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Topic: [SPAM]Re: percents
Replies: 2   Last Post: Sep 27, 2013 3:31 PM

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Clyde Greeno @ MALEI

Posts: 220
Registered: 9/13/10
Re: [SPAM]Re: percents
Posted: Sep 27, 2013 2:04 AM
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As "code" (a numeral), "3/9" normally is first learned as the quantity,
3(9ths) ... numerator, 3, denominaTION, 9ths ... as in "pie slices." In
that evolution, it can be quite a jump to recognize the theorem that
3(9ths) is the quotient from dividing 3(wholes) by 9.

Another underlying theorem is that the quotient from 3 divided by 9 can be
got by carrying out the "long/short division" process. We can bully students
into believing it, or guide them to conclude it. Only by putting those two
under-standing theorems together can one conclude that "3/9, say, is just
code, an instruction to divide 9 into 3."

Apart from calculations, there is something to be said for approaching the
decimal-point symbols, alphabetically (or "library-wise") ... as is done
also with the construction of scaled tapes/rulers. Begin with the "primary
school" scale, [0,1,2,3,4,...] Then insert the 1-decimal-place "codes" , [0,
0.1, 0.2. ... 0.9, 1, 1.1, ... ] ... then the 2-decimal-place codes, etc.

The alphabetized family of all such (finite) decimal-points is dense.
Allowing also the infinite ones yields a continuum. All of that can easily
be done *without* regarding the decimal-point codes as representing
"numbers." The cognitive gain is that students thus can perceive the
alphabetic ordering of those points *prior* to development of the decimal
numbers. [A one-time viewing of a simple video should suffice.]

As for 13 one-$ bills, there is nothing "wrong" with 13W(ashingtons) ...
other than it is an unnecessarily cumbersome stack of bills. That is why it
is an *improper* quantity ... whose *proper combination* is 1H(amilton)+3W.
The 13W --> 1H+3W "carrying" conversion of the decimal-currency vectors is
far from "meaningless." Just like conversions among equivalent fractions,
carrying and borrowing among equivalent money-vectors are fundamental
"reduction" operations of vector arithmetic.
http://sections.maa.org/okar/papers/2005/greeno1.pdf

With W as "singles", the decimal borrowings of the proper (0, 0, 1, 3, 0, 0,
0) .... into the improper (0, 0, 0, 13, 0, 0, 0) and
(0, 0, 0, 0, 130, 0, 0) and (0, 0, 0, 0, 0, 1300, 0) and (0, 0, 0, 0, 0, 0,
13000) ... make real-world sense with physical money. The carryings of (0,
0, 1, 3, 0, 0, 0) into (0, 0, 1.3, 0, 0, 0, 0) and (0, 0.13, 0, 0, 0, 0, 0)
and (0.013, 0, 0, 0, 0, 0, 0) are steps into "accounting space." (:-)>

Cordially,
Clyde

- -------------------------------------------------
From: "Alain Schremmer" <schremmer.alain@gmail.com>
Sent: Thursday, September 26, 2013 5:54 AM
To: <mathedcc@mathforum.org>
Subject: [SPAM]Re: percents

>
> On Sep 25, 2013, at 10:42 PM, Alain Schremmer wrote:
>

>> both
>>

>>> Two-place decimals were
>>> introduced next, followed by 3- and 1-place decimals.

>>
>> and
>>

>>> Fractional
>>> notation was introduced last, as an alternative form for representing
>>> decimals. ...

>>
>>
>> look to me to be typical educando: completely beyond the pale.

>
> (-1) 3/9, say, is just code, an instruction to divide 9 into 3. Period.
> When we actually carry out the division, we can get any one of the
> following depending on the required precision. :
>
> 0 + [...]
> 0.3 + [...]
> 0.33 + [...]
> 0.333 + [...]
> etc
>
> (I use [...] to mean "something too small to matter here".)
>
> So, introducing decimals one at a time is idiotic. Here is the way I go
> about it.
>
> (0) By itself, 3.27 is meaningless in the sense that 3.27 does not
> represent anything in the real world. 3.27 is just a *numerator* that
> says HOW MANY and we also need a *denominator* to know WHAT we are
> talking about.
>
> (1) Say we use the digits 1, 2, ... 9. Then, if we have a real-world
> collection of three one-dollar-bills, we can represent it on paper by
> writing the number-phrase "3 Washingtons".
>
> (2) But if we have a real-world collection of thirteen one-dollar- bills,
> we cannot represent it on paper. What we must do is collect TEN of these
> real-world one-dollar bills and change them for one ten- dollar bill. We
> now have one ten-dollar-bill and three one-dollar-bill which we represent
> by writing the combination (aka vector) "1 Hamilton & 3 Washingtons".
>
> (3) The next step is to use a header under which to enter the numerators
> and 0 elsewhere:
>
> Clevelands, Franklins, Hamiltons, Washingtons, Dimes, Pennies, Mills
> 0 0 1 3
> 0 0 0
>
> Headers are in fact what we use when we add and, incidentally, it shows
> that the so-called, meaningless, "carry over" is just the result of
> having to change TEN real-world bills for one real-world bill of the next
> higher ... denomination. And, of course, same with the so- called,
> meaningless, "borrowing". (Anyone got change for a ten?)
>
> (4) The final step is to select a denominator and mark the corresponding
> numerator with a decimal point. Thus, we can represent one
> ten-dollar-bill and three one-dollar-bill by writing any of the following
> decimal number phrases:
>
> 0. 013 Cleveland
> 0. 13 Franklins
> 1. 3 Hamiltons
> 13. Washingtons
> 130. Dimes
> 1300. Cents
> 13000. Mills
>
> For more, see <http://www.freemathtexts.org/Standalones/RDA/Downloads.php

> > and/or <http://www.ams.org/notices/201303/rnoti-p340.pdf>
>
> Regards
> --schremmer
>
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