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Re: Please remind me why 3^2 = 9
Posted:
Nov 18, 2012 2:48 AM


73^2 means 7+3^2 means 7+1*3^2
The point is that it is all convention, there are reasons for choosing the conventions we do.
I am one who usually uses the "flying hyphen" to indicate a negative or opposite sign.
I have seen one elementary curriculum that used a hat to indicate negative numbers.
Beth
On Nov 15, 2012, at 1:36 AM, "Clyde Greeno" <clydegreeno@cox.net> wrote:
Beth, thanks for responding!
My #s are below
  From: "Beth Hentges" <Beth.Hentges@century.edu> Sent: Wednesday, November 14, 2012 10:15 PM To: "Clyde Greeno" <greeno@malei.org> Cc: "Wayne Ford Mackey" <wmackey@uark.edu>; "Alain Schremmer" <schremmer.alain@gmail.com>; "Philip Mahler" <mahlerp@middlesex.mass.edu>; <mathedcc@mathforum.org>; "Clyde Greeno @ MALEI" <greeno@malei.org> Subject: Re: Please remind me why 3^2 = 9
> (3)^2 is clear > > (3^2) is clear > > We have to choose which we mean when we write the following. > > 3^2 > # So far, so good!
> The reason we choose 3^2 to mean (3^2) is because we do powers before multiplication and because a = 1*a. # And the reasons for that rule and for that equation are ???? It seems as though you are trying to *conclude* what is actually a grammarrule of the written language. The "powers before multiplication" is just such a grammar rule, and it gains no rational support from the equation. Nonetheless, I can believe that your use of that equation very well might help train students to calculate, better ... even without grasping the commonsensibility of it all.
# The mathematical issue with your equation is that it does the usual curricular "slight of hand" by ambiguously using "" with two differing meanings ... while many students use even a third meaning. Within the context of linear number systems. "1" means "1 below 0" ... while "a" means "the negative/opposite of a" ... while students routinely take "" to mean "subtract a."
# In the case of 73^2, many students take the "" to mean subtraction. Then their question becomes, "Does it mean (73)^2? [as 4^2] ... or 7(3^2) [as 79]?" Would your equation lead some to interpret it as 7(1)*(3^2)? Of course an instructor might simply *define" subtraction so that the meaning of "73" is 7+(1)*3. But would that be commonsensible to students?
# So the instructological question is whether or not your students actually grasp and apply your theorem: neg of (a) = (1below0)*a. Even if not, they still might use your symbolswitching device for more systematically calculating, correctly.
# But as yet, I fail to perceive how your equation makes algebraic formulas any more commonsensible than would a good treatment (much better than usual) of how to use parentheses in accord with the syntaxconventions that currently are in commonplace use throughout the field (and in most calculators).
Cordially, Clyde
> > (3)^2 = (1*3)^2 , and the parentheses tell us to do the multiplication first. > > (3^2) = 1*(3^2), and the parentheses tell us to do the power first. > > 3^2 = 1*3^2, and we do powers before multiplication. > > > As for PEMDAS, I use PEMA. Otherwise, students think PEMDAS says to do multiplication before division. Also, when I write it in words, for the E for exponents (which really should be another P for powers), I write, "Do exponents and roots from left to right." So, even if I did use PEMDAS it would be PERMDAS. Then, we also have to be careful with absolute value as well. > > Beth in MN
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