Algebra II : Unit #7 - Rational Functions : Lesson #2

The Cell Phone Problem, Day 2

Objective: SWBAT model average monthly costs using rational functions and make comparisons between several different functions.
Standards: HSA-CED.A.1 HSA-CED.A.2 MP2 MP4
Subject(s): Math
60 minutes
1 Getting Started - 5 minutes

In the previous lesson, we learned about cell phone signal strength and saw that it depended on how close we were to a cell site.  Today, we'll take a look at a different aspect of cell phone usage - its cost in dollars!

Everyone wants the latest phone, but it's so much more expensive than the older models!  What if you find a good deal on a service plan?  You'd have to pay more up front, but could it possibly be cheaper in the long run?  Does six months count as "long run", or six years?

This calls for a math model - let's get started!

(By the way, as a teacher I was fascinated by a recent article from the Wall Street Journal.  It isn't part of the lesson for students, but it made me reconsider just how "real world" this problem really is: Inside the Phone-Plan Pricing Puzzle - WSJ.com)

2 Considering the First Option - 10 minutes

Pass out The Cell Phone Problem, Part 2 to the class and instruct everyone to begin by working individually.  In these 10 minutes, each student should do the following:

  • Make sense of the situation
  • Determine the total cost of Phone 1 over a 2-year period
  • Determine the average monthly cost of Phone 1 over two years
  • Describe how the average monthly cost of Phone 1 depends on the number of months

The aim is for each student to carefully consider the cost of Phone 1 alone.  This gives each student the opportunity to individually reason through the concept of average monthly cost and begin to model its relationship to the number of months.

For an alternative opening to this lesson, see the video, Alternate Opening.

The Alternate Opening
Discourse and Questioning

I chose to use the alternate opening when I taught this class recently.  I'd been having trouble keeping the class together lately; stronger students were racing through problems without noticing the deeper implications, and weaker students were getting completely bogged down somewhere in the middle.  The alternate opening gave me more control over the pacing of the lesson and allowed me to keep everyone on track.

In conversation, I found that everyone seemed to clearly understand the notion of an average total monthly cost.  They could even see that we were simply spreading out the initial device cost more and more thinly over the time span.

However, when it came time to express the average total monthly cost as an equation, many got lost.  I found a number of students simply calculating the total cost and graphing a straight line.  Others were mixing up years and months in their equations and coming up with some strange results.  Often the other students in their groups couldn't tell what they were doing wrong, so I would have to intervene to help them.

By the end, everyone was on track, and I had managed to keep the class together.  No one was bored, but no one was hopelessly lost.  That's some kind of success!

3 Comparing Plans over Two Years - 10 minutes

Working individually again, students should now determine the average monthly costs for Phones 2 and 3 over the same two-year period.  All other things being equal (that phrase is worth discussion!), this should allow them to decide which phone is the best deal for a two-year plan.  (This is problem 4 on the handout, if you're using it.)

Since the calculations for Phones 2 and 3 are structurally the same as the ones just completed for Phone 1, this section provides a quick formative assessment for the teacher.  In most cases, I think you'll find that almost everyone can easily reason through the problem, but creating an equation is much more difficult.  This means that almost all students can see that the total cost is the device cost plus 24 times the monthly cost, and they know that if they divide this by 24 they'll get the average.  But for some reason, you'll find that many students make serious mistakes if they try to first write an equation.  In this case, encourage them to stop, think it through first, and then try to capture in an equation what they're actually doing.

As individual students finish their cost comparison, encourage them to quietly check with their classmates.  Are they in agreement on the cheapest phone?  Did they come up with the same average monthly rates?  If not, it's crucial to figure out what went wrong!

Once everyone's on board, it's time to move to the next section of the lesson.

4 Cost Comparisons over Time - 20 minutes

Working in small groups, students should now create explicit mathematical models to compare the average monthly cost of the three phones both graphically and analytically.  Up to this point we have assumed that the phone would be used for exactly two years, but now we're going to allow the number of months to vary.  (This is problem 5 & 6 on the handout, if you're using it.)  Personally, I like to motivate the questions with dialogue:

"Let's be realistic: I don't know exactly how long I'm going to keep the phone.  Which phone should I buy?"  

  • Well, it depends on how long you're going to use it!  


"Give me some specifics.  How many months would I have to use it to make Phone 1 the best deal?  What about Phones 2 and 3?  I'm a visual person; can you show me a graph of the costs?"

  • Give us some time to think about it, and we'll get back to you.


"Great!  You get to work, and we'll talk about your results at the beginning of tomorrow's lesson!"