Math / 8th Grade / Unit 4: 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
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In Unit 4, eighth-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, coins and the number of minutes of parking at a meter, 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). 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 students 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). 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), setting them up for Unit 5. Lastly, students make connections between stories and graphs by modeling situations like distance or speed over time.
In sixth and seventh 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, eighth-grade 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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The following assessments accompany Unit 4.
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.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment
Post-Unit Assessment Answer Key
Post-Unit Self-Assessment
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Unit Launch
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
Degrees Fahrenheit is a function of degrees Celsius
$$F=\frac{9}{5}C+32$$
Temperature is a function of time.
The central mathematical concepts that students will come to understand in this unit
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 8th Grade Course Material Overview.
Terms and notation that students learn or use in the unit
function
input/output
initial value
linear function
nonlinear function
rate of change
To see all the vocabulary for Unit 4, view our 8th Grade Vocabulary Glossary.
Topic A: Defining Functions
Define and identify functions.
8.F.A.1
Use function language to describe functions. Identify function rules.
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Topic B: Representing and Interpreting Functions
Identify properties of functions represented in tables, equations, and verbal descriptions. Evaluate functions.
8.F.A.1 8.F.A.2 8.F.B.4
Represent functions with equations.
8.F.A.1 8.F.B.4
Read inputs and outputs in graphs of functions. Determine if graphs are functions.
Identify properties of functions represented in graphs.
Topic C: Comparing Functions
Define and graph linear and nonlinear functions.
8.F.A.3
Determine if functions are linear or nonlinear when represented as tables, graphs, and equations.
8.F.A.1 8.F.A.3
Compare functions represented in different ways (Part 1).
8.F.A.2
Compare functions represented in different ways (Part 2).
Topic D: Describing and Drawing Graphs of Functions
Describe functions by analyzing graphs. Identify intervals of increasing, decreasing, linear, or nonlinear activity.
8.F.B.5
Sketch graphs of functions given qualitative descriptions of the relationship.
Key
Major Cluster
Supporting Cluster
Additional Cluster
The content standards covered in this unit
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.
8.F.A.2 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3 — Interpret the equation 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.
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.
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.
Standards covered in previous units or grades that are important background for the current unit
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.
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.
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."
7.RP.A.2 — Recognize and represent proportional relationships between quantities.
7.RP.A.2.B — Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of 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.
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.
Standards in future grades or units that connect to the content in this unit
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.
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.
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.
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).
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.
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.
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.
Unit 3
Transformations and Angle Relationships
Unit 5
Linear Relationships