ICL: Reading Response 6

Readings: Standards Deconstructed: High School Functions (located in Course Materials module) – pp. 16-37

Reading Response 6:

Respond to the following questions.

The Standards Deconstructed document provides more detailed information about the CCSSM. To respond to this week’s reading, you will do a 3-2-1.

1. Provide at least three specific examples of how the Standards Deconstructed document supported your understanding of the Standards. In other words, what are three things that you learned about the meaning of the Standards from reading the Deconstructed document?

2. Describe at least two ideas related to the teaching of the Standards that you take away from the reading. In other words, if you were responsible for teaching the standards covered in the reading, what are two ideas that you would implement in your instruction?

3. What is one question that you still have about these Standards?

COMMON
CORE

MATHEMATICS

855.809.7018 | www.commoncoreinstitute.com

HIGH SCHOOL
FUNCTIONS

DECONSTRUCTED for
CLASSROOM IMPACT

State Standards

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 3

MATHEMATICS

Introduction
The Common Core Institute is pleased to offer this grade-level tool for educators who are teaching with the
Common Core State Standards.

The Common Core Standards Deconstructed for Classroom Impact is designed for educators by educators
as a two-pronged resource and tool 1) to help educators increase their depth of understanding of the
Common Core Standards and 2) to enable teachers to plan College & Career Ready curriculum and
classroom instruction that promotes inquiry and higher levels of cognitive demand.

What we have done is not all new. This work is a purposeful and thoughtful compilation of preexisting
materials in the public domain, state department of education websites, and original work by the Center
for College & Career Readiness. Among the works that have been compiled and/or referenced are the
following: Common Core State Standards for Mathematics and the Appendix from the Common Core State
Standards Initiative; Learning Progressions from The University of Arizona’s Institute for Mathematics and
Education, chaired by Dr. William McCallum; the Arizona Academic Content Standards; the North Carolina
Instructional Support Tools; and numerous math practitioners currently in the classroom.

We hope you will find the concentrated and consolidated resource of value in your own planning. We also
hope you will use this resource to facilitate discussion with your colleagues and, perhaps, as a lever to help
assess targeted professional learning opportunities.

Understanding the Organization

The Overview acts as a quick-reference table of contents
as it shows you each of the domains and related clusters
covered in this specific grade-level booklet. This can help
serve as a reminder of what clusters are part of which
domains and can reinforce the specific domains for each
grade level.

Critical Areas of Focus is designed to help you begin to
approach how to examine your curriculum, resources,
and instructional practices. A review of the Critical Areas
of Focus might enable you to target specific areas of
professional learning to refresh, as needed.

For each domain is the domain itself and the associated
clusters. Within each domain are sections for each of the
associated clusters. The cluster-specific content can take
you to a deeper level of understanding. Perhaps most
importantly, we include here the Learning Progressions. The Learning Progressions provide context for
the current domain and its related standards. For any grade except Kindergarten, you will see the domain-
specific standards for the current

Math Fluency Standards
K Add/subtract within 5

1 Add/subtract within 10

2 Add/subtract within 20
Add/subtract within 100 (pencil & paper)

3 Multiply/divide within 100
Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6 Multi-digit division
Multi-digit decimal operations

7 Solve px + q = r, p(x + q) = r

8 Solve simple 2 x 2 systems by inspection

O
V

ER
V

IE
W

4 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

grade in the center column. To the left are the domain-specific standards for the preceding grade and to the
right are the domain-specific standards for the following grade. Combined with the Critical Areas of Focus,
these Learning Progressions can assist you in focusing your planning.

For each cluster, we have included four key sections: Description, Big Idea, Academic Vocabulary, and
Deconstructed Standard.

The cluster Description offers clarifying information, but also points to the Big Idea that can help you focus
on that which is most important for this cluster within this domain. The Academic Vocabulary is derived
from the cluster description and serves to remind you of potential challenges or barriers for your students.

Each standard specific to that cluster has been deconstructed. There Deconstructed Standard for each
standard specific to that cluster and each Deconstructed Standard has its own subsections, which can
provide you with additional guidance and insight as you plan. Note the deconstruction drills down to the
sub-standards when appropriate. These subsections are:

� Standard Statement

� Standard Description

� Essential Question(s)

� Mathematical Practice(s)

� DOK Range Target for Learning and Assessment

� Learning Expectations

� Explanations and Examples

As noted, first are the Standard Statement and Standard Description, which are followed by the Essential
Question(s) and the associated Mathematical Practices. The Essential Question(s) amplify the Big Idea,
with the intent of taking you to a deeper level of understanding; they may also provide additional context
for the Academic Vocabulary.

The DOK Range Target for Learning and Assessment remind you of the targeted level of cognitive
demand. The Learning Expectations correlate to the DOK and express the student learning targets for
student proficiency for KNOW, THINK, and DO, as appropriate. In some instances, there may be no learning
targets for student proficiency for one or more of KNOW, THINK or DO. The learning targets are expressions
of the deconstruction of the Standard as well as the alignment of the DOK with appropriate consideration of
the Essential Questions.

The last subsection of the Deconstructed Standard includes Explanations and Examples. This subsection
might be quite lengthy as it can include additional context for the standard itself as well as examples of
what student work and student learning could look like. Explanations and Examples may offers ideas for
instructional practice and lesson plans.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 5

MATHEMATICS

O
V

ER
V

IE
W

PRACTICE EXPLANATION

MP.1 Make sense
and persevere in
problem solving.

Mathematically proficient students start by explaining to themselves the meaning
of a problem and looking for entry points to its solution. They analyze givens,
constraints, relationships, and goals. They make conjectures about the form and
meaning of the solution and plan a solution pathway rather than simply jumping
into a solution attempt. They consider analogous problems, and try special cases
and simpler forms of the original problem in order to gain insight into its solution.
They monitor and evaluate their progress and change course if necessary. Older
students might, depending on the context of the problem, transform algebraic
expressions or change the viewing window on their graphing calculator to get
the information they need.
Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important
features and relationships, graph data, and search for regularity or trends. Younger
students might rely on using concrete objects or pictures to help conceptualize
and solve a problem. Mathematically proficient students check their answers to
problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving
complex problems and identify correspondences between different approaches.

MP.2 Reason
abstractly and
quantitatively.

Mathematically proficient students make sense of the quantities and their
relationships in problem situations. Students bring two complementary
abilities to bear on problems involving quantitative relationships: the ability to
decontextualize—to abstract a given situation and represent it symbolically and
manipulate the representing symbols as if they have a life of their own, without
necessarily attending to their referents—and the ability to contextualize, to pause
as needed during the manipulation process in order to probe into the referents
for the symbols involved. Quantitative reasoning entails habits of creating a
coherent representation of the problem at hand; considering the units involved;
attending to the meaning of quantities, not just how to compute them; and
knowing and flexibly using different properties of operations and objects.

Standards for Mathematical Practice in High School
Mathematics Courses
The Standards for Mathematical Practice describe varieties of expertise that
mathematics educators at all levels should seek to develop in their students.
These practices rest on important “processes and proficiencies” with longstanding
importance in mathematics education. The first of these are the NCTM process
standards of problem solving, reasoning and proof, communication, representation,
and connections. The second are the strands of mathematical proficiency specified
in the National Research Council’s report Adding It Up: adaptive reasoning, strategic
competence, conceptual understanding (comprehension of mathematical concepts,
operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition (habitual
inclination to see mathematics as sensible, useful, and worthwhile, coupled with a
belief in diligence and one’s own efficacy).

6 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

PRACTICE EXPLANATION

MP.3 Construct
viable arguments
and critique the
reasoning of
others.

Mathematically proficient students understand and use stated assumptions,
definitions, and previously established results in constructing arguments. They
make conjectures and build a logical progression of statements to explore
the truth of their conjectures. They are able to analyze situations by breaking
them into cases, and can recognize and use counterexamples. They justify their
conclusions, communicate them to others, and respond to the arguments of
others.
They reason inductively about data, making plausible arguments that take
into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments,
distinguish correct logic or reasoning from that which is flawed, and—if there is a
flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as
objects, drawings, diagrams, and actions. Such arguments can make sense and be
correct, even though they are not generalized or made formal until later grades.
Later, students learn to determine domains to which an argument applies.
Students at all grades can listen or read the arguments of others, decide whether
they make sense, and ask useful questions to clarify or improve the arguments.

MP.4 Model with
mathematics.

Mathematically proficient students can apply the mathematics they know to
solve problems arising in everyday life, society, and the workplace. In early grades,
this might be as simple as writing an addition equation to describe a situation.
In middle grades, a student might apply proportional reasoning to plan a school
event or analyze a problem in the community. By high school, a student might
use geometry to solve a design problem or use a function to describe how one
quantity of interest depends on another. Mathematically proficient students
who can apply what they know are comfortable making assumptions and
approximations to simplify a complicated situation, realizing that these may need
revision later.
They are able to identify important quantities in a practical situation and map
their relationships using such tools as diagrams, 2-by-2 tables, graphs, flowcharts
and formulas. They can analyze those relationships mathematically to draw
conclusions.
They routinely interpret their mathematical results in the context of the situation
and reflect on whether the results make sense, possibly improving the model if it
has not served its purpose.

MP.5 Use
appropriate
tools
strategically.

Mathematically proficient students consider the available tools when solving a
mathematical problem. These tools might include pencil and paper, concrete
models, ruler, protractor, calculator, spreadsheet, computer algebra system,
statistical package, or dynamic geometry software. Proficient students are
sufficiently familiar with tools appropriate for their grade or course to make sound
decisions about when each of these tools might be helpful, recognizing both the
insight to be gained and their limitations. For example, mathematically proficient
high school students interpret graphs of functions and solutions generated using
a graphing calculator.
They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions,
explore consequences, and compare predictions with data. Mathematically
proficient students at various grade levels are able to identify relevant external
mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to
explore and deepen their understanding of concepts.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 7

MATHEMATICS

PRACTICE EXPLANATION

MP.6 Attend to
precision.

Mathematically proficient students try to communicate precisely to others. They
try to use clear definitions in discussion with others and in their own reasoning.
They state the meaning of the symbols they choose, are careful about specifying
units of measure, and labeling axes to clarify the correspondence with quantities
in a problem.
They express numerical answers with a degree of precision appropriate for the
problem context. In the elementary grades, students give carefully formulated
explanations to each other.
By the time they reach high school they have learned to examine claims and
make explicit use of definitions.

MP.7 Look for
and make use of
structure.

Mathematically proficient students look closely to discern a pattern or structure.
Young students, for example, might notice that three and seven more is the
same amount as seven and three more, or they may sort a collection of shapes
according to how many sides the shapes have.
Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the expression x2 + 9x
+ 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7.
They recognize the significance of an existing line in a geometric figure and can
use the strategy of drawing an auxiliary line for solving problems.
They also can step back for an overview and shift perspective.
They can see complicated things, such as some algebraic expressions, as single
objects or as composed of several objects. For example, they can see 5 – 3(x – y)2
as 5 minus a positive number times a square and use that to realize that its value
cannot be more than 5 for any real numbers x and y.

MP.8 Look for
and express
regularity
in repeated
reasoning.

Mathematically proficient students notice if calculations are repeated, and look
both for general methods and for shortcuts. Upper elementary students might
notice when dividing 25 by 11 that they are repeating the same calculations over
and over again, and conclude they have a repeating decimal. By paying attention
to the calculation of slope as they repeatedly check whether points are on the line
through (1, 2) with slope 3, middle school students might abstract the equation
(y – 2)(x – 1) = 3.

O
V

ER
V

IE
W

8 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

OVERVIEW

Functions describe situations where one quantity determines another. For example, the return on
$10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time the
money is invested. Because we continually make theories about dependencies between quantities in
nature and society, functions are important tools in the construction of mathematical models.

In school mathematics, functions usually have numerical inputs and outputs and are often defined by
an algebraic expression. For example, the time in hours it takes for a car to drive 100 miles is a function
of the car’s speed in miles per hour, v; the rule T(v) = 100/v expresses this relationship algebraically and
defines a function whose name is T.

The set of inputs to a function is called its domain. We often infer the domain to be all inputs for which
the expression defining a function has a value, or for which the function makes sense in a given context.

A function can be described in various ways, such as by a graph (e.g., the trace of a seismograph); by a
verbal rule, as in, “I’ll give you a state, you give me the capital city;” by an algebraic expression like f(x) = a
+ bx; or by a recursive rule. The graph of a function is often a useful way of visualizing the relationship of
the function models, and manipulating a mathematical expression for a function can throw light on the
function’s properties.

Functions presented as expressions can model many important phenomena. Two important families
of functions characterized by laws of growth are linear functions, which grow at a constant rate, and
exponential functions, which grow at a constant percent rate. Linear functions with a constant term of
zero describe proportional relationships.

A graphing utility or a computer algebra system can be used to experiment with properties of these
functions and their graphs and to build computational models of functions, including recursively
defined functions.

Connections to Expressions, Equations, Modeling, and Coordinates.

Determining an output value for a particular input involves evaluating an expression; finding inputs that
yield a given output involves solving an equation. Questions about when two functions have the same
value for the same input lead to equations, whose solutions can be visualized from the intersection of
their graphs. Because functions describe relationships between quantities, they are frequently used in
modeling. Sometimes functions are defined by a recursive process, which can be displayed effectively
using a spreadsheet or other technology.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 9

MATHEMATICS
OVERVIEW

Interpreting Functions (F-IF)
� Understand the concept of a function and use function notation

� Interpret functions that arise in applications in terms of the context

� Analyze functions using different representations

Building Functions (F-BF)
� Build a function that models a relationship between two quantities

� Build new functions from existing functions

Linear, Quadratic, and Exponential Models (F-LE)
� Construct and compare linear, quadratic, and exponential models and solve problems

� Interpret expressions for functions in terms of the situation they model

Trigonometric Functions (F-TF)
� Extend the domain of trigonometric functions using the unit circle

� Model periodic phenomena with trigonometric functions

� Prove and apply trigonometric identities

Mathematical Practices (MP)
MP 1. Make sense of problems and persevere in solving them.

MP 2. Reason abstractly and quantitatively.

MP 3. Construct viable arguments and critique the reasoning of others.

MP 4. Model with mathematics.

MP 5. Use appropriate tools strategically.

MP 6. Attend to precision.

MP 7. Look for and make use of structure.

MP 8. Look for and express regularity in repeated reasoning.

O
V

ER
V

IE
W

The high school standards specify the mathematics that all students should study in order to be college
and career ready. Additional mathematics that students should learn in order to take advanced courses
such as calculus, advanced statistics, or discrete mathematics is indicated by (+), as in this example:
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and
imaginary numbers).

All standards without a (+) symbol should be in the common mathematics curriculum for all college and
career ready students. Standards with a (+) symbol may also appear in courses intended for all students.

10 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

Domain HS Algebra I Mathematics I Geometry

N
um

be
r &

Q
ua

nt
it

The Real Number System (RN) RN, 1, 2, 3

Quantities (Q) Q.1, 2, 3 Q.1, 2, 3

The Complex Number System (CN)

Vector Quantities and Matrices (VM)

A
lg

eb
ra

Seeing Structure in Expressions (SSEE) SSE.1a, 1b, 2, 3a, 3b, 3c SSE.1a, 1b

Arithmetic with Polynomials and
Rational Expressions (APR) APR .1

Creating Equations (CED) CED. 1, 2, 3, 4 CED. 1, 2, 3, 4

Reasoning with Equations and
Inequalities (REI)

REI. 1, 3, 4a, 4b, 5, 6, 7,
10, 11, 12 REI. 1, 3, 5, 6, 10, 11, 12

Fu
nc

ti
on

Interpreting Functions (IF) IF, 1, 2, 3, 4, 5, 6, 7a, 7b,
7c, 8a, 8b, 9

IF. 1, 2, 3, 4, 5, 6, 7a,
7c, 9

Building Functions (BF) BF. 1a, 1b, 2, 3, 4a BF. 1a, 1b, 2, 3

Linear, Quadratic, and Exponential
Models (LE) LE. 1a, 1b, 1c, 2, 3, 5 LE. 1a, 1b, 1c, 2, 3, 5

Trigonometric Functions (TF)

G
eo

m
et

Congruence (CO) CO. 1, 2, 3, 4, 5, 6, 7, 8,
12, 13 CO. 1-13

Similarity, Right Triangles, and
Trigonometry (SRT) SRT. 1-11

Circle (C) C. 1-5

Expressing Geometric Properties with
Equations (GPE) GEP. 4, 5, 7 GPE. 1, 2, 4-7

Geometric Measurement and
Dimension (GMD) GMD. 1, 3, 4

Modeling with Geometry (MG) MG. 1, 2, 3

St
at

is
ti

c
an

d
Pr

ob
ab

ili
ty

Interpreting Categorical and
Quantitative Data (D) ID. 1-3, 5-9 ID. 1-3, 5-9

Making Interference and Justifying
Conclusions (IC)

Conditional Probability and the Rules
of Probability (CP) CP. 1-9

Using Probability to Make Decisions
(MD) MD. 6-7

High School Pathways for Traditional and Integrated Courses

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 11

MATHEMATICS

Mathematics II HS Algebra II Mathematics III 4th Courses (T) 4th Courses (I)

N
um

be
r &

Q
ua

nt
it

RN, 1, 2, 3

CN.1, 2, 7, 8, 9 CN.1, 2, 7, 8, 9 CN. 8, 9 CN. 3, 4, 5, 6 CN. 3, 4, 5, 6

VM. 1, 2, 3, 4 (a-c), 5(a-
b), 6, 7, 8, 9, 10, 11, 12

A
lg

eb
ra

SSE.1a, 1b, 2, 3a, 3b, 3c SSE. 1, 1b, 2, 4 SSE. 1, 1b, 2, 4

APR .1 APR .1-7 APR .1-7

CED. 1, 2, 4 CED. 1, 2, 3, 4 CED. 1, 2, 4

REI. 4a, 4b, 7 REI. 2, 11 REI. 2, 11 REI. 8, 9 REI. 8, 9

Fu
nc

ti
on

IF. 4, 5, 6, 7a, 7b, 8a,
8b, 9 IF. 4, 5, 6, 7b, 7c, 7e, 8, 9 IF. 4, 5, 6, 7b, 7c, 7e, 8, 9 IF. 7d IF. 7d

BF. 1a, 1b, 3, 4a BF. 1b, 3, 4a BF. 1b, 3, 4a BF. 1c, 4b, 4c, 4d, 5 BF. 1c, 4b, 4c, 4d, 5

LE. 3 LE. 4 LE. 4

TF. 8 TF. 1, 2, 5, 8 TF. 1, 2, 5 TF. 3, 4, 6, 7, 9 TF. 3, 4, 6, 7, 9

G
eo

m
et

CO. 9, 10, 11

SRT. 1a, 1b, 2, 3, 4, 5,
6, 7, 8 SRT. 9, 10, 11

C. 1-5

GPE. 1, 2, 4 GPE. 3 GPE. 3

GMD. 1, 3 GMD. 4 GMD. 2 GMD. 2

MG. 1, 2, 3

St
at

is
ti

c
an

d
Pr

ob
ab

ili
ty

ID. 4 ID. 4

IC. 1-6 IC. 1-6

CP. 1-9

MD. 6-7 MD. 6, 7 MD. 6-7 MD. 1-5 MD. 1-5

High School Pathways for Traditional and Integrated Courses O
V

ER
V

IE
W

12 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

Functions
Math 1 Math 2 Math 3 Math 4

Interpreting Functions (IF)

F.IF.1

F.IF.2

F.IF.3

F.IF.4 F.IF.4 F.IF.4

F.IF.5 F.IF.5 F.IF.5

F.IF.6 F.IF.6 F.IF.6

F.IF.7(a,e) F.IF.7(a,b) F.IF.7(b,c,e) F.IF.7(d)

F.IF.8(a-b) F.IF.8

F.IF.9 F.IF.9 F.IF.9

Building Functions (IF)
F.BF.1(a-b) F.BF.1(a-b) F.BF.1(b) F.BF.1 c

F.BF.2

F.BF.3 F.BF.3 F.BF.3

F.BF.4a F.BF.4a F.BF.4(b,c,d)

F.BF.5

Linear, Quadratic, and Exponential Models (LE)

F.LE.1(a-c)

F.LE.2

F.LE.3 F.LE.3

F.LE.4

F.LE.5

Trigonometric Functions (TF)

F.TF.8 F.TF.1

F.TF.2

F.TF.3

F.TF.4

F.TF.5

F.TF.6

F.TF.7

F.TF.9

Learning Progressions for Integrated Courses

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 13

MATHEMATICS

O
V

ER
V

IE
W

Functions
Algebra 1 (Grade 9) Geometry (Grade 10) Algebra 2 (Grade 11) PreCalculus (Grade 12)

Interpreting Functions (IF)

F.IF.1

F.IF.2

F.IF.3

F.IF.4 F.IF.4

F.IF.5 F.IF.5

F.IF.6 F.IF.6

F.IF.7(a,b,e) F.IF.7(b,c,e) F.IF.7(d)

F.IF.8(a-b) F.IF.8

F.IF.9 F.IF.9

Building Functions (IF)

F.BF.19a-b) F.BF.1(b) F.BF.1 c

F.BF.2

F.BF.3 F.BF.3

F.BF.4(a) F.BF.4a F.BF.4(b-c)

F.BF.5

Linear, Quadratic, and Exponential Models (LE)

F.LE.1(a-c)

F.LE.2

F.LE.3

F.LE.4

F.LE.5

Trigonometric Functions (TF)

F.TF.1

F.TF.2

F.TF.3

F.TF.4

F.TF.5

F.TF.6

F.TF.7

F.TF.8

F.TF.9

Learning Progressions for Traditional Courses

MATHEMATICS

INTERPRETING
FUNCTIONS

(F-IF)

DOMAIN:

HIGH SCHOOL
FUNCTIONS

16 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

DOMAIN Interpreting Functions

CLUSTERS
1. Understand the concept of a function and use function notation.
2. Interpret functions that arise in applications in terms of the context.
3. Analyze functions using different representations.

ACADEMIC
VOCABULARY

domain, range, function notation, fibonacci sequence, recursive process, intercepts, increasing intervals,
decreasing intervals, positive intervals, negative intervals, relative maximum, relative minimum, symmetries, end
behavior, periodicity, rate of change, step function, absolute value function, asymptote, exponential function
logarithmic function, trigonometric function, period, midline, amplitude, exponential growth, exponential decay,
constant function, arithmetic sequence, geometric sequence, invertible function, radian measure, arc, sine cosine,
tangent

CLUSTER 1. Understand the concept of a function and use function notation.

BIG IDEA
• Functions describe actions we do everyday (real-world situations).

• Relationships between two sets of numbers can be described by mathematical rules, where a
function is a unique rule that has a one-to-one correspondence.

STANDARD AND DECONSTRUCTION

F.IF.1

Understand that a function from one set (called the domain) 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 f(x) denotes the output of f corresponding to the input x. The graph of
f is the graph of the equation y = f(x).

DESCRIPTION F.IF.1 Use the definition of a function to determine whether a relationship is a function given a table, graph or
words.

F.IF.1 Given the function f(x), identify x as an element of the domain, the input, and f(x) is an element in the range,
the output.

F.IF.1 Know that the graph of the function, f, is the graph of the equation y=f(x).

ESSENTIAL
QUESTION(S)

• What do the symbols (parentheses, brackets, braces) represent when evaluating an expression?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

DOK Range Target
for Instruction &

Assessment
T 1 o 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Identify the domain and range of a
function.

Determine if a relation is a function.

Determine the value of the function
with proper notation.

Evaluate functions for given values
of x.

EXPLANATIONS
AND EXAMPLES

A function is like any other system. What you get out of the system depends on what you put into it. Think of the
human body. We put food into our digestive system and we get something very different out. (Gross.) What we put
into our bodies affects what comes out and if you don’t believe that, try eating beets or asparagus.

Students should understand that functions do the exact same thing, only with numbers. (Maybe not the exact
same thing.) They’re all about describing relationships between two sets of numbers. These two sets of numbers
must have the condition that each item from the first set of numbers pairs with exactly one item from the second
set of numbers.

In other words, for every input, there is exactly one output. That’s like the functions’ motto.

Students should know that functions can be expressed as a pair of input and output values. A relation is a set of
pairs of input and output values, usually represented in ordered pairs. For instance, the ordered pair (1, 2) means
that for an input value of 1, we get an output value of 2. The ordered pair (2, 3) means that if we input 2, we get 3
out. Typically, we represent this as (x, y).

The domain is the set of inputs in a relation, also called the x-coordinates of an ordered pair. The range is the set
of outputs in a relation, also called the y-coordinates of an ordered pair. If students have a hard time remembering
which is which, tell them to think alphabetically. Since D comes before R in the alphabet, the domain has to come
before the range. If that doesn’t work, the acronym “DIXROY” might. (Domain, input, x; Range, output, y.)

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 17

MATHEMATICS

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

STANDARD AND DECONSTRUCTION

F.IF.1

DESCRIPTION F.IF.1 Use the definition of a function to determine whether a relationship is a function given a table, graph or
words.

F.IF.1 Given the function f(x), identify x as an element of the domain, the input, and f(x) is an element in the range,
the output.

F.IF.1 Know that the graph of the function, f, is the graph of the equation y=f(x).

ESSENTIAL
QUESTION(S)

• What do the symbols (parentheses, brackets, braces) represent when evaluating an expression?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

DOK Range Target
for Instruction &

Assessment
T 1 o 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Identify the domain and range of a
function.

Determine if a relation is a function.

Determine the value of the function
with proper notation.

Evaluate functions for given values
of x.

EXPLANATIONS
AND EXAMPLES

In other words, for every input, there is exactly one output. That’s like the functions’ motto.

18 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

STANDARD AND DECONSTRUCTION

F.IF.2 Use function notations, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.

DESCRIPTION F.IF.2 When a relation is determined to be a function, use f(x) notation.

F.IF.2 Evaluate functions for inputs in their domain.

F.IF.2 Interpret statements that use function notation in terms of the context in which they are used.).

ESSENTIAL
QUESTION(S)

• What is a function, how is it written and interpreted?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Identify mathematical relationships
and express them using function
notation.

Define a reasonable domain, which
depends on the context and/
or mathematical situation, for a
function focusing on linear and
exponential functions.

Evaluate functions at a given input
in the domain, focusing on linear
and exponential functions.

Interpret statements that use
functions in terms of real world
situations, focusing on linear and
exponential functions.

EXPLANATIONS
AND EXAMPLES

The domain of a function given by an algebraic expression, unless otherwise specified, is the largest possible
domain.

Examples:

• If f(x) = x2 – 12, find f(2).

• Let f(x) = 2(x + 3)2. Find f(3), f(a), and f(a – h)

If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = 487,000 and P(10)-P(9) = 5,900.

EXPLANATIONS
AND EXAMPLES

(continued)

To start with, students can represent functions as several ordered pairs in the form of a table. One column will be
the input, or x values, and the other will be the output, or y values. For instance, we can rewrite the three points of
a function (-2, 3), (0, 4), and (1, -3) in the following table.

Here, we have our clearly defined domain (D: x = -2, 0, 1) and range (R: y = 3, 4, -3). Any table with the same x value
resulting in multiple y values is not a function. Remember the functions’ motto?

When domains and ranges cover more than just a few select points, they’re often included in parenthesis or
brackets. Parenthesis indicate that the point on that end is not included, while brackets indicate that it is included.
When the ∞ symbol is used, we use parentheses. Makes sense, since infinity isn’t really a number and can’t actually
be reached.

As useful as tables are, many functions have domains and ranges that extend to positive and negative infinity.
When the students’ data tables start getting longer than their arms, we recommend switching to graphs. Spare a
headache and a few trees, while they’re at it.

A graph is a visual representation of relations. We plot the input values as x and the output values as y and treat
the ordered pairs (x, y) as points on the coordinate plane. For the function above, we could plot the points as (-2, 3),
(0, 4), and (1, -3).

Students should also know that functions can be represented by curves on the coordinate plane as y = f(x)
where f(x) is some function of x. These are basically a bunch of points that are so close together that they form a
continuous curve. For instance, these points could be part of a larger function shown by the graph below.

If students aren’t sure whether they’re looking at a function or not, they should perform the vertical line test: if
they draw a vertical line on a graph of a relation and it intersects with the curve more than once, the relation is not
a function.

The key concept to remember is that functions are systems in which one input corresponds to one output. Just like
the human body is a system in which every meal corresponds to one trip to the bathroom. Or something like that.

(Source: www.shmoop.com)

http://www.shmoop.com

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 19

MATHEMATICS
STANDARD AND DECONSTRUCTION

F.IF.2 Use function notations, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.

DESCRIPTION F.IF.2 When a relation is determined to be a function, use f(x) notation.

F.IF.2 Evaluate functions for inputs in their domain.

F.IF.2 Interpret statements that use function notation in terms of the context in which they are used.).

ESSENTIAL
QUESTION(S)

• What is a function, how is it written and interpreted?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Identify mathematical relationships
and express them using function
notation.

Define a reasonable domain, which
depends on the context and/
or mathematical situation, for a
function focusing on linear and
exponential functions.

Evaluate functions at a given input
in the domain, focusing on linear
and exponential functions.

Interpret statements that use
functions in terms of real world
situations, focusing on linear and
exponential functions.

EXPLANATIONS
AND EXAMPLES

The domain of a function given by an algebraic expression, unless otherwise specified, is the largest possible
domain.

Examples:

• If f(x) = x2 – 12, find f(2).

• Let f(x) = 2(x + 3)2. Find f(3), f(a), and f(a – h)

If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = 487,000 and P(10)-P(9) = 5,900.

f(- )
1_
2

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

20 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

STANDARD AND DECONSTRUCTION

F.IF.3
Recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci
sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n – 1) for n ≥ 1.

DESCRIPTION F.IF.3 Recognize that sequences, sometimes defined recursively, are functions whose domain is a subset of the set of
integers.

ESSENTIAL
QUESTION(S)

• What is a function, how is it written and interpreted?

MATHEMATICAL
PRACTICE(S)

HS.MP.8. Look for and express regularity in repeated reasoning.

DOK Range Target
for Instruction &

Assessment
T 1 o 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Recognize that sequences are
functions, sometimes defined
recursively, whose domain is a
subset of the integers.

EXPLANATIONS
AND EXAMPLES

When handling functions (careful, they’re fragile!), it’s easy to see patterns emerge when comparing the x and y
values to each other. Students should know that these patterns are not coincidences and, unlike certain patterned
wallpaper, they won’t make your kitchen look like it belongs in the 1970s.

Students should know that these patterns can be thought of as sequences, or a list of numbers. Sequences can
be either arithmetic (where the same number is added or subtracted) or geometric (where the same number is
multiplied or divided).

Arithmetic sequences can be converted into functions of the form A(n) = A(1) + (n – 1)d where A(n) is the value of
the nth term, A(1) is the value of the first term, n is the term number, and d is the common difference between the
terms. So the sequence 3, 8, 13, 18… can be thought of as the function A(n) = 3 + 5(n – 1). That way, the nth term of
the sequence will have the value A(n).

Geometric series take a similar form: G(n) = G(1) × rn – 1 where G(n) is the value of the nth term, G(1) is the value of
the first term, n is the term number, and r is the common ratio between the terms. So the sequence 3, 6, 12, 24, 48…
can be thought of as the function G(n) = 3 × 2n – 1. That way, the nth term of the sequence will have the value G(n).

Students should know that sequences can be defined recursively, or using previous terms to define future terms.
For instance, The Fibonacci sequence is a list of numbers where each term is the sum of the two before it. As such,
we end up with 1, 1, 2, 3, 5, 8, 13, and so on. We can define this recursively as f(n + 1) = f(n) + f(n – 1). Once the first
two terms are defined, (f(0) = f(1) = 1), the sequence can just keep on going, not unlike the Energizer bunny.

Students should be so skilled in seeing the functions in sequences that you should find them accidentally
converting other things into functions, too. In fact, it should become second nature.

We’re talking analyzing the number of breaths they take as a function of time. The number of times they sneeze as
a function of their allergies. The number of detentions they get as a function of how often they kick soccer balls at
Coach Gibson. Sorry, Coach.

(Source: www.shmoop.com)

CLUSTER 2. Interpret functions that arise in applications in terms of the context.

BIG IDEA • Graphs and tables represent functions in context and application.

http://www.shmoop.com

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 21

MATHEMATICS
CLUSTER 2. Interpret functions that arise in applications in terms of the context.

BIG IDEA • Graphs and tables represent functions in context and application.

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

22 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

STANDARD AND DECONSTRUCTION

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.
Key features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.

DESCRIPTION
F.IF.4 Given a function, identify key features in graphs and tables including: intercepts; intervals where the function
is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and
periodicity.
F.IF.4 Given the key features of a function, sketch the graph.

ESSENTIAL
QUESTION(S)

• How are functions interpreted when used in applications?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.4. Model with mathematics.

HS.MP.5. Use appropriate tools strategically.

HS.MP.6. Attend to precision.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Define and recognize the key
features in tables and graphs of
linear and exponential functions:
intercepts; intervals where the
function is increasing, decreasing,
positive, or negative, and end
behavior.

Interpret key features of graphs
and tables of functions in the terms
of the contextual quantities each
function represents.

Sketch graphs showing the key
features of a function, modeling
a relationship between two
quantities, given a verbal
description of the relationship.

EXPLANATIONS
AND EXAMPLES

Students may be given graphs to interpret or produce graphs given an expression or table for the function, by
hand or using technology.

Examples:

• A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is
given by h = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured
in feet.

� What is a reasonable domain restriction for t in this context?

� Determine the height of the rocket two seconds after it was launched.

� Determine the maximum height obtained by the rocket.

� Determine the time when the rocket is 100 feet above the ground.

� Determine the time at which the rocket hits the ground.

� How would you refine your answer to the first question based on your response to the second and fifth
questions?

• Compare the graphs of y = 3×2 and y = 3×3.

(Source: www.shmoop.com)

EXPLANATIONS
AND EXAMPLES

(continued)

• Compare the graphs of y = 3×2 and y = 3×3.

• Let . Graph the function and identify end behavior and any intervals of constancy, increase,

and decrease.

It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the storm was over, with a total
rainfall of 3 inches. It didn’t rain for the rest of the day. Sketch a possible graph for the number of inches of rain as a
function of time, from

(Source: www.shmoop.com)

http://www.shmoop.com

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 23

MATHEMATICS
EXPLANATIONS
AND EXAMPLES

(continued)

• Compare the graphs of y = 3×2 and y = 3×3.

• Let . Graph the function and identify end behavior and any intervals of constancy, increase,

and decrease.

(Source: www.shmoop.com)

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

http://www.shmoop.com

24 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

STANDARD AND DECONSTRUCTION

F.IF.5
Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a factory, then
the positive integers would be an appropriate domain for the function.

DESCRIPTION F.IF.5 Given the graph of a function, determine the practical domain of the function as it relates to the numerical
relationship it describes.

ESSENTIAL
QUESTION(S)

• How are functions interpreted when used in applications?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.4. Model with mathematics.

HS.MP.6. Attend to precision.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Identify and describe the domain
of a function, given the graph or
a verbal/written description of a
function.

Identify an appropriate domain
based on the unit, quantity, and
type of function it describes.

Relate the domain of a function to
its graph and to the quantitative
relationship it describes, where
applicable.

Explain why a domain is
appropriate for a given situation.

EXPLANATIONS
AND EXAMPLES

Domain is to a function as trails are to a mountain. Certain trails get to certain places on the mountaintop, just like
certain inputs can give you certain outputs for the function. The domain of a mountain is the set of all the possible
trails we can take. Mathematically, the domain (n) of a function f(n), is the collection of all the different x values we
can input into our function.

You could arguably say that any function can take any domain. After all, the x-axis does extend to infinity and
beyond.

But we all know there aren’t an infinite number of marked trails on the mountain. There may be an infinite number
of ways to get to the top of the mountain (bungee jumping and teleportation come to mind), but there are
limitations as to how many trails we can have. For instance, we need a whole number of trails because half a trail
can’t get us anywhere. Not to mention the negative numbers (what would a negative number of trails mean?).

So, here we are, trying to make our way up the metaphorical mountain, and our domain of trails shrunk from
infinity down to positive whole numbers, probably not exceeding half a dozen or so. While this is a very simplistic
way to look at domain, it’s the right idea to get your students thinking about it. You may need to scale it up for
more complex problems, though.

Although there are some functions whose domains are limited (like f(x) = 1⁄x, where x ≠ 0), many domains are
restricted because they don’t make sense in terms of a particular context (like having a negative number of trails
that lead to a mountaintop). Students should be able to determine the appropriate domains based on the context
of the problem.

Hopefully faster than it takes to climb that mountain.

(Source: www.shmoop.com)

STANDARD AND DECONSTRUCTION

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.

DESCRIPTION
F.IF.6 Calculate the average rate of change over a specified interval of a function presented symbolically or in a table.
F.IF.6 Estimate the average rate of change over a specified interval of a function from the function’s graph.
F.IF.6 Interpret, in context, the average rate of change of a function over a specified interval.

ESSENTIAL
QUESTION(S)

• How are functions interpreted when used in applications?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.4. Model with mathematics.

HS.MP.5. Use appropriate tools strategically.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Recognize slope as an average rate
of change.

Calculate 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
linear or exponential graph.

Interpret the average rate of
change of a function (presented
symbolically or as a table) over a
specified interval.

EXPLANATIONS
AND EXAMPLES

The average rate of change of a function y = f(x) over an interval [a,b] is . In addition to finding average rates of
change from functions given symbolically, graphically, or in a table, Students may collect data from experiments
or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the
situation.

Examples:

• Use the following table to find the average rate of change of g over the intervals [-2, -1] and [0,2]:

• The table below shows the elapsed time when two different cars pass a 10, 20, 30, 40 and 50 meter mark on
a test track.

� For car 1, what is the average velocity (change in distance divided by change in time) between the 0 and
10 meter mark? Between the 0 and 50 meter mark? Between the 20 and 30 meter mark? Analyze the data
to describe the motion of car 1.

� How does the velocity of car 1 compare to that of car 2?.

http://www.shmoop.com

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 25

MATHEMATICS
STANDARD AND DECONSTRUCTION

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.

ESSENTIAL
QUESTION(S)

• How are functions interpreted when used in applications?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.4. Model with mathematics.

HS.MP.5. Use appropriate tools strategically.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Recognize slope as an average rate
of change.

Calculate 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
linear or exponential graph.

Interpret the average rate of
change of a function (presented
symbolically or as a table) over a
specified interval.

EXPLANATIONS
AND EXAMPLES

Examples:

• Use the following table to find the average rate of change of g over the intervals [-2, -1] and [0,2]:

• The table below shows the elapsed time when two different cars pass a 10, 20, 30, 40 and 50 meter mark on
a test track.

� How does the velocity of car 1 compare to that of car 2?.

x g(x)
-2 2
-1 -1
0 -4
2 -10

Car 1 Car 2
d t t

10 4.472 1.742
20 6.325 2.899
30 7.746 3.831
40 8.944 4.633
50 10 5.348

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

26 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

STANDARD AND DECONSTRUCTION

F.IF.6
(MODELING)

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.

DESCRIPTION
F.IF.6 Calculate the rate of change in a quadratic function over a given interval from a table or equation.

F.IF.6 Compare rates of change in quadratic functions with those in linear or exponential functions.

ESSENTIAL
QUESTION(S)

• How are functions interpreted when used in applications?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.4. Model with mathematics.

HS.MP.5. Use appropriate tools strategically.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Recognize slope as an average rate
of change.

Calculate 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
linear or exponential graph.

Interpret the average rate of
change of a function (presented
symbolically or as a table) over a
specified interval.

EXPLANATIONS
AND EXAMPLES

Who doesn’t love ice cream? As much as humans love ice cream, we all know that it’s a penguin’s favorite dessert.
They eat, breathe, and sleep ice cream. Well, actually they just eat it. If they breathed ice cream, they’d probably be
extinct by now.

The point is that penguins eat ice cream right out of the carton—and fast! Let’s say h(t) = h0 – At represents the
height of the ice cream in its carton after t hours, where h0 is the initial ice cream height. A is the amount of ice
cream that a penguin eats every hour.

In that case, the average rate of change between two values is just the change in the ice cream level (the values of
the function) over the change in time (the values of the domain). In other words:

Students should know that, in general, we can write this for any function f(n):

For instance, if a penguin eats ice cream at a rate of A = 1 inch per hour and the initial ice cream level is 8 inches,
the rate of change between time t1 = 3 hours and t2 = 5 hours, we can calculate the average rate of change:

If we plotted h(t), we’d see that this rate of change is just the slope of the line since h(t) is in the form y = mx + b. We
also had this information right from the start, since A is the slope of the linear equation, and A was already given as
1 inch per hour.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 27

MATHEMATICS
EXPLANATIONS
AND EXAMPLES

(continued)

Students should know that we can find the rate of change given a table of x and y values. Really, it’s the exact same
thing as finding the slope:

However, for non-linear functions, the rate of change between any two points might not be the same as the rate of
change of the overall function. For instance, let’s say we have the following table of data.

In this case, we note that the slope of the line, or the rate of change between the measurements is not constant.
This is why we define rate of change over a specified range, say, between each pair of adjacent points.

In this case, we don’t have the same slope for all points on the line. That’s not a mistake. That’s because life is
sometimes unpredictable, and changes at odd rates. If we’re more interested in the overall meaning of life (it’s 42,
by the way) we could calculate the overall rate of change (in this case, between x = 0 and x = 7) and this would give
us a slightly bigger picture.

Students should know that we can calculate the average rate of change for any function. Having a linear
relationship is super nice, but not necessary; as long as we have initial and final values for x and f(x), then they
shouldn’t have any problems. If they’re confused or struggling, we recommend relating rate of change to the slope
of a line. Every function has a “slope” between any two points, and the rate of change is how we find that slope.

Of course, we can expand the idea of rate of change to non-linear functions and real-life situations, but they all
come back to the same basic principle: ice cream.

(Source: www.shmoop.com)

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

http://www.shmoop.com

28 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

CLUSTER 3. Analyze functions using different representations.

BIG IDEA
• Equations and inequalities can be solved using arithmetic and algebraic rules and equivalence.

• Using technology and Graphing functions provides a simplified representation.

STANDARD AND DECONSTRUCTION

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

DESCRIPTION

F.IF.7 Graph functions expressed symbolically and show key features of the graph. Graph simple cases by hand, and
use technology to show more complicated cases including:

F.IF.7a Linear functions showing intercepts, quadratic functions showing intercepts, maxima, or minima.

F.IF.7b Square root, cube root, and piecewise-defined functions, including step functions and absolute value
functions.

F.IF.7c Polynomial functions, identifying zeros when factorable, and showing end behavior.

F.IF.7d (+) Rational functions, identifying zeros and asymptotes when factorable, and showing end behavior.

F.IF.7e Exponential and logarithmic functions, showing intercepts and end behavior.

F.IF.7e Trigonometric functions, showing period, midline, and amplitude.

ESSENTIAL
QUESTION(S)

• How are functions interpreted when used in applications?
• What essential information is indicated when graphing linear and quadratic functions?
• What essential information is indicated when graphing square root, cube root and piecewise-defined functions?
• What essential information is indicated when graphing polynomial functions?
• What essential information is indicated when graphing rational functions?
• What essential information is indicated when graphing exponential, logarithmic and trigonometric functions?

MATHEMATICAL
PRACTICE(S)

HS.MP.5. Use appropriate tools strategically.

HS.MP.6. Attend to precision.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

SUBSTANDARD
DECONSTRUCTED

F.IF.7 a. Graph linear and quadratic functions and show intercepts, maxima,
and minima.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Graph linear functions by hand in
simple cases or using technology
for more complicated cases and
show/label intercepts of the graph.

Determine the differences
between simple and complicated
linear, exponential and quadratic
functions and know when the use
of technology is appropriate.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 29

MATHEMATICS
STANDARD AND DECONSTRUCTION

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

DESCRIPTION

F.IF.7 Graph functions expressed symbolically and show key features of the graph. Graph simple cases by hand, and
use technology to show more complicated cases including:

F.IF.7a Linear functions showing intercepts, quadratic functions showing intercepts, maxima, or minima.

F.IF.7b Square root, cube root, and piecewise-defined functions, including step functions and absolute value
functions.

F.IF.7c Polynomial functions, identifying zeros when factorable, and showing end behavior.

F.IF.7d (+) Rational functions, identifying zeros and asymptotes when factorable, and showing end behavior.

F.IF.7e Exponential and logarithmic functions, showing intercepts and end behavior.

F.IF.7e Trigonometric functions, showing period, midline, and amplitude.

ESSENTIAL
QUESTION(S)

MATHEMATICAL
PRACTICE(S)

HS.MP.5. Use appropriate tools strategically.

HS.MP.6. Attend to precision.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

SUBSTANDARD
DECONSTRUCTED

F.IF.7 a. Graph linear and quadratic functions and show intercepts, maxima,
and minima.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Graph linear functions by hand in
simple cases or using technology
for more complicated cases and
show/label intercepts of the graph.

Determine the differences
between simple and complicated
linear, exponential and quadratic
functions and know when the use
of technology is appropriate.

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

30 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

SUBSTANDARD
DECONSTRUCTED

F.IF.7b. Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Graph square root, cube root,
and piecewise-defined functions,
including step functions and
absolute value functions, by hand
in simple cases or using technology
for more complicated cases, and
show/label key features of the
graph.

Determine the difference between
simple and complicated linear,
quadratic, square root, cube root,
and piecewise-defined functions
including step functions and
absolute value functions and know
when the use of technology is
appropriate.

Compare and contrast absolute
value, step- and piecewise-defined
functions with linear, quadratic, and
exponential functions.

Compare and contrast the domain
and range of absolute value,
step- and piecewise-defined
functions with linear, quadratic, and
exponential function.

Analyze the difference between
simple and complicated linear,
quadratic, square root, cube root,
piecewise-defined, exponential,
logarithmic, and trigonometric
functions, including step and
absolute value functions.

Select the appropriate type of
function, taking into consideration
the key features, domain, and
range, to model a real-world
situation.

SUBSTANDARD
DECONSTRUCTED

F.IF.7c. Graph polynomial functions, identifying zeros when suitable
factorizations are available, and showing end behavior.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Graph polynomial functions, by
hand in simple cases or using
technology for more complicated
cases, and show/label maxima and
minima of the graph, identify zeros
when suitable factorizations are
available, and show end behavior.

Determine the difference between
simple and complicated polynomial
functions.

Relate the relationship between
zeros of quadratic functions
and their factored forms to the
relationship between polynomial
functions of degrees greater than
two.

SUBSTANDARD
DECONSTRUCTED

F.IF.7d. (+) Graph rational functions, identifying zeros and asymptotes
when suitable factorizations are available, and showing end behavior.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Graph rational functions, by hand
in simple cases or using technology
for more complicated cases, and
show/label the graph, identify zeros
when suitable factorizations are
available, and show end behavior.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 31

MATHEMATICS

SUBSTANDARD
DECONSTRUCTED

F.IF.7d. (+) Graph rational functions, identifying zeros and asymptotes
when suitable factorizations are available, and showing end behavior.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

SUBSTANDARD
DECONSTRUCTED

F.IF.7e. Graph exponential and logarithmic functions, showing intercepts
and end behavior, and trigonometric functions, showing period, midline,
and amplitude.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Graph exponential functions, by
hand in simple cases or using
technology for more complicated
cases, and show intercepts and end
behavior.

Graph exponential, logarithmic, and
trigonometric functions, by hand in
simple cases or using technology
for more complicated cases. For
exponential and logarithmic
functions, show: intercepts and
end behavior; for trigonometric
functions, show: period, midline,
and amplitude.

Determine the differences between
simple and complicated linear and
exponential functions and know
when the use of technology is
appropriate.

Compare and contrast the
domain and range of exponential,
logarithmic, and trigonometric
functions with linear, quadratic,
absolute value, step- and
piecewise-defined functions.

Select the appropriate type of
function, taking into consideration
the key features, domain, and
range, to model a real-world
situation.

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

32 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

EXPLANATIONS
AND EXAMPLES

Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and
asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to
graph functions.

Examples:.

Describe key characteristics of the graph of

f(x) = | x – 3 | + 5.

Sketch the graph and identify the key characteristics of the function described below.

Graph the function f(x) = 2x by creating a table of values. Identify the key characteristics of the graph.

Graph f(x) = 2 tan x – 1. Describe its domain, range, intercepts, and asymptotes.

Draw the graph of f(x) = sin x and f(x) = cos x. What are the similarities and differences between the two graphs?

(Source: www.shmoop.com)

STANDARD AND DECONSTRUCTION

F.IF.8 Write a function defined by an expression in different but equivalent forms
to reveal and explain different properties of the function.

DESCRIPTION

F.IF.8 Write a function in equivalent forms to show different properties of the function.

F.IF.8 Explain the different properties of a function that are revealed by writing a function in equivalent forms.

F.IF.8a 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.

F.IF.8b Use the properties of exponents to interpret expressions for percent rate of change, and classify them as
growth or decay.

ESSENTIAL
QUESTION(S)

• How can multiple representations of functions help reveal and explain different properties of the function?
• How can a quadratic function be interpreted when factoring and completing the square?
• How can the properties of exponents be used to interpret and classify exponential functions?

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.7. Look for and make use of structure.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

http://www.shmoop.com

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 33

MATHEMATICS
STANDARD AND DECONSTRUCTION

F.IF.8 Write a function defined by an expression in different but equivalent forms
to reveal and explain different properties of the function.

DESCRIPTION

F.IF.8 Write a function in equivalent forms to show different properties of the function.

F.IF.8 Explain the different properties of a function that are revealed by writing a function in equivalent forms.

F.IF.8a 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.

F.IF.8b Use the properties of exponents to interpret expressions for percent rate of change, and classify them as
growth or decay.

ESSENTIAL
QUESTION(S)

MATHEMATICAL
PRACTICE(S)

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.7. Look for and make use of structure.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

SUBSTANDARD
DECONSTRUCTED

F.IF.8 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.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Identify different forms of a
quadratic expression.

Identify zeros, extreme values,
and symmetry of the graph of a
quadratic function.

Identify how key features of a
quadratic function relate to its
characteristics in a real-world
context.

Write functions in equivalent forms
using the process of factoring.

Interpret different yet equivalent
forms of a function defined by an
expression in terms of a context.

Given the expression of a quadratic
function, interpret zeros, extreme
values, and symmetry of the graph
in terms of a real-world context.

Write a quadratic function defined
by an expression in different but
equivalent forms to reveal and
explain different properties of the
function and determine which
form of the quadratic is the most
appropriate for showing zeros and
symmetry of a graph in terms of a
real-world context.

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.

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

34 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

SUBSTANDARD
DECONSTRUCTED

F.IF.8 b. Use the properties of exponents to interpret expressions for
exponential functions. For example, identify percent rate of change
in functions such as y = , y =, y =, y =, and classify them as representing
exponential growth or decay.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Classify the exponential function
as exponential growth or decay by
examining the base.

Identify how key features of an
exponential function relate to
its characteristics in a real-world
context.

Use the properties of exponents
to interpret expressions for
exponential functions in a real-
world context.

Given the expression of an
exponential function, interpret the
expression in terms of a real-world
context, using the properties of
exponents.

Write an exponential function
defined by an expression in
different but equivalent forms
to reveal and explain different
properties of the function, and
determine which form of the
function is the most appropriate
for interpretation in a real-world
context.

EXPLANATIONS
AND EXAMPLES

Students should be able to find the x-intercepts of a quadratic function using both factoring and completing the
square. All that should be given to students is the standard form of a quadratic equation in the form y = ax2 + bx + c.

When a = 1, factoring is fairly easy. The equation can be factored into the form y = (x + p)(x + q), where p + q = b and
pq = c. For example, if given the equation y = x2 – 9x + 18, we’d need p and q values such that p + q = -9 and pq = 18.
A quick check will tell us that p = -3 and q= -6 are the values that make sense. So our factored equation is y = (x – 3)
(x – 6).

Since y = 0 for x-intercepts, we can set the factored form of our equation to equal zero. The entire equation will be
zero when either (or both) of the factors are zero. We can find the roots by solving each factor for x. The factors x – 3 =
0 and x – 6 = 0 mean that our x-intercepts are 3 and 6.

Completing the square is another way to factor. Starting with the standard form of the equation y = ax2 + bx + c, we
can use the following steps to create a perfect square trinomial and solve for the x-intercepts that way.

1. Set y = 0.

2. Divide through by a (the coefficient in front of x2 must be 1).

3. Subtract the constant factor from both sides.

4. Rewrite the function in the form x2 + 2hx + h2.

5. Rewrite the function in the form (x + h)2.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 35

MATHEMATICS
EXPLANATIONS
AND EXAMPLES

(continued)

6. Take the square root of both sides.

It turns out that completing the square is a straightforward way to derive the quadratic equation. Pretty sweet,
right?

We can also describe extreme values of the function, or the vertex of the parabola. If we return to the original form
of the equation, y = ax2 + bx + c, the coordinates of the vertex can be written as

If we look at the other form of the equation, y = (x + h)2, we can say that the centerline (or line of symmetry) lies at x
= -h.

So much information can be extracted from the different forms of writing a quadratic equation. If students are lost
and confused within the many terms of a quadratic equation, the standard form is their landmark. From there, they
can get wherever they need to go and find whatever they need to find.

If we have a function of the form y = abx, we can either be describing exponential growth or exponential decay. If a
> 0 and 0 < b < 1, the equation represents decay. If a > 0 and b > 1, the equation represents growth.

Equations like y = a(1 + c)x can represent the balance of a savings account after x years with starting balance a and
interest rate c. If the annual interest rate is 2% and we start with $100, how much money will we have in 10 years?
Just substitute in our values, and we’re good to go.

y = a(1 + c)x = 100(1 + 0.02)10 = $121.90

Not too bad, but don’t go spending it all on bubble gum. We may want to save up for a new iPod.

(Source: www.shmoop.com)

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

http://www.shmoop.com

36 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

STANDARD AND DECONSTRUCTION

F.IF.9
Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.

DESCRIPTION F.IF.9 Compare the key features of two functions represented in different ways. For example, compare the end
behavior of two functions, one of which is represented graphically and the other is represented symbolically.

ESSENTIAL
QUESTION(S)

• How can multiple representations of functions help reveal and explain different properties of the function?

MATHEMATICAL
PRACTICE(S)

HS.MP.6. Attend to precision.

HS.MP.7. Look for and make use of structure.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Identify types of functions based
on verbal, numerical, algebraic, and
graphical descriptions and state key
properties.

Differentiate between exponential
and linear functions using a variety
of descriptors (graphical, verbal,
numerical, and algebraic).

Differentiate between two types
of functions using a variety of
descriptors (graphical, verbal,
numerical, algebraic).

Use a variety of function
representations (algebraic,
graphical, numerical in tables, or
by verbal descriptions) to compare
and contrast properties of two
functions.

EXPLANATIONS
AND EXAMPLES

In a perfect world, we would have explicit equations for all relationships (and we don’t just mean the Facebook
official ones). We’d just plug in our effort and calculate our reward. But as you may have realized, life is much less
predictable and far from perfect. Even still, that doesn’t mean we can’t make sense out of it.

Students should be able to compare two functions even when they’re both represented differently. To do this
successfully, they have to be able to translate between an equation, a graph, a bunch of words, and a table of
values, and understand how certain aspects of one representation impact the rest.

Generally, students should start by knowing the difference between polynomial, linear, quadratic, exponential, and
rational functions, and be able to identify them by equation and by graph. This means that given a parabolic curve,
students should automatically look for equations in the form of y = ax2 + bx + c.

More specifically, a function f(x) that has a y-intercept of 4 would need to have an equation such that f(0) = 4.
Similarly, a table of values for this function would be expected to have the point (0, 4).

The struggles students might face could be traced back to the different representations of functions. Students may
have particular difficulty with one type of representation and as such, may have trouble with conversion. If this
becomes a problem, try going over each type of representation side by side, highlighting the corresponding parts
of each and matching them like a giant game of connect-the-dots.

Students should also know the accuracy of each representation. For instance, a table of values can’t conclusively
define a certain type of function, while a graph can’t pinpoint intercepts with certainty. An equation is the most
accurate and useful when defining a function, and students should make use of that.

If they can transform any graph, table of values, or description into a mathematical equation that describes the
function, they should be good to go.

(Source: www.shmoop.com)

http://www.shmoop.com

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 37

MATHEMATICS
EXPLANATIONS
AND EXAMPLES

(continued)

Example:

Examine the functions below. Which function has the larger maximum? How do you know?

IN
TE

RP
RE

TI
N

G
F

U
N

CT
IO

N
S

MATHEMATICS

BUILDING
FUNCTIONS

(F-BF)

DOMAIN:

HIGH SCHOOL
FUNCTIONS

40 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

DOMAIN Building Functions (F-BF)

CLUSTERS
1. Build a function that models a relationship between two quantities
2. Build new functions from existing functions

ACADEMIC
VOCABULARY

domain, range, function notation, fibonacci sequence, recursive process, intercepts, increasing intervals,
decreasing intervals, positive intervals, negative intervals, relative maximum, relative minimum, symmetries, end
behavior, periodicity, rate of change, step function, absolute value function, asymptote, exponential function,
logarithmic function, trigonometric function, period, midline, amplitude, exponential growth, exponential decay,
constant function, arithmetic sequence, geometric sequence, invertible function, radian measure, arc, sine, cosine,
tangent

CLUSTER 1. Build a function that models a relationship between two quantities

BIG IDEA • Relationships between two sets of numbers can be described by mathematical rules, where a
function is a unique rule that has a one-to-one correspondence.

STANDARD AND DECONSTRUCTION

F.BF.1 Write a function that describes a relationship between two quantities.

DESCRIPTION F.BF.1a From context, either write an explicit expression, define a recursive process, or describe the calculations
needed to model a function between two quantities.

F.BF.1b. Combine standard function types, such as linear and exponential, using arithmetic operations.

F.BF.1c Compose functions.

SUBSTANDARD
DECONSTRUCTED

F.BF.1 a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Define explicit function and
recursive process.

Write a function that describes
a relationship between two
quantities by determining an
explicit expression, a recursive
process, or steps for calculation
from a context.

SUBSTANDARD
DECONSTRUCTED

F.BF.1 b. Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of a cooling body
by adding a constant function to a decaying exponential, and relate these
functions to the model.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Combine two functions using the
operations of addition, subtraction,
multiplication, and division.

Evaluate the domain of the
combined function.

Given a real-world situation or
mathematical problem, build
standard functions to represent
relevant relationships/ quantities.

Given a real-world situation or
mathematical problem, determine
which arithmetic operation
should be performed to build the
appropriate combined function.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 41

MATHEMATICS
STANDARD AND DECONSTRUCTION

F.BF.1 Write a function that describes a relationship between two quantities.

DESCRIPTION F.BF.1a From context, either write an explicit expression, define a recursive process, or describe the calculations
needed to model a function between two quantities.

F.BF.1b. Combine standard function types, such as linear and exponential, using arithmetic operations.

F.BF.1c Compose functions.

ESSENTIAL
QUESTION(S)

• What strategy can be used to write a function that describes a relationship between two quantities?

MATHEMATICAL
PRACTICE(S)

HS.MP.1. Make sense of problems and persevere in solving them.

HS.MP.2. Reason abstractly and quantitatively.

HS.MP.4. Model with mathematics.

HS.MP.5. Use appropriate tools strategically.

HS.MP.6. Attend to precision.

HS.MP.7. Look for and make use of structure.

HS.MP.8. Look for and express regularity in repeated reasoning.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

SUBSTANDARD
DECONSTRUCTED

F.BF.1 a. Determine an explicit expression, a recursive process, or steps for
calculation from a context.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Define explicit function and
recursive process.

Write a function that describes
a relationship between two
quantities by determining an
explicit expression, a recursive
process, or steps for calculation
from a context.

SUBSTANDARD
DECONSTRUCTED

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Combine two functions using the
operations of addition, subtraction,
multiplication, and division.

Evaluate the domain of the
combined function.

Given a real-world situation or
mathematical problem, build
standard functions to represent
relevant relationships/ quantities.

Given a real-world situation or
mathematical problem, determine
which arithmetic operation
should be performed to build the
appropriate combined function.

BU
IL

D
IN

G
F

U
N

CT
IO

N
S

42 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

HIGH SCHOOL FUNCTIONS
LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L,

11TH – 12TH GRADES: 1185L TO 1385L

SUBSTANDARD
DECONSTRUCTED

F.BF.1 c. (+) Compose functions. For example, if T(y) is the temperature in
the atmosphere as a function of height, and h(t) is the height of a weather
balloon as a function of time, then T(h(t)) is the temperature at the location of
the weather balloon as a function of time.

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning. Tasks assessing modeling/applications.

Students should
be able to:

Given a real-world situation or
mathematical problem, relate the
combined function to the context
of the problem.

Compose functions.

EXPLANATIONS
AND EXAMPLES

Students will analyze a given problem to determine the function expressed by identifying patterns in the function’s
rate of change. They will specify intervals of increase, decrease, constancy, and, if possible, relate them to the
function’s description in words or graphically. Students may use graphing calculators or programs, spreadsheets, or
computer algebra systems to model functions.

Examples:

• You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly
payments of $250. Express the amount remaining to be paid off as a function of the number of months,
using a recursion equation.

• A cup of coffee is initially at a temperature of 93º F. The difference between its temperature and the room
temperature of 68º F decreases by 9% each minute. Write a function describing the temperature of the
coffee as a function of time.

• The radius of a circular oil slick after t hours is given in feet by , for 0 ≤ t ≤ 10. Find the area of the oil slick as a
function of time.

(Source: www.shmoop.com)

STANDARD AND DECONSTRUCTION

F.BF.2
Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the
two forms.

DESCRIPTION

F.BF.2 Write arithmetic sequences recursively and explicitly, use the two forms to model a situation, and translate
between the two forms.

F.BF.2 Write geometric sequences recursively and explicitly, use the two forms to model a situation, and translate
between the two forms.

F.BF.2 Understand that linear functions are the explicit form of recursively-defined arithmetic sequences and that
exponential functions are the explicit form of recursively-defined geometric sequences.

ESSENTIAL
QUESTION(S)

• What arithmetic and geometric sequences can be used to model situations?

MATHEMATICAL
PRACTICE(S)

HS.MP.4. Model with mathematics.

HS.MP.5. Use appropriate tools strategically.

HS.MP.8. Look for and express regularity in repeated reasoning.

DOK Range Target
for Instruction &

Assessment
T 1 T 2 o 3 o 4

Learning Expectations Know: Concepts/Skills Think Do

Assessment Types Tasks assessing concepts, skills, and
procedures.

Tasks assessing expressing mathematical
reasoning.

Tasks assessing modeling/
applications.

Students should
be able to:

Identify arithmetic and geometric
patterns in given sequences.

Generate arithmetic and geometric
sequences from recursive and
explicit formulas.

Given an arithmetic or geometric
sequence in recursive form,
translate into the explicit formula.

Given an arithmetic or geometric
sequence as an explicit formula,
translate into the recursive form.

Use given and constructed arithmetic
and geometric sequences, expressed
both recursively and with explicit
formulas, to model real-life situations.

Determine the recursive rule given
arithmetic and geometric sequences.

Determine the explicit formula given
arithmetic and geometric sequences.

Justify the translation between the
recursive form and explicit formula for
arithmetic and geometric sequences.

EXPLANATIONS
AND EXAMPLES

An explicit rule for the nth term of a sequence gives an as an expression in the term’s position n; a recursive rule
gives the first term of a sequence, and a recursive equation relates an to the preceding term(s). Both methods of
presenting a sequence describe an as a function of n.

Examples:

• Generate the 5th-11th terms of a sequence if A1= 2 and

• Use the formula: An= A1 + d(n – 1) where d is the common difference to generate a sequence whose first
three terms are: -7, -4, and -1.

• There are 2,500 fish in a pond. Each year the population decreases by 25 percent, but 1,000 fish are added to
the pond at the end of the year. Find the population in five years. Also, find the long-term population.

• Given the formula An= 2n – 1, find the 17th term of the sequence. What is the 9th term in the sequence 3, 5, 7,
9, …? Given a1 = 4 and an = an-1 + 3, write the explicit formula.

(Source: www.shmoop.com)

http://www.shmoop.com

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 43