# Lesson 8

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.

### Glossary Entries

• linear relationship

A linear relationship between two quantities means they are related like this: When one quantity changes by a certain amount, the other quantity always changes by a set amount. In a linear relationship, one quantity has a constant rate of change with respect to the other.

The relationship is called linear because its graph is a line.

The graph shows a relationship between number of days and number of pages read.

When the number of days increases by 2, the number of pages read always increases by 60. The rate of change is constant, 30 pages per day, so the relationship is linear.

• vertical intercept

The vertical intercept is the point where the graph of a line crosses the vertical axis.

The vertical intercept of this line is $$(0,\text-6)$$ or just -6.