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*To*: "Hasty, James E." <hastyj@clarke.k12.ga.us>*Subject*: Re: [xmca] a minus times a plus*From*: Mike Cole <lchcmike@gmail.com>*Date*: Mon, 27 Apr 2009 19:43:28 -0700*Cc*: "eXtended Mind, Culture,Activity" <xmca@weber.ucsd.edu>*Delivered-to*: xmca@weber.ucsd.edu*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:reply-to:in-reply-to :references:date:message-id:subject:from:to:cc:content-type; bh=J7w2av1Y0LRXkoECoQH2qBay0kIBineM+RnsymyfNvc=; b=O+p3cQiOJkgKi9JgAQLTeclSB231u4reod9/vXEZ63+ejtjpTsVhgISjfNjIdXU7ng /jzidcLsWM6cwA6ppKX/RSochPgxsPctnI+c0W6yYON4hdFlbmHSqm1f3ILwNQqcy9nH gKS2515CXt6I85SVO16mffIn+WL+A1wiWAtEU=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:reply-to:in-reply-to:references:date:message-id :subject:from:to:cc:content-type; b=AtdCVWjppraXNsDRJgXJ4Fa9EcW9sOJ6Oslnm9Z/6mbCbU2hM06rtZjmMVXEcR1N4W ltxSmxlOgj2X9OTpLVdrvDTiWx+xa5on8IErI6eHRWUYUDJqq59laQQRN/SmctzPNpYx pqjWT+anvh0y4BGTXYygMj+th5mYFh99umnpM=*In-reply-to*: <B56F52E76D330648A67ADB3E7B1BB4DB013EEC4E@datsrv019.clarke.k12.ga.us>*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> <B56F52E76D330648A67ADB3E7B1BB4DB013EEC4E@datsrv019.clarke.k12.ga.us>*Reply-to*: mcole@weber.ucsd.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu

That is COOL, Eric. I will try to implement this in the afterschool environment we work in and see if we can get it into the discourse. Speaking of discourse, arent there people who study math understand as the aquisition of new discourse skills. What do they have to help me out here?? mike On Mon, Apr 27, 2009 at 6:03 PM, Hasty, James E. <hastyj@clarke.k12.ga.us>wrote: > 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 > > _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

**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>

**RE: [xmca] a minus times a plus***From:*"Hasty, James E." <hastyj@clarke.k12.ga.us>

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