# Re: [xmca] a minus times a plus

```Mike

```
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.
```
Ed

```
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.
```
Ed

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?

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
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