Lesson 8

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. 

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).