SWBAT use a rational function to model two quantities that are inversely proportional.

Real world modeling of rational functions. Cell phone signal strength, can you hear me now?

5 minutes

I'll begin with a brief conversation to introduce the notion that cell phones use a radio signal with a signal strength that isn't always the same and when the signal gets too weak the call is dropped. We then will have a class discussion where I'll ask students to suggest situations/objects that contribute to weak signals. With the right prompting distance from cell site should be emphasized as the focus of today's lesson. See Getting Started for more details.

10 minutes

I'll hand out *The Cell Phone Problem, Part 1 *and ask students to work individually at first. Their aim should be to get into problem 3 by the time these 10 minutes are up.

The first question is a simple one, but it's important to make sure that students have a general, intuitive notion of what an inverse relationship is. The second question gets into the details of that relationship and allows students to take the first steps toward creating an equation to model it. Finally, the third problem asks explicitly for students to construct a mathematical model and to represent it with both an equation and a graph.

In my experience, the most challenging aspect of this problem is in reading the verbal description of the relationship between signal strength and distance. Too many students pay too little attention to the carefully chosen words, and this presents an opportunity to stress the importance of precision in mathematics. (**MP 6**)

20 minutes

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.

10 minutes

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:

- 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.
- 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.

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