Lesson 3 | Functions | 8th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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.

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.
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.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.

Foundational Standards

The foundational standards covered in this lesson

6.EE.A.2.C

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.
6.RP.A.2

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."
7.RP.A.2.B

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Find the rate of change of a function represented in a table, equation, and verbal description.
Find the initial value of a function represented in a table, equation, and verbal description.
Evaluate a function formula for given values.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Lessons 3 and 4 focus on functions represented in tables, equations, and verbal descriptions. In Lesson 3, students analyze these representations to make sense of the relationships between the quantities (MP.2). They use their understanding of unit rate and constant of proportionality to determine the rate of change between quantities. They consider what a starting or initial value means in context of a situation, and how to determine these values from a table or equation.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Determine the rate of change in each situation. Then describe the relationship using function language (for example, ____ is a function of ____ because _____.)

Amt of watermelon in pounds, $$x$$	0	0.25	0.5	1.5	2	3	5
Cost of watermelon in dollars, $$y$$	0	0.35	0.70	2.10	2.80	4.20	7.00

b. $${d=25t}$$, where $$d$$ is distance in miles and $$t$$ is time in hours

c. Giovanni can decorate $$2$$ cakes, $$c$$, every $$5$$ hours, $$h$$.

Guiding Questions

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Problem 2

Determine the starting or initial output value in each situation.

Distance traveled in cab in miles, $$x$$	0	1	3	4
Cost of cab ride in dollars, $$y$$	2.50	4.75	9.25	11.50

Distance traveled in cab in miles, $$x$$	1	2	4	5
Cost of cab ride in dollars, $$y$$	4.50	7.00	12.00	14.50

c. $${p=16d+20}$$ where $$p$$ is the total number of pages read in a book and $$d$$ is the number of days

Guiding Questions

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Problem 3

The function formula below shows the relationship between temperature in degrees Fahrenheit and degrees Celsius. °F is a function of °C.

$${F=\frac{9}{5}C+32}$$

a. Determine what 28°C is in °F.

b. Determine what 23°F is in °C.

Guiding Questions

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Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

The formula below represents simple interest, $$I$$, as a function of the principle amount, $$P$$, the simple interest rate, $$R$$, and the time of the investment, $$T$$.

$$I=PRT$$

Determine the principle amount, $$P$$, when $${$8.50}$$ of interest has been earned after $${1.25}$$ years with an annual rate of $$2$$%.

Problem 2

The table below shows the number of miles Alan biked over time. Alan biked at a constant speed. Find the rate of change.

Time in hours, $$x$$	0	0.5	2	4	4.5	6
Distance biked in miles, $$y$$	0	7.5	30	60	67.5	90

Student Response

An example response to the Target Task at the level of detail expected of the students.

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Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include additional items similar to Anchor Problems 2 and 3, where students are given functions (as tables, as equations, as verbal representations) and are asked to find the rate of change and initial value.
Include problems where students evaluate function formulas, such as
- circumference/perimeter
- area/volume
- distance

Smarter Balanced Assessment Consortium: Sample Items Item #3587

Lesson 2

Lesson 4

Lesson Map

Topic A: Defining Functions

Define and identify functions.

8.F.A.1

Use function language to describe functions. Identify function rules.

8.F.A.1

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.

8.F.A.1

Identify properties of functions represented in graphs.

8.F.A.1 8.F.B.4

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).

8.F.A.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.

8.F.B.5

Functions

Objective

Common Core Standards

Core Standards

Functions

Functions

Functions

Foundational Standards

Expressions and Equations

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Problem 1

Problem 2

Student Response

Additional Practice

Lesson Map

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