### 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?

February 4, 2011 at 10:03 pm

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.

February 7, 2011 at 7:30 pm

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.

February 7, 2011 at 7:36 pm

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”).

February 9, 2011 at 12:56 am

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?”