the skin of our teeth

tonight we looked at a *great deal*
of material-from-the-syllabus.

too much by almost any standard,
i think. and obviously, missing a day…
half a week, really… doesn’t help.

and yet. it’s built into the course.
never mind snow days: if you’re getting
most of your ideas about What Algebra Is
*or* How To Do It, *from this presentation*,
then you’re almost sure to be left behind
(& very quickly) in the nature of the case.
this is College Math: “way too late
and much too fast”.

(meanwhile. *read*, dammit! and *talk* to each other.
for hecksake! do you think you’ll be young *forever*?)

okay. everybody gets their expectations
pushed-and-pulled around (and otherwise distorted)
and somehow we work out some way of getting along;
some way of talking about what it all might mean.
if there’s scribbling-on-the-board involved…
if there’s *symbolism*… then that’s what i call
“Doing The Math”. okay okay. don’t hate me for it.

quadratic equations, *again*? well, yes.
maybe you’ve missed something.

it’s much better if *you* take a piece
of chalk: *what* were you saying, again?

why have the deepest-thinking sages
returned to suchlike issues again and again
throughout all recorded time? (no… really:
why?)

because, yeah, duh: “theology”, so called.
“ontology”, forsooth.

it’s mostly transparent, though, if we agree
that “appeals to authority” are even more
contemptable than “outright lies”.
funding, funding, funding. everybody
talks about the weather. fuck the
god damn weather.

philosophy. feh.

quadratic equations.

*every* quadratic equation…

Ax^2 + Bx + C = 0…

can be solved (in the appropriate “domain”)
by the famous “Quadratic Formula”
(cf: QF Lore [a popular piece
from my blogging heyday]). To wit.

let D = B^2 – 4AC.

(D… or \sqrt{D}… i forget…
is the *discriminant* of our function…
we *were* talking about a function, right?
let’s see. let f(x) = Ax^2 + Bx + C, where
A, B, and C denote “numbers” [i.e., elements
of the Domain of Discourse] and “x” is
an “indeterminant”.)

we are not data.

*obviously* there isn’t-and-can-never-be
any “quadratic formula” for life itself:
“if you act *this* way, life will
work itself out in *that* way!”…
and it’s halfpast time we stopped
thinking that “math’, all by itself,
could ever fool anybody into thinking
that it’ll even ever’ve been a good
idea to *try*…

still. dammit.

suppose A \not= 0.
(otherwise, our equation
isn’t quadratic at all
[rightly so-called] but
merely linear; refer to
some already-well-understood
theory, duh). then 1/A
is a number. multiply
both sides of Ax^2 + Bx + C = 0
to obtain (the “monic” equation…
leading coefficient equal to 1…)
x^2 + (B/A)x + (C/A) = 0.

one “easily” applies the technique
called “completing the square”
at this point (add-and-subtract
the square of “half the middle coefficient”;
“regroup” the pieces and rewrite
the appropriate bit of code
as a “perfect square”).

clean it up and show
(at some appropriate level
of rigor) that the equation
Ax^2 + Bx + C = 0
(by the godlike authority
of faith-in-perfect-clarity
[work it out!]) is equivalent
(when A \not= 0; when the domain
of discourse allows the relevant
operations) to
x \in { (-B +\sqrt{D})/(2A), (-B -\sqrt{D})/(2A) }.

nobody wants to talk about the set theory here.
wait. that’s false. *i* want to talk about
the set theory here. can i get a witness?

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  1. Have you seen Tanton’s complete the square video? (http://www.youtube.com/watch?v=OZNHYZXbLY8) I like it, and I think my students last semester ‘got’ completing the square, and the proof of the qf much better than usual, when I used his method.

  2. good to know. tanton’s a great source.
    recently i got the (from his twitter feed,
    reproduced at _math_mama_) observation
    that in the

    2 9 4
    7.5 3
    6 1 8

    magic square, a random choice of
    A from the first row, B from the second,
    and C from the third gives
    P(A<B) = P(B<C) = P(C<A) = 5/9.

    weird-seeming and easily memorized
    (there's essentially
    only one way to *make* a 3-by-3
    magic square [and, check this
    out, the same trick works
    for *columns* as rows).
    a great little exercise.

    we haven't done completing the square yet
    this go-round (and this ramble isn't much
    of a commercial for it as i expect… just, for now,
    a place to drop my old links while i debriefed myself
    from the… much more coherent on on-topic,
    i assure you!… lecture of a few hours before).

    *when* i present it, i'll be winging it as usual
    and'll allot time (and enthusiasm) based in large part
    on "perceived audience interest". if i'm connecting
    with even a few, i'll hit it pretty hard.

    i remember it as a pretty big moment in my own
    growth-in-confidence-in-algebra.

  3. oh. p.s.
    x wasn’t an “indeterminant” up there,
    never *mind* what i said. x was an
    “indeterminate” (as th’ wiki has it,
    “a symbol that doesn’t stand for
    anything but itself”).

  4. lots of fun today: students working at the board,
    one of me favorite things. but *i* did the
    “complete the square” stuff, lecture-style, solo.
    way too fast of course but there was apparently
    some “buy in” with the vertex-and-axis connection.

    (the x-coordinate of the vertex is “halfway between”
    the “plus” and “minus” bits of a certain expression
    involving “plus or minus”, etcetera).

    the stuff about discriminants and domains?
    not so popular here (and no wonder).
    “what am *i* gonna do with it?”




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