Lesson 9

Two Graphs for Each Relationship

In this lesson students focus on the relationship between the graph and the equation of a proportional relationship. They start with an activity designed to help them see all the different ways in which the graph and the equation are connected, for example the relation between a point \((a,b)\) on the graph and the constant of proportionality \(k=\frac{b}{a}\) in the equation and the fact that the point \((1,k)\) on the graph tells you the constant of proportionality. This prepares them for the next two activities where they see two ways to graph a proportional relationship, depending on which quantity goes on which axis. This connects with previous work with tables and equations, and gives students an opportunity to remember the fact that the constants of proportionality in the two ways are reciprocals.

As students connect the structure of an equation with features of the graph, they engage in MP7.

Coordinate (orally and in writing) tables, graphs, and equations that represent the same proportional relationship.
Interpret two different graphs that represent the same proportional relationship, but have reversed which quantity is represented on each axis.
Write an equation to represent a proportional relationship given only one pair of values or one point on the graph.

Let’s use tables, equations, and graphs to answer questions about proportional relationships.