## Archive for the ‘Cool Tricks’ Category

(because violet blue red… but that’s

off-topic [and off-color]…)

three green-peppers; none of ’em green.

it’s even beautiful in its package and,

of course, even more so the more they’re

played with.

as shown here, the tops have been cut off

and served (with their middles cut out)

with ranch. the next thing that happened

was that a “ring” was cut off at the top

of each and the whole set got bagged and

put away.

the rings, in their turn, were opened up

into long slices and split down their

middles, the long way, with a steak knife.

finally (thus far), diced fine and stirred

into a chicken salad (along with some

carrots, also chopped fine, and, obviously,

some chicken—one big american breast).

add mayo to taste; mix; serve. (serving

suggestion: ritz crackers. we’ve got

lettuce & tomato, though, so actual sand-

wiches aren’t out of the question.)

ABCDEFG

(seven chords seven ways, part mercury)

ten for maria.

1 – There isn’t enough user-generated content or “making your own math artifacts.”

equations, most likely, first.

but wait. zero-th.

by-hand copies of the *symbols*

for the material at hand.

“the student learns essentially

nothing until the student’s

pencil makes marks on the page”

is a pretty good first approximation

a lot of the time… or anyhow,

i’m far from the only teacher

given to *saying* stuff like this.

i’ve got plenty to say, too, *about*

this but i’m hoping for a list of ten

in under 2^12 characters (for a little

longer; i’ve begun to despair already

at least a little though if you want

to know the truth).

“unions” should look different from “u” ‘s

as an example more or less at random.

*our medium is handwriting.*

first-and-a-half.

out-loud discussion of and…

second.

…written sentences *about*

those equations. written

at leisure without the

instructor (or fellow student).

third.

similar or exact versions of such equations,

repeated, or, much better of course,

improvised, in a “public” setting

with small or, slightly better i

suppose, large *groups* of fellow

students. oral presentation of

the sentences themselves is not

only okay here but much to be

preferred (the board should not

be littered with sentences).

the “correctness” of the sentences

should nonetheless be at issue

throughout the presentation.

said “correctness” is to refer

explicitly to “code”…

utilizing (hey! ed jargon!)

the symbols from our step zero.

it does not escape my attention

that the “artifacts” created by

the student presentations i here

imagine are scribbles of chalk

on a board, soon erased. so be it.

…

leaving some out…

sixth

yick, computer code.

seventh

student-designed exercises,

exam templates, lesson plans…

eighth

songs and other verse, games,

comics and other graphics,

something to astonish even me.

ninth

blogs.

tenth

fanzines.

Reply

Sue VanHattumMarch 7, 2010 at 7:08 PM

Maria, I loved your list.

Owen wrote:

>”the student learns essentially

nothing until the student’s

pencil makes marks on the page”

Maybe for higher math, but not at all for young kids. The mathematical issues they’re working on don’t usually require pen(cil) and paper.

My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)

Reply

AnonymousMarch 8, 2010 at 6:57 PM

My son is thinking so much about what I’d call place value these days. “60 and 60 is 120, right?” “Yep.” No writing – at home, anyway. Lots of mathematical thought.

I’ve long been puzzled by your emphasis on “the code”. Maybe someday I’ll get it… :^)

—sue v.

maybe today!

the “places” of “place value” are

places *in* certain symbol strings!

it sure doesn’t matter that you

*speak* of such strings without

having actual *written* code

in front of your actual eyes…

that’s not what i’m always

going on about at all…

60+60=120

presumably gets its interest

from 6+6=12,

together with, right,

the “place value” concept…

*as it manifests in base ten*.

now of course you and your kid

don’t have to have spoken of

bases-other-than-ten for

the essential *role* of “ten”

in discussions of place value

to have become quite clear

all around.

“what’s so special about ten?”

i can now imagine asking

some kid of the same age

if i were lucky enough to

know any…

and i’d sure enough expect

(maybe with a *little*

stack-the-deck prompting

from me) pretty soon to

start hearing about the

role of *zero* (in, again,

certain symbol-strings).

and when our conversations

*without* written work begin

to break down… and if we

still *care*… why then,

we’ll break out some *pencils*

and take a look:

“what do you mean, *precisely*?”.

we’ve been talking about code all along.

tangent.

calculating with numbers

is the very *model* of

one-right-answer-ism:

3*4=13

is just flat-out wrong.

and this is our greatest strength.

in principle, anything worth

talking about passionately

in a math class should have

the *same* character:

there *is* a right answer

if we could only find it.

in order to have this happen,

we have to agree on things.

we *can’t* agree… and be

*sure* we agree… and be *right*…

without certain so-called “rigorous

definitions”: marks on paper

(generally; otherwise

*verbatim verbal formulas*

memorized syllable-for-syllable

[mostly… i don’t seek a

“rigorous” definition of “rigor”…

“one is *this* many”

and its ilk (so-called “ostensive

definitions”) are all the rigor

we can *get* sometimes]).

generally the “rigor” one speaks of

is… i think… pretty *close* to the

being-able-to-calculate-it-out-like-a-computer

thing i spoke of (with reference to

elementary arithmetic) a moment ago.

and this comes from “code”.

again. our power in mathematics

comes to an amazing extent from

being-able-in-principle to emulate

some doesn’t-know-anything-*but*-code

*machine*.

now i’m as much of a luddite as the next

guy, if the next guy figures the wrong turn

was somewhere around “domesticated animals”.

but one *glaring* benefit of computers

in math ed is that students will work

for *hours* on getting code letter-perfect

(if they know no human being can see

their failures happening), that wouldn’t put

in five *minutes* of homework on paper

without getting so frantic about each

“move” that they fall apart before even

getting started. it’s that “interactivity”.

this used to break my heart but it’s true.

if schools were for clarity,

command-line programming

would begin in about first grade.

it’s much *easier* than almost

any other thing you can do

with a computer (which is why

it emerged much *earlier*

than the hugely-user-unfriendly

[from a “code” point of view]

*graphical* interfaces that

erased it from the national

consciousness in around 1984).

(somebody mention “logo”.)

math *is* hard.

but it’s much easier than anything else.

because we’ve got *all* the certainty.

(programming on this model

is of course a subset of math).

ot

three newish guitar-stands arranged in such a way

that any two will fall down without the third.

and madeline’s “three women” statue, having a

similar property. blessings from our happy home

to yours if you’ve got one; double blessings if

you’re doing without. happy “black friday”.

the linking and not-linking rings

(blogpost of 05/03/014).

w’edia.

flickr shot.

exercise: draw the other six lines.

hint.

about time these guys got some names.

ROY (this one) is obvious.

let’s see.

green blend.

bgy byg gby gyb ybg ygb;

GABY, then.

secondaries.

gop gpo ogp opg pgo pog;

PEGGIE-OH, then, maybe.

the orange blur.

bmo, bom, mbo, mob, obm, omb.

JIMBO comes to mind.

submissions welcome. yes, it’s the great

name-the-lines contest of ought-fifteen!

i’ll be forced for the sake of my own peace

of mind to type out the rest of the exercise

and make up three more names. “the purple

blur” isn’t much of a *name* for a line.

more like a *secret identity*.

formed by my right hand. i got a new shipment

of strings today… thanks, madeline!… so i

(finally!) tuned up one of my guitars “lefty”.

if i try to *play* it that way, it sounds like

day one. but if i play it *right* handed, i

can make it sound like music. the chords and

the dynamics are different, though. so, cool

trick, it sounds like somebody else playing

(while still sounding a lot like me).

or maybe that’s the cough syrup talking. whee!

saturday night i colored in the corners of this cube.

the underlying black-and-white is based on a work of

the great dutch artist m.~c.~escher. the cardboard

cut-out version is from a collection by the american

mathematician doris schattschneider.

(_m.c._escher_kaleidocycles_).

anyhow, i’ve had the whole “5 platonic solids” set

from this work on display in the front room at home

for a while. the others are in color already, right

out of the book. i’ve had *another* set of these,

too: it’s a great “book” and might still be in print

for all i know. i had two editions, from years apart,

years ago.

i took this one to church on sunday and used it in my talk.

there wasn’t time to explain why i’d colored it the way i

did. but i *did* count the symmetry group of the cube,

two ways. any talk by me should have a theorem in it;

i’m happy to count that as a theorem.

24 because any of the 6 faces can be “face down”,

and each such choice-of-face allows for any of 4

remaining faces then to “face front” (all but the

“face-up” one *opposite* to our “chosen” face).

but also 24 because

1 identity

6 180-degree “flips” that fix two edges

(one for each pair of opposite edges)

8 120-degree “corner-turns”

(fix a pair of opposite corners;

there are 4 such; one may “turn”

right or left)

6 90-degree “face turns”

(fix opposite faces—3 ways; again,

one can go “right” or “left”)

3 180-degree face turns.

and this messy version is actually quite clear when

one is actually holding up an actual cube and pointing

at the drawing on the board. or in this case, at one’s

own sweatshirt. canvas makes a good whiteboard.

you can do it all without even mentioning “group theory”…

and i *sure* didn’t get to prove that this set of 24

“moves” gives a version of “the symmetric group on 4

objects”. anyway, part of the point is that one need

not have introduced any “math code” into the discussion

at all to arrive some some *very* useful results.

i learned the symmetry-groups-of-solids trick from an

old master. i mentioned this book in the service.

but mostly i talked about stuff like bible studies and music.

stuff that people actually show up at church *for*. bring what

you love to church and share it. food and money are particularly

welcome.

Introduction

(Lines 1—40)

God (as I choose to call my higher power)

Grant me an audience for half an hour

And I will, if I can, do all the rest.

My subject is the story I know best;

I mean my own. It starts in a motel,

The night of my divorce. I felt like Hell.

Think of a pilot, learning how to fly,

Who, though he should know better, flies too high,

Then falls in the Atlantic and is drowned.

His body and the plane are never found.

There’s something like our marriage in that story,

The way it shoots to misery from glory.

The similarity might not be strong,

But, as to suffering, I’m never wrong:

Divorce is brutal. Trust me when I say

I’d rather be that pilot any day.

Lisa, in a voice that tore my heart,

Had told me, “From now on, we’ll live apart.

I’ll keep your stuff till you’ve got your new place.

The First Street house is mine. I want my space.”

And so, a stranger in my own home town,

I left my room to have a look around.

Across the street, a Big Red liquor store

And Waffle House. A porno shop next door.

“The restaurant then. For now, I’ll do what’s right.

I’ve got no strength for sex and drugs tonight.”

The waitress, call her Ruby, perked me up.

I never saw the bottom of my cup.

A refill and a smile, and off she’d glide;

She wore her sixty years with grace and pride.

“Why look upon myself and curse my fate:

I couldn’t stand to only serve and wait.

I’ll bet that woman’s life is harder still

Than mine, by far. And, if I only will,

I could throw all my misery away

And love my life the way it is today!”

If that was true—and I don’t think it was—

I proved myself an awful fool, because

For years I didn’t love my life at all.

The story starts with my decline and fall.

dick’s *exegesis*.

inspiring visions of an inspired visionary.

6.1.5

Let a, b, c, & d

be integers satisfying

b > 0, d > 0, and

ad – bc = 1.

Let n = max{b,d}.

Then a/b and c/d are consecutive

fractions in the n^th Farey Sequence.

Proof:

Since ad-bc=1, we have

ad/(bd) = (1+bc)/(bd), hence

a/b = 1/(bd) + c/d and so

a/b > c/d.

Note that (cf. Corollary 6.2)

ad-bc = 1 also implies

(a,b) = 1 and (c,d) = 1;

our fractions are in

“lowest terms” (and hence

appear in F_n).

Now suppose c/d & a/b are *not* adjacent.

Then there is at least one fraction,

which we may assume is in “lowest terms”,

“p/q”, say, with (p,q) = 1, such that

p/q is *between* c/d and a/b

in the Farey sequence F_n:

F_n = { …, c/d, …, p/q, …, a/b, …}.

Then

q = (ad-bc)q

= adq ___________ – bqc

= adq – bdp + bdp – bcq

= d(aq-bp) + b(dp-cq)

(*).

But a/b > p/q implies

aq – pb > 0, and so

(since the variables are integers)

we have that

aq – bp is greater-or-equal to 1

(aq – bp \ge 1).

The same computation with different

letters shows that

[p/q > c/d] \implies [dp – cq \ge 1].

These inequalities together with (*) give

q = d(aq-bp) + b(dp-cq)

q \ge d + b.

Recall that n = max{b,d}.

It follows that d + b > n.

This gives us q > n, a contradiction

(“p/q” cannot be in F_n with q > n).

Hence a/b and c/d *are* adjacent in F_n; done.

— the number of “permutations of k things

from a set of n” (“en-permute-kay”, in its

most convenient say-it-out-loud version);

the number of ways to make an (ordered)

*list* of k (distinct) things chosen from

a set of n things. one has known this from,

well, time immemorial. but never written

it out in straight-up *set* notation

until today.

now it’s on page 5 of my “bogart”.

(_introductory_combinatorics_, kenneth

p.~bogart, 1983; mine’s a recent acquisition

from the math complex, once of the

“bertha halley ross collection” according

to its bookplate. that turns out to have

been mrs. *arnold* ross, late of the OSU

[an all-time giant in recruiting new talent

into mathematics, founder of the “ross program”].

you can see from the edges of the pages that

somebody read it up to chapter six and stopped.

they didn’t write on the pages though.

[but that’s being put right now.])

and cranking out suchlike “set code” *is* fun

and easy… for me. showing somebody how to

*read* it? fun but *not* easy. showing some-

body how to *write* it? well… how would i

*know*? (how would *anyone*?)