# A unit on rational functions?

I couldn’t find a book with a unit on rational functions. So I made one up.

This year we decided to revamp precalculus. We have three groups mixed in precalc. Many seniors take the course, and we want them ready for calculus, if they go there, next year. But juniors also take precalc after algebra II/Trig, the ones who will take calc (with me?!) next year. And a small group of juniors, enrolled in the second term of trig, will take precalc concurrently (just one term) and jump to calculus next year (also with me?).

So we rearranged and expanded to make the second term of precalc be “what we want kids to get right before calculus.” It forced us to consider which particular skills were necessary, which needed reemphasis, and which had been arbitrarily thrown in.

Our precalc book, Advanced Mathematics by Richard Brown, is fine for most of this. But we made a note about finding material for Rational Functions. And I have been lead on this course, so that meant me. And I forgot to look. Until we got close. And I could not find a unit on rational functions in any of the Algebra, Algebra II, College Algebra, or Precalculus books on my shelf or at school. Maybe I examined 30 – 40 books? Those that had something generally had, as part of a chapter on “functions,” one section on rational functions. There were a few precalc books with two sections, the second focusing on asymptotes. And one, Hostetler’s Algebra II, had three sections – one on the graphs, one on the algebra, and one on partial fractions.

In addition, Functions and Graphs (it’s $9, every math teacher should have a copy) Gelfand/Glagoleva/Schnol has two extended sections about producing graphs of rational functions. This is from a Soviet correspondence course, and does not look like our text books at all. The answers are in the margins, and the questions are “how do we graph this?” and “how can we analyze the algebra to graph this?”

Why rational functions? Why a full unit? Asymptotes. End behavior. Algebraic Manipulation. Algebraic Skill practice. Transformation of Functions.

I am curious what you think about the idea (2+ weeks of this). Later on, I’ll post some of the worksheets I threw together. (they need considerable refinement, if we are to make this a permanent unit)

Funnily enough, with the very high-achieving further maths group I mentioned here before I spent about two weeks doing something not far from this – it wasn’t restricted to rational functions, but 90% or more of what we did probably was rational function work. The pretence was that we were doing this to solve inequalities, but I made it very clear that what I was mostly interested in sketching and asymptotics.

I was just going to offer to translate some Russian stuff for you when I read further – Gelfand, of course!

I consider the topic to be valuable and very beautiful. It unites many conceptual ideas in neat heady packages.

However, you should be warned. Russian curricula spend a lot of time emphasizing “number as measure” – that is, continuum models of number, including number lines and linear measurements. This results in people being somewhat more comfortable with fractions. US curricula emphasize “number as quantity” models. As a result, many students are uncomfortable with fractions, at a rather fundamental level. They don’t have the essential fraction-supporting metaphors, and have to work much harder with anything that involves a fraction-related concept. You may want to go over the essentials, such as the behavior of 1/x, in more detail than Gelfand assumes.

As an aside, I just spent a lot of time, on two parallel projects, with Common Core standards for K-3 AND with a precalc college course I am helping to develop. The painful steps the precalc course has to take around rational expressions resonated with the rarity of “number as measure” experiences in the elementary standards. The first time the word “ratio” appears is Grade 6 standards, I believe.

The biggest difference between my students and regular students (as far as math) is that they arrive at 9th grade being able to handle fractions.

That being said, this topic stretches them. I would like to plan it better, and work in more interconnections, next time. But I am not teaching this again next year…

Good for your students! I am really looking forward to seeing your materials. I assume you will publish them here on your blog.

I was going to suggest Gelfand, Functions and Graphs as well. When I had to take a refresher in Precalculus for qualifying exams, I had a Russian professor who literally changed my relationship to functions, analysis, and their graphs. I have since used his approach with my Algebra 2 students to great success.

For rational functions, I used the material from Stewart, Redlin, and Watson Precalculus (or College Algebra, same material) and found it very effective with my students. Very similar to the Russian approach, but broken down into smaller bites, including the idea of moving from 1/x onward towards more complicated rational functions.

The way everybody else in my department teaches rational expressions did not give them the deep conceptual understanding my kids developed with regard to asymptotes. My colleagues were blown away by my skills quiz on rational functions and the careful, beautiful analysis and graphs the kids did. As a side benefit, the work on vertical asymptotes gave students a mind-boggling appreciation for the relationship between the function and infinity, which led to some wonderful discussions and writing. I showed them some M.C. Escher lithographs (the one hand drawing a hand that is drawing the first hand) and it connected their relationship with mathematics to their relationships to their humanities and arts classes. Very rich and rewarding.

– Elizabeth (aka @cheesemonkeysf on Twitter)

There is a lot here. My first try has been far from perfect. I regret not teaching the same course next year… But when I get back to it, I will pick up the rational functions and refine them.

The best piece, I think, is slowing it down and looking at them from several perspectives. I also think the 2-3 days of fraction skills were well worth the time.

So we have a graph, and we look at end behavior and asymptotes, and describe, at least loosely, the function.

We have an equation, a ratio of polynomials, and we can predict end behavior and asymptotes. Or we can decompose it by long division and or partial fractions, and look at the pieces.

We can look at and describe what each constant does. And we can extend by adding mx…

There’s analysis here that they are doing, beyond what they knew how to do when the unit started. And it has to do with more than just curvy lines.