Date: Nov 18, 2012 2:48 AM
Author: Guest
Subject: Re: Please remind me why -3^2 = -9

7-3^2 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.


On Nov 15, 2012, at 1:36 AM, "Clyde Greeno" <> wrote:

Beth, thanks for responding!

My #s are below

- --------------------------------------------------
From: "Beth Hentges" <>
Sent: Wednesday, November 14, 2012 10:15 PM
To: "Clyde Greeno" <>
Cc: "Wayne Ford Mackey" <>; "Alain Schremmer" <>; "Philip Mahler" <>; <>; "Clyde Greeno @ MALEI" <>
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).


> (-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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