏𝟏𝟐𝟐.𝟒𝟒𝟓𝟓𝟖𝟖+ (𝟕𝟕.𝟑𝟑𝟒𝟒𝟓𝟓)𝒙𝒙.

d. Based on the context of the problem, interpret in words the intercept and slope of the line you found in
part (c).

Answers will vary based on part (c). The intercept says that babies born in New York City in Year 𝟕𝟕 should
expect to live around −𝟓𝟓𝟑𝟑𝟕𝟕 years! Be sure students actually say that this is an unrealistic result and that
interpreting the intercept is meaningless in this problem. Regarding the slope, for an increase of 𝟏𝟏 in the year
of birth, predicted life expectancy increases by 𝟕𝟕.𝟑𝟑𝟑𝟑𝟕𝟕𝟓𝟓 year, which is a little over four months.

e. Use your line to predict life expectancy for babies born in New York City in 2010.

Answers will vary based on part (c).

−𝟓𝟓𝟑𝟑𝟕𝟕.𝟒𝟒𝟑𝟑𝟖𝟖+ (𝟕𝟕.𝟑𝟑𝟑𝟑𝟕𝟕𝟓𝟓)(𝟐𝟐𝟕𝟕𝟏𝟏𝟕𝟕) = 𝟖𝟖𝟕𝟕.𝟑𝟑

Using the line calculated in part (c), the predicted life expectancy for babies born in New York City in 2010
is 𝟖𝟖𝟕𝟕.𝟑𝟑 years, which is also the value given on the website.

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Problem Set Sample Solutions

1. From the United States Bureau of Census website, the population sizes (in millions of people) in the United States
for census years 1790–2010 are as follows.

Year 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890
Population
Size 𝟑𝟑.𝟑𝟑 𝟓𝟓.𝟑𝟑 𝟕𝟕.𝟐𝟐 𝟑𝟑.𝟔𝟔 𝟏𝟏𝟐𝟐.𝟑𝟑 𝟏𝟏𝟕𝟕.𝟏𝟏 𝟐𝟐𝟑𝟑.𝟐𝟐 𝟑𝟑𝟏𝟏.𝟒𝟒 𝟑𝟑𝟖𝟖.𝟔𝟔 𝟓𝟓𝟕𝟕.𝟐𝟐 𝟔𝟔𝟑𝟑.𝟕𝟕

Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
Population
Size 𝟕𝟕𝟔𝟔.𝟐𝟐 𝟑𝟑𝟐𝟐.𝟐𝟐 𝟏𝟏𝟕𝟕𝟔𝟔.𝟕𝟕 𝟏𝟏𝟐𝟐𝟑𝟑.𝟐𝟐 𝟏𝟏𝟑𝟑𝟐𝟐.𝟐𝟐 𝟏𝟏𝟓𝟓𝟏𝟏.𝟑𝟑 𝟏𝟏𝟕𝟕𝟑𝟑.𝟑𝟑 𝟐𝟐𝟕𝟕𝟑𝟑.𝟑𝟑 𝟐𝟐𝟐𝟐𝟔𝟔.𝟓𝟓 𝟐𝟐𝟒𝟒𝟖𝟖.𝟕𝟕 𝟐𝟐𝟖𝟖𝟏𝟏.𝟒𝟒 𝟑𝟑𝟕𝟕𝟖𝟖.𝟕𝟕

a. If you wanted to be able to predict population size in a given year, which variable would be the independent
variable, and which would be the dependent variable?

Population size (dependent variable) is being predicted based on year (independent variable).

b. Draw a scatter plot. Does the relationship between year and population size appear to be linear?

The relationship between population size and year of birth is definitely nonlinear. Note that investigating
nonlinear relationships is the topic of the next two lessons.

c. Consider the data only from 1950 to 2010. Does the relationship between year and population size for these
years appear to be linear?

Drawing a scatter plot using the 1950–2010 data indicates that the relationship between population size and
year of birth is approximately linear, although some students may say that there is a very slight curvature to
the data.

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d. One line that could be used to model the relationship between year and population size for the data from
1950 to 2010 is 𝒚𝒚 = −𝟒𝟒𝟖𝟖𝟕𝟕𝟓𝟓.𝟕𝟕𝟐𝟐𝟏𝟏 + 𝟐𝟐.𝟓𝟓𝟕𝟕𝟖𝟖𝒙𝒙. Suppose that a sociologist believes that there will be negative

consequences if population size in the United States increases by more than 𝟐𝟐𝟑𝟑𝟒𝟒 million people annually.
Should she be concerned? Explain your reasoning.

This problem is asking students to interpret the slope. Some students will no doubt say that the sociologist
need not be concerned, since the slope of 𝟐𝟐.𝟓𝟓𝟕𝟕𝟖𝟖 million births per year is smaller than her threshold value of
𝟐𝟐.𝟕𝟕𝟓𝟓 million births per year. Other students may say that the sociologist should be concerned, since the
difference between 𝟐𝟐.𝟓𝟓𝟕𝟕𝟖𝟖 and 𝟐𝟐.𝟕𝟕𝟓𝟓 is only 𝟏𝟏𝟕𝟕𝟐𝟐,𝟕𝟕𝟕𝟕𝟕𝟕 births per year.

e. Assuming that the linear pattern continues, use the line given in part (d) to predict the size of the population
in the United States in the next census.

The next census year is 2020.

−𝟒𝟒𝟖𝟖𝟕𝟕𝟓𝟓.𝟕𝟕𝟐𝟐𝟏𝟏+ (𝟐𝟐.𝟓𝟓𝟕𝟕𝟖𝟖)(𝟐𝟐𝟕𝟕𝟐𝟐𝟕𝟕) = 𝟑𝟑𝟑𝟑𝟐𝟐.𝟓𝟓𝟑𝟑𝟑𝟑

The given line predicts that the population then will be 𝟑𝟑𝟑𝟑𝟐𝟐.𝟓𝟓𝟑𝟑𝟑𝟑 million people.

2. In search of a topic for his science class project, Bill saw an interesting YouTube video in which dropping mint
candies into bottles of a soda pop caused the soda pop to spurt immediately from the bottle. He wondered if the
height of the spurt was linearly related to the number of mint candies that were used. He collected data using 𝟏𝟏, 𝟑𝟑,
𝟓𝟓, and 𝟏𝟏𝟕𝟕 mint candies. Then, he used two-liter bottles of a diet soda and measured the height of the spurt in
centimeters. He tried each quantity of mint candies three times. His data are in the following table.

Number of Mint
Candies 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝟑𝟑 𝟑𝟑 𝟑𝟑 𝟓𝟓 𝟓𝟓 𝟓𝟓 𝟏𝟏𝟕𝟕 𝟏𝟏𝟕𝟕 𝟏𝟏𝟕𝟕

Height of Spurt
(centimeters) 𝟒𝟒𝟕𝟕 𝟑𝟑𝟓𝟓 𝟑𝟑𝟕𝟕 𝟏𝟏𝟏𝟏𝟕𝟕 𝟏𝟏𝟕𝟕𝟓𝟓 𝟑𝟑𝟕𝟕 𝟏𝟏𝟕𝟕𝟕𝟕 𝟏𝟏𝟔𝟔𝟕𝟕 𝟏𝟏𝟖𝟖𝟕𝟕 𝟒𝟒𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑𝟕𝟕 𝟒𝟒𝟐𝟐𝟕𝟕

a. Identify which variable is the independent variable and which is the dependent
variable.

Height of spurt is the dependent variable, and number of mint candies is the
independent variable because height of spurt is being predicted based on number of
mint candies used.

b. Draw a scatter plot that could be used to determine whether the relationship
between height of spurt and number of mint candies appears to be linear.

Scaffolding:
 The word spurt may need

to be defined for English
language learners.

 A spurt is a sudden stream
of liquid or gas forced out
under pressure. Showing
a visual aid to accompany
this exercise may help
student comprehension.

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c. Bill sees a slight curvature in the scatter plot, but he thinks that the relationship between the number of mint
candies and the height of the spurt appears close enough to being linear, and he proceeds to draw a line.
His eyeballed line goes through the mean of the three heights for three mint candies and the mean of the
three heights for 𝟏𝟏𝟕𝟕 candies. Bill calculates the equation of his eyeballed line to be

𝒚𝒚 = −𝟐𝟐𝟕𝟕.𝟔𝟔𝟏𝟏𝟕𝟕+ (𝟒𝟒𝟑𝟑.𝟕𝟕𝟑𝟑𝟓𝟓)𝒙𝒙,

where the height of the spurt (𝒚𝒚) in centimeters is based on the number of mint candies (𝒙𝒙). Do you agree
with this calculation? He rounded all of his calculations to three decimal places. Show your work.

Yes, Bill’s equation is correct.

The slope of the line through (𝟑𝟑,𝟏𝟏𝟕𝟕𝟏𝟏.𝟔𝟔𝟔𝟔𝟕𝟕) and (𝟏𝟏𝟕𝟕,𝟒𝟒𝟕𝟕𝟑𝟑.𝟑𝟑𝟑𝟑𝟑𝟑) is
𝟒𝟒𝟕𝟕𝟑𝟑.𝟑𝟑𝟑𝟑𝟑𝟑 − 𝟏𝟏𝟕𝟕𝟏𝟏.𝟔𝟔𝟔𝟔𝟕𝟕

𝟏𝟏𝟕𝟕 − 𝟑𝟑
= 𝟒𝟒𝟑𝟑. 𝟕𝟕𝟑𝟑𝟓𝟓.

The intercept could be found by solving 𝟒𝟒𝟕𝟕𝟑𝟑.𝟑𝟑𝟑𝟑𝟑𝟑 = 𝒂𝒂 + (𝟒𝟒𝟑𝟑.𝟕𝟕𝟑𝟑𝟓𝟓)(𝟏𝟏𝟕𝟕) for 𝒂𝒂, which yields
𝒂𝒂 = −𝟐𝟐𝟕𝟕.𝟔𝟔𝟏𝟏𝟕𝟕.

So, a possible prediction line is 𝒚𝒚 = −𝟐𝟐𝟕𝟕.𝟔𝟔𝟏𝟏𝟕𝟕+ (𝟒𝟒𝟑𝟑.𝟕𝟕𝟑𝟑𝟓𝟓)𝒙𝒙.

d. In the context of this problem, interpret in words the slope and intercept for Bill’s line. Does interpreting the
intercept make sense in this context? Explain.

The slope is 𝟒𝟒𝟑𝟑.𝟕𝟕𝟑𝟑𝟓𝟓, which means that for every mint candy dropped into the bottle of soda pop, the height
of the spurt increases by 𝟒𝟒𝟑𝟑.𝟕𝟕𝟑𝟑𝟓𝟓 𝐜𝐜𝐜𝐜.

The 𝒚𝒚-intercept is (𝟕𝟕,−𝟐𝟐𝟕𝟕.𝟔𝟔𝟏𝟏𝟕𝟕). This means that if no mint candies are dropped into the bottle of soda pop,
the height of the spurt is −𝟐𝟐𝟕𝟕.𝟔𝟔𝟏𝟏𝟕𝟕 𝐟𝐟𝐟𝐟. This does not make sense within the context of the problem.

e. If the linear trend continues for greater numbers of mint candies, what do you predict the height of the spurt
to be if 𝟏𝟏𝟓𝟓 mint candies are used?

−𝟐𝟐𝟕𝟕.𝟔𝟔𝟏𝟏𝟕𝟕+ (𝟒𝟒𝟑𝟑.𝟕𝟕𝟑𝟑𝟓𝟓)(𝟏𝟏𝟓𝟓) = 𝟔𝟔𝟏𝟏𝟖𝟖.𝟖𝟖𝟕𝟕𝟖𝟖

The predicted height would be 𝟔𝟔𝟏𝟏𝟖𝟖.𝟖𝟖𝟕𝟕𝟖𝟖 𝐜𝐜𝐜𝐜, which is slightly over 𝟐𝟐𝟕𝟕 𝐟𝐟𝐟𝐟.

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Lesson 12: Nonlinear Models in a Data Context (Optional)

Student Outcomes

 Students give verbal descriptions of how 𝑦𝑦 changes as 𝑥𝑥 changes given the graph of a nonlinear function.
 Students draw nonlinear functions that are consistent with a verbal description of a nonlinear relationship.

Lesson Notes
The Common Core Standards do not require that eighth-grade students fit curves to nonlinear data. This lesson is
included as an optional extension to provide a deeper understanding of the key features of linear relationships in
contrast to nonlinear ones.

Previous lessons focused on finding the equation of a line and interpreting the slope and intercept for data that followed
a linear pattern. In the next two lessons, the focus shifts to data that do not follow a linear pattern. Instead of drawing
lines through data, a curve is used to describe the relationship observed in a scatter plot.

In this lesson, students calculate the change in height of plants grown in beds with and without compost. The change in
growth in the non-compost beds approximately follows a linear pattern. The change in growth in the compost beds
follows a curved pattern rather than a linear pattern. Students are asked to compare the growth changes and recognize
that the change in growth for a linear pattern shows a constant change, while nonlinear patterns show a rate of growth
that is not constant.

Classwork

Example 1 (3 minutes): Growing Dahlias

Present the experiment for the two methods of growing dahlias. One method
was to plant eight dahlias in a bed of soil that has no compost. The other was to
plant eight dahlias in a bed of soil that has been enriched with compost. Explain
that the students measured the height of each plant at the end of each week and
recorded the median height of the eight dahlias.

Before students begin Example 1, ask the following:

 Is there a pattern in the median height of the plants?

 The median height is increasing every week by about 3.5 inches.

Scaffolding:
 An image of a growth experiment

may help English language learners
understand the context of the
example.

 The words compost and bed may
be unfamiliar to students in this
context.

 Compost is a mixture of decayed
plants and other organic matter
used by gardeners to enrich soil.

 Bed has multiple meanings. In this
context, bed refers to a section of
ground planted with flowers.

 Showing visuals of these terms to
accompany the exercises aids in
student comprehension.

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Example 1: Growing Dahlias

A group of students wanted to determine whether or not compost is beneficial in plant growth. The students used the
dahlia flower to study the effect of composting. They planted eight dahlias in a bed with no compost and another eight
plants in a bed with compost. They measured the height of each plant over a 𝟗𝟗-week period. They found the median
growth height for each group of eight plants. The table below shows the results of the experiment for the dahlias grown
in non-compost beds.

Exercises 1–7 (13 minutes)

This exercise set is designed as a review of the previous lesson on fitting a line to data.

The scatter plot shows that a line fits the data reasonably well. Exercise 3 asks students to find only the slope of the line.
Consider having students write the equation of the line. They could then use this equation to help answer Exercise 7.

As students complete the table in Exercise 4, emphasize how the values of the change in height are all approximately
equal and that they center around the value of the slope of the line that they have drawn.

Allow students to work in small groups to complete the exercises. Discuss the answers as a class.

Exercises 1–15

1. On the grid below, construct a scatter plot of non-compost height versus week.

Week Median Height in
Non-Compost Bed (inches)

𝟏𝟏 𝟗𝟗.𝟎𝟎𝟎𝟎
𝟐𝟐 𝟏𝟏𝟐𝟐.𝟕𝟕𝟕𝟕
𝟑𝟑 𝟏𝟏𝟏𝟏.𝟐𝟐𝟕𝟕
𝟒𝟒 𝟏𝟏𝟗𝟗.𝟕𝟕𝟎𝟎
𝟕𝟕 𝟐𝟐𝟑𝟑.𝟎𝟎𝟎𝟎
𝟏𝟏 𝟐𝟐𝟏𝟏.𝟕𝟕𝟕𝟕
𝟕𝟕 𝟑𝟑𝟎𝟎.𝟎𝟎𝟎𝟎
𝟖𝟖 𝟑𝟑𝟑𝟑.𝟕𝟕𝟕𝟕
𝟗𝟗 𝟑𝟑𝟕𝟕.𝟐𝟐𝟕𝟕

Scaffolding:
Median is developed in Grades
6 and 7 as a measure of center
that is used to identify a typical
value for a skewed data
distribution.

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2. Draw a line that you think fits the data reasonably well.

3. Find the rate of change of your line. Interpret the rate of change in terms of growth (in height) over time.

Most students should have a rate of change of approximately 𝟑𝟑.𝟕𝟕 inches per week. A rate of change of 𝟑𝟑.𝟕𝟕 means
that the median height of the eight dahlias increased by about 𝟑𝟑.𝟕𝟕 inches each week.

4. Describe the growth (change in height) from week to week by subtracting the previous week’s height from the
current height. Record the weekly growth in the third column in the table below. The median growth for the
dahlias from Week 1 to Week 2 was 𝟑𝟑.𝟕𝟕𝟕𝟕 inches (i.e., 𝟏𝟏𝟐𝟐.𝟕𝟕𝟕𝟕 − 𝟗𝟗.𝟎𝟎𝟎𝟎 = 𝟑𝟑.𝟕𝟕𝟕𝟕).

Week
Median Height in
Non-Compost Bed

(inches)

Weekly Growth
(inches)

𝟏𝟏 𝟗𝟗.𝟎𝟎𝟎𝟎 −
𝟐𝟐 𝟏𝟏𝟐𝟐.𝟕𝟕𝟕𝟕 𝟑𝟑.𝟕𝟕𝟕𝟕
𝟑𝟑 𝟏𝟏𝟏𝟏.𝟐𝟐𝟕𝟕 𝟑𝟑.𝟕𝟕
𝟒𝟒 𝟏𝟏𝟗𝟗.𝟕𝟕𝟎𝟎 𝟑𝟑.𝟐𝟐𝟕𝟕
𝟕𝟕 𝟐𝟐𝟑𝟑.𝟎𝟎𝟎𝟎 𝟑𝟑.𝟕𝟕
𝟏𝟏 𝟐𝟐𝟏𝟏.𝟕𝟕𝟕𝟕 𝟑𝟑.𝟕𝟕𝟕𝟕
𝟕𝟕 𝟑𝟑𝟎𝟎.𝟎𝟎𝟎𝟎 𝟑𝟑.𝟐𝟐𝟕𝟕
𝟖𝟖 𝟑𝟑𝟑𝟑.𝟕𝟕𝟕𝟕 𝟑𝟑.𝟕𝟕𝟕𝟕
𝟗𝟗 𝟑𝟑𝟕𝟕.𝟐𝟐𝟕𝟕 𝟑𝟑.𝟕𝟕

5. As the number of weeks increases, describe how the weekly growth is changing.

The growth each week remains about the same—approximately 𝟑𝟑.𝟕𝟕 inches.

6. How does the growth each week compare to the slope of the line that you drew?

The amount of growth per week varies from 𝟑𝟑.𝟐𝟐𝟕𝟕 to 𝟑𝟑.𝟕𝟕𝟕𝟕 but centers around 𝟑𝟑.𝟕𝟕, which is the slope of the line.

7. Estimate the median height of the dahlias at 𝟖𝟖𝟏𝟏𝟐𝟐 weeks. Explain how you made your estimate.

An estimate is 𝟑𝟑𝟕𝟕.𝟕𝟕 inches. Students can use the graph, the table, or the equation of their line.

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Exercises 8–14 (13 minutes)

These exercises present a set of data that do not follow a linear pattern. Students are asked to draw a curve through the
data that they think fits the data reasonably well. Students may want to connect the ordered pairs, but encourage them
to draw a smooth curve. A piece of thread or string can be used to sketch a smooth curve rather than connecting the
ordered pairs. In this lesson, it is not expected that students find a function (nor are they given a function) that would fit
the data. The main focus is that the rate of growth is not a constant when the data do not follow a linear pattern.

Allow students to work in small groups to complete the exercises. Then, discuss answers as a class.

The table below shows the results of the experiment for the dahlias grown in compost beds.

Week
Median Height in

Compost Bed (inches)
𝟏𝟏 𝟏𝟏𝟎𝟎.𝟎𝟎𝟎𝟎
𝟐𝟐 𝟏𝟏𝟑𝟑.𝟕𝟕𝟎𝟎
𝟑𝟑 𝟏𝟏𝟕𝟕.𝟕𝟕𝟕𝟕
𝟒𝟒 𝟐𝟐𝟏𝟏.𝟕𝟕𝟎𝟎
𝟕𝟕 𝟑𝟑𝟎𝟎.𝟕𝟕𝟎𝟎
𝟏𝟏 𝟒𝟒𝟎𝟎.𝟕𝟕𝟎𝟎
𝟕𝟕 𝟏𝟏𝟕𝟕.𝟎𝟎𝟎𝟎
𝟖𝟖 𝟖𝟖𝟎𝟎.𝟕𝟕𝟎𝟎
𝟗𝟗 𝟗𝟗𝟏𝟏.𝟕𝟕𝟎𝟎

8. Construct a scatter plot of height versus week on the grid below.

9. Do the data appear to form a linear pattern?

No, the pattern in the scatter plot is curved.

MP.7

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10. Describe the growth from week to week by subtracting the height from the previous week from the current height.
Record the weekly growth in the third column in the table below. The median weekly growth for the dahlias from
Week 1 to Week 2 is 𝟑𝟑.𝟕𝟕 inches. (i.e., 𝟏𝟏𝟑𝟑.𝟕𝟕 − 𝟏𝟏𝟎𝟎 = 𝟑𝟑.𝟕𝟕).

Week
Compost Height

(inches)
Weekly Growth

(inches)
𝟏𝟏 𝟏𝟏𝟎𝟎.𝟎𝟎𝟎𝟎 −
𝟐𝟐 𝟏𝟏𝟑𝟑.𝟕𝟕𝟎𝟎 𝟑𝟑.𝟕𝟕𝟎𝟎
𝟑𝟑 𝟏𝟏𝟕𝟕.𝟕𝟕𝟕𝟕 𝟒𝟒.𝟐𝟐𝟕𝟕
𝟒𝟒 𝟐𝟐𝟏𝟏.𝟕𝟕𝟎𝟎 𝟑𝟑.𝟕𝟕𝟕𝟕
𝟕𝟕 𝟑𝟑𝟎𝟎.𝟕𝟕𝟎𝟎 𝟗𝟗.𝟎𝟎
𝟏𝟏 𝟒𝟒𝟎𝟎.𝟕𝟕𝟎𝟎 𝟏𝟏𝟎𝟎.𝟎𝟎
𝟕𝟕 𝟏𝟏𝟕𝟕.𝟎𝟎𝟎𝟎 𝟐𝟐𝟒𝟒.𝟕𝟕
𝟖𝟖 𝟖𝟖𝟎𝟎.𝟕𝟕𝟎𝟎 𝟏𝟏𝟕𝟕.𝟕𝟕𝟎𝟎
𝟗𝟗 𝟗𝟗𝟏𝟏.𝟕𝟕𝟎𝟎 𝟏𝟏𝟏𝟏.𝟎𝟎

11. As the number of weeks increases, describe how the growth changes.

The amount of growth per week varies from week to week. In Weeks 1 through 4, the growth is around 𝟒𝟒 inches
each week. From Weeks 5 to 7, the amount of growth increases, and then the growth slows down for Weeks 8
and 9.

12. Sketch a curve through the data. When sketching a curve, do not connect the ordered pairs, but draw a smooth
curve that you think reasonably describes the data.

13. Use the curve to estimate the median height of the dahlias at 𝟖𝟖𝟏𝟏𝟐𝟐 weeks. Explain how you made your estimate.

Answers will vary. A reasonable estimate of the median height at 𝟖𝟖𝟏𝟏𝟐𝟐 weeks is approximately 𝟖𝟖𝟕𝟕 inches. Starting at

𝟖𝟖𝟏𝟏𝟐𝟐 on the 𝒙𝒙-axis, move up to the curve and then over to the 𝒚𝒚-axis for the estimate of the height.

14. How does the weekly growth of the dahlias in the compost beds compare to the weekly growth of the dahlias in the
non-compost beds?

The growth in the non-compost is about the same each week. The growth in the compost starts the same as the
non-compost, but after four weeks, the dahlias begin to grow at a faster rate.

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Exercise 15 (7 minutes)

15. When there is a car accident, how do the investigators determine the speed of the cars involved? One way is to
measure the skid marks left by the cars and use these lengths to estimate the speed.

The table below shows data collected from an experiment with a test car. The first column is the length of the skid
mark (in feet), and the second column is the speed of the car (in miles per hour).

Skid-Mark Length
(feet)

Speed (miles per hour)

𝟕𝟕 𝟏𝟏𝟎𝟎
𝟏𝟏𝟕𝟕 𝟐𝟐𝟎𝟎
𝟏𝟏𝟕𝟕 𝟒𝟒𝟎𝟎
𝟏𝟏𝟎𝟎𝟕𝟕 𝟕𝟕𝟎𝟎
𝟐𝟐𝟎𝟎𝟕𝟕 𝟕𝟕𝟎𝟎
𝟐𝟐𝟏𝟏𝟕𝟕 𝟖𝟖𝟎𝟎

Data Source: http://forensicdynamics.com/stopping-braking-distance-calculator

(Note: Data has been rounded.)

a. Construct a scatter plot of speed versus skid-mark length on the grid below.

b. The relationship between speed and skid-mark length can be described by a curve. Sketch a curve through
the data that best represents the relationship between skid-mark length and the speed of the car. Remember
to draw a smooth curve that does not just connect the ordered pairs.

See the plot above.

c. If the car left a skid mark of 𝟏𝟏𝟎𝟎 𝐟𝐟𝐟𝐟., what is an estimate for the speed of the car? Explain how you determined
the estimate.

The speed is approximately 𝟑𝟑𝟖𝟖 𝐦𝐦𝐦𝐦𝐦𝐦. Using the graph, for a skid mark of 𝟏𝟏𝟕𝟕 𝐟𝐟𝐟𝐟., the speed was 𝟒𝟒𝟎𝟎 𝐦𝐦𝐦𝐦𝐦𝐦, so
the estimate is slightly less than 𝟒𝟒𝟎𝟎 𝐦𝐦𝐦𝐦𝐦𝐦.

d. A car left a skid mark of 𝟏𝟏𝟕𝟕𝟎𝟎 𝐟𝐟𝐟𝐟. Use the curve you sketched to estimate the speed at which the car was
traveling.

𝟏𝟏𝟐𝟐.𝟕𝟕 𝐦𝐦𝐦𝐦𝐦𝐦

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e. If a car leaves a skid mark that is twice as long as another skid mark, was the car going twice as fast? Explain.

No. When the skid mark was 𝟏𝟏𝟎𝟎𝟕𝟕 𝐟𝐟𝐟𝐟. long, the car was traveling 𝟕𝟕𝟎𝟎 𝐦𝐦𝐦𝐦𝐦𝐦. When the skid mark was 𝟐𝟐𝟎𝟎𝟕𝟕 𝐟𝐟𝐟𝐟.
long (about twice the 𝟏𝟏𝟎𝟎𝟕𝟕 𝐟𝐟𝐟𝐟.), the car was traveling 𝟕𝟕𝟎𝟎 𝐦𝐦𝐦𝐦𝐦𝐦, which is not twice as fast.

Closing (1 minute)

Review the Lesson Summary with students.

Exit Ticket (8 minutes)

Lesson Summary

When data follow a linear pattern, they can be represented by a linear function whose rate of change can be used
to answer questions about the data. When data do not follow a linear pattern, then there is no constant rate of
change.

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Name Date

Lesson 12: Nonlinear Models in a Data Context (Optional)

Exit Ticket

The table shows the population of New York City from 1850 to 2000 for every 50 years.

Year Population
Population Growth

(change over 𝟕𝟕𝟎𝟎-year
time period)

1850 515,547 −

1900 3,437,202

1950 7,891,957

2000 8,008,278

Data Source: www.census.gov

1. Find the growth of the population from 1850 to 1900. Write your answer in the table in the row for the year 1900.

2. Find the growth of the population from 1900 to 1950. Write your answer in the table in the row for the year 1950.

3. Find the growth of the population from 1950 to 2000. Write your answer in the table in the row for the year 2000.

4. Does it appear that a linear model is a good fit for the data? Why or why not?

5. Describe how the population changes as the years increase.

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6. Construct a scatter plot of time versus population on the grid below. Draw a line or curve that you feel reasonably
describes the data.

7. Estimate the population of New York City in 1975. Explain how you found your estimate.

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Exit Ticket Sample Solutions

The table shows the population of New York City from 1850 to 2000 for every 𝟕𝟕𝟎𝟎 years.

Year Population
Population Growth

(change over 𝟕𝟕𝟎𝟎-year
time period)

1850 𝟕𝟕𝟏𝟏𝟕𝟕,𝟕𝟕𝟒𝟒𝟕𝟕 −

1900 𝟑𝟑,𝟒𝟒𝟑𝟑𝟕𝟕,𝟐𝟐𝟎𝟎𝟐𝟐 𝟐𝟐,𝟗𝟗𝟐𝟐𝟏𝟏,𝟏𝟏𝟕𝟕𝟕𝟕

1950 𝟕𝟕,𝟖𝟖𝟗𝟗𝟏𝟏,𝟗𝟗𝟕𝟕𝟕𝟕 𝟒𝟒,𝟒𝟒𝟕𝟕𝟒𝟒,𝟕𝟕𝟕𝟕𝟕𝟕

2000 𝟖𝟖,𝟎𝟎𝟎𝟎𝟖𝟖,𝟐𝟐𝟕𝟕𝟖𝟖 𝟏𝟏𝟏𝟏𝟏𝟏,𝟑𝟑𝟐𝟐𝟏𝟏

Data Source: www.census.gov

1. Find the growth of the population from 1850 to 1900. Write your answer in the table in the row for the year 1900.

2. Find the growth of the population from 1900 to 1950. Write your answer in the table in the row for the year 1950.

3. Find the growth of the population from 1950 to 2000. Write your answer in the table in the row for the year 2000.

4. Does it appear that a linear model is a good fit for the data? Why or why not?

No, a linear model is not a good fit for the data. The rate of population growth is not constant; the values in the
change in population column are all different.

5. Describe how the population changes as the years increase.

As the years increase, the population increases.

6. Construct a scatter plot of time versus population on the grid below. Draw a line or curve that you feel reasonably
describes the data.

Students should sketch a curve. If students use a straight line, point out that the line does not reasonably describe
the data, as some of the data points are far away from the line.

7. Estimate the population of New York City in 1975. Explain how you found your estimate.

It is approximately 𝟖𝟖,𝟎𝟎𝟎𝟎𝟎𝟎,𝟎𝟎𝟎𝟎𝟎𝟎. An estimate can be found by recognizing that the growth of the city did not change
very much from 1950 to 2000. The mean of the 1950 population and the 2000 population could also be found.

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Problem Set Sample Solutions

1. Once the brakes of the car have been applied, the car does not stop immediately. The distance that the car travels
after the brakes have been applied is called the braking distance. The table below shows braking distance (how far
the car travels once the brakes have been applied) and the speed of the car.

Speed (miles per hour) Braking Distance (feet)
𝟏𝟏𝟎𝟎 𝟕𝟕
𝟐𝟐𝟎𝟎 𝟏𝟏𝟕𝟕
𝟑𝟑𝟎𝟎 𝟑𝟑𝟕𝟕
𝟒𝟒𝟎𝟎 𝟏𝟏𝟕𝟕
𝟕𝟕𝟎𝟎 𝟏𝟏𝟎𝟎𝟕𝟕
𝟏𝟏𝟎𝟎 𝟏𝟏𝟕𝟕𝟎𝟎
𝟕𝟕𝟎𝟎 𝟐𝟐𝟎𝟎𝟕𝟕
𝟖𝟖𝟎𝟎 𝟐𝟐𝟏𝟏𝟕𝟕

Data Source: http://forensicdynamics.com/stopping-braking-distance-calculator

(Note: Data has been rounded.)

a. Construct a scatter plot of braking distance versus speed on the grid below.

b. Find the amount of additional distance a car would travel after braking for each speed increase of 𝟏𝟏𝟎𝟎 𝐦𝐦𝐦𝐦𝐦𝐦.
Record your answers in the table below.

Speed (miles per hour) Braking Distance (feet) Amount of Distance Increase
𝟏𝟏𝟎𝟎 𝟕𝟕 −
𝟐𝟐𝟎𝟎 𝟏𝟏𝟕𝟕 𝟏𝟏𝟐𝟐
𝟑𝟑𝟎𝟎 𝟑𝟑𝟕𝟕 𝟐𝟐𝟎𝟎
𝟒𝟒𝟎𝟎 𝟏𝟏𝟕𝟕 𝟐𝟐𝟖𝟖
𝟕𝟕𝟎𝟎 𝟏𝟏𝟎𝟎𝟕𝟕 𝟒𝟒𝟎𝟎
𝟏𝟏𝟎𝟎 𝟏𝟏𝟕𝟕𝟎𝟎 𝟒𝟒𝟕𝟕
𝟕𝟕𝟎𝟎 𝟐𝟐𝟎𝟎𝟕𝟕 𝟕𝟕𝟕𝟕
𝟖𝟖𝟎𝟎 𝟐𝟐𝟏𝟏𝟕𝟕 𝟏𝟏𝟎𝟎

c. Based on the table, do you think the data follow a linear pattern? Explain your answer.

No. If the relationship is linear, the values in the Amount of Distance Increase column would be
approximately equal.

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d. Describe how the distance it takes a car to stop changes as the speed of the car increases.

As the speed of the car increases, the distance it takes the car to stop also increases.

e. Sketch a smooth curve that you think describes the relationship between braking distance and speed.

f. Estimate braking distance for a car traveling at 𝟕𝟕𝟐𝟐 𝐦𝐦𝐦𝐦𝐦𝐦. Estimate braking distance for a car traveling at
𝟕𝟕𝟕𝟕 𝐦𝐦𝐦𝐦𝐦𝐦. Explain how you made your estimates.

For 𝟕𝟕𝟐𝟐 𝐦𝐦𝐦𝐦𝐦𝐦, the braking distance is about 𝟏𝟏𝟏𝟏𝟕𝟕 𝐟𝐟𝐟𝐟.

For 𝟕𝟕𝟕𝟕 𝐦𝐦𝐦𝐦𝐦𝐦, the braking distance is about 𝟐𝟐𝟑𝟑𝟎𝟎 𝐟𝐟𝐟𝐟.

Both estimates can be made by starting on the 𝒙𝒙-axis, moving up to the curve, and then moving over to the
𝒚𝒚-axis.

2. The scatter plot below shows the relationship between cost (in dollars) and radius length (in meters) of fertilizing
different-sized circular fields. The curve shown was drawn to describe the relationship between cost and radius.

a. Is the curve a good fit for the data? Explain.

Yes, the curve fits the data very well. The data points lie close to the curve.

b. Use the curve to estimate the cost for fertilizing a circular field of radius 𝟑𝟑𝟎𝟎 𝐦𝐦. Explain how you made your
estimate.

Using the curve drawn on the graph, the cost is approximately $𝟐𝟐𝟎𝟎𝟎𝟎–$𝟐𝟐𝟕𝟕𝟎𝟎.

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c. Estimate the radius of the field if the fertilizing cost was $𝟐𝟐,𝟕𝟕𝟎𝟎𝟎𝟎. Explain how you made your estimate.

Using the curve, an estimate for the radius is approximately 𝟗𝟗𝟒𝟒 𝐦𝐦. Locate the approximate cost of $𝟐𝟐,𝟕𝟕𝟎𝟎𝟎𝟎.
The approximate radius for that point is 𝟗𝟗𝟒𝟒 𝐦𝐦.

3. Suppose a dolphin is fitted with a GPS that monitors its position in relationship to a research ship. The table below
contains the time (in seconds) after the dolphin is released from the ship and the distance (in feet) the dolphin is
from the research ship.

Time (seconds) Distance from the Ship
(feet)

Increase in Distance
from the Ship

𝟎𝟎 𝟎𝟎 −

𝟕𝟕𝟎𝟎 𝟖𝟖𝟕𝟕 𝟖𝟖𝟕𝟕

𝟏𝟏𝟎𝟎𝟎𝟎 𝟏𝟏𝟗𝟗𝟎𝟎 𝟏𝟏𝟎𝟎𝟕𝟕

𝟏𝟏𝟕𝟕𝟎𝟎 𝟑𝟑𝟗𝟗𝟖𝟖 𝟐𝟐𝟎𝟎𝟖𝟖

𝟐𝟐𝟎𝟎𝟎𝟎 𝟕𝟕𝟕𝟕𝟕𝟕 𝟏𝟏𝟕𝟕𝟗𝟗

𝟐𝟐𝟕𝟕𝟎𝟎 𝟖𝟖𝟕𝟕𝟑𝟑 𝟐𝟐𝟕𝟕𝟏𝟏

𝟑𝟑𝟎𝟎𝟎𝟎 𝟏𝟏,𝟏𝟏𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏𝟗𝟗

a. Construct a scatter plot of distance versus time on the grid below.

b. Find the additional distance the dolphin traveled for each increase of 𝟕𝟕𝟎𝟎 seconds. Record your answers in the
table above.

See the table above.

c. Based on the table, do you think that the data follow a linear pattern? Explain your answer.

No, the change in distance from the ship is not constant.

d. Describe how the distance that the dolphin is from the ship changes as the time increases.

As the time away from the ship increases, the distance the dolphin is from the ship is also increasing.
The farther the dolphin is from the ship, the faster it is swimming.

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e. Sketch a smooth curve that you think fits the data reasonably well.

f. Estimate how far the dolphin will be from the ship after 𝟏𝟏𝟖𝟖𝟎𝟎 seconds. Explain how you made your estimate.

About 𝟕𝟕𝟎𝟎𝟎𝟎 𝐟𝐟𝐟𝐟. Starting on the 𝒙𝒙-axis at approximately 𝟏𝟏𝟖𝟖𝟎𝟎 seconds, move up to the curve and then over to
the 𝒚𝒚-axis to find an estimate of the distance.

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8
G R A D E

New York State Common Core

Mathematics Curriculum
GRADE 8 • MODULE 6

Topic D

Bivariate Categorical Data

8.SP.A.4

Focus Standard: 8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way table. Construct and
interpret a two-way table summarizing data on two categorical variables collected from
the same subjects. Use relative frequencies calculated for rows or columns to describe
possible association between the two variables. For example, collect data from
students in your class on whether or not they have a curfew on school nights and
whether or not they have assigned chores at home. Is there evidence that those who
have a curfew also tend to have chores?

Instructional Days: 2

Lesson 13: Summarizing Bivariate Categorical Data in a Two-Way Table (P)1

Lesson 14: Association Between Categorical Variables (P)

Topic D extends the concept of a relationship between variables to bivariate categorical data. In Lesson 13,
students organize bivariate categorical data into a two-way table (8.SP.A.4). They calculate row and column
relative frequencies and interpret them in the context of a problem. They informally decide if there is an
association between two categorical variables by examining the differences of row or column relative
frequencies. They interpret association between two categorical variables as knowing the value of one of the
variables provides information about the likelihood of the different possible values of the other variable.

1Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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Lesson 13: Summarizing Bivariate Categorical Data in a

Two-Way Table

Student Outcomes

 Students organize bivariate categorical data into a two-way table.
 Students calculate row and column relative frequencies and interpret them in context.

Lesson Notes
In this lesson, students first organize data from a survey on a single categorical variable (i.e., a univariate categorical
data) into a one-way frequency table. Some questions review content on random and representative samples that
students first encountered in Grade 7. Then, they organize data on two categorical variables (i.e., bivariate categorical
data) into two-way frequency tables. This lesson also introduces students to relative frequencies (e.g., row and column
relative frequencies). Students then interpret relative frequencies in context.

Classwork

Exercises 1–5 (3–5 minutes)

Read the opening scenario to the class. Allow students a few minutes to choose their favorite ice cream flavors.
Consider also asking students to raise their hands for each flavor preference and having them record the class data in the
table provided for Exercise 1.

Exercises 1–8

On an upcoming field day at school, the principal wants to provide ice cream during lunch. She offers three flavors:
chocolate, strawberry, and vanilla. She selected your class to complete a survey to help her determine how much of each
flavor to buy.

1. Answer the following question. Wait for your teacher to count how many students selected each flavor. Then,
record the class totals for each flavor in the table below.

“Which of the following three ice cream flavors is your favorite: chocolate, strawberry, or vanilla?”

Answers will vary. One possibility is shown below.

Ice Cream Flavor Chocolate Strawberry Vanilla Total

Number of Students 𝟏𝟏𝟏𝟏 𝟒𝟒 𝟏𝟏 𝟐𝟐𝟐𝟐

2. Which ice cream flavor do most students prefer?

Students should respond with the most-selected flavor. For the data set shown here, that is chocolate.

3. Which ice cream flavor do the fewest students prefer?

Students should respond with the least-selected flavor. For the data set shown here, that is strawberry.

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4. What percentage of students preferred each flavor? Round to the nearest tenth of a percent.

Answers will vary based on the data gathered in Exercise 1.

Chocolate:
𝟏𝟏𝟏𝟏
𝟐𝟐𝟐𝟐

≈ 𝟔𝟔𝟔𝟔. 𝟏𝟏%

Strawberry:
𝟒𝟒
𝟐𝟐𝟐𝟐

≈ 𝟏𝟏𝟒𝟒. 𝟑𝟑%

Vanilla:
𝟏𝟏
𝟐𝟐𝟐𝟐

= 𝟐𝟐𝟐𝟐%

5. Do the numbers in the table in Exercise 1 summarize data on a categorical variable or a
numerical variable?

The numbers in this table summarize data on a categorical variable—the preferred flavor of
ice cream.

Exercises 6–8 (5 minutes)

Let students work with a partner to discuss and answer Exercises 6–8. These exercises review the concepts of random
samples and representative samples from Grade 7. These exercises may also be used to structure a class discussion.

6. Do the students in your class represent a random sample of all the students in your school? Why or why not?
Discuss this with your neighbor.

No, because there is no indication that the students were selected randomly.

7. Is your class representative of all the other classes at your school? Why or why not? Discuss this with your
neighbor.

This class might be representative of the other eighth-grade classes but might not be representative of sixth- and
seventh-grade classes.

8. Do you think the principal will get an accurate estimate of the proportion of students who prefer each ice cream
flavor for the whole school using only your class? Why or why not? Discuss this with your neighbor.

It is unlikely to give a good estimate. It would depend on how representative the class is of all of the students at the
school.

Example 1 (3–5 minutes)

In this example, be sure that students understand the vocabulary. Univariate means one
variable. Thus, univariate categorical data means that there are data on one variable that
are categorical, such as favorite ice cream flavor. A one-way frequency table is typically
used to summarize values of univariate categorical data. When the data are categorical, it
is customary to convert the table entries to relative frequencies instead of frequencies.

In other words, the fraction
frequency

total
should be used, which is the relative frequency or

proportion for each possible value of the categorical variable.

Scaffolding:
 Point out the prefix uni –

means one. So, univariate
means one variable.

 Some students may
recognize the word table
but may not yet know the
mathematical meaning of
the term. Point out that
this lesson defines table as
a tool for organizing data.

Scaffolding:
Categorical variables are
variables that represent
categorical data. Data that
represent specific descriptions
or categories are called
categorical data.

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Students in another class were asked the same question about their favorite ice cream
flavors. In this particular class of 25 students, 11 preferred chocolate, 4 preferred
strawberry, and 10 preferred vanilla. Thus, the relative frequency for chocolate is
11
25

= 0.44. The interpretation of this value is “44% of the students in this class prefer

chocolate ice cream.” Students often find writing interpretations to be difficult. Explain
why this is not the case in this example.

Example 1

Students in a different class were asked the same question about their favorite ice cream flavors. The table below shows
the ice cream flavors and the number of students who chose each flavor for that particular class. This table is called a
one-way frequency table because it shows the counts of a univariate categorical variable.

This is the univariate categorical
variable.

These are the counts for each
category.

We compute the relative frequency for each ice cream flavor by dividing the count by the total number of observations.

𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 𝐟𝐟𝐫𝐫𝐫𝐫𝐟𝐟𝐟𝐟𝐫𝐫𝐟𝐟𝐟𝐟𝐟𝐟 =
𝐟𝐟𝐜𝐜𝐟𝐟𝐟𝐟𝐫𝐫 𝐟𝐟𝐜𝐜𝐫𝐫 𝐫𝐫 𝐟𝐟𝐫𝐫𝐫𝐫𝐫𝐫𝐜𝐜𝐜𝐜𝐫𝐫𝐟𝐟

𝐫𝐫𝐜𝐜𝐫𝐫𝐫𝐫𝐫𝐫 𝐟𝐟𝐟𝐟𝐧𝐧𝐧𝐧𝐫𝐫𝐫𝐫 𝐜𝐜𝐟𝐟 𝐜𝐜𝐧𝐧𝐨𝐨𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐜𝐜𝐟𝐟𝐨𝐨

Since 𝟏𝟏𝟏𝟏 out of 𝟐𝟐𝟐𝟐 students answered chocolate, the relative frequency would be
𝟏𝟏𝟏𝟏
𝟐𝟐𝟐𝟐

= 𝟔𝟔. 𝟒𝟒𝟒𝟒. This relative frequency

shows that 𝟒𝟒𝟒𝟒% of the class prefers chocolate ice cream. In other words, the relative frequency is the proportional value
that each category is of the whole.

Exercises 9–10 (3 minutes)

Let students work independently and confirm their answers with a neighbor.

Exercises 9–10

Use the table for the preferred ice cream flavors from the class in Example 1 to answer the following questions.

9. What is the relative frequency for the category strawberry?

Relative frequency = 𝟒𝟒
𝟐𝟐𝟐𝟐 = 𝟔𝟔.𝟏𝟏𝟔𝟔

10. Write a sentence interpreting the relative frequency value in the context of strawberry ice cream preference.

𝟏𝟏𝟔𝟔% of the students in this class prefer strawberry ice cream.

Ice Cream
Flavor Chocolate Strawberry Vanilla Total

Number of
Students 𝟏𝟏𝟏𝟏 𝟒𝟒 𝟏𝟏𝟔𝟔 𝟐𝟐𝟐𝟐

Scaffolding:
The word relative has multiple
meanings, such as a family
member. In this context, it
refers to a measure that is
compared to something else.
Making this distinction clear
aids in comprehension.

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Example 2 (3–5 minutes)

Read through the example as a class. In this example, the focus shifts to bivariate categorical data. The prefix bi- means
two, so these data contain values for two variables that are both categorical, such as favorite ice cream flavor and
gender.

Example 2

The principal also wondered if boys and girls have different favorite ice cream flavors. She decided to redo the survey by
taking a random sample of students from the school and recording both their favorite ice cream flavors and their genders.
She asked the following two questions:

 “Which of the following ice cream flavors is your favorite: chocolate, strawberry, or vanilla?”

 “What is your gender: male or female?”

The results of the survey are as follows:

 Of the 𝟑𝟑𝟔𝟔 students who prefer chocolate ice cream, 𝟐𝟐𝟐𝟐 are males.

 Of the 𝟐𝟐𝟐𝟐 students who prefer strawberry ice cream, 𝟏𝟏𝟐𝟐 are females.

 Of the 𝟐𝟐𝟏𝟏 students who prefer vanilla ice cream, 𝟏𝟏𝟑𝟑 are males.

The values of two variables, which were ice cream flavor and gender, were recorded in this survey. Since both of the
variables are categorical, the data are bivariate categorical data.

Exercises 11–17 (10 minutes)

Present Exercises 11 and 12 to the class one at a time.

Exercises 11–17

11. Can we display these data in a one-way frequency table? Why or why not?

No, a one-way frequency table is for univariate data. Here we have bivariate data, so we would need to use a
two-way table.

12. Summarize the results of the second survey of favorite ice cream flavors in the following table:

Favorite Ice Cream Flavor

Chocolate Strawberry Vanilla Total

G
en

de
r Male 𝟐𝟐𝟐𝟐 𝟏𝟏𝟔𝟔 𝟏𝟏𝟑𝟑 𝟒𝟒𝟐𝟐

Female 𝟐𝟐 𝟏𝟏𝟐𝟐 𝟏𝟏𝟒𝟒 𝟑𝟑𝟏𝟏

Total 𝟑𝟑𝟔𝟔 𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏 𝟐𝟐𝟐𝟐

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Next, remind students how to calculate relative frequencies. Give students a few minutes to calculate the approximate
relative frequencies and to write them in the table. A cell relative frequency is a cell frequency divided by the total
number of observations. Let students work independently on Exercises 13–17. Discuss and confirm the answers to
Exercises 16 and 17 as a class.

13. Calculate the relative frequencies of the data in the table in Exercise 12, and write them in the following table.

Favorite Ice Cream Flavor

Chocolate Strawberry Vanilla Total

G
en

de
r Male ≈ 𝟔𝟔.𝟐𝟐𝟏𝟏 ≈ 𝟔𝟔.𝟏𝟏𝟐𝟐 ≈ 𝟔𝟔.𝟏𝟏𝟔𝟔 ≈ 𝟔𝟔.𝟐𝟐𝟐𝟐

Female ≈ 𝟔𝟔.𝟏𝟏𝟔𝟔 ≈ 𝟔𝟔.𝟏𝟏𝟐𝟐 ≈ 𝟔𝟔.𝟏𝟏𝟏𝟏 ≈ 𝟔𝟔.𝟒𝟒𝟐𝟐

Total ≈ 𝟔𝟔.𝟑𝟑𝟏𝟏 ≈ 𝟔𝟔.𝟑𝟑𝟔𝟔 ≈ 𝟔𝟔.𝟑𝟑𝟑𝟑 𝟏𝟏.𝟔𝟔

Use the relative frequency values in the table to answer the following questions:

14. What is the proportion of the students who prefer chocolate ice cream?

𝟔𝟔.𝟑𝟑𝟏𝟏

15. What is the proportion of students who are female and prefer vanilla ice cream?

𝟔𝟔.𝟏𝟏𝟏𝟏

16. Write a sentence explaining the meaning of the approximate relative frequency 𝟔𝟔.𝟐𝟐𝟐𝟐.

Approximately 𝟐𝟐𝟐𝟐% of students responding to the survey are males.

17. Write a sentence explaining the meaning of the approximate relative frequency 𝟔𝟔.𝟏𝟏𝟔𝟔.

Approximately 𝟏𝟏𝟔𝟔% of students responding to the survey are females who prefer chocolate ice cream.

Example 3 (3–5 minutes)

In this example, students learn that they can also use row and column totals to calculate
relative frequencies. This concept provides a foundation for future work with conditional
relative frequencies in Algebra I.

Point out that students need to carefully decide which total (i.e., table total, row total, or
column total) they should use.

Scaffolding:
 English language learners

may need a reminder
about the difference
between columns and
rows.

 A column refers to a
vertical arrangement, and
a row refers to a
horizontal arrangement in
the table.

 Keeping a visual aid posted
that labels these parts aids
in comprehension.

MP.6

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Example 3

In the previous exercises, you used the total number of students to calculate relative frequencies. These relative
frequencies were the proportion of the whole group who answered the survey a certain way. Sometimes we use row or
column totals to calculate relative frequencies. We call these row relative frequencies or column relative frequencies.

Below is the two-way frequency table for your reference. To calculate “the proportion of male students who prefer
chocolate ice cream,” divide the 𝟐𝟐𝟐𝟐 male students who preferred chocolate ice cream by the total of 𝟒𝟒𝟐𝟐 male students.

This proportion is
𝟐𝟐𝟐𝟐
𝟒𝟒𝟐𝟐

≈ 𝟔𝟔. 𝟒𝟒𝟒𝟒. Notice that you used the row total to make this calculation. This is a row relative

frequency.

Favorite Ice Cream Flavor

Chocolate Strawberry Vanilla Total

G
en

de
r Male 𝟐𝟐𝟐𝟐 𝟏𝟏𝟔𝟔 𝟏𝟏𝟑𝟑 𝟒𝟒𝟐𝟐

Female 𝟐𝟐 𝟏𝟏𝟐𝟐 𝟏𝟏𝟒𝟒 𝟑𝟑𝟏𝟏

Total 𝟑𝟑𝟔𝟔 𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏 𝟐𝟐𝟐𝟐

Exercises 18–22 (8–10 minutes)

Discuss Exercise 18 as a class. When explaining the problem, try covering the unused part of the table with paper to
focus attention on the query at hand.

Exercises 18–22

In Exercise 13, you used the total number of students to calculate relative frequencies. These relative frequencies were
the proportion of the whole group who answered the survey a certain way.

18. Suppose you are interested in the proportion of male students who prefer chocolate ice cream. How is this value
different from “the proportion of students who are male and prefer chocolate ice cream”? Discuss this with your
neighbor.

The proportion of students who are male and prefer chocolate ice cream is
𝟐𝟐𝟐𝟐
𝟐𝟐𝟐𝟐

≈ 𝟔𝟔. 𝟐𝟐𝟏𝟏. This proportion uses all

𝟐𝟐𝟐𝟐 students. The proportion of male students who prefer chocolate ice cream is
𝟐𝟐𝟐𝟐
𝟒𝟒𝟐𝟐

≈ 𝟔𝟔. 𝟒𝟒𝟒𝟒. This proportion uses

only the 𝟒𝟒𝟐𝟐 male students as its total.

Now, allow students time to answer Exercises 19–22. Discuss student answers stressing which total was used in the
calculation.

19. Use the table provided in Example 3 to calculate the following relative frequencies.

a. What proportion of students who prefer vanilla ice cream are female?

𝟏𝟏𝟒𝟒
𝟐𝟐𝟏𝟏

≈ 𝟔𝟔.𝟐𝟐𝟐𝟐

b. What proportion of male students prefer strawberry ice cream? Write a sentence explaining the meaning of
this proportion in the context of this problem.

𝟏𝟏𝟔𝟔
𝟒𝟒𝟐𝟐

≈ 𝟔𝟔. 𝟐𝟐𝟐𝟐 Twenty-two percent of male students in this survey prefer strawberry ice cream.

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c. What proportion of female students prefer strawberry ice cream?

𝟏𝟏𝟐𝟐
𝟑𝟑𝟏𝟏

≈ 𝟔𝟔.𝟒𝟒𝟏𝟏

d. What proportion of students who prefer strawberry ice cream are female?

𝟏𝟏𝟐𝟐
𝟐𝟐𝟐𝟐

≈ 𝟔𝟔.𝟔𝟔𝟔𝟔

20. A student is selected at random from this school. What would you predict this student’s favorite ice cream to be?
Explain why you chose this flavor.

I would predict that the student’s favorite flavor is chocolate because more students chose chocolate in the survey.

21. Suppose the randomly selected student is male. What would you predict his favorite flavor of ice cream to be?
Explain why you chose this flavor.

I would predict his favorite flavor to be chocolate because more male students chose chocolate in the survey.

22. Suppose the randomly selected student is female. What would you predict her favorite flavor of ice cream to be?
Explain why you chose this flavor.

I would predict her favorite flavor to be strawberry because more female students chose strawberry in the survey.

Closing (2 minutes)

Review the Lesson Summary with students.

Exit Ticket (5 minutes)

Lesson Summary
 Univariate categorical data are displayed in a one-way frequency table.

 Bivariate categorical data are displayed in a two-way frequency table.

 Relative frequency is the frequency divided by a total ( ).

 A cell relative frequency is a cell frequency divided by the total number of observations.

 A row relative frequency is a cell frequency divided by the row total.

 A column relative frequency is a cell frequency divided by the column total.

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Name Date

Lesson 13: Summarizing Bivariate Categorical Data in a Two-Way

Table

Exit Ticket

1. Explain what the term bivariate categorical data means.

2. Explain how to calculate relative frequency. What is another word for relative frequency?

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3. A random group of students are polled about how they get to school. The results are summarized in the table
below.

School Transportation Survey
Walk Ride Bus Carpool Total

G
en

de
r Male

9 26 9 44

Female
8 26 24 58

Total
17 52 33 102

a. Calculate the relative frequencies for the table above. Write them as a percent in each cell of the table.
Round to the nearest tenth of a percent.

b. What is the relative frequency for the Carpool category? Write a sentence interpreting this value in the
context of school transportation.

c. What is the proportion of students who are female and walk to school? Write a sentence interpreting this

value in the context of school transportation.

d. A student is selected at random from this school. What would you predict this student’s mode of school
transportation to be? Explain.

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Exit Ticket Sample Solutions

1. Explain what the term bivariate categorical data means.

Bivariate categorical data means that the data set comprises data on two variables that are both categorical.

2. Explain how to calculate relative frequency. What is another word for relative frequency?

Relative frequency is calculated by dividing a frequency by the total number of observations. Another word for
relative frequency is proportion.

3. A random group of students are polled about how they get to school. The results are summarized in the table
below.

School Transportation Survey

Walk Ride Bus Carpool Total

G
en

de
r Male 𝟒𝟒

≈ 𝟐𝟐.𝟐𝟐%
𝟐𝟐𝟔𝟔

≈ 𝟐𝟐𝟐𝟐.𝟐𝟐%
𝟒𝟒

≈ 𝟐𝟐.𝟐𝟐%
𝟒𝟒𝟒𝟒

≈ 𝟒𝟒𝟑𝟑.𝟏𝟏%

Female 𝟏𝟏
≈ 𝟔𝟔.𝟒𝟒%

𝟐𝟐𝟔𝟔
≈ 𝟐𝟐𝟐𝟐.𝟐𝟐%

𝟐𝟐𝟐𝟐
≈ 𝟐𝟐𝟒𝟒.𝟐𝟐%

𝟐𝟐𝟐𝟐
≈ 𝟐𝟐𝟔𝟔.𝟒𝟒%

Total 𝟏𝟏𝟔𝟔
≈ 𝟏𝟏𝟐𝟐.𝟏𝟏%

𝟐𝟐𝟐𝟐
≈ 𝟐𝟐𝟏𝟏.𝟔𝟔%

𝟑𝟑𝟒𝟒
≈ 𝟑𝟑𝟑𝟑.𝟑𝟑%

𝟏𝟏𝟔𝟔𝟐𝟐
𝟏𝟏𝟔𝟔𝟔𝟔.𝟔𝟔%

a. Calculate the relative frequencies for the table above. Write them as a percent in each cell of the table.
Round to the nearest tenth of a percent.

See the completed table above.

b. What is the relative frequency for the Carpool category? Write a sentence interpreting this value in the
context of school transportation.

The relative frequency is 𝟔𝟔.𝟑𝟑𝟑𝟑𝟑𝟑, or 𝟑𝟑𝟑𝟑.𝟑𝟑%. Approximately 𝟑𝟑𝟑𝟑.𝟑𝟑% of the students surveyed use a carpool to
get to school.

c. What is the proportion of students who are female and walk to school? Write a sentence interpreting this
value in the context of school transportation.

The proportion is 𝟔𝟔.𝟔𝟔𝟔𝟔𝟒𝟒, or 𝟔𝟔.𝟒𝟒%. Approximately 𝟔𝟔.𝟒𝟒% of the students surveyed are female and walk to
school.

d. A student is selected at random from this school. What would you predict this student’s mode of school
transportation to be? Explain.

I would predict the student would ride the bus because more students in the survey chose this mode of
transportation.

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Problem Set Sample Solutions

Every student at Abigail Douglas Middle School is enrolled in exactly one extracurricular activity. The school counselor
recorded data on extracurricular activity and gender for all 𝟐𝟐𝟐𝟐𝟒𝟒 eighth-grade students at the school.

The counselor’s findings for the 𝟐𝟐𝟐𝟐𝟒𝟒 eighth-grade students are the following:

 Of the 𝟐𝟐𝟔𝟔 students enrolled in band, 𝟒𝟒𝟐𝟐 are male.

 Of the 𝟔𝟔𝟐𝟐 students enrolled in choir, 𝟐𝟐𝟔𝟔 are male.

 Of the 𝟐𝟐𝟐𝟐 students enrolled in sports, 𝟑𝟑𝟔𝟔 are female.

 Of the 𝟐𝟐𝟏𝟏 students enrolled in art, 𝟒𝟒 are female.

1. Complete the table below.

Extracurricular Activities

Band Choir Sports Art Total

G
en

de
r Female 𝟑𝟑𝟐𝟐 𝟒𝟒𝟐𝟐 𝟑𝟑𝟔𝟔 𝟒𝟒 𝟏𝟏𝟐𝟐𝟐𝟐

Male 𝟒𝟒𝟐𝟐 𝟐𝟐𝟔𝟔 𝟐𝟐𝟐𝟐 𝟏𝟏𝟐𝟐 𝟏𝟏𝟑𝟑𝟐𝟐

Total 𝟐𝟐𝟔𝟔 𝟔𝟔𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏 𝟐𝟐𝟐𝟐𝟒𝟒

2. Write a sentence explaining the meaning of the frequency 𝟑𝟑𝟐𝟐 in this table.

The frequency of 𝟑𝟑𝟐𝟐 represents the number of eighth-grade students who are enrolled in band and are female.

Use the table provided above to calculate the following relative frequencies.

3. What proportion of students are male and enrolled in choir?

𝟐𝟐𝟔𝟔
𝟐𝟐𝟐𝟐𝟒𝟒

≈ 𝟔𝟔.𝟔𝟔𝟐𝟐

4. What proportion of students are enrolled in a musical extracurricular activity (i.e., band or choir)?

𝟐𝟐𝟔𝟔 + 𝟔𝟔𝟐𝟐
𝟐𝟐𝟐𝟐𝟒𝟒

≈ 𝟔𝟔.𝟐𝟐𝟏𝟏

5. What proportion of male students are enrolled in sports?

𝟐𝟐𝟐𝟐
𝟏𝟏𝟑𝟑𝟐𝟐

≈ 𝟔𝟔.𝟒𝟒𝟒𝟒

6. What proportion of students enrolled in sports are male?

𝟐𝟐𝟐𝟐
𝟐𝟐𝟐𝟐

≈ 𝟔𝟔.𝟔𝟔𝟔𝟔

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Pregnant women often undergo ultrasound tests to monitor their babies’ health. These tests can also be used to predict
the gender of the babies, but these predictions are not always accurate. Data on the gender predicted by ultrasound and
the actual gender of the baby for 𝟏𝟏,𝟔𝟔𝟔𝟔𝟔𝟔 babies are summarized in the two-way table below.

Predicted Gender

Female Male

Ac
tu

G
en

de
r Female 𝟒𝟒𝟑𝟑𝟐𝟐 𝟒𝟒𝟐𝟐

Male 𝟏𝟏𝟑𝟑𝟔𝟔 𝟑𝟑𝟒𝟒𝟔𝟔

7. Write a sentence explaining the meaning of the frequency 𝟏𝟏𝟑𝟑𝟔𝟔 in this table.

The frequency of 𝟏𝟏𝟑𝟑𝟔𝟔 represents the number of babies who were predicted to be female but were actually male
(i.e., the ultrasound prediction was not correct for these babies).

Use the table provided above to calculate the following relative frequencies.

8. What is the proportion of babies who were predicted to be male but were actually female?

𝟒𝟒𝟐𝟐
𝟏𝟏𝟔𝟔𝟔𝟔𝟔𝟔

= 𝟔𝟔.𝟔𝟔𝟒𝟒𝟐𝟐

9. What is the proportion of incorrect ultrasound gender predictions?

𝟏𝟏𝟑𝟑𝟔𝟔+ 𝟒𝟒𝟐𝟐
𝟏𝟏𝟔𝟔𝟔𝟔𝟔𝟔

= 𝟔𝟔.𝟏𝟏𝟏𝟏𝟐𝟐

10. For babies predicted to be female, what proportion of the predictions were correct?

𝟒𝟒𝟑𝟑𝟐𝟐
𝟐𝟐𝟔𝟔𝟐𝟐

≈ 𝟔𝟔.𝟏𝟏𝟔𝟔𝟒𝟒

11. For babies predicted to be male, what proportion of the predictions were correct?

𝟑𝟑𝟒𝟒𝟔𝟔
𝟒𝟒𝟑𝟑𝟐𝟐

≈ 𝟔𝟔.𝟐𝟐𝟒𝟒𝟔𝟔

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NYS COMMON CORE MATHEMATICS CURRICULUM 8•6 Lesson 14

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Lesson 14: Association Between Categorical Variables

Student Outcomes

 Students use row relative frequencies or column relative frequencies to informally determine whether there is
an association between two categorical variables.

Lesson Notes
In this lesson, students consider whether conclusions are reasonable based on a two-way table. Students think about
what it means to have similar row relative frequencies for all rows in a table or to have similar column relative
frequencies for all columns in a table. They also consider what it means to have row relative frequencies that are not
similar for all rows in the table. Students study the meaning of association between two categorical variables. For
example, students are asked to predict the favorite type of movie of a person whose gender is not known, and then they
are asked if knowing that the person is female would change their predictions. This lesson provides a foundation for
more detailed coverage of association in Algebra I.

This lesson is designed to have students work in groups of two to three. Prior to class, prepare the list of students in
each group, and arrange desks or tables to allow for group work.

Classwork

Example 1 (2–3 minutes)

Let students compare the two tables. Use the following questions to lead into a discussion about association. Some
students may calculate row relative frequencies to justify their answers.

 What are the variables being recorded?

 Smartphone use, gender, and age

 What can you conclude about the table Smartphone Use and Gender?
 Answers will vary. Possible responses: 75% of those surveyed use

smartphones. The percentage is the same for males and females, which
is 75%.

 What can you conclude about the table Smartphone Use and Age?

 Answers will vary. Possible responses: 75% of those surveyed use smartphones. However, a larger
percentage of those under 40 years old use a smartphone (90%) compared to the percentage of those
40 or older (60%).

 If you knew that someone was 20 years old, would you expect that person to use a smartphone? Explain.

 Yes. Possible explanation: One would expect a young person to use a smartphone based on the results
in the table because 90% of people under 40 use smartphones.

Scaffolding:
Some English language learners
may need to learn the word
smartphone. Consider
providing a visual aid.

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Example 1

Suppose a random group of people are surveyed about their use of smartphones. The results of the survey are
summarized in the tables below.

Smartphone Use and Gender

Use a

Smartphone
Do Not Use a
Smartphone

Total

Male 𝟑𝟑𝟑𝟑 𝟏𝟏𝟑𝟑 𝟒𝟒𝟑𝟑

Female 𝟒𝟒𝟒𝟒 𝟏𝟏𝟒𝟒 𝟔𝟔𝟑𝟑

Total 𝟕𝟕𝟒𝟒 𝟐𝟐𝟒𝟒 𝟏𝟏𝟑𝟑𝟑𝟑

Smartphone Use and Age

Use a

Smartphone
Do Not Use a
Smartphone

Total

Under 𝟒𝟒𝟑𝟑
Years of Age 𝟒𝟒𝟒𝟒 𝟒𝟒 𝟒𝟒𝟑𝟑

𝟒𝟒𝟑𝟑 Years of
Age or Older 𝟑𝟑𝟑𝟑 𝟐𝟐𝟑𝟑 𝟒𝟒𝟑𝟑

Total 𝟕𝟕𝟒𝟒 𝟐𝟐𝟒𝟒 𝟏𝟏𝟑𝟑𝟑𝟑

Example 2 (2 minutes)

Read the beginning of Example 2 to the class. Ask students:

 What are the variables being recorded?

 Movie preference and teacher or student status

Example 2

Suppose a sample of 𝟒𝟒𝟑𝟑𝟑𝟑 participants (teachers and students) was randomly selected from the middle schools and high
schools in a large city. These participants responded to the following question:

Which type of movie do you prefer to watch?

1. Action (The Avengers, Man of Steel, etc.)

2. Drama (42 (The Jackie Robinson Story), The Great Gatsby, etc.)

3. Science Fiction (Star Trek Into Darkness, World War Z, etc.)

4. Comedy (Monsters University, Despicable Me 2, etc.)

Movie preference and status (teacher or student) were recorded for each participant.

Exercises 1–7 (12–15 minutes)

Have students work in small groups. Give groups one to two minutes to answer Exercise 1, and then confirm their
answers as a class.

Students should read the results of the survey. Remind them that a row relative frequency is the cell frequency divided
by the corresponding row total. Allow groups to answer Exercises 2–5, and then confirm answers as a class. Give groups
adequate time to discuss Exercises 6 and 7, and then discuss as a class.

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Exercises 1–7

1. Two variables were recorded. Are these variables categorical or numerical?

Both variables are categorical.

2. The results of the survey are summarized in the table below.

Movie Preference
Action Drama Science Fiction Comedy Total
Student 𝟏𝟏𝟐𝟐𝟑𝟑 𝟔𝟔𝟑𝟑 𝟑𝟑𝟑𝟑 𝟗𝟗𝟑𝟑 𝟑𝟑𝟑𝟑𝟑𝟑
Teacher 𝟒𝟒𝟑𝟑 𝟐𝟐𝟑𝟑 𝟏𝟏𝟑𝟑 𝟑𝟑𝟑𝟑 𝟏𝟏𝟑𝟑𝟑𝟑
Total 𝟏𝟏𝟔𝟔𝟑𝟑 𝟖𝟖𝟑𝟑 𝟒𝟒𝟑𝟑 𝟏𝟏𝟐𝟐𝟑𝟑 𝟒𝟒𝟑𝟑𝟑𝟑

a. What proportion of participants who are teachers prefer action movies?

𝟒𝟒𝟑𝟑
𝟏𝟏𝟑𝟑𝟑𝟑

= 𝟑𝟑.𝟒𝟒𝟑𝟑

b. What proportion of participants who are teachers prefer drama movies?

𝟐𝟐𝟑𝟑
𝟏𝟏𝟑𝟑𝟑𝟑

= 𝟑𝟑.𝟐𝟐𝟑𝟑

c. What proportion of participants who are teachers prefer science fiction movies?

𝟏𝟏𝟑𝟑
𝟏𝟏𝟑𝟑𝟑𝟑

= 𝟑𝟑.𝟏𝟏𝟑𝟑

d. What proportion of participants who are teachers prefer comedy movies?

𝟑𝟑𝟑𝟑
𝟏𝟏𝟑𝟑𝟑𝟑

= 𝟑𝟑.𝟑𝟑𝟑𝟑

The answers to Exercise 2 are called row relative frequencies. Notice that you divided each cell frequency in the Teacher
row by the total for that row. Below is a blank relative frequency table.

Table of Row Relative Frequencies

Movie Preference

Action Drama Science Fiction Comedy

Student 𝟑𝟑.𝟒𝟒𝟑𝟑 𝟑𝟑.𝟐𝟐𝟑𝟑 𝟑𝟑.𝟏𝟏𝟑𝟑 𝟑𝟑.𝟑𝟑𝟑𝟑

Teacher (a) 𝟑𝟑.𝟒𝟒𝟑𝟑 (b) 𝟑𝟑.𝟐𝟐𝟑𝟑 (c) 𝟑𝟑.𝟏𝟏𝟑𝟑 (d) 𝟑𝟑.𝟑𝟑𝟑𝟑

Write your answers from Exercise 2 in the indicated cells in the table above.

3. Find the row relative frequencies for the Student row. Write your answers in the table above.

a. What proportion of participants who are students prefer action movies?

b. What proportion of participants who are students prefer drama movies?

c. What proportion of participants who are students prefer science fiction movies?

d. What proportion of participants who are students prefer comedy movies?

See the table above.

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4. Is a participant’s status (i.e., teacher or student) related to what type of movie he would prefer to watch? Why or
why not? Discuss this with your group.

No. Teachers are just as likely to prefer each movie type as students are, according to the row relative frequencies.

5. What does it mean when we say that there is no association between two variables? Discuss this with your group.

Answers will vary. No association means that knowing the value of one variable does not tell anything about the
value of the other variable.

6. Notice that the row relative frequencies for each movie type are the same for both the
Teacher and Student rows. When this happens, we say that the two variables, movie
preference and status (student or teacher), are not associated. Another way of thinking
about this is to say that knowing if a participant is a teacher (or a student) provides no
information about his movie preference.

What does it mean if row relative frequencies are not the same for all rows of a two-way
table?

It means that there is an association or a tendency between the two variables.

7. You can also evaluate whether two variables are associated by looking at column relative frequencies instead of row
relative frequencies. A column relative frequency is a cell frequency divided by the corresponding column total.

For example, the column relative frequency for the Student/Action cell is
𝟏𝟏𝟐𝟐𝟑𝟑
𝟏𝟏𝟔𝟔𝟑𝟑

= 𝟑𝟑. 𝟕𝟕𝟒𝟒.

a. Calculate the other column relative frequencies, and write them in the table below.

Table of Column Relative Frequencies

Movie Preference
Action Drama Science Fiction Comedy

Student 𝟑𝟑.𝟕𝟕𝟒𝟒 𝟑𝟑.𝟕𝟕𝟒𝟒 𝟑𝟑.𝟕𝟕𝟒𝟒 𝟑𝟑.𝟕𝟕𝟒𝟒

Teacher 𝟑𝟑.𝟐𝟐𝟒𝟒 𝟑𝟑.𝟐𝟐𝟒𝟒 𝟑𝟑.𝟐𝟐𝟒𝟒 𝟑𝟑.𝟐𝟐𝟒𝟒

b. What do you notice about the column relative frequencies for the four columns?

The column relative frequencies are equal for all four columns.

c. What would you conclude about association based on the column relative frequencies?

Because the column relative frequencies are the same for all four columns, we would conclude that there is no
association between movie preference and status.

In this part of the lesson, students should understand that there is a mathematical way to determine if there is no
association between two categorical variables. Students can look to see if the row relative frequencies are the same (or
approximately the same) for each row in the table. Discuss the mathematical method for determining if there is no
association between two categorical variables.

Scaffolding:
For English language learners,
the concept of no association
may be difficult. However, for
students working in groups,
consider explicitly modeling
the thinking employed in
Exercise 6.

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Example 3 (2 minutes)

Introduce the data in Example 3. Give students a moment to read the results. Take a quick movie preference poll in
class. Ask the following:

 Who likes action movies?

 Do you think movie preference is equal among males and females?
 Answers will vary. Encourage students to explain why they think the preferences might be equal or

different.

Example 3

In the survey described in Example 2, gender for each of the 𝟒𝟒𝟑𝟑𝟑𝟑 participants was also recorded. Some results of the
survey are given below:

 𝟏𝟏𝟔𝟔𝟑𝟑 participants preferred action movies.

 𝟖𝟖𝟑𝟑 participants preferred drama movies.

 𝟒𝟒𝟑𝟑 participants preferred science fiction movies.

 𝟐𝟐𝟒𝟒𝟑𝟑 participants were females.

 𝟕𝟕𝟖𝟖 female participants preferred drama movies.

 𝟑𝟑𝟐𝟐 male participants preferred science fiction movies.

 𝟔𝟔𝟑𝟑 female participants preferred action movies.

Exercises 8–11 (8–10 minutes)

Let students work with their groups on Exercises 8–10, and then confirm answers as a class. Give students two to three
minutes to complete Exercise 11.

Exercises 8–15

Use the results from Example 3 to answer the following questions. Be sure to discuss these questions with your group
members.

8. Complete the two-way frequency table that summarizes the data on movie preference and gender.

Movie Preference
Action Drama Science Fiction Comedy Total
Female 𝟔𝟔𝟑𝟑 𝟕𝟕𝟖𝟖 𝟖𝟖 𝟗𝟗𝟒𝟒 𝟐𝟐𝟒𝟒𝟑𝟑
Male 𝟏𝟏𝟑𝟑𝟑𝟑 𝟐𝟐 𝟑𝟑𝟐𝟐 𝟐𝟐𝟔𝟔 𝟏𝟏𝟔𝟔𝟑𝟑
Total 𝟏𝟏𝟔𝟔𝟑𝟑 𝟖𝟖𝟑𝟑 𝟒𝟒𝟑𝟑 𝟏𝟏𝟐𝟐𝟑𝟑 𝟒𝟒𝟑𝟑𝟑𝟑

9. What proportion of the participants are female?

𝟐𝟐𝟒𝟒𝟑𝟑
𝟒𝟒𝟑𝟑𝟑𝟑

= 𝟑𝟑.𝟔𝟔𝟑𝟑

10. If there was no association between gender and movie preference, should you expect more females than males or
fewer females than males to prefer action movies? Explain.

If there was no association between gender and movie preference, then I would expect more females than males to
prefer action movies just because there are more females in the sample. However, if there was an association
between gender and movie preference, then I would expect either fewer females than males who prefer action
movies or considerably more females than males who prefer action movies.

MP.2

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11. Make a table of row relative frequencies of each movie type for the Male row and the Female row. Refer to
Exercises 2–4 to review how to complete the table below.

Movie Preference
Action Drama Science Fiction Comedy
Female 𝟑𝟑.𝟐𝟐𝟒𝟒 𝟑𝟑.𝟑𝟑𝟐𝟐𝟒𝟒 𝟑𝟑.𝟑𝟑𝟑𝟑𝟑𝟑 𝟑𝟑.𝟑𝟑𝟗𝟗𝟐𝟐
Male 𝟑𝟑.𝟔𝟔𝟐𝟐𝟒𝟒 𝟑𝟑.𝟑𝟑𝟏𝟏𝟐𝟐𝟒𝟒 𝟑𝟑.𝟐𝟐 𝟑𝟑.𝟏𝟏𝟔𝟔𝟐𝟐𝟒𝟒

Exercises 12–15 (12–15 minutes)

Read the next instructions. Make sure that students understand that 1 of the 400 participants is randomly selected.
Allow groups about five minutes to discuss and answer Exercises 12 and 13.

Then, discuss as a class what association means. Allow students three minutes to answer Exercise 14.

Allow five minutes for groups to discuss whether the statements in Exercise 15 are correct. Call on groups to share their
answers.

Suppose that you randomly pick 𝟏𝟏 of the 𝟒𝟒𝟑𝟑𝟑𝟑 participants. Use the table of row relative frequencies on the previous
page to answer the following questions.

12. If you had to predict what type of movie this person chose, what would you predict? Explain why you made this
choice.

The participant likely prefers action movies because the largest proportion of participants preferred action movies.

13. If you know that the randomly selected participant is female, would you predict that her favorite type of movie is
action? If not, what would you predict, and why?

No. A female participant is more likely to prefer comedy since it has the greatest row relative frequency in the
Female row.

14. If knowing the value of one of the variables provides information about the value of the other variable, then there is
an association between the two variables.

Is there an association between the variables gender and movie preference? Explain.

Yes. The row relative frequencies are not the same (not even close) in each row in the table.

15. What can be said when two variables are associated? Read the following sentences. Decide if each sentence is a
correct statement based upon the survey data. If it is not correct, explain why not.

a. More females than males participated in the survey.

Correct

b. Males tend to prefer action and science fiction movies.

Correct

c. Being female causes one to prefer drama movies.

Incorrect Association does not imply a cause-and-effect relationship.

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Closing (3 minutes)

Read through the Lesson Summary with students.

If time allows, have students refer back to Example 1 and calculate row relative frequencies for each table to determine
if there is evidence of association between variables.

Exit Ticket (5 minutes)

Lesson Summary

 Saying that two variables are not associated means that knowing the value of one variable provides no
information about the value of the other variable.

 Saying that two variables are associated means that knowing the value of one variable provides
information about the value of the other variable.

 To determine if two variables are associated, calculate row relative frequencies. If the row relative
frequencies are about the same for all of the rows, it is reasonable to say that there is no association
between the two variables that define the table.

 Another way to decide if there is an association between two categorical variables is to calculate
column relative frequencies. If the column relative frequencies are about the same for all of the
columns, it is reasonable to say that there is no association between the two variables that define the
table.

 If the row relative frequencies are quite different for some of the rows, it is reasonable to say that there
is an association between the two variables that define the table.

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Lesson 14: Association Between Categorical Variables

Exit Ticket

A random sample of 100 eighth-grade students are asked to record two variables: whether they have a television in
their bedrooms and if they passed or failed their last math test. The results of the survey are summarized below.

 55 students have a television in their bedrooms.

 35 students do not have a television in their bedrooms and passed their last math test.

 25 students have a television and failed their last math test.

 35 students failed their last math test.

1. Complete the two-way table.

Pass Fail Total

Television in
the Bedroom

No Television
in the
Bedroom

Total

2. Calculate the row relative frequencies, and enter the values in the table above. Round to the nearest thousandth.

3. Is there evidence of association between the variables? If so, does this imply there is a cause-and-effect

relationship? Explain.

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Exit Ticket Sample Solutions

A random sample of 𝟏𝟏𝟑𝟑𝟑𝟑 eighth-grade students are asked to record two variables: whether they have a television in their
bedrooms and if they passed or failed their last math test. The results of the survey are summarized below.

 𝟒𝟒𝟒𝟒 students have a television in their bedrooms.

 𝟑𝟑𝟒𝟒 students do not have a television in their bedrooms and passed their last math test.

 𝟐𝟐𝟒𝟒 students have a television and failed their last math test.

 𝟑𝟑𝟒𝟒 students failed their last math test.

1. Complete the two-way table.

Pass Fail Total

Television in the
Bedroom

𝟑𝟑𝟑𝟑
≈ 𝟑𝟑.𝟒𝟒𝟒𝟒𝟒𝟒

𝟐𝟐𝟒𝟒
≈ 𝟑𝟑.𝟒𝟒𝟒𝟒𝟒𝟒

𝟒𝟒𝟒𝟒
= 𝟏𝟏.𝟑𝟑𝟑𝟑𝟑𝟑

No Television in the
Bedroom

𝟑𝟑𝟒𝟒
≈ 𝟑𝟑.𝟕𝟕𝟕𝟕𝟖𝟖

𝟏𝟏𝟑𝟑
≈ 𝟑𝟑.𝟐𝟐𝟐𝟐𝟐𝟐

𝟒𝟒𝟒𝟒
= 𝟏𝟏.𝟑𝟑𝟑𝟑𝟑𝟑

Total 𝟔𝟔𝟒𝟒
= 𝟑𝟑.𝟔𝟔𝟒𝟒𝟑𝟑

𝟑𝟑𝟒𝟒
= 𝟑𝟑.𝟑𝟑𝟒𝟒𝟑𝟑

𝟏𝟏𝟑𝟑𝟑𝟑
= 𝟏𝟏.𝟑𝟑𝟑𝟑𝟑𝟑

2. Calculate the row relative frequencies, and enter the values in the table above. Round to the nearest thousandth.

The row relative frequencies are displayed in the table above.

3. Is there evidence of association between the variables? If so, does this imply there is a cause-and-effect
relationship? Explain.

Yes, there is evidence of association between the variables because the relative frequencies are different among the
rows. However, this does not necessarily imply a cause-and-effect relationship. The fact that a student has a
television in his room does not cause the student to fail a test. Rather, it may be that the student is spending more
time watching television or playing video games instead of studying.

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Problem Set Sample Solutions

A sample of 𝟐𝟐𝟑𝟑𝟑𝟑 middle school students was randomly selected from the middle schools in a large city. Answers to
several survey questions were recorded for each student. The tables below summarize the results of the survey.

For each table, calculate the row relative frequencies for the Female row and for the Male row. Write the row relative
frequencies beside the corresponding frequencies in each table below.

1. This table summarizes the results of the survey data for the two variables, gender and which sport the students
prefer to play. Is there an association between gender and which sport the students prefer to play? Explain.

Sport

Football Basketball Volleyball Soccer Total

G
en

de
r Female 𝟐𝟐

≈ 𝟑𝟑.𝟑𝟑𝟐𝟐𝟏𝟏
𝟐𝟐𝟗𝟗

≈ 𝟑𝟑.𝟐𝟐𝟗𝟗𝟗𝟗
𝟐𝟐𝟖𝟖

≈ 𝟑𝟑.𝟐𝟐𝟖𝟖𝟗𝟗
𝟑𝟑𝟖𝟖

≈ 𝟑𝟑.𝟑𝟑𝟗𝟗𝟐𝟐 𝟗𝟗𝟕𝟕

Male 𝟑𝟑𝟒𝟒
≈ 𝟑𝟑.𝟑𝟑𝟒𝟒𝟑𝟑

𝟑𝟑𝟔𝟔
≈ 𝟑𝟑.𝟑𝟑𝟒𝟒𝟑𝟑

𝟖𝟖
≈ 𝟑𝟑.𝟑𝟑𝟕𝟕𝟖𝟖

𝟐𝟐𝟒𝟒
≈ 𝟑𝟑.𝟐𝟐𝟑𝟑𝟑𝟑 𝟏𝟏𝟑𝟑𝟑𝟑

Total 𝟑𝟑𝟕𝟕 𝟔𝟔𝟒𝟒 𝟑𝟑𝟔𝟔 𝟔𝟔𝟐𝟐 𝟐𝟐𝟑𝟑𝟑𝟑

Yes, there appears to be an association between gender and sports preference. The row relative frequencies are not
the same for the Male and the Female rows, as shown in the table above.

2. This table summarizes the results of the survey data for the two variables, gender and the students’ T-shirt sizes.
Is there an association between gender and T-shirt size? Explain.

School T-Shirt Sizes

Small Medium Large X-Large Total

G
en

de
r Female 𝟒𝟒𝟕𝟕

≈ 𝟑𝟑.𝟒𝟒𝟖𝟖𝟒𝟒
𝟑𝟑𝟒𝟒

≈ 𝟑𝟑.𝟑𝟑𝟔𝟔𝟏𝟏
𝟏𝟏𝟑𝟑

≈ 𝟑𝟑.𝟏𝟏𝟑𝟑𝟒𝟒
𝟐𝟐

≈ 𝟑𝟑.𝟑𝟑𝟐𝟐𝟏𝟏 𝟗𝟗𝟕𝟕

Male 𝟏𝟏𝟏𝟏
≈ 𝟑𝟑.𝟏𝟏𝟑𝟑𝟕𝟕

𝟒𝟒𝟏𝟏
≈ 𝟑𝟑.𝟑𝟑𝟗𝟗𝟖𝟖

𝟒𝟒𝟐𝟐
≈ 𝟑𝟑.𝟒𝟒𝟑𝟑𝟖𝟖

𝟗𝟗
≈ 𝟑𝟑.𝟑𝟑𝟖𝟖𝟕𝟕 𝟏𝟏𝟑𝟑𝟑𝟑

Total 𝟒𝟒𝟖𝟖 𝟕𝟕𝟔𝟔 𝟒𝟒𝟒𝟒 𝟏𝟏𝟏𝟏 𝟐𝟐𝟑𝟑𝟑𝟑

Yes, there appears to be an association between gender and T-shirt size. The row relative frequencies are not the
same for the Male and the Female rows, as shown in the table above.

3. This table summarizes the results of the survey data for the two variables, gender and favorite type of music.
Is there an association between gender and favorite type of music? Explain.

Favorite Type of Music

Pop Hip-Hop Alternative Country Total

G
en

de
r Female 𝟑𝟑𝟒𝟒
≈ 𝟑𝟑.𝟑𝟑𝟔𝟔𝟏𝟏

𝟐𝟐𝟖𝟖
≈ 𝟑𝟑.𝟐𝟐𝟖𝟖𝟗𝟗

𝟏𝟏𝟏𝟏
≈ 𝟑𝟑.𝟏𝟏𝟏𝟏𝟑𝟑

𝟐𝟐𝟑𝟑
≈ 𝟑𝟑.𝟐𝟐𝟑𝟑𝟕𝟕 𝟗𝟗𝟕𝟕

Male 𝟑𝟑𝟕𝟕
≈ 𝟑𝟑.𝟑𝟑𝟒𝟒𝟗𝟗

𝟑𝟑𝟑𝟑
≈ 𝟑𝟑.𝟐𝟐𝟗𝟗𝟏𝟏

𝟏𝟏𝟑𝟑
≈ 𝟑𝟑.𝟏𝟏𝟐𝟐𝟔𝟔

𝟐𝟐𝟑𝟑
≈ 𝟑𝟑.𝟐𝟐𝟐𝟐𝟑𝟑 𝟏𝟏𝟑𝟑𝟑𝟑

Total 𝟕𝟕𝟐𝟐 𝟒𝟒𝟖𝟖 𝟐𝟐𝟒𝟒 𝟒𝟒𝟔𝟔 𝟐𝟐𝟑𝟑𝟑𝟑

No, there does not appear to be an association between gender and favorite type of music. The row relative
frequencies are about the same for the Male and Female rows, as shown in the table above.

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Module 6: Linear Functions

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Name Date

1. The Kentucky Derby is a horse race held each year. The following scatter plot shows the speed of the
winning horse at the Kentucky Derby each year between 1875 and 2012.

Data Source: http://www.kentuckyderby.com/

(Note: Speeds were calculated based on times given on website.)

a. Is the association between speed and year positive or negative? Give a possible explanation in the
context of this problem for why the association behaves this way considering the variables involved.

b. Comment on whether the association between speed and year is approximately linear, and then
explain in the context of this problem why the form of the association (linear or not) makes sense
considering the variables involved.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

http://www.kentuckyderby.com/

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

191

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c. Circle an outlier in this scatter plot, and explain, in context, how and why the observation is unusual.

2. Students were asked to report their gender and how many times a day they typically wash their hands.
Of the 738 males, 66 said they wash their hands at most once a day, 583 said two to seven times per day,
and 89 said eight or more times per day. Of the 204 females, 2 said they wash their hands at most once
a day, 160 said two to seven times per day, and 42 said eight or more times per day.

a. Summarize these data in a two-way table with rows corresponding to the three different frequency-

of-hand-washing categories and columns corresponding to gender.

b. Do these data suggest an association between gender and frequency of hand washing? Support your
answer with appropriate calculations.

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8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

192

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3. Basketball players who score a lot of points also tend to be strong in other areas of the game such as
number of rebounds, number of blocks, number of steals, and number of assists. Below are scatter plots
and linear models for professional NBA (National Basketball Association) players last season.

𝑦𝑦 = 21.54 + 3.833𝑥𝑥 𝑦𝑦 = 294.9 + 22.45𝑥𝑥

𝑦𝑦 = 98.03 + 9.558𝑥𝑥 𝑦𝑦 = 166.2 + 2.256𝑥𝑥

a. The line that models the association between points scored and number of rebounds is
𝑦𝑦 = 21.54 + 3.833𝑥𝑥, where 𝑦𝑦 represents the number of points scored and 𝑥𝑥 represents the number
of rebounds. Give an interpretation, in context, of the slope of this line.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

193

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b. The equations on the previous page all show the number of points scored (𝑦𝑦) as a function of the
other variables. An increase in which of the variables (rebounds, blocks, steals, and assists) tends to
have the largest impact on the predicted points scored by an NBA player?

c. Which of the four linear models shown in the scatter plots on the previous page has the worst fit to
the data? Explain how you know using the data.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

194

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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

A Progression Toward Mastery

Assessment
Task Item

STEP 1
Missing or incorrect
answer and little
evidence of
reasoning or
application of
mathematics to
solve the problem

STEP 2
Missing or incorrect
answer but
evidence of some
reasoning or
application of
mathematics to
solve the problem

STEP 3
A correct answer
with some
evidence of
reasoning or
application of
mathematics to
solve the problem,
OR an incorrect
answer with
substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem

STEP 4
A correct answer
supported by
substantial
evidence of solid
reasoning or
application of
mathematics to
solve the problem

8.SP.A.1

Student does not use
the data in the scatter
plot or context to
answer the question.

Student discusses horses
getting faster with
newer training methods
but does not discuss the
data in the scatter plot.

Student discusses the
overall increase of
speeds but does not
discuss how those data
imply horses getting
faster over time.

Student discusses the
overall increase of
speeds and how those
data imply horses
getting faster over time.

8.F.B.5

Student does not use
the data in the scatter
plot or context to
answer the question.

Student does not
recognize the nonlinear
nature of the data.

Student discusses the
nonlinear nature of the
data but does not relate
to the context.

Student discusses the
curvature in the data,
which indicates that
speeds should level off.

8.SP.A.2

Student does not use
the data in the scatter
plot or context to
answer the question.

Student picks the year
with the fastest or
lowest speed of the
winning horse and does
not explain the choice.

Student picks the year
with the lowest speed of
the winning horse but
does not interpret the
negative residual.

Student picks the year
with the lowest speed of
the winning horse and
states that the speed is
much lower than
expected for that year.

8.SP.A.4

Student does not use
the data given in the
stem.

Student gives the tallies
of the two distributions
separately without
looking at the cross-
tabulation.

Student constructs the
table but uses gender as
the row variable.

Student constructs a
3 × 2 two-way table,
including appropriate
labels.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

195

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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

8.SP.A.4

Student answer is based
only on context without
references to the data.

Student gives some
information about the
association but does not
back it up numerically.
OR
Student says the results
cannot be compared
because the numbers of
males and females are
not equal.

Student attempts to
calculate the six
conditional proportions
but compares them
inappropriately.
OR
Student does not
correctly complete all
the calculations.

Student calculates the
six conditional
proportions, compares
them, and draws an
appropriate comparison
(e.g., 20% of females
wash eight or more
times compared to 12%
of males).

8.F.B.4

Student cannot identify
the slope from the stem.

Student interprets the
slope incorrectly.

Student interprets the
slope correctly but not
in context or not in
terms of the model
estimation.

Student interprets the
slope correctly and
predicts that on
average, for each
additional rebound, an
increase of 3.833 points
is scored.

8.SP.A.3

Student does not relate
to the functions
provided above the
scatter plots.

Student focuses on the
strength of the
association in terms of
how close the dots fall
to the regression line.

Student appears to
relate the question to
the slope of the
equation but cannot
make a clear choice of
which variable has the
largest impact or does
not provide a complete
justification.

Student relates the
question to the slope
and identifies the
number of blocks as the
variable with the largest
impact.

8.SP.A.2

Student does not use
the data in the scatter
plots to answer the
question.

Student focuses only on
the slope of the line or
on one or two values
that are not well
predicted.

Student focuses on the
vertical distances of the
dots from the line but is
not able to make a clear
choice.

Student selects the
number of blocks based
on the additional spread
of the dots about the
regression line in that
scatter plot compared to
the other variables.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

196

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

Name Date

1. The Kentucky Derby is a horse race held each year. The following scatter plot shows the speed of the
winning horse at the Kentucky Derby each year between 1875 and 2012.

Data Source: http://www.kentuckyderby.com/

(Note: Speeds were calculated based on times given on website.)

a. Is the association between speed and year positive or negative? Give a possible explanation in the

context of this problem for why the association behaves this way considering the variables involved.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

http://www.kentuckyderby.com/

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

197

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

c. Circle an outlier in this scatter plot, and explain, in context, how and why the observation is unusual.

2. Students were asked to report their gender and how many times a day they typically wash their hands.

Of the 738 males, 66 said they wash their hands at most once a day, 583 said two to seven times per day,
and 89 said eight or more times per day. Of the 204 females, 2 said they wash their hands at most once
a day, 160 said two to seven times per day, and 42 said eight or more times per day.

a. Summarize these data in a two-way table with rows corresponding to the three different frequency-

of-hand-washing categories and columns corresponding to gender.

b. Do these data suggest an association between gender and frequency of hand washing? Support your
answer with appropriate calculations.

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

198

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

𝑦𝑦 = 21.54 + 3.833𝑥𝑥 𝑦𝑦 = 294.9 + 22.45𝑥𝑥

𝑦𝑦 = 98.03 + 9.558𝑥𝑥 𝑦𝑦 = 166.2 + 2.256𝑥𝑥

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

8•6 End-of-Module Assessment Task NYS COMMON CORE MATHEMATICS CURRICULUM

Module 6: Linear Functions

199

This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

c. Which of the four linear models shown in the scatter plots on the previous page has the worst fit to
the data? Explain how you know using the data

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US

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