Lesson 2

Representing Proportional Relationships

Lesson Narrative

The purpose of this lesson is for students to understand that there are many successful ways to set up and scale axes in order to graph a proportional relationship. Sometimes, however, we choose specific ranges for the axes in order to see specific information (MP5). This lesson presents a few options with regard to which activities are used depending on what students need to work on.

If students need additional practice identifying features of the graphs of proportional relationships, such as the unit rate, or writing equations for a proportional relationship given the graph, use the optional activity Card Sort: Proportional Relationships.

In the following activity, students graph a proportional relationship representing water filling a tank on two differently scaled axes. Then they compare their graph to a graph of a nonproportional relationship representing water filling a different tank and answer questions about the situation. By looking at the same two relationships graphed at different scales, students see how much effect the scale of the axes has on the information we can figure out.

Next, students create representations of proportional relationships when given two to start from. For each representation, they identify key features such as the constant of proportionality and relate how they know that each representation is for the same situation.

If students demonstrate a strong understanding of graphing proportional relationships, consider using the activity Info Gap: Proportional Relationships instead of Representations of Proportional Relationships. For this info gap, the student with the problem card needs to graph a proportional relationship on an empty pair of axes that includes a specific point. In order to do so, they need to request information about the proportional relationship as well as calculate the specific point. The focus here is on the graphs students create and their decisions on how to scale the axes in an appropriate manner for the situation.

Learning Goals

Teacher Facing

• Compare graphs that represent the same proportional relationship using differently scaled axes.
• Create an equation and a graph to represent proportional relationships, including an appropriate scale and axes.
• Determine what information is needed to create graphs that represent proportional relationships. Ask questions to elicit that information.

Student Facing

Let's graph proportional relationships.

Required Preparation

This lesson includes two optional activities that each require pre-printed slips cut from copies of the blackline master. Prepare these sets of slips if you plan on using the optional activities. Provide access to straightedges.

Student Facing

• I can graph a proportional relationship from an equation.
• I can scale and label coordinate axes in order to graph a proportional relationship.
• I can tell when two graphs are of the same proportional relationship even if the scales are different.

Building On

Building Towards

Glossary Entries

• rate of change

The rate of change in a linear relationship is the amount $$y$$ changes when $$x$$ increases by 1. The rate of change in a linear relationship is also the slope of its graph.

In this graph, $$y$$ increases by 15 dollars when $$x$$ increases by 1 hour. The rate of change is 15 dollars per hour.