Lesson plan

Lesson 8: Translating to y=mx+b

teaches Common Core State Standards 8.EE.B http://www.corestandards.org/the-standards
teaches Common Core State Standards MP4 http://corestandards.org/Math/Practice/MP4
teaches Common Core State Standards MP8 http://corestandards.org/Math/Practice/MP8
teaches Common Core State Standards 8.G.A.1 http://corestandards.org/Math/Content/8/G/A/1

Lesson 8: Translating to y=mx+b

This lesson develops a third way to understand an equation for a line in the coordinate plane. In previous lessons, students wrote an equation of a line by generalizing from repeated calculations using their understanding of similar triangles and slope (MP8). They have also written an equation of a linear relationship by reasoning about initial values and rates of change and have graphed the equation as a line in the plane. This lesson introduces the idea that any line in the plane can be considered a vertical translation of a line through the origin.

In the previous lesson, the terms in the expression are more likely to be arranged \(b+mx\) because the situation involves a starting amount and then adding on a multiple. In this lesson, \(mx+b\) is more likely because the situation involves starting with a relationship that includes (0,0) and shifting up or down. Students continue to only consider lines with positive slopes, but in this lesson, the notion of a negative \(y\)-intercept (not in a context) is introduced.

In addition, students match lines presented in many different forms: equation, graph, description, table. This combines much of what they have learned about lines in this unit, including slope and vertical intercept.

Lesson overview

  • 8.1 Warm-up: Lines that Are Translations (5 minutes)
  • 8.2 Activity: Increased Savings (15 minutes)
    • There is a digital applet in this activity. 
  • 8.3 Activity: Translating a Line (15 minutes)
    • Includes "Are you Ready for More?" extension problem.
    • There is a digital applet in this activity.
  • Lesson Synthesis
  • 8.4 Cool-down: Similarities and Differences in Two Lines (5 minutes)

Learning goals:

  • Coordinate (orally) features of the equation \(y=b+mx\) to the graph, including lines with a negative \(y\)-intercept.
  • Create and compare (orally and in writing) graphs that represent linear relationships with the same rate of change but different initial values.

Learning goals (student facing):

  • Let’s see what happens to the equations of translated lines.

Learning targets (student facing):

  • I can explain where to find the slope and vertical intercept in both an equation and its graph.
  • I can write equations of lines using \(y=mx+b\).

Required materials:

  • Geometry toolkits
  • Pre-printed cards, cut from copies of the blackline master

Required preparation:

  • Print and cut up slips from the Translating a Line blackline master.
  • Prepare 1 set of cards for every 2 students (this is not needed if doing the digital version).


  • Access the complete Grade 8 glossary.






IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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