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Math Tools Grades 3–12 (2nd edition)

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Math(s) tools are ways to build mathematical practices, differentiate instruction and increase student engagement.

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Home > Topdrawer > Fractions > Good teaching > Dividing > Fractions of collections

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.

Three groups of four counters, each group a different colour.

Using 12 counters to model division.

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.

A total of twelve counters arranged in a 3 by 4 array next to a 3 by 4 blank grid.

An array of 12 and a grid of 12.

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 \(4\frac{1}{3}\).

Three groups of four counters, each group a different colour. The thirteenth counter is divided into three equal parts.

Using 13 counters to model division.

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.