Linear Relationships

Students compare proportional relationships, define and identify slope from various representations, graph linear equations in the coordinate plane, and write equations for linear relationships.

Unit Summary

In Unit 5, eighth-grade students zoom into linear functions, extending several ideas they learned in the previous unit on Functions. They begin the unit by investigating and comparing proportional relationships, bridging concepts from seventh grade, such as constant of proportionality and unit rate, to new ideas in eighth grade, such as slope. Students formally define slope and learn how to identify slope in various representations including graphs, tables, equations, and coordinate points. Investigating slope is an opportunity for students to engage in MP.8, as they use the repeated reasoning of vertical change over horizontal change to strengthen their understanding of what slope is and what it looks like in different functions. Lastly, students will spend time writing equations for linear relationships, and they’ll use equations as tools to model real-world situations and interpret features in context (MP.4).

Just as in Unit 4, students will draw on previous understandings from sixth and seventh grades related to rates and proportional relationships, and the equations and graphs that represent these relationships.

The concepts and skills students learn in this unit are foundational to the next unit on systems of linear equations. In Unit 6, students will investigate what happens when two linear equations are considered simultaneously. In high school, students will continue to build on their understanding of linear relationships and extend this understanding to graphing solutions to linear inequalities as half-planes in the coordinate plane.

Pacing: 19 instructional days (15 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.

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Assessment

This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

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, 6-8, Expressions and Equations for the relevant standards for this unit.
Read the Progressions for the Common Core State Standards in Mathematics, Grade 8, High School, Functions for the relevant standards for this unit.

Essential Understandings

A proportional relationship can be represented by the equation $${ y=mx}$$ and by a straight line in the coordinate plane passing through the origin. A non-proportional linear relationship can be represented by the equation $${y=mx+b}$$ and by a straight line in the coordinate plane that crosses the $$y$$-axis at point $${(0, b)}$$.
The slope of a non-vertical line is the measure of vertical change over the measure of horizontal change between any two points on the line. In a proportional relationship, the slope of the graph is the same as the unit rate.
Linear relationships can be represented and compared by writing equations, drawing graphs, and identifying the slope and $$y$$-intercept. Linear functions can be used to model and make sense of real-world situations.

Vocabulary

initial value

y-intercept

unit rate

linear equation

rate of change

slope

zero slope

undefined slope

proportional relationship

table of values

Related Teacher Tools:

8th Grade Vocabulary Glossary

Unit Materials, Representations and Tools

$${y=mx+b}$$
Graph paper

Lesson Map

Topic A: Comparing Proportional Relationships

1

8.EE.B.5

Review representations of proportional relationships.

2

8.EE.B.5

Graph proportional relationships and interpret slope as the unit rate.

3

8.EE.B.5

Compare proportional relationships represented as graphs.

4

8.EE.B.5

Compare proportional relationships represented in different ways.

Topic B: Slope and Graphing Linear Equations

5

8.F.B.4

Graph a linear equation using a table of values.

6

8.EE.B.6

Define slope and determine slope from graphs.

7

8.EE.B.6

8.F.B.4

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

8

8.EE.B.6

8.F.A.3

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

9

8.EE.B.6

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

Topic C: Writing Linear Equations

10

8.EE.B.6

8.F.B.4

Write linear equations from graphs in the coordinate plane.

11

8.EE.B.6

8.F.B.4

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

12

8.EE.B.6

8.F.B.4

Write linear equations using two given points on the line.

13

8.EE.B.6

Write linear equations for parallel and perpendicular lines.

14

8.F.A.2

8.F.B.4

Compare linear functions represented in different ways.

15

8.F.B.4

Model real-world situations with linear relationships.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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.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.

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.

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.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.

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.
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

Expressions and Equations

6.EE.C.9

Expressions and Equations

6.EE.C.9 — Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
7.EE.B.4

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.
8.EE.C.7

Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.

Geometry

8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.2

Geometry

8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.4

Geometry

8.G.A.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.A.5

Geometry

8.G.A.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Ratios and Proportional Relationships

7.RP.A.2

Ratios and Proportional Relationships

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

The Number System

7.NS.A.1

The Number System

7.NS.A.1 — Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.2.B

The Number System

7.NS.A.2.B — Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.

Future Standards

Creating Equations

A.CED.A.2

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Expressions and Equations

8.EE.C.8

Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

Reasoning with Equations and Inequalities

A.REI.D.10

Reasoning with Equations and Inequalities

A.REI.D.10 — Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

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.

Linear Relationships

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Vocabulary

Related Teacher Tools:

Unit Materials, Representations and Tools

Lesson Map

Topic A: Comparing Proportional Relationships

1

2

3

4

Topic B: Slope and Graphing Linear Equations

5

6

7

8

9

Topic C: Writing Linear Equations

10

11

12

13

14

15

Common Core Standards

Core Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Functions

Functions

Functions

Functions

Foundational Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Geometry

Geometry

Geometry

Geometry

Geometry

Ratios and Proportional Relationships

Ratios and Proportional Relationships

The Number System

The Number System

The Number System

Future Standards

Creating Equations

Creating Equations

Expressions and Equations

Expressions and Equations

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Standards for Mathematical Practice

Unit 4

Functions

Unit 6

Systems of Linear Equations