How to Teach Content You Don't Like (Part 2)

This post is the second post in a series. Check out the first post here.

I’ve been teaching Precalculus since I first started teaching. It was a particularly loathed class among students at my school, but I enjoyed the challenge. Precalculus gave us an opportunity to dive deeper into the various function types and also to hit on other topics such as sequences and series, polar graphs, and partial fractions. There were interesting things in the curriculum, and I would work hard to create opportunities for my students to see the beauty in what they were learning. But I’ll be honest. The level of abstraction we were dealing with could get to be a little much sometimes, and the “Do we really have to know this stuff?” look of disdain etched on my students’ faces was sometimes tiring to curb with my enthusiasm. What I didn’t confess to my students is that sometimes I agreed with them! And nowhere did that feel more true than in our annual slog through rational functions. But boy was I wrong about those!

When AP Precalculus launched, I was back in lesson writing mode and spent many hours studying the CED, then imagining creative ways to help students fall in love with (or at least appreciate (or at least not hate)) rational functions. Even now, after the AP Precalculus lessons are written and being used in classrooms, I find myself still getting curious about rational functions, still wondering if there’s a whole other angle to them that I hadn’t yet considered, still building my thesis around why rational functions, like everything else in the math curriculum, can really be boiled down to one of the four or five big ideas of mathematics.

So here’s the big epiphany I had about rational functions:

Rational functions are connected to ratios. Before you scoff and say “duh, it’s right in the name”, hear me out. Yes, I knew that rational functions were quotients of two functions. But I did not immediately consider that everything we had taught students about rates, ratios, and proportional reasoning (topics I absolutely loved, see Exhibit A and Exhibit B) could be applied here. When I realized this about rational functions, it was a game changer. Rational functions tell you, in essence, how many times greater the numerator polynomial is than the denominator polynomial at any x-value. 

An Intuitive Introduction

To introduce this idea, consider the following prompt, accessible to students from late elementary school all the way through high school:

Which is the most impressive?

• Jeanna read 42 books last year and read 63 books this year.

• Paul read 10 books last year and read 18 books this year.

• Samantha read no books last year and read 1 book this year.

There are many ways to analyze this of course. Jeanna definitely read the most books, which is impressive. She increased her reading by 21 books which is more than the other two.

But Jeanna only increased her reading by 50%, whereas Paul increased his reading by 80%, so relatively, Paul improved more, and growth is impressive. And on that train of thought, Samantha increased her reading by…an infinite percent!! Just by reading one book! So is Samantha the most impressive after all? 

The underlying idea here is that:

Going from nothing to anything blows everything else out of the water. 

In other words, anything is infinitely times larger than nothing. So why does this matter?

Short-Run Behavior of Rational Functions

When we’re studying rational functions, we’re asking about the ratio of outputs between the polynomial in the numerator and the polynomial in the denominator.  What if instead of graphing the rational function right away, we actually graphed both polynomials separately and studied the relationship between their outputs at various x-values? As an example, let’s consider the two quadratics shown below where the blue curve is given by g(x)=⅙(x-6)(x+2) and the red curve is given by f(x)=(x+4)(x+2). 

• At x=-6, f(x)=g(x) so the ratio is 1 and thus h(-6)=1.

• At x=-4, f(x) is 0 and g(x) is 10/3. How much of 10/3 is 0? Well, 0! So h(-4)=0.

• At x=0, the ratio of f(x) to g(x) is 8/-2 or -4, so h(0)=-4.

• At x=6, we get the ratio of 80/0 which means f(x) is infinitely times bigger than g(x) at x=6, blowing all other comparisons out of the water. This gives students an accessible and logical explanation for why the graph of h, the function that represents the ratio of f(x) to g(x),  has a vertical asymptote at x=6.The ratio goes to infinity! 

• But what about x=-2? How many times bigger is 0 than 0? What’s the ratio of 0/0? Is it 1 since both values are the same? Is it infinity because anything is infinitely times larger than nothing? Or is it 0 since 0 is 0% of anything. The answer is unclear. We literally can’t determine the answer. For our AP Calc friends out there, this is why we call 0/0 an indeterminate form. Since we can’t determine the ratio of f(x) to g(x) at x=-2, the graph of h will have a hole at x=-2, denoting that the value is undefined.

Check out "The Hole Truth" to see our EFFL lesson on these short-run function features.

Long-Run Behavior of Rational Functions

So that’s short run behavior, but what about end behavior? How do we analyze which function “wins out” in the long run?

If f grows a lot quicker than g, then that ratio is going to infinity. The function g can not keep up with f’s growth rate and the ratio of outputs just keeps getting bigger and bigger, increasing without bound.

If f grows a lot more slowly than g, then that ratio is going to 0. The function f can not keep up with g’s growth rate and f’s outputs will become a smaller and smaller fraction of g’s, tending towards 0. 

But what if f and g are pretty equally matched? If their outputs grow at the same rate, then the ratio between them stays roughly constant. Maybe f’s outputs are always roughly 3 times bigger than g’s outputs, or maybe f’s outputs are roughly half of g’s outputs. Either way, if they’re progressing at the same rate, that ratio will stay consistent whether we’re comparing outputs at x=1000 or x=10000000. 

What determines that growth rate? The degree of the polynomial. Or, in AP Calculus, by the polynomial’s derivative which is most influenced by its degree.

End behavior is about zooming out on our graph, thinking about how our two polynomials are faring in the competition at x-values far, far away from zero.

What happens when you calculate those ratios? You get values fairly close to 6, that’s what!

We can zoom out even further to consider even larger values of x. The results are consistent though. The ratio of f(x) to g(x) is approximately 6 to 1.

Rational functions are about the relationship between the numerator and denominator functions. Suppose you knew that f(14000)=196,084,008. Could you make a pretty good guess about what g(14000) is? If I were a betting woman, I would say approximately 32,680,668, or about ⅙ of f(14000). (The actual answer is 32,657,331.3333, so for numbers of that magnitude, our rough estimate was actually pretty close).

Principles of Lesson Design

This approach to teaching rational functions is an example of “Make it new” one of the lesson design principles we use when writing EFFL lessons. We’re offering an alternative way to conceptualize rational functions, by first studying the numerator and denominator polynomials separately, before graphing their ratio. We’re also applying “Make it connect” by helping students see rational functions as an extension of what they already know about ratios and proportions.

What else might we unpack about rational functions with this new lens? Check out our AP Precalculus lesson “Let’s Be Rational!” to see how equivalent forms of rational functions are not only useful, but also connected to what students already know about dividends, divisors, and quotients.