Interpreting division

Division is basically an inverse operation of multiplication, therefore it's all about scaling a value by another value.

$c = \frac{a}{b}$

But how to interpret it mentally, there are a few ways of going about it:

• division is a measure how many times $a$ is bigger than $b$ (how many times b can be "fitted" into a)

• division can also be said to be a way of dividing a value in $b$ equal parts. This is exactly how we think about density (dividing mass equally across units of volume or getting the value of mass per unit of volume) and also average.

These two are a bit bothering me because the seem to clash in my head, I understand they are inherently true. But how can I think about it in the "second way", since when calculating, we are always thinking about how many times a is bigger than b, not how we can split a in b parts.

Due to the commutative property of multiplication, it is indeed true:

$c = \frac{a}{b}$ => $b = \frac{a}{c}$

Therefore the previous quotient is the number we were "looking for", the value that splits a into b parts. What I am trying to say, if we have a box filled with 12 elements of the size 3, we can have 3 elements of the size 12, right?

$5 = \frac{10}{2}$ divides 10 into 2 parts -> $2 = \frac{10}{5}$

Most basic question if I didn't make sense up here, how can I interpret division in a way where $a$ is divided into $b$ equal parts $while$ calculating, images of cake being cut up in n parts is trivial and not the point. It is true, but how to think about it?

Is it simply a matter of: if we can fit 50 items of size 1, then we can fit 1 item of size 50?

• Think of the multiplication $ab$ as the total size of $a$ groups each of size $b$. Then $c/d$ would be ''how many groups, each of size $d$, would give $c$?" Would it be obvious to you that this is equivalent to asking "if you split $c$ into groups each of size $d$, how many groups would you have?". Sorry if this is vague... – David Mitra Nov 9 '11 at 12:36
• Yeah, that's somewhat the picture in my head. Thank you, it's not vague. Perhaps someone will elaborate even further. Lately, I've been quite interested in the very foundations of mathematics/arithmetic – Curiosity Nov 9 '11 at 12:48

Think about distributing $a$ balls into $b$ boxes. (I assume $a$ is a multiple of $b$.)

First, I prepare $b$ boxes. In the first round, I distribute one ball to each box, so $b$ balls in total. In the second round, I again distribute one ball to each box, so $b$ balls in total. (...)

$c=a/b$ means exactly $c$ rounds are needed to distribute all the balls into the boxes. And the number of rounds is equal to the number of balls in each box.

• A really nice analogy, it helped a lot! – Curiosity Nov 9 '11 at 13:55

To really get into this, you should look into the "quotative model" and "partitive model" of division. To use the balls/boxes metaphor from @pharmine's answer: when you divide $b$ into $a$, you are either:

• using $a$ boxes, and counting how many balls are in each box; or
• putting $a$ balls into each box, and counting how many boxes you use.

They seem about the same, but psychologically they're quite different for students just learning it.

• Thank you very much for the recommendation, I will definitely look into it! – Curiosity Nov 9 '11 at 13:55