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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}.
Part and wholes
Students develop strategies for working out the total number in a collection, given the number of items in the fractional part.
Divide it up
Sharing-as-division problems may give a result with a remainder. In some contexts it makes sense to divide the remainder into equal parts (fractions) and continue the sharing process.