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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?

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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 $\mathbf N.$

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  • $\begingroup$ @BPP It doesn't, not as it stands, since that was not the question (OP wasn't asking how to explain commutativity); but if what you meant is, How would I suggest explaining (or rather, motivating the axiom) of the commutative property of multiplication of rationals to OP's target audience, then I would still just use the multiplication-in-$\mathbf N$-as-iterated-addition idea to show why it is true in $\mathbf N,$ and then simply note that we would like to keep this property whenever it is possible to do so as we move into $\mathbf Q,$ and so on. $\endgroup$ – Allawonder Apr 5 at 18:22
  • $\begingroup$ @BPP For the second time, I was not trying to explain the commutativity of multiplication, as that's not what OP is about. $\endgroup$ – Allawonder Apr 5 at 19:50
  • $\begingroup$ @BPP Of course I read it, and I don't know how you translated that into commutativity. OP was obviously asking (as is explained in the body of the OQ too) how to justify the interpretation of of as $\times.$ I hope you now understand this problem. Suffice it to say OP accepted my answer as most relevant. Apparently, I must have understood his question better. $\endgroup$ – Allawonder Apr 6 at 5:13
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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.

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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.

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    $\begingroup$ The word "of" means ×. While I agree with it in this context, I can't help but fear having to explain statements such as "We had 8 games this season, and we won 5 of them. So, we won 5$\times$8 of them." I prefer to say "Often, the word of can be understood as multiply. For example...". $\endgroup$ – Nick C Apr 1 at 19:32
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    $\begingroup$ @NickC You're right but I think "$k$ of $\ell$" almost always mean $k\ell$ when it isn't part of a longer sentence. $\endgroup$ – Paracosmiste Apr 1 at 20:05
  • $\begingroup$ What? "10 of 12" means "120"? Not to me. I would say "half of 12" means "6", that is multiply. But I probably would not say "three halves of 12" at all; nor any other multiplier larger than 1. $\endgroup$ – Gerald Edgar Apr 1 at 21:08
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    $\begingroup$ @GeraldEdgar $10$ of $12$ could mean multiplication; for example $1000\%=10$ of $12$ means $120$. $k$ shouldn't have the same "units" as $\ell$ . For example $10$ of the $12$ apples doesn't imply multiplication because $10$ and $12$ have the same "unit" apples. $\endgroup$ – Paracosmiste Apr 1 at 22:35
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    $\begingroup$ "10 of 12" definitely means 120 to me. $\endgroup$ – Rusty Core Apr 1 at 23:44

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