# Lesson 7

Translating to $y=mx+b$

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

• 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.

### Student Facing

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

### 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).

### 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.