Quotition and partition

In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotition division one asks, "how many parts are there?"; while in partition division one asks, "what is the size of each part?".

For example, the expression ${\displaystyle 6\div 2}$ can be constructed of either of two ways:

1. "How many parts of the size of 2 must be added to get the amount of 6?" (Quotition division)

One can write

${\displaystyle 6=\underbrace {2+2+2} _{\text{3 parts}}.}$

Since it takes 3 parts, the conclusion is that

${\displaystyle 6\div 2=3.}$

1. "What is the size of 2 equal parts whose sum is that of 6?". (Partition division)

One can write

${\displaystyle 6=\underbrace {3+3} _{\text{2 parts}}.}$

Since the size of each part is 3, the conclusion is that

${\displaystyle 6\div 2=3.}$

It is a fact of elementary theoretical mathematics that the numerical answer is always the same no matter which way you put it,  6 ÷ 2 = 3. This is essentially equivalent to the commutativity of multiplication in multiplication arithmetic.

Division involves thinking about a whole in terms of its parts. One frequent division special case, is that of a natural number (positive integers) of equal parts, is known to teachers as a partition or sharing: the whole entity becomes an integer number with equal parts. What quotition focuses on, is explained by removing the word integer in the last sentence. Allow the number to be any fraction and you may have a quotition instead of a partition.