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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Unit Prep

Essential Understandings

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  • 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

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initial value

y-intercept

unit rate

linear equation

rate of change

slope

zero slope

undefined slope

proportional relationship

table of values

Related Teacher Tools:

Unit Materials, Representations and Tools

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  • $${y=mx+b}$$
  • Graph paper

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

Assessment

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

Lesson Map

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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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.

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.

  • 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

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Expressions and Equations
  • 6.EE.C.9

  • 7.EE.B.4

  • 8.EE.C.7

Geometry
  • 8.G.A.1

  • 8.G.A.2

  • 8.G.A.4

  • 8.G.A.5

Ratios and Proportional Relationships
  • 7.RP.A.2

The Number System
  • 7.NS.A.1

  • 7.NS.A.2.B

Future Standards

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Creating Equations
  • A.CED.A.2

Expressions and Equations
  • 8.EE.C.8

Reasoning with Equations and Inequalities
  • A.REI.D.10

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.