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)