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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
>>
>>
>>
>
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