Your approach is not unlike the one I used, a very long time ago.
I started by generating a case for negative numbers - debt (in the red),
temperatures (easier if you use the Celsius scale), counting back, above and
below sea level, etc, ...
I then made (common) sense of, e.g., 4x(-3), using repeated addition, and
then (-3)x4, invoking the commutative law - with a nod to the concrete
referents.
Then I would ask the question "What will I do with (-3)x(-6)?" and watch the
fun.
The "friends and enemies" is a simple aid which helps to fix this in
memory.
It always seemed to me that in teaching mathematics, I should make concepts
and procedures reasonable and then I should follow through with the
consequences - appealing to common sense and, if you will, abstract
'modelling'. Thus, mathematics develops from our experiences into
abstractions of these experiences and then into relationships between and
within these experiences. I accept that many areas of mathematics have long
since escaped their experiential origins, and that our pupils should
appreciate this, but our introduction to this important discipline must
acknowledge its historical development - if only because its history is the
record of a great learning experience.
As I read this, it seems a little earnest and, maybe, boring, but I am
convinced that any approach to mathematics based on procedures and tools for
solving problems or one based solely on structural and logical (whatever
that means to a 15 year old) considerations is woefully deficient. Remember
E T Bell - "Mathematics - The Queen and the Servant of Sciences" - I hope I
got that right.
David J McLaren
----- Original Message -----
From: Bret Taylor <bret@IAG.NET>
To: <mathedcc@archives.math.utk.edu>
Sent: Thursday, November 11, 1999 11:22 AM
Subject: [MATHEDCC] Multiplying two negatives
> I appreciate the thoughts expressed here. Some I have seen (baloons and
> sand bags) others I haven't (friends and enemies).
>
> But, I aslo appreciate those who say we should not be afraid to teach math
> for math's sake.
>
> One of the things I tell my student's over and over again is:
>
> There is good news and bad news. The good news is - the rules never
change.
> the bad news is - we never let you forget anything.
>
> Now, using the good news: We define zero, negative and rational exponents
> the way we do because we want the rules that hold for natural number
> exponents to hold for all exponents. So, we either define them the way we
> do, or we have to change the rules for non-natural exponents.
>
> The same is true for multiplying two negatives:
>
> Consider (-4)(-3) + (-4)(3)
>
> Using the distributive property and the property of additive inverses, we
> can see that this sum must be zero.
>
> We know that (-4)(3) = -12
>
> Therefore, (-4)(-3) must be the additve inverse of -12, which is 12. So,
> unless we want to change all our mathematical rules when multiplying two
> negative numbers, then the product of two negatives must be positive.
>
> Then I usually ask them, "I'll let you decide. Do you want to define the
> product of two negatives to be something else and then make up a whole new
> set of rules, or do you just want to use the same rules we already know
and
> let the product of two negatives be positive?"
>
>
>
> Bret Taylor "It matters not the subject taught,
> Lake-Sumter Community College nor all the books on all the shelves.
> Leesburg, FL What matters more, yes most of all,
> John 3:3^3+3 is what the teachers are themselves."
> John Wooden
>
>
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