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Re: Please remind me why -3^2 = -9
Posted:
Nov 15, 2012 2:36 AM
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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
grammar-rule 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 7-3^2, many students take the "-" to mean subtraction. Then
their question becomes, "Does it mean (7-3)^2? [as 4^2] ... or 7-(3^2) [as
7-9]?" 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 "7-3" is 7+(-1)*3. But would that be common-sensible to students?
# So the instructological question is whether or not your students actually
grasp and apply your theorem: neg of (a) = (1-below-0)*a. Even if not, they
still might use your symbol-switching 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 syntax-conventions 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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