The Cell Phone Problem, Day 1

For this next section I have students work in groups of three, and they should produce professional-looking graphs, correct equations, and answers to the questions posed in problems 4 and 5. I encourage the students to provide both graphical and analytic solutions to these problems (MP1).  Since this problem is intended as an introduction to the study of rational functions, pay careful attention to the way students approach the analytic solution!  This is your chance to see what they already know and what they'll need support with throughout the unit.

With regard to problem 5, I do not expect students to be able to explain why their model is unreliable, but they should be able to see clearly that it is.  The physics of the situation is really complicated, and it's enough to simply assert that the model is reliable when the cell phone is a reasonable distance from the tower.

First, I use a document camera to display one group's graph and equation for a whole class discussion.  Hopefully, all groups will have the same graph, keeping in mind scaling may affect the appearance.  If I encounter this case, I will ask how different groups decided on the domain and range for their graph.

Once we have had the overall conversation it is important to make sure that students have drawn important mathematical conclusions and can interpret them in context:

1. Once the graph drops below a certain level, it will never rise again.
• This means that the signal strength continues to diminish the further away you go.  It approaches zero, which is reasonable.
2. The graph has a vertical asymptote at d = 0.
• The signal strength "approaches infinity" the closer you get to the source.  This is nonsense!

It isn't necessary to go into the physics of the situation, but help your students to see that the model is unreliable when the distance is too small.