Table of Contents Appendix E-examples Resources

Learning about Rate of Change in Linear Functions Using Interactive Graphs: Changing 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 the first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In this second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls. Understanding 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.

Task

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

Quik-Talk advertises monthly cellular-phone service for $0.50 per minute for the first 60 minutes of calls, but only $0.10 per minute for each additional minute thereafter.

During your interaction with the graphs, notice how changing the cost per minute shown in the first graph affects the total cost shown in the second graph. What is represented in each graph? What is the relationship between the graphs?

[How to Use the Interactive Figure]

[Stand-alone applet]

Discussion

Students may have to think hard to understand what is represented by the two graphs. Note that the first graph depicts number of minutes (on the x-axis) versus cost per minute (on the y-axis) and that two different costs per minute are represented. Then in the second graph, the number of minutes remains on the x-axis but the y-axis represents the total cost of calls and the graph "bends" in the middle. To help students think about the differences in the two graphs, the teacher may ask questions such as, 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? How can we tell from each of the graphs that the pricing scheme changes after a certain number of minutes?

Students may also have to think hard to understand how the first graph is related to the second graph. The teacher may prompt students with questions such as, Why are the segments in the first graph horizontal? Why does changing the height of the horizontal segments in the first graph affect the slopes of the line segments in the second graph? How can you determine the slopes of the line segments in the second graph (total cost function)? How do these numbers appear in the first graph (cost-per-minute function)? Thinking about these ideas will contribute to students' understanding of slope and rate of change. Other questions that a teacher can ask about the second graph include Why is the y-intercept zero? Can you think of a pricing scheme for which the y-intercept would not be zero? Why is the left part of the total-cost function steeper than the right part? Will it always appear that way? Can you make the right part steeper than the left part?

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 does the activity described in this example help students with their understanding of slope and y-intercept? How does the dynamic nature of the graphs in this example help students explore the relationships between symbolic expressions and graphs of lines?

Constant Cost per Minute	Changing Cost per Minute

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