8th Grade Math | Linear Relationships

Unit Summary

In Unit 5, 8th 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 7th 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 the value of 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 6th grade and 7th grade 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)

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Assessment

The following assessments accompany Unit 5.

Pre-Unit

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.

Mid-Unit

Have students complete the Mid-Unit Assessment after lesson 9.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Unit Launch

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

Internalization of Standards via the Post-Unit Assessment

Take the Post-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 the Unit Summary.
Notice the progression of concepts through the unit using the Lesson Map.
Do all Target Tasks. Annotate the Target Tasks for:
- Essential Understandings
- Connection to Post-Unit Assessment questions
Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on 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

The central mathematical concepts that students will come to understand in this unit

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

Terms and notation that students learn or use in the unit

initial value

linear equation

proportional relationship

rate of change

slope

table of values

unit rate

undefined slope

y-intercept

zero slope

To see all the vocabulary for Unit 5, view our 8th Grade Vocabulary Glossary.

Materials

The materials, representations, and tools teachers and students will need for this unit

Graph Paper (2-3 sheets per student)
Ruler (1 per student)
Patty paper (transparency paper) (1 sheet per student)
Optional: Matching game (1 per pair of students)

To see all the materials needed for this course, view our 8th Grade Course Material Overview.

Lesson Map

Topic A: Comparing Proportional Relationships

1

Review representations of proportional relationships.

8.EE.B.5

2

Graph proportional relationships and interpret slope as the unit rate.

8.EE.B.5

3

Compare proportional relationships represented as graphs.

8.EE.B.5

4

Compare proportional relationships represented in different ways.

8.EE.B.5

Topic B: Slope and Graphing Linear Equations

5

Graph a linear equation using a table of values.

8.F.B.4

6

Define slope and determine slope from graphs.

8.EE.B.6

7

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

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

8

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

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

9

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

8.EE.B.6

Topic C: Writing Linear Equations

10

Write linear equations from graphs in the coordinate plane.

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

11

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

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

12

Write linear equations using two given points on the line.

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

13

Write linear equations for parallel and perpendicular lines.

8.EE.B.6

14

Compare linear functions represented in different ways.

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

15

Model real-world situations with linear relationships.

8.F.B.4

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

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

Standards covered in previous units or grades that are important background for the current unit

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

Standards in future grades or units that connect to the content in this unit

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

Pre-Unit

Mid-Unit

Post-Unit

Unit Prep

Intellectual Prep

Internalization of Standards via the Post-Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Vocabulary

Materials

Lesson Map

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

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