Turns out an earlier edition of this text had a whole chapter on Conic Sections. One section each for Parabolas, Ellipses, and Hyperbolas and a final section on Rotations. With the “Reflective Property of a Parabola” theorem—a “Problem Solving” problem of our text… one I’ve been looking at a little bit with the class—worked out in proof-from-the-book perfection. It even fits exactly onto a single page. Construct a quick triangle; show two sides match; conclude that the angles we want to match, do match. You can bet I’ll be zapping off copies for the class.
I’d looked at a couple of “high tech” approaches that I’m still not sure can be made to work. One student stayed a little after class last time and we worked through some calculations together; the end appeared to be in sight. I was deliberately holding back of course; I’ll probably look at this approach some more this weekend.
I’d decided to see what would happen by using an idea from an early example in the text: use slope as a parameter. Because then one of the numbers in the problem comes with a very simple representation. After fiddling, I’d decided I wanted the focus, not the vertex, at the origin (the idea here is again to make a certain number appear with an “easy” algebraic representation). The student in question took it from there (I’d showed my ideas to the whole class) and had worked out a few appropriate formulas; these matched mine from the night before… and so we were “on the same page” as we worked out the next couple of moves. Just how it oughta go; I’ll bet he’s got it finished up one way or another by Monday.
Anyhow, next I guessed: oh, hey. I’ll bet this Polar Co-ordinates stuff that we’re handling so badly in this version of the course could be the best way to go. And it very well may be one very good way to go. I think I know the next “formula” I’d need to work out and have spotted an exercise earlier on that might be a really useful hint. Then… let no one else’s work evade your eyes… I looked to the books on the shelf.
And right there, first one I looked at (Edwards & Penny… another two-editions-ago freebie of course) had a sort of a neat one (with an unusual twist in the logic that I thought I’d rather avoid). And I found the proof-from-the-book ideal next in the older Larson. And now I’m about to look on the net.
Because I sure as heck put in the instructions that looking things up counts as work in this context and should be presented proudly. Then, like a fool, when I was trying to get everybody to volunteer in good order to work up front I somehow more-or-less forgot that we’ve got a computer hook-up where the whole class can see what the speaker does with the computer up front and I’ve used it—and on that day—to get up on the net for illustrations and whatnot and had meant to ask a student to look up some leads online as part of the “quiz” which consists of “everybody shows everybody what they can do”.
So Monday maybe somebody’ll show us some browsing skills. Here’s some of mine while we’re waiting. No, wait. They’ll’ve appeared as links (not yet found as I write these words; nor as I first post ’em momentarily). God bless this World Wide Web.