Functions

Students learn how to represent, interpret, and analyze functions in various forms, leading to understanding features such as rates of change, initial values, and intervals of increase and decrease.

Math

Unit 4

8th Grade

Unit Summary

In Unit 4, 8th grade students are introduced to the concept of a function that relates inputs and outputs. They begin by investigating all types of relationships between sets, such as students and their number of siblings, people and the number of languages they speak, distance and time spent running, etc. They learn how to represent and interpret functions in various forms, including tables, equations, graphs, and verbal descriptions MP.2 Reason abstractly and quantitatively. . As students progress through the unit, they analyze functions to better understand features such as rates of change, initial values, and intervals of increase or decrease, which in turn enables them to make comparisons across functions even when they are not represented in the same format. Students analyze real-world situations for rates of change and initial values and use these features to construct equations to model the function relationships MP.4 Model with mathematics. . Students also spend time comparing linear functions to nonlinear functions, building an understanding of the underlying structure of a function that makes it linear MP.7 Look for and make use of structure. , setting them up for Unit 5. Lastly, students make connections between stories and graphs by modeling situations like distance or speed over time.

In 6th grade and 7th grade, students studied rate and constant of proportionality in proportional relationships. They developed an understanding of how one quantity changes in relationship to another. Students draw on that knowledge as they investigate how quantities are related in tables, equations, and graphs, and as they investigate linear vs. nonlinear relationships.

Immediately following this unit, students will begin a unit on linear relationships. In that unit, they will revisit and extend on many of the topics introduced in this Functions unit. Students will interpret rate of change as slope and initial value as the $$y-$$intercept of a linear equation $$y=mx+b$$. In high school, the study of functions extends across multiple topics and fields of study, including quadratic, exponential, and trigonometric functions.

Pacing: 16 instructional days (12 lessons, 3 flex days, 1 assessment day)

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Assessment

The following assessments accompany Unit 4.

Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Mid-Unit

Have students complete the Mid-Unit Assessment after Lesson 6.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Unit Launch

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

Internalization of Standards via the Post-Unit Assessment

Take the Post-Unit Assessment. Annotate for:
- Standards that each question aligns to
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that Assessment points to

Internalization of Trajectory of Unit

Read and annotate the Unit Summary.
Notice the progression of concepts through the unit using the Lesson Map.
Do all Target Tasks. Annotate the Target Tasks for:
- Essential Understandings
- Connection to Post-Unit Assessment questions
Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Read the Progressions for the Common Core State Standards in Mathematics, Grade 8, High School, Functions for the standards relevant to this unit.
It is also helpful to read the Progressions for the Common Core State Standards in Mathematics, 6-8, Expressions and Equations, in particular for 8.EE.5 and 8.EE.6, as they are very connected to the function standards.

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

A function is a rule that assigns exactly one output to each input. Functions can be represented as tables, equations, graphs, and verbal descriptions.
Linear functions consist of ordered pairs that when graphed lie on a straight line; nonlinear functions consist of ordered pairs that when graphed do not lie on a straight line.
Functions can be analyzed to understand their rate of change, their initial value, or intervals where they may be increasing or decreasing, linear or nonlinear.
Functions can be used to model relationships between values and shown as equations, graphs, tables, or qualitative descriptions.

Materials

The materials, representations, and tools teachers and students will need for this unit

Optional: Calculators (1 per student)
Optional: Graph Paper
Optional: Ruler
Optional: Dry erase marker (1 per student)
Optional: White board (1 per student)
Function Sentence Stems
Graphing technology — Graphing calculator or graphing application on a computer

To see all the materials needed for this course, view our 8th Grade Course Material Overview.

Vocabulary and Models

Unit Vocabulary

function

input/output

initial value

linear function

nonlinear function

rate of change

Foundational Vocabulary

dependent variable

independent variable

proportional relationship

proportional

rate

unit rate

To see all the vocabulary for Unit 4, view our 8th Grade Vocabulary Glossary.

Access foundational vocabulary for Unit 4 in the same 8th Grade Vocabulary Glossary.

Models

Models and representations that students learn or use in the unit

Model	Example
input/output table of functions
equation of function	Degrees Fahrenheit is a function of degrees Celsius $$F=\frac{9}{5}C+32$$
graph of function	Temperature is a function of time.
verbal representation of function	The total distance a runner has traveled is a function of time spent running.

Lesson Map

Topic A: Defining Functions

Lesson

Define and identify functions.

Standards

8.F.A.1

Lesson

Use function language to describe functions. Identify function rules.

Standards

8.F.A.1

Topic B: Representing and Interpreting Functions

Lesson

Identify properties of functions represented in tables, equations, and verbal descriptions.

Standards

8.F.A.18.F.A.28.F.B.4

Lesson

Represent linear functions with equations.

Standards

8.F.A.18.F.B.4

Lesson

Read inputs and outputs in graphs of functions. Determine if graphs are functions.

Standards

8.F.A.1

Lesson

Identify properties of functions represented in graphs.

Standards

8.F.A.18.F.B.4

Topic C: Comparing Functions

Lesson

Define and graph linear and nonlinear functions.

Standards

8.F.A.3

Lesson

Determine if functions are linear or nonlinear when represented as tables, graphs, and equations.

Standards

8.F.A.18.F.A.3

Lesson

Compare functions represented in different ways.

Standards

8.F.A.2

Lesson

Compare functions represented in different ways. (Part 2)

Standards

8.F.A.2

Topic D: Describing and Drawing Graphs of Functions

Lesson

Describe functions by analyzing graphs. Identify intervals of increasing, decreasing, linear, or nonlinear activity.

Standards

8.F.B.5

Lesson

Sketch graphs of functions given qualitative descriptions of the relationship.

Standards

8.F.B.5

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Functions

Functions

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

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

8.F.A.3 — Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Functions

8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Functions

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

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Expressions and Equations

Expressions and Equations

6.EE.A.2.C — Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.
Expressions and Equations

7.EE.B.4 — Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Operations and Algebraic Thinking

Operations and Algebraic Thinking

4.OA.A.3 — Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Operations and Algebraic Thinking

4.OA.C.5 — Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Ratios and Proportional Relationships

Ratios and Proportional Relationships

6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."
Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.
Ratios and Proportional Relationships

7.RP.A.2.B — Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Ratios and Proportional Relationships

7.RP.A.2.C — Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
Ratios and Proportional Relationships

7.RP.A.2.D — Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Future Standards

Standards in future grades or units that connect to the content in this unit

Building Functions

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Expressions and Equations

Expressions and Equations

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

8.EE.B.6 — Use 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.

Interpreting Functions

Interpreting Functions

F.IF.A.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).
Interpreting Functions

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

F.IF.B.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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

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