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.
Unit Summary
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 will 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 will 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 will 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)
For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 8th Grade Scope and Sequence Recommended Adjustments.
Assessment
This assessment accompanies Unit 4 and should be given on the suggested assessment day or after completing the unit.
- Download Unit Assessment
- Download Unit Assessment Answer Key
- Download Student Self-Assessment
Unit Prep
Intellectual Prep
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Internalization of Standards via the Unit Assessment
- Take 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 “Unit Summary.”
- Notice the progression of concepts through the unit using “Unit at a Glance.”
- Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions
- Identify key opportunities to engage students in academic discourse. Read through our Guide to 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.
- Read the following table that includes models used throughout the unit.
Input/output table of functions Example:
Equation of function Example: Degrees Fahrenheit is a function of degrees Celsius
$$F=\frac{9}{5}C+32$$
Graph of function Example: Temperature is a function of time.
Verbal representation of function Example: The total distance a runner has traveled is a function of time spent running. Essential Understandings
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- A function is a rule that assigns to each input exactly one output. 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.
Vocabulary
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function
input/output
initial value
rate of change
linear function
nonlinear function
Related Teacher Tools:
Unit Materials, Representations and Tools
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- Input/output tables
- Equations
- Graphs
- Verbal representations
Lesson Map
Topic A: Defining Functions
1
8.F.A.1
Define and identify functions.
2
8.F.A.1
Use function language to describe functions. Identify function rules.
Topic B: Representing and Interpreting Functions
3
8.F.A.1
8.F.A.2
8.F.B.4
Identify properties of functions represented in tables, equations, and verbal descriptions. Evaluate functions.
4
8.F.A.1
8.F.B.4
Represent functions with equations.
5
8.F.A.1
Read inputs and outputs in graphs of functions. Determine if graphs are functions.
6
8.F.A.1
8.F.B.4
Identify properties of functions represented in graphs.
Topic C: Comparing Functions
7
8.F.A.3
Define and graph linear and nonlinear functions.
8
8.F.A.1
8.F.A.3
Determine if functions are linear or nonlinear when represented as tables, graphs, and equations.
9
8.F.A.2
Compare functions represented in different ways (Part 1).
10
8.F.A.2
Compare functions represented in different ways (Part 2).
Topic D: Describing and Drawing Graphs of Functions
Common Core Standards
Key: Major Cluster Supporting Cluster Additional Cluster
Core Standards
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Functions
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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.
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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.
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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.
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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.
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8.F.B.5 — Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Foundational Standards
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Expressions and Equations
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6.EE.A.2.C
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7.EE.B.4
Ratios and Proportional Relationships
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6.RP.A.2
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7.RP.A.2
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7.RP.A.2.B
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7.RP.A.2.C
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7.RP.A.2.D
Future Standards
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Building Functions
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F.BF.A.1
Expressions and Equations
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8.EE.B.5
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8.EE.B.6
Interpreting Functions
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F.IF.A.1
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F.IF.A.2
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F.IF.B.4
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.
