[IAEP] multiplication and division as scaling (was Re: [NaturalMath] Looking for Concrete "Fraction Experiences")

[David Corking]

Maria Droujkova  wrote:
> I think scaling can correspond both to division and to multiplication. You can scale up and
> down - also by numbers over and under one if you think more algebraically.

Absolutely. In my observation, many 7-year-olds are able to think
algebraically about multiplication and division of natural numbers
(1,2,3,...). Since they simultaneously discover common fractions,
place value, and decimal money, it won't be long before they have all
the building blocks in place to confidently multiply by various kinds
of fractions.

I would like to learn more about the Proportion Bar that Rebecca
Hanson mentions in her blog article.

Moving on, perhaps Maria's comment gives me the opportunity to explain
how I interpret this as an opportunity to intrigue rather than
frighten students the first time they come across a MULTIPLICATION
that MAKES THINGS SMALLER. By the way, I don't mean to open a
'repeated addition' debate, which is another topic entirely.

a x b = c

can be said to mean "a scaled by b is equal to c"

(Teachers use other symbols according to local conventions. For an
international audience, I will use x and / as operators.)

When b is more than one, we all know what happens.

But, when b is say, a half, students may be surprised that
multiplication works in just the same way: c gets smaller in a linear
or proportional fashion.

For example, b may be the number of cups of sugar in a recipe
comprising 4 cups of dry ingredients. a is 4. When b is 1, the total
recipe, c, is 4 cups, when b is 2, c is 8, and when b is a half, c is
2, when b is 0.1, c is 0.4, and so on according to your arithmetic
ability, or until you run out of ingredients. (Since we adults all
know scaling when we see it, we will spot that when b gets less than
one, c becomes less than a.)

When b is the scale factor, what does division mean?

Firstly, c / a = b.  Here, division is a tool to find the scale
factor, when you know the number that was scaled, and the result. For
another example, when investigating Hooke's law on a spring, c is the
tension or weight, a is the extension, and b is the spring constant, a
kind of scale factor.

Secondly, c / b = a.

When we divide the volume of the final recipe, c, by the scale factor,
b, the result is a, the volume of the original recipe. In other words,
division allows us to reverse, undo or INVERT the original scaling
operation, to get from the final value, c, back to the original value,
a.

Eventually, when we have divided by a scale factor, b, of a half, to
get from c=2 to a=4, we have reached Steve Thomas's original challenge
to find a concrete experience of division by a common fraction. And,
hopefully, if we do this and other scaling tasks enough times, we are
not too surprised that this division calculation has a result of a
being bigger than c.

I guess all I am trying to say is, firstly, that the idea of scaling
is a natural part of everyday life, and not something dreamed up by
pure mathematicians to make life hard for parents and educators.

And secondly, I claim that it is more easy, for me, to understand
scaling PURELY as multiplication, while treating division as the
inverse, a way of undoing a scaling operation, which matches its
mathematical definition. To me, it seems to be an unhelpful
complication to view division as another way of scaling a quantity,
even though we sometimes choose to do this in adult life.

I hope my interpretation proves interesting to some of you.

David