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.
May 3, 2009 at 1:38 pm
The Lehrer link I can only get to play the New Math song (and I think subtracting base 8 is an ok thing to teach). Had to go to Youtube to listen to him singing about Metro-Goldwyn-Moskva (or I could buy a turntable. I still have the vinyl)
May 4, 2009 at 11:06 am
i linked there for the lyrics; the soundtrack
was an unexpected bonus. wow. lehrer on vinyl.
i’ve got _the_remains_of_ on this computer.
the disks themselves i’ve lent out but with
a pretty good chance of actually seeing again.
of *course* base addition base eight is worth teaching.
the fact that certain clueless pedagogues consider such
calculations the very symbol of the uselessness of
“new math” is just more evidence if any were needed
*of* their cluelessness. it becomes much *easier*
to talk about, say, “place value” in the *un*familiar
setting… just as “parts of speech” (verbs, nouns, whateveh)
become interesting when studying *foreign* language
(i [rightly] refused to learn any of this terminology
in *english* class since it was clearly useless…
i could already generate correct english sentences
effortlessly *without* it…).
May 5, 2009 at 2:06 pm
I agree that learning subtraction in another base deepens our understanding. I’m wondering what makes this song so funny? (I love it!) Partly it’s just the fun music behind it. But there’s more. I think we’re partly laughing at the teacher who’s acting like something hard for the student ought to be easy.
I’m not sure I’m saying this at all well…
May 5, 2009 at 5:16 pm
tom lehrer in VME; followup.
May 5, 2009 at 5:30 pm
my (ragged as hell) PDF lecture notes have
(section 2.2) binary, octal, and hex conversions;
this is such a good thing i did it again:
bases three, nine, and twenty-seven (section 2.5).
on actual “base n calculations” i punted
(though a display on the first page of this section
is what but a base-eight subtraction problem).
the *fact* that “seven from eleven is four!”
is one i hear in my head more or less
in tom lehrer’s voice dozens of times a year.
just the *way* he sings is indeed funny.
a few of the songs i loved well before i knew
what the heck they were even about really.