Fractions of collections

Finding one-third of a number is the same as dividing by three, and can be represented with materials in a similar way.

For example, \(\frac{1}{3}\) of 12 and 12 ÷ 3 could both be modelled using 12 counters partitioned into three equal groups of four.

The mathematical relations suggested by the three equal groups of four counters are:

3 \(\times\) 4	= 12
12 ÷ 3	= 4
\(\frac{1}{3}\) of 12	= 4

Working with fractions of collections is helped by a sound knowledge of factors and multiples.

Using arrays and area grids strengthens the relationships between multiplication, division and fractions, by making the inverse relations more apparent.

3 \(\times\) 4	= 12	4 \(\times\) 3	= 12
12 ÷ 3	= 4	12 ÷ 4	= 3
\(\frac{1}{3}\) of 12	= 4	\(\frac{1}{4}\) of 12	= 3

Fractions also appear in whole-number division when remainders occur.

For example, \(\frac{1}{3}\) of 13 (or 13 ÷ 3) results in 4 remainder 1. The remainder 1 can be partitioned into three equal parts and the sharing process continued, leading to the mixed-number answer \(4\frac{1}{3}\).