The unit and equivalent fractions

Name the whole before the fraction

Half of a small ribbon can be shorter than one quarter of a much longer ribbon. A fraction names a part relative to a chosen whole. It does not name a length on its own.

When one ribbon is the whole, divide it into four equal lengths. Each length is one quarter of that ribbon. Three of those lengths make three quarters. In 3/4, the denominator tells us the size of the unit (quarters); the numerator counts three of those units.

Drawing four unequal pieces and shading three does not show three quarters. Equal-sized units are what make the fraction useful.

Check your understanding 1 Half of which ribbon?

Ribbon A is 12 cm long. Ribbon B is 20 cm long. Is half of each ribbon the same length? Find both lengths and explain.

The two lengths and why they differ

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Half of A is 6 cm; half of B is 10 cm. Both are one half, but the wholes differ. Compare your explanation with the named wholes, not just the two numbers.

Rename an amount without changing it

Take one whole strip. Shade one half. Split each half into three equal pieces. The whole now contains six equal pieces, and the shaded amount contains three of them:

Both the number of shaded units and the number of units in the whole were multiplied by three. The shaded amount did not grow. This is why multiplying numerator and denominator by the same nonzero number preserves a fraction’s value.

Reversing that regrouping gives 6/8 = 3/4: every two eighths form one quarter. You can divide both numerator and denominator by two because it groups all the equal units consistently.

Division can give a fraction

Three identical loaves are shared equally among four people. Each person receives one quarter of each loaf: three quarters of a loaf in total. Hence

The answer’s whole is one loaf, not the three-loaf collection. Four shares of 3/4 loaf give all three loaves back.

A fraction can also be written as a decimal by division. For 3/4, use 3÷4 = 0.75. Check with hundredths: 3/4 = 75/100.

Check your understanding 2 Sharing five pizzas

Five identical pizzas are shared equally among four people. How much pizza does each person receive? Give a mixed number and a decimal.

Share per person in both forms

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5÷4 = 5/4 = 11/4 = 1.25 pizzas per person. Each person receives one pizza and one quarter of the remaining pizza. Four such shares use five pizzas. The fraction exceeds one because each share exceeds one pizza.

Keep the unit when the number exceeds one

A mixed number names whole units and a fractional part. 23/5 means two whole strips and three fifths of another equal strip. Each whole has five fifths, so

The numerator can exceed the denominator. It counts fractional units across several wholes. It does not mean a whole was split into thirteen equal parts.

Before you move on

Without drawing every piece, explain why 9/6 = 11/2 = 1.5. Name the whole, the fractional unit and the number of those units. If one of these is unclear, draw the wholes before using a conversion rule.