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*To*: <mcole@weber.ucsd.edu>, "eXtended Mind, Culture,Activity" <xmca@weber.ucsd.edu>*Subject*: RE: [xmca] a minus times a plus*From*: "Hasty, James E." <hastyj@clarke.k12.ga.us>*Date*: Mon, 27 Apr 2009 21:03:54 -0400*Cc*:*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=KqgyJlx417t+qkGIzGVIyxPYz+MURgmb5JdgzrS8mzpnnPKrMwid5FDDg0WDDi2iA PwD/IQozOK13MdTlVOQKw==*List-archive*: <http://dss.ucsd.edu/mailman/private/xmca>*List-help*: <mailto:xmca-request@weber.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca.weber.ucsd.edu>*List-post*: <mailto:xmca@weber.ucsd.edu>*List-subscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=subscribe>*List-unsubscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=unsubscribe>*References*: <30364f990904271547o5b4df21eifca69bf8318483f2@mail.gmail.com><2C46D7A7-AD94-441C-AABD-269045835E3D@umich.edu> <30364f990904271706l114497cax5f814ffa09a51893@mail.gmail.com>*Reply-to*: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu*Thread-index*: AcnHlaMQulzwtY42Tsuw9v6/rB5Y9gAA7PKi*Thread-topic*: [xmca] a minus times a plus

Ok, we start with the number line again only we have a negative number in the right column 4 * -5 = -20 3 * -5 = -15 2 * -5 = -10 1 * -5 = -5 0 * -5 = 0 Once we get here we talk about how the number is getting larger by 5 at each step, so we continue the number line -1 * -5 = 5 -2 * -5 = 10 -3 * -5 = 15 and so on. A coworker showed this to me by having his students walk a number line. He put a number line from -10 to +10 on the board and had a student stand at 0 and face toward the +10 representing a positive orientation. He has the student walk 3 steps back and shows 1 * -3 = -3 He then has the student return to zero, turn around and face -10 representing a negative orientation, and take 3 steps back representing -1 * -3 = 3 He then doubles the number of students (2 * -3 = -6) and then turn around and do it again (-2 * -3 = 6) eric Great!! Thanks Ed and Eric and please, anyone else with other ways of explaining the underlying concepts. Now, we appear to have x and y coordinates here. If I am using a number line that ranges along both x and y axes from (say) -10 to +10 its pretty easy of visualize the relations involved. And there are games that kids can play that provide them with a lot of practice in getting a strong sense of how positive and negative positions along these lines work. What might there be of a similar nature that would help kids and old college professors understand why -8*8=64 while -8*-8=64? Might the problem of my grand daughter, doing geometry, saying, "Well, duh, grandpa, its just a fact!) arise from the fact (is it a fact?) that they learn multiplication "facts" before they learn about algebra and grokable explanations that involve even simple equations such as y+a=0 are unintelligible have become so fossilized that the required reorganization of understanding is blocked? mike On Mon, Apr 27, 2009 at 4:16 PM, Ed Wall <ewall@umich.edu> wrote: > Mike > > It is simply (of course, it isn't simple by the way) because, the > negative integers (and, if you wish, zero) were added to the natural numbers > in a way that preserves (in a sense) their (the natural numbers) usual > arithmetical regularities. It would be unfortunate if something that was > true in the natural numbers was no longer true in the integers, which is a > extension that includes them. Perhaps the easiest way to the negative x > positive business is as follows (and, of course, this can be made opaquely > precise - smile): > > 3 x 1 = 3 > 2 x 1 = 2 > 1 x 1 = 1 > 0 x 1 = 0 > > so what, given regularity in the naturals + zero) do you think happens > next? This thinking works for, of course, for negative times negative. The > opaque proof is more or less as follows. > > Negative numbers are solutions to natural number equations of the form (I'm > simplifying all this a little) > > x + a = 0 ('a' a natural number) > > and likewise positive numbers are solutions to natural number equations of > the form > > y = b ('b' a natural number) > > > Multiplying these two equations in the usual fashion within the natural > numbers gives > > > xy + ay = 0 > > or substituting for y > > > xy + ab = 0 > > so, by definition, xy is a negative number. > > Notice how all this hinges on the structure of the natural numbers (which > I've somewhat assumed in all this). > > Ed > > > > > On Apr 27, 2009, at 6:47 PM, Mike Cole wrote: > > Since we have some mathematically literate folks on xmca, could someone >> please post an explanation of why >> >> multiplying a negative number by a positive numbers yields a negative >> number? What I would really love is an explanation >> that is representable in a manner understandable to old college professors >> and young high school students alike. >> >> mike >> _______________________________________________ >> xmca mailing list >> xmca@weber.ucsd.edu >> http://dss.ucsd.edu/mailman/listinfo/xmca >> >> >> > _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

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**Follow-Ups**:**Re: [xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**References**:**[xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Ed Wall <ewall@umich.edu>

**Re: [xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

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