How to explain to pupils that "$\frac n{100}$ OF $a$" is equivalent to "$a$ TIMES $\frac{n}{100}$"?

Question

How to explain to pupils that "$\frac n{100}$ OF $a$" is equivalent to "$\frac{n}{100}\times a$"?

There is some difficulty in explaining that the first sentence, containing "OF" (which could suggest something like "taking a part of" and therefore something like division) has to be mathematically translated by the second sentence (with its "TIMES").

What is the most efficient method to show that the "TIMES" translation is correct?

Mathematically speaking, is the intuition that "of" suggests a division totally wrong?

Answer 1

One can only see that this is true just as one saw what the symbols $1,2,3,\cdots$ meant. What's happening in this case is that we are seeing a ratio. Suppose we want $n/100$ of $Q.$ Then you may explain this to them as

Divide $Q$ into $100$ equal parts, and take $n$ of such parts of $100.$ That is, we're taking $n$ of something. But this simply means we add $n$ such things together, which means we are multiplying such things by $n.$

NB. The explanation above will be clearer if you've previously introduced positive integer multiplication as a short notation for iterated addition, and will make sense therefore only if $n$ is a positive integer. Using this convention more generally is just a matter of agreement, as with most other operations outside $\mathbf N.$ However, the explanation for $n\in\mathbf N$ should be a good motivation as to why this convention of thinking of 'of' as some type of multiplication (later when they learn about functions and the notation $f(x),$ they're going to have a nod of déjà vu!) is most useful, convenient, and agrees with the fact in $\mathbf N.$

Answer 2

The earliest mathematical insight I remember from childhood is that the word "of" almost always means "times."

Half of a dozen $= \frac12 \cdot 12 = 6$

Three-fourths of a mile $= \frac34 \cdot 5280$ feet $= 3960$ feet.

I'll take 6 of those thousand-count boxes $= 6 \cdot 1000 = 6000$.

I remember feeling like I had secret knowledge that no one else had.

Showing children this "secret" might give some of them the same moment I still remember to this day.

a one-hundredth of six $= \frac{1}{100} \cdot 6 = 0.06$.

The fact that "a one-hundredth of six" is the same thing as "six hundredths" is not a trivial fact.

Answer 3

The word "of" means $\times$. For example "half of $a$" means $\frac12\times a$. Note that the product of $a$ by a number isn't necessarily less than $a$ ; it'll be less than $a$ iff the number is less than $1$ ; in that case we can write the number as $\frac{m}{n}$ ($m<n$) so that $\frac{m}{n}\times a$ means to divide $a$ into $n$ equal parts and take $m$ parts of them.

If we take a rectangle of dimensions $a$ and $n$, its area is $na$ (you could partition it into $na$ unit squares). If we divide $a$ by $10$ we obtain $10$ smaller identical rectangles so the area of each one is $n\times \frac{a}{10}$ which is also the tenth of the initial rectangle $\frac{na}{10}$. If we divide $n$ by $10$ instead of $a$ we get $\frac{n}{10}\times a=\frac{na}{10}$. Then $\frac{n}{10}\times a=n\times\frac{a}{10}$. Of course we can "prove" more generally that $ab=ba$ for any $a$ and $b$ : $ba$ is the area of the rectangle obtained by rotating the rectangle $ab$ by a quarter revolution.