[IAEP] Looking for Concrete "Fraction Experiences"

[David Corking]

Maria, your exposition is excellent, and I strongly agree.  In my own
experience as an engineer, I am nearly always happy to picture in my
head multiplying by the reciprocal, rather than create any mental
picture of dividing by a fraction. On reflection I probably think
about it as the inverse of a scaling operation: the quotient is what I
must have started with if I scaled the divisor by the dividend.

However, I wonder if you would allow me to clarify the following
point, as it seems to me that division involving non-natural numbers
is important outside pure math.

> In practice, nurses, pizza cooks, carpenters and so
> on don't "really" divide by fractions - they work with numerators and
> denominators separately.

This may not be the case when dealing with decimal fractions or
percentages, when they might wish to crank a formula, or hit the
division key on a calculator. Therefore I think it is worthwhile to
give students a chance to develop a mental picture of division by a
fraction, even if they choose to forget it later, and rely on the
algorithm.

I see from your wiki page that you might not agree with me that
percentages and decimals are special cases of fractions. However, at
least in some curricula, fraction arithmetic is taught first, and the
others follow.

A couple of examples:
(1) if a toy car completes a five foot track in three-quarters of a
second, what is its speed?
(2) if you want to take 20% sales tax, or value added tax, off an
invoice to find the pre-tax price, then you divide by 1.2
(3) devise a formula to convert lap times measured in problem (1) into
speeds in miles per hour, or kilometres if you prefer.

I think we will find numerous other examples in science, engineering
and finance. (I think we could describe some of these calculations as
non-integer ratios, or as the need to reverse or invert a
multiplication.)

I think authentic and concrete concepts of division of all real
numbers can be useful to help us to use calculators confidently, and
to use, develop, and adapt formulae that include division.

(Aspiring electrical technicians will probably also want to extend the
concept to complex numbers.)

Thanks for your patience. David