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?