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# The Cell Phone Problem, Day 1

Lesson 1 of 17

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

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

*45 minutes*

#### Getting Started

*5 min*

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.

#### Resources

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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**)

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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.

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#### Summary Discussion

*10 min*

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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Hi Jacob,

This is a great idea for a lesson. I might make a few modifications for my class but definitely expect to use it. I found a pretty good slide show about wireless networks at http://www.cs.cmu.edu/afs/cs/academic/class/15441-s06/lectures/L18_Wireless.pdf" target="_blank" >http://www.cs.cmu.edu/afs/cs/academic/class/15441-s06/lectures/L18_Wireless.pdf">http://www.cs.cmu.edu/afs/cs/academic/class/15441-s06/lectures/L18_Wireless.pdf It might be beyond the reach of most our students (I work with 8th graders) but some of the graphics are good.

A couple of modifications I'm considering, mostly as extensions: I had to read the first paragraph on the constancy of the product of s and d^2 to realize you** weren't** saying that strength is proportional to d^2 instead of inversely proportional. It's clear in the problems and graphs and intuitively makes sense that you're not going to get a stronger signal from farther away, but I did think, "Wait, that's not right!" a couple of times.

Also, I think I might do a dimensional analysis of the constant of proportionality. If I'm right it turns out to be (mass)(length)^4/(time)^2.

There are lots of good ways to illustrate the inverse square law - for example watching a flashlight's beam expand as you walk away from a wall. There's also lot's of good illustrations available.

Finally, I really like the question at the end about the signal strength at the tower itself - a great example of perils of dividing by 0!

Thanks again for this great idea.

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: The Cell Phone Problem, Day 1
- LESSON 2: The Cell Phone Problem, Day 2
- LESSON 3: The Cell Phone Problem, Day 3
- LESSON 4: Discontinuity in Rational Functions
- LESSON 5: Simplifying Rational Expressions Day 1
- LESSON 6: Simplifying Rational Expressions, Day 2
- LESSON 7: Rational Expressions & Equations
- LESSON 8: Solving Rational Equations
- LESSON 9: Solving Rational Equations, Day 2
- LESSON 10: Asymptotic Behavior, Day 1 of 2
- LESSON 11: Asymptotic Behavior, Day 2 of 2
- LESSON 12: Practice with Asymptotes
- LESSON 13: Egyptian Fractions
- LESSON 14: Combined Fuel Economy, Day 1 of 2
- LESSON 15: Combined Fuel Economy, Day 2 of 2
- LESSON 16: The Tin Can Model
- LESSON 17: A Medical Model