### 5/8 of MEdZ 0.3

“algebra 1” students in enormous numbers

are confronted every semester (or quarter)

with a survey-of-number-systems.

, and

…(to give them their standard symbols…

as the textbooks, more or less of course,

do *not* do [effectively; sometimes

they *gesture* at these symbols])…

denote the sets of

Natural numbers, integer numberZ,

rational (Quotient) numbers,

& (so-called) Real numbers.

the texts then go on to *ignore* their own “survey”.

& students get *worse-than-nothing* out of it.

in many cases, they’ll have seen this treatment

*many* times (and become *worse* prepared to

think about the rest of the course material

[about “graphing” and “factoring” and so on]

*every* time).

then there’re these exercises. wherein one is

required to say, about *each* of a (given) collection

of numbers, which of our “number-sets” include it

(and which don’t)… for instance,

“pi” is real but not natural, integer, or rational

“-11” is integer, rational, and real, but not natural

etcetera. right on the front page of the final…

the same damn *problem*, verbatim, for years,

in at least one instance known to me… and yet

substantial subsets of *every* class of students

miss these incredibly-easy-if-the-mere-vocab-

ulary-is-understood “exercises”.

because the one thing they know for sure

after all this time is “i will never understand

what any of these words mean” (so “just show me

how to ‘do’ the problems”).

whereas.

rational number arithmetic is routinely taught

to children in functional schools (and families).

with a certain amount of effort, of course.

but it’s “easy” enough if it’s done *clearly*.

so here’s something that *ought* to be in the text.

not necessarily *instead* of

“ and are integers “,

but certainly somewhere *nearby*

(if the deliberately-obfuscatory

high-theory “set-builder” thingum

*must* somehow be included in our treatment).

namely, a table (showing possible numerators

and denominators & their Quotients), such that

is “all the numbers in the table

(and their opposites)”.

(the little “” icon at upper-left

is meant to indicate that, e.g. “-3/2”

(the “opposite” [or “additive inverse”] of “3/2”)

should be considered to be “in the table”…

this is a nuance, improvised for the zine

[and if i were giving this lecture today

at the blackboard, i’d very likely draw

out another part of the table and include

some negative rational numbers explicitly].)

there it is. we’re looking right at it.

*now* we can talk about it.

(and, say, how various “decimals”

do or don’t “give us” a

“number on the table”.)

(secretly, of course, “we” know this means

“a number that can be represented as [an]

integer-over-rational-number”… but

it does no good to keep on *saying so*

once we know darn well we’re being

“tuned out”.)

now. about those dotted-line circles.

(… get the student to talk…)

hey, “lowest terms”. wow, cool.

now about this funky *graph* over here.

same idea, different picture.

each “lowest terms” ratio appears

as the *nearest point to zero*

on a “line” having the “slope”

represented by that number.

(by “nearest” [point to zero],

one here means “nearest on the

‘integer lattice’ “…

but of course one will not

[necessarily] *say so* explicitly;

this is… or was… a “lecture

without words” more-or-less precisely

*because* the real work is getting

the *student* to do a lot of the

talking [and pencil-moving];

“nearest *on the diagram* is sufficient

for our purpose [a clear view

of the set-of-rational-numbers,

in case you’ve forgotten].)

you’ve gotta give ’em hope.

May 6, 2015 at 12:26 pm

don’t point that thing at me

unless you intend to use it.