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?
