Finding onethird 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 wholenumber 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 mixednumber answer \(4\frac{1}{3}\).