### Learning about Rate of Change in Linear Functions Using Interactive Graphs: Constant Cost per Minute

 Constant Cost Per Minute Changing Cost Per Minute

In this two-part example, users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). In this first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls. Beginning to understand the relationship between change and accumulation is a precursor to understanding calculus. This example illustrates the use of dynamic graphs to learn about change and linear relationships, as described in the Algebra Standard.

The interactive figures below depict two graphical representations derived from the following situation:

 ChitChat, a cellular-phone-service provider, has no monthly fee for cellular-phone service but does charge a \$0.45 per minute usage fee.

Your task is to analyze and interpret the two graphs, and then to determine how the two graphs are related. First, drag the slider on the second graph (Total Cost). Note what happens both on the graphs and in the Total Cost box below. Now do the same thing for the slider on the first graph (Cost per Minute). As you use the graphs, notice how changing the cost per minute shown in the first graph affects the total cost shown in the second graph.

### Discussion

Students should be asked to examine the relationships depicted in the two graphs, focusing on the similarities and differences between the two graphs. They should consider the shape of each graph—specifically slope and linearity—and the quantities whose relationship is graphed in each case. Note that the first graph depicts the number of minutes (on the x-axis) versus the cost per minute (on the y-axis). Students should be asked to explain why the line in the first graph is horizontal. Then in the second graph, the number of minutes remains on the x-axis, but the y-axis now represents the total cost of calls. Focusing students' attention on the coordinates of the points on the graphs will assist them in learning about the differences in the two graphs. This can be done by asking questions like, How should we interpret the coordinates of any point on the second graph? How is that different from the interpretation of any point on the first graph?

Students may also have to consider and reflect on how the first graph relates to the second graph. The teacher can prompt students with questions such as, Why does changing the y-intercept of the first graph affect the slope of the line in the second graph? What is the slope of the line in the second graph (total-cost function) and what does it represent in this context? How does this number appear in the first graph (cost-per-minute function)? Thinking about and discussing these ideas can contribute to students' understanding of slope and rate of change.

 Take Time to Reflect The Algebra Standard states that "in grades 6–8 all students should . . . explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope." How would this activity help students understand slope and y-intercept? How would the dynamic nature of the graphs in this activity help students explore the relationships between symbolic expressions and graphs of lines?

 Constant Cost Per Minute Changing Cost Per Minute

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