Comparing Cell Phone Plans

Three cell phone plans are compared. It turns out that each is the best deal for a different person, depending on how much they talk on the phone each month. It is fairly straightforward to answer the question for specific examples, but that is just the opening. You should gear the exploration towards the more general question:

What range of minutes would make each plan the least expensive option?

A full answer requires the use of graphs and the solving of linear equations.

All the print components in one zipped file.

Students should have been exposed to the concept of variable, and they should be able to create and interpret expressions involving simple arithmetic operations.

Familiarity with one or more of the following technologies is helpful, but these skills can also be learned during the lesson: electronic grapher; spreadsheet; computer algebra system.

Two short periods, or one "block" period. Perhaps longer if you want to use this lesson to introduce basic spreadsheet techniques or some other technology that is new to the students, or if you want them to make graphs by hand.

If you show the introductory slides (PDF | on line), stop at the question "What information do you need?" Solicit ideas from the students about what information is relevant.

Show the final slide, and/or hand out the student worksheet to launch the exploration. Make sure the students understand that while the initial question is about specific cases, they are to find a general answer to the question of what range of minutes makes each plan the least expensive option.

Students should work in small groups (four tends to be optimal) and encouraged to attack the above questions with little initial guidance, so as to promote student creativity. After a while, you can provide these hints, as appropriate to each group, or to the whole class if more than one group needs the help.

Hint 1: See how much each of the three customers would be charged by each plan. You need to convert the hours to minutes to simplify the calculations.

Hint 2: Which plan is best for very few minutes? Which is best for very many minutes?

Hint 3: If your students have access to computers, and you want them to use a spreadsheet in this investigation, you might find an opportunity to interrupt their investigation, and introduce basic spreadsheet techniques. Or you could use graphing technology or a calculator/computer algebra system. (See Tech Support.)

Hint 4: There is a “break-even point”, a number of minutes where two plans cost the same thing. Find this point for each pair of plans (A and B, B and C, A and C.)

Note that the break-even points can of course be calculated using algebra, but we recommend first allowing students to use trial and error and numerical calculations to find them. It is also possible to use a graphical approach with the software to identify those points. This can then be followed up by the algebraic approach, which will have that much more meaning for the students, and which will be shown to be the most efficient way to go. (Of course, if students spontaneously think of using algebra, do not discourage them!)

To wrap up the exploration, it is useful to show a graph of the three plans, as it makes it very clear what is going on, including the key concept of the break-even points. You may project this graph, or hand it out, if the students haven't made their own graphs using technology. It is of course possible for them to make the graphs by hand on graph paper, but that would be time-consuming and shift the focus of the activity.

Here are some prompts for the discussion of the graph:

What variable is shown on each axis?

What color is associated with each plan?

What are the equations for the functions represented here?

What is the significance of the lines' intersections?

It is most likely that students will have used minutes, rather than hours, to solve the problem. They should be encouraged to also give approximate final answer in hours, which is easier to get a handle on for most people. Part of doing math is being able to communicate your results effectively!

A worthwhile closing discussion can be had with the following prompts:

Why would the Tick Talk Telephone company offer these three plans?

Which is the best deal for the company?

Would they want their customers to know enough math to figure out the best plan for them?

What sort of "real world" problems can be solved using the approaches we used for this problem?

This is of course a real world problem even though the Tick Talk Telephone company is fictional. The approaches used in solving it apply to such diverse questions as comparing car rental plans, or determining if a discount / membership card is worth purchasing.