ICL 7053: Week 12 Reading Discussion

Read Progressions document in Module XII content.

As noted previously, each Progressions document describes the ways the topic develops across multiple grades.

After reading the Progressions document on Functions, respond to the following questions:

Imagine that you teach Algebra I,

Identify one idea from the Progressions document about elementary and middle school (grades 6 & 7) that would be important for you to know as an Algebra I teacher? Why would this be important?
Identify one idea from the Progressions document about the concept of Functions in 8th grade that would be important for you to know as an Algebra I teacher? Why would this be important?
Identify one idea from the Progressions document about the teaching of high school Functions? Why is this important?

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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