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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