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:firstname.lastname@example.org>email@example.com >www.devmathrevival.net > >From: firstname.lastname@example.org >[mailto:email@example.com] On Behalf Of Wayne Ford Mackey >Sent: Thursday, October 18, 2012 12:11 PM >To: Guy Brandenburg; firstname.lastname@example.org; 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: email@example.com >[firstname.lastname@example.org] on behalf of Guy >Brandenburg [email@example.com] >Sent: Thursday, October 18, 2012 6:04 AM >To: firstname.lastname@example.org; 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: "email@example.com" <firstname.lastname@example.org> >To: Philip Mahler <email@example.com> >Cc: mathedcc <firstname.lastname@example.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" <email@example.com> >To: "mathedcc" <firstname.lastname@example.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 >