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Fractions as division

Anyone who has studied secondary school mathematics would probably be comfortable with the convention of 'a over b' meaning 'a divided by b'.
That is, \(\frac{a}{b}\) = a ÷ b.

For example, the fraction two-thirds means 2 divided by 3.
That is, \(\frac{2}{3}\) = 2 ÷ 3.

We use this idea when we convert an improper fraction to a mixed numeral.
For example, \(\frac{7}{4}\) = 7 ÷ 4 = 1\(\frac{3}{4}\).

We also accept that a fraction expresses division when converting a fraction to a decimal.
For example, \(\frac{3}{8}\) = 3 ÷ 8 = 0.375.

People are perhaps less likely to think of the reverse situation, particularly in a real-life problem. For example, if asked to share two chocolate bars equally among three people, would you automatically think that each person would get two-thirds of a chocolate bar?

2 ÷ 3 = \(\frac{2}{3}\)

In this situation, the fraction is the outcome of the division process, or the quotient. That is why this way of working with fractions is sometimes called the 'quotient' model.

For further exploration of these ideas, read the article Fractions as Division: The Forgotten Notion? on the AAMT website.

Using the division model

Concrete models and problem-solving contexts can help to develop understanding of fractions as an expression of division.