 Lesson plan

# Lesson 8: Translating to y=mx+b

teaches Alabama State Standards 8-9.d.
teaches Alabama State Standards 8-9.c.
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
teaches Minnesota State Standards 8.2.4.3.
teaches Minnesota State Standards 8.2.4.1.
teaches Minnesota State Standards 8.2.2.2.
teaches Minnesota State Standards 8.2.2.1.
teaches New York State Standards NY-8.G.1.
teaches Pennsylvania State Standards CC.2.2.8.B.2.

# 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).

Glossary:

• 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/.