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Re: [SPAM]Re: percents
Posted:
Sep 27, 2013 2:04 AM


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 understanding 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 decimalpoint symbols, alphabetically (or "librarywise") ... 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 1decimalplace "codes" , [0, 0.1, 0.2. ... 0.9, 1, 1.1, ... ] ... then the 2decimalplace codes, etc.
The alphabetized family of all such (finite) decimalpoints is dense. Allowing also the infinite ones yields a continuum. All of that can easily be done *without* regarding the decimalpoint 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 onetime 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 decimalcurrency vectors is far from "meaningless." Just like conversions among equivalent fractions, carrying and borrowing among equivalent moneyvectors 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 realworld 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 >> >>> Twoplace decimals were >>> introduced next, followed by 3 and 1place 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 realworld > collection of three onedollarbills, we can represent it on paper by > writing the numberphrase "3 Washingtons". > > (2) But if we have a realworld collection of thirteen onedollar bills, > we cannot represent it on paper. What we must do is collect TEN of these > realworld onedollar bills and change them for one ten dollar bill. We > now have one tendollarbill and three onedollarbill 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 socalled, meaningless, "carry over" is just the result of > having to change TEN realworld bills for one realworld 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 > tendollarbill and three onedollarbill 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/rnotip340.pdf> > > Regards > schremmer > > **************************************************************************** > * To post to the list: email mathedcc@mathforum.org * > * To unsubscribe, email the message "unsubscribe mathedcc" to > majordomo@mathforum.org * > * Archives at http://mathforum.org/kb/forum.jspa?forumID=184 * > ****************************************************************************
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