||Changing Cost per
The cost per minute for phone use changes after the first sixty minutes of call..
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
To change the cost per minute
for the first sixty minutes of cellular-phone service, drag the vertical blue
slider on the left. To change the cost per minute for each additional minute
after sixty minutes of phone service, drag the vertical pink slider on the left.
To change the total minutes of phone service, drag the horizontal blue slider
at the lower right.
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?
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
- How does the dynamic
nature of the graphs in this example help students explore the relationships
between symbolic expressions and graphs of lines?
- 6.2 Learning about Rate of Change in Linear Functions Using Interactive Graphs
- 6.2.1 Constant Cost per Minute