Quotition and partition

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

In general, a quotient $Q=N/D,$ where Q, N, and D are integers, can be conceived of in either of 2 ways:

1. Quotition: "How many parts of size D must be added to get a sum of N?" $$N=Q\times D=\underbrace {D+D+\cdots +D} _{\text{Q parts}}.$$
2. Partition: "What is the size of each of D equal parts whose sum is N?" $$N=D\times Q=\underbrace {Q+Q+\cdots +Q} _{\text{D parts}}.$$

In the rational number system used in elementary mathematics, the numerical answer is always the same no matter which way you put it, as a consequence of the commutativity of multiplication.

For example, the quotient $6/2=3$ can be conceived of as representing either of the decompositions:

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

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.

See also

• List of partition topics

References

• Klapper, Paul (1916). The teaching of arithmetic: A manual for teachers. p. 202.
• Solomon, Pearl Gold (2006). The math we need to know and do in grades preK–5 : concepts, skills, standards, and assessments (2nd ed.). Thousand Oaks, Calif.: Corwin Press. pp. 105–106. ISBN 9781412917209.

External links

• A University of Melbourne web page shows what to do when the fraction is a ratio of integers or rational.