In the previous lesson, students worked in small groups to compare the average monthly costs of three different phones for a variable number of months. Today, they should be ready to discuss the results of that comparison.
First, I like to ask informally which phone students think I should buy. Most students have eliminated one of the phones, then the choice between the other two depends on the number of months you expect to own it. The tipping point is between two and three years, so I hope this question generates some debate!
Using the document camera, I'll have one group share their equations, graphs, and conclusions with the class. I've also included some representative samples of student work with my comments.
Since the purpose of this problem is to introduce rational functions, I will pay special attention to the following:
To really address the third point, it's important to ask a student how to find the intersection of two of these curves analytically or algebraically. This should lead to some discussion of numerators and denominators, factoring, simplifying, cross-multiplying, and the rest. I'll be using the discussion to carefully probe just how much the class already knows about solving rational equations.
Once there is agreement on which phone is the best deal and at what times, I like to push my students to think a little more deeply about the model they've created. The two questions I have in mind can be found on the Going Deeper document.
As an introduction to rational functions, these two questions are crucial. The first question deals with the asymptotic end behavior of the function, while the second deals with the asymptotic behavior of the function around a discontinuity. The aim here is not to define these concepts formally, but to describe them in the context of this problem. A possible extension here is to consider the function on the negative numbers and comment on its symmetry. See the video for some thoughts on this section.
Now it's time to take a step back from the situation we've been modeling and consider rational functions more generally. We'll do this primarily by comparing the examples of rational functions we've encountered so far to the polynomial functions with which we're so familiar.
I'll ask the class, "How would you say that a rational function is different from a polynomial?"
I expect the responses to fall into two categories: differences in the equation and differences in the graph. In the first case, students will probably recognize that polynomials never involve division by a variable, but these rational functions do. With regard to the graph, students should recognize in some way that polynomials are always continuous and the end behavior always includes a tendency toward infinity. Neither of these seems true of rational functions.
After this brief overview, I like to end class by giving a quick preview of the sorts of things we'll be learning about rational functions. They are: