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Re: Please remind me why -3^2 = -9
Posted:
Oct 18, 2012 1:26 PM
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Although I agree with Jack Rotman about the
damage inflicted by PEMDAS, I'm not sure
. . ."the expression -3^2 deals with the order of operations"
gets to the heart of the problem. The reason is,
in this case, the little horizontal bar in front
of the 3 could be a part of the number's name. So
"-3^2" is not two operations, just like "43^2" is
not two operations. In the latter, the 4 is part
of the number's name, it is not a multiplier.
This is the problem, in this case, and so I think
Guy Brandenburg is right. Writing -3^2 is just asking for trouble.
But, Guy, writing -x^2 is not. This is clearly two operations!
Paul Hertzel
NIACC
Mason City, IA
At 11:38 AM 10/18/2012, Jack Rotman wrote:
>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, Ive posted some
>anti-PEMDAS comments on my blog
>(<http://www.devmathrevival.net>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 <mailto:rotmanj@lcc.edu>rotmanj@lcc.edu
>www.devmathrevival.net
>
>From: owner-mathedcc@mathforum.org
>[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; 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
>
>
>----------
>From: 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; 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
>============================
>
>From: "john.peterson20@comcast.net" <john.peterson20@comcast.net>
>To: Philip Mahler <mahlerp@middlesex.mass.edu>
>Cc: mathedcc <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
>
>
>----------
>From: "Philip Mahler" <mahlerp@middlesex.mass.edu>
>To: "mathedcc" <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 couldnt connect at all
>for the first week or so. I was ready to believe
>I couldnt 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 didnt know there was a rule for it.
>Since that discovery Ive 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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