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Discussion: Traffic Jam Applet tool
Topic: exponents
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Subject:   RE: exponents
Author: Susan
Date: Oct 19 2004
On Oct 18 2004, Mathman wrote:
> On Oct 18 2004, Susan wrote:
> Even after teaching the students why
> x^-1 is 1/x in the same fashion
> as Mathman does, two months
> later, even after completley
> understanding the derivation,
> students will write 2^-1 = -2.

For what it's worth:

Some get
> confused over much less than that, and in much shorter a period of
> time.  Perhaps we might reflect on the number who do not get
> confused, and compare numbers to see whether or not any partiular
> approach is to be considered effective.  Also, we might reflect upon
> our own education, and note our own success or failure to understand
> then and now.

Consider my argument again:

Empirically [if we
> do the calculations], it is found that 10^5 / 10^3 = 10^2.  Several
> other examples, such as 5^7 / 5^4 = 5^3, lead to the inevitable
> conclusion that in general we subtract exponents when dividing.
> That is all that needs to be remembered, as well as the fact that in
> each case, the exponent is a convenient notation for common numbers.
> So we carry that argument for any numbers, and find again
> empirically, as well as from the general rule already established
> [that is, in the same way] that a^x / a^y = a^(x-y).  It then
> follows that this same exponent notation works for other values
> without loss of generality. So a^(x-y) can be negative, and the
> question is then what does this mean.  Also, that value was obtained
> from an expression like 5^4/5^7. That is, using the rule already
> established, we find that...

5^7/5^4 = 78125/625 = 125

We also
> note that 5^4/5^7 = 625/78125 = 1/125 making them reciprocal to each
> other.

We note again that 5^7/5^4=125=5^3, and that 5^4/5^7=5^-3
> from the rule for division for exponents, and this is the same as
> 1/125.

Once a rule is established, it needs only to be memorised
> and put to further use.  Forgetfulness by the student does not
> diminish its correctness or usefulness.  If there is a better, or
> equally efficient way to do this study I would be grateful to know
> of it and pass it along.  As they say ...whatever works, and we
> really do learn something new every day.  Students will often make
> up their own rules such as, "If x occurs twice on the same page,
> cancel."  We've usually seen it all by the time we retire.

> regard to memorisation:  Some "tricks" and devices like SOHCAHTOA to
> memorise the trig functions, or FOIL for binary multiplication  are
> useful, but the real learning occurs when the student applies the
> function to definition for angles and solution of triangles in any
> position with much practice.  It is the practice and use that embed
> the ideas more than any other device, and young people today get
> neither sufficient sample, nor variety, having to spend more time
> with other associated studies in the same subject.


Thanks David, and again, I've used exactly the same procedures for teaching this
concept.  I guess that the students understand why it works, but when it comes
to looking at 2^-1 the "why" gets lost.  You are giving them the last step of
the "why" presentation and expecting them to work backwards, or to remember what
happened to get to that last step.  I've tried lots of ways to help them
remember--for example, I have tried to rename the negative exponent as a
"flipper" .  I think they get confused because -2 means the opposite of 2, but
2^-1 means the reciprocal of 2, so the negative plays a different role in each
context.  I try to emphasize this difference by giving them lots of practice,
but I think it is confusing from the student perspective.

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