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RE: Please remind me why -3^2 = -9
Posted:
Oct 18, 2012 2:24 PM
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Part of why students are confused about -3^2 is that our standard teaching about order of operations is incomplete. We don't generally specify where negation fits into the order of operations. In fact, PEMDAS (if not banned) should be taught as P.E.N.MD.AS, with negation inserted after exponentiation and before multiplication/division. Then it would be clear that in -3^2, the exponentiation operation is performed before the negation. (I realize that some would say that negation is just multiplying by -1, so would have the same precedence as any multiplication, but clearly many students don't see that.) But unfortunately P.E.N.MD.AS just doesn't make a nice, easy-to-remember mnemonic.
Peter Collinge, Professor
Department of Mathematics
Monroe Community College, Rochester NY 14623
E-mail: pcollinge@monroecc.edu<BLOCKED::mailto:pcollinge@monroecc.edu>
Voice: (585)292-2943
[Description: cid:image001.jpg@01CC9D61.57D754A0]
From: owner-mathedcc@mathforum.org [mailto:owner-mathedcc@mathforum.org] On Behalf Of Jack Rotman
Sent: Thursday, October 18, 2012 12:39 PM
To: Wayne Ford Mackey; Guy Brandenburg; john.peterson20@comcast.net; Philip Mahler
Cc: mathedcc
Subject: RE: Please remind me why -3^2 = -9
Phil and all:
The expression "-3^2" deals with the order of operations; the most advanced operations are always done first unless a grouping symbol 'forces' a lower prior operation to be done first. Since exponentiation is more advanced than the sign of a number, the exponent only applies to one symbol (the 3) when there are no grouping symbols.
Instead of banning this type of problem, I believe that we should ban "PEMDAS" or anything like it. The use of overly simplistic rules (often stated as a sequence of nouns) discourages learning and encourages memorization. If a large rate of correct answers is the only criteria, just have students use a calculator and train them on use of parentheses. If we are teaching mathematics, we should focus on understanding the priority of operations. What Phil internalized was this understanding; saying 'PEMDAS' does not provide any of the understanding to our students [I normally spend about a tenth of my time in class trying to undo the damage of PEMDAS. Undoing partially correct information is terribly difficult!]
Even if we never showed "-3^2", students would still be evaluating x^2 for x=-3; knowing that this means squaring a negative is a part of basic literacy in mathematics.
For those with an interest, I've posted some "anti-PEMDAS" comments on my blog (www.devmathrevival.net<http://www.devmathrevival.net> ). You can use the search box on the site to find them.
Jack Rotman
Professor, Mathematics Department
Lansing Community College
(517)483-1079 rotmanj@lcc.edu<mailto:rotmanj@lcc.edu>
www.devmathrevival.net<http://www.devmathrevival.net>
From: owner-mathedcc@mathforum.org<mailto:owner-mathedcc@mathforum.org> [mailto:owner-mathedcc@mathforum.org]<mailto:[mailto:owner-mathedcc@mathforum.org]> On Behalf Of Wayne Ford Mackey
Sent: Thursday, October 18, 2012 12:11 PM
To: Guy Brandenburg; john.peterson20@comcast.net<mailto:john.peterson20@comcast.net>; Philip Mahler
Cc: mathedcc
Subject: RE: Please remind me why -3^2 = -9
It should be read as the opposite of 3 squared. Since 3 squared is 9, the opposite is -9. The "-" sign is used in 3 different ways. In front of a natural number it means negative or minus, in front of anything else it means opposite and between two things it means add the opposite.
wayne
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From: owner-mathedcc@mathforum.org<mailto:owner-mathedcc@mathforum.org> [owner-mathedcc@mathforum.org] on behalf of Guy Brandenburg [gfbrandenburg@yahoo.com]
Sent: Thursday, October 18, 2012 6:04 AM
To: john.peterson20@comcast.net<mailto:john.peterson20@comcast.net>; Philip Mahler
Cc: mathedcc
Subject: Re: Please remind me why -3^2 = -9
It's a convention. In a case like that, one really ought to use parentheses to make the meaning clear, since a lot of people, not just youngsters, will get confused.
If one intends to say (-3)*(-3), then write (-3)^2. If one means - (3)*(3), then write - (3^2).
Writing -3^2 is simply asking for confusion.
Guy Brandenburg, Washington, DC
http://gfbrandenburg.wordpress.com/
http://home.earthlink.net/~gfbranden/GFB_Home_Page.html
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From: "john.peterson20@comcast.net<mailto:john.peterson20@comcast.net>" <john.peterson20@comcast.net<mailto:john.peterson20@comcast.net>>
To: Philip Mahler <mahlerp@middlesex.mass.edu<mailto:mahlerp@middlesex.mass.edu>>
Cc: mathedcc <mathedcc@mathforum.org<mailto:mathedcc@mathforum.org>>
Sent: Thursday, October 18, 2012 6:05 AM
Subject: Re: Please remind me why -3^2 = -9
Phil,
-3 means -1 x 3, so -3^2 is (-1)(3^2) = (-1)(9) = -9.
John Peterson
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From: "Philip Mahler" <mahlerp@middlesex.mass.edu<mailto:mahlerp@middlesex.mass.edu>>
To: "mathedcc" <mathedcc@mathforum.org<mailto:mathedcc@mathforum.org>>
Sent: Thursday, October 18, 2012 5:37:07 AM
Subject: Please remind me why -3^2 = -9
I have been teaching a long time, and I know from experience that 50% of students will tell me that -3^2 = +9 on a test or a final, despite having discussed it a few times in a course.
When I first started teaching I taught calculus and precalc. Piece of cake. Then I started with an Algebra I class and couldn't connect at all for the first week or so. I was ready to believe I couldn't teach. I simply could not explain how I got the right answers when evaluating expressions... Then I discovered the order of operations (PEMDAS to some). A definition of the order of operations which I had so internalized that I didn't know there was a rule for it. Since that discovery I've been a wonderful teacher. :-)
So... I must be missing something that so many of my students think -3^2 is +9. What is the rule I have never discovered?
Full disclosure: I think -k^2, k a constant, should be banned from mathematics texts and tests. -x^2, x a variable, evaluated for say -3, absolutely (no pun intended) but not -3^2.
Phil
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