Lesson 14 | Linear Relationships | 8th Grade Mathematics

Objective

Compare linear functions represented in different ways.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Criteria for Success

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

Identify slope and $$y$$-intercept of lines represented as graphs, tables, and equations.
Compare the slopes and $$y$$-intercepts of linear functions.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In Lessons 9 and 10 of Unit 4, students compared functions by looking at the rates of change and initial values of those functions. This lesson revisits some of those concepts, but looks specifically at linear functions and describes the graphs of these functions by the slope and $$y$$-intercept of the line.

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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

Three linear functions are represented below.

Function A:

Function B:

Function C:

$$y=x$$

$$x$$	$$y$$
-3	1
6	4
9	5

a. Order the functions by the slope of the line that represents each function, from least to greatest.

b. Order the functions by the $$y$$-intercept of the line that represents each function, from least to greatest.

Guiding Questions

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Problem 2

Three friends participate in the loyalty program at a local coffee shop. A participant in the program earns points for each visit that can be used to get drinks and food for free. The graph below shows the point balances of the three friends over several weeks.

a. Describe what each line tells you about the friends’ point balance. What can you tell about how they earn and use their points?

b. Who has the most points at the start?

c. Write an equation for the point balance for each friend.

d. A fourth friend has a point balance represented by the equation $${y=12x+50}$$. Does this friend earn points at a faster or slower rate than Adan?

Guiding Questions

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Problem 3

Line $$L$$ goes through the points (4, 9) and (-4, -7). Line $$M$$ has a $$y-$$intercept that is greater than line $$L$$.

What could be the equation for line $$M$$?

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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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Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

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

Linear function $$K$$ is represented by the equation $${y={1\over2}x-2}$$.

Draw the graph of a line that has a greater slope than function $$K$$ and the same $$y$$-intercept as function $$K$$.

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 multiple-choice problems where students are presented with two linear functions and determine true statements that compare slope and/or $$y$$-intercepts of the lines.

MARS Formative Assessment Lesson for Grade 8 Comparing Lines and Linear Equations — Great activity where students are required to pay close attention to how graphs and equations represent different rates.
EngageNY Mathematics Grade 8 Mathematics > Module 5 > Topic A > Lesson 7 — Revisit the Problem Set; this was included as a resource in Unit 4 Lesson 9 but may be worthwhile returning to in light of the new concepts learned in this unit.

Lesson 13

Lesson 15

Lesson Map

Topic A: Comparing Proportional Relationships

Review representations of proportional relationships.

8.EE.B.5

Graph proportional relationships and interpret slope as the unit rate.

8.EE.B.5

Compare proportional relationships represented as graphs.

8.EE.B.5

Compare proportional relationships represented in different ways.

8.EE.B.5

Topic B: Slope and Graphing Linear Equations

Graph a linear equation using a table of values.

8.F.B.4

Define slope and determine slope from graphs.

8.EE.B.6

Determine slope from coordinate points. Find slope of horizontal and vertical lines.

8.EE.B.6 8.F.B.4

Graph linear equations using slope-intercept form $${y = mx + b}$$.

8.EE.B.6 8.F.A.3

Write equations into slope-intercept form in order to graph. Graph vertical and horizontal lines.

8.EE.B.6

Topic C: Writing Linear Equations

Write linear equations from graphs in the coordinate plane.

8.EE.B.6 8.F.B.4

Write linear equations using slope and a given point on the line.

8.EE.B.6 8.F.B.4

Write linear equations using two given points on the line.

8.EE.B.6 8.F.B.4

Write linear equations for parallel and perpendicular lines.

8.EE.B.6

Compare linear functions represented in different ways.

8.F.A.2 8.F.B.4

Model real-world situations with linear relationships.

8.F.B.4

Linear Relationships

Objective

Common Core Standards

Core Standards

Functions

Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Student Response

Additional Practice

Lesson Map

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