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Subject:   RE: Walking a Number Line
Author: Mathman
Date: Jun 16 2005
On Jun 16 2005, Suzanne wrote:
>and she was not convinced that a "negative times a
>negative" was a positive. We
> usually communicated in email but this time she called me and I
> explained how I would have my students physically walk on a number
> line that I made with masking tape in the room.

As they say, "Whatever works."  But i odn't know if this is any different
really, or if it makes the difference in the number who catch on at the end of
the lesson.  A tough one to call.

You say:
*********
If I represent 3 x -2 using the number line, it means
    going three spaces, but the minus sign before the 2
    tells me that I have to be going the opposite direction.
*********
Which is "convenient", but does not follow directly from what you had stated
just before that.  There, your second number has nothing to do with "direction",
but to do with the number of times a certain number of paces is take first to
the right and then to the left.  For consistency, it should retain the meaning,
OR it should be mentioned at the start that this number controls direction. Yet,
you used the first number for that purpose already when you stated that 3 x 2
was three paces "to the right" taken twice, and that -3x2 was three paces to
the left, taken twice.  As I say, it is convenient, but still begs the question
students love to ask, "Sez who?".

I will accept argument, but I think that the idea has been proposed, and is a
good one, that the property be studied by sample series, such as one reduced by
3, as follows:

3x4=12
3x3=9
3x2=6
3x1=3
3x0=0
3x-1=-3
3x-2=-6
... Reducing the multiplier by 1, there is good reason to continue reducing the
result by 3 into the negatives.  A second series can be set up to include
negatives in the multiplicand instead.

A study of several such series should establish the rule, which should then be
memorised.

David.

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