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Re: [xmca] a minus times a plus


I'm not using x and y to indicate co-ordinates, by the way so there really isn't, in how I'm thinking about the situation, a number line connection.

This for me, at least, is a very complicated question and while I have some beginnings on a sort of answer, I may not be able to get to that in any detail for awhile. Let begin though with your mention of multiplication facts. In a sense the important single digit multiplication facts are:

                                          0 x a = 0
                                          1 x a = a
                                           a x b = b x a
                                           a x (b x c)  = (a x b) x c
                                           a x (b + c ) = a x b + a x c
(a + 1) x b = ab + a (an important instance of the just above)

and then
2 x a = a + a (skip counting)
                                           3 x a = 2 x a + 1 x a
                                           4 x a = 2 x (2 x a)
5 x a = a + a + a + a + (money)
                                           8 x a = 2 x ( 2 x (2 x a)))

7 x
6 x
9 x

are a little less obvious although I'm sure there are things one could remember and, of course, if you wish

9 x a = [a-1][9- (a-1)] (where [][] indicates a two digit number)

and 10 x a which results in a shift one place to the left - the paramount fact in our Western base-ten system.


The multiplication fact I somewhat cringe about (not because I don't respect the children that can do such, but because of the amount a garbage that mounts up in the brain) are illustrated by the following once proudly said by a child who was struggling with his tables:

Goin' fishing, got my bait. Six times eight is forty-eight.

Anyway, my answer in part is, children need to be given some time exploring and playing with the structure of the number system. There is a quite old game called equations by Allen (there is a sort of new version which I don't like) and there is the game called the Broken Calculator. These with some thoughtful teaching have been helpful.


On Apr 27, 2009, at 8:06 PM, Mike Cole wrote:

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?


On Mon, Apr 27, 2009 at 4:16 PM, Ed Wall <ewall@umich.edu> wrote:


   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).


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.

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