Re: Explaining a common fraction

On Thu, 26 Jan 2006 14:22:40 +1300, Barb Knox <see@xxxxxxxxx> wrote:

>>Here's another one I've mentioned before: When you divide one proper
>>fraction by another, you invert and multiply. Why? Almost all can do
>>it without knowing in the slightest as to why. There are many such
>>questions like that, and if the course was stopped to answer, they
>>would only bog down a practical [non-theoretical] study.
>
>Maybe in the limit (e.g., having to derive everything from Peano's
>axioms), but IMO in the case you cite there is educational benefit in
>seeing that things actually do hang together rather than being just a
>large collection of arbitrary rules.
>
> (a/b) / (c/d) = (a/b) * 1/(c * 1/d)

I'd rather look at it this way:

Equivalent fractions are formed by multiplying both numerator and
denominator by the same quantity: a/b = ma/mb. Choose one suitable
to the occasion, (d/c). Then (a/b)/(c/d)=
[(a/b)*(d/c)]/[(c/d)*(d/c)]. This was chosen in order to make the
denominator of the major fraction be 1. That also relies upon more
basic rules such as commutativity, but I'd avoid that also in an
introduction to division of fractions to the very young. How can they
at the same time absorb that (c/d)*(d/c) = (cd)/dc) = (cd)/(cd) = 1?
Some rules must be arbitrary until a better occasion arises [maturity
of age and in study] for more introspection and generalisation. It
worked for us, I think.

.

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