# Lesson 9

Slopes Don't Have to be Positive

### Lesson Narrative

In previous lessons, students have arrived at an equation for a line in three ways:

• By reasoning about similarity of slope triangles on a line
• By reasoning about starting values and rates of change in a linear relationship
• By reasoning about vertical translations of lines through the origin

Students encountered linear relationships with positive rates of change and either positive or negative vertical intercepts. The graphs of these relationships all had an uphill appearance.

In this lesson, students get their first glimpse of lines that visually slope downhill as well as a “flat” line or line with 0 slope. After reflecting on commonalities and differences between lines that slope in different directions, students explore a situation in which one quantity decreases at a constant rate in relation to a second quantity. They interpret a graph of the situation and reason that it makes sense for the slope to be negative in terms of the context. The scenario is then extended to consider a quantity that does not change with respect to another, and students realize that a flat graph has a slope of zero.

### Learning Goals

Teacher Facing

• Create a graph of a line representing a linear relationship with a non-positive rate of change.
• Interpret the slope of a non-increasing line in context.

### Student Facing

Let’s find out what a negative slope means.

### Student Facing

• I can give an example of a situation that would have a negative slope when graphed.
• I can look at a graph and tell if the slope is positive or negative and explain how I know.