Fractions

Fractions become much easier when you keep asking one question: what does each number describe in the same whole? In this P5 lesson, you will match piece sizes for mixed-number addition and subtraction, then use repeated groups or an overlap model for multiplication.

Parents: ask your child to predict the next line before revealing each worked panel. The page stays within the P5 fraction-operation families, while the linked notes handle earlier foundations and division.

By the end you'll be able to add and subtract mixed numbers, multiply the required proper and improper fraction families, and multiply a mixed number by a whole number.

Add and subtract mixed numbers

A mixed number names whole units and a fractional part of one more whole. For addition or subtraction, the fractional parts must name equal-sized pieces. Find a common denominator, then operate on the numerators. If the fraction being subtracted is larger, use regrouping.

For addition, $\frac{1}{4}$ is $\frac{3}{12}$ and $\frac{2}{3}$ is $\frac{8}{12}$.

Add mixed numbers

$$\frac{1}{4} = \frac{3}{12}$$

$$\frac{2}{3} = \frac{8}{12}$$

$$1\frac{1}{4} + 2\frac{2}{3} = 3\frac{11}{12}$$

For regrouping subtraction, exchange one of the three wholes for $\frac{12}{12}$ before subtracting.

Regroup before subtracting

$$3\frac{1}{4} = 3\frac{3}{12} = 2\frac{15}{12}$$

$$1\frac{2}{3} = 1\frac{8}{12}$$

$$2\frac{15}{12} - 1\frac{8}{12} = 1\frac{7}{12}$$

Proper or improper fraction × whole number

Multiplying a fraction by a whole number makes repeated equal groups. Multiply the numerator by the whole number and retain the denominator because the piece size has not changed. The result may be greater than one whole, so simplify or convert to a mixed number when the question asks for it. The P5 source specifies without calculator for this multiplication family; the written steps below show the hand method.

Proper fraction × whole

$$\frac{3}{5} \times 4 = \frac{12}{5}$$

$$\frac{12}{5} = 2\frac{2}{5}$$

Improper fraction × whole

$$\frac{7}{4} \times 3 = \frac{21}{4}$$

$$\frac{21}{4} = 5\frac{1}{4}$$

The next family starts in the same way: a mixed number can be rewritten as an improper fraction before it is multiplied by a whole number.

See the mixed-number bridge

$$2\frac{1}{3} = \frac{7}{3}$$

$$\frac{7}{3} \times 3 = \frac{21}{3} = 7$$

Proper fraction × proper or improper fraction

For a product of two fractions, name the parent whole before you calculate. An area grid makes both factors visible: the first factor selects horizontal pieces and the second selects vertical pieces. Heavy unit boundaries keep complete parent wholes visible even when an improper factor crosses more than one whole. Multiply numerators and denominators, then simplify.

Proper × proper

$$\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$$

For a proper fraction multiplied by an improper fraction, keep the same named whole and allow the result to be less than or greater than one as the givens decide.

Proper × improper

$$\frac{2}{5} \times \frac{7}{3} = \frac{14}{15}$$

The word “of” needs its quantity attached. For example, “$\frac{2}{3}$ of $\frac{3}{4}$ of one metre” means $\frac{2}{3}$ of the stated $\frac{3}{4}$-metre amount, with one metre still the measurement unit.

Two improper fractions

An improper fraction can contain more than one whole. When both factors are improper, the product can contain several wholes too. Keep the parent whole visible while you count the repeated equal parts.

Improper × improper

$$\frac{7}{4} \times \frac{5}{3} = \frac{35}{12}$$

$$\frac{35}{12} = 2\frac{11}{12}$$

🔬

Fraction multiplication: overlap the same whole

🤔 Predict first: For $\frac{3}{4}$ × $\frac{2}{3}$, how many of 12 equal parts are selected?

6 twelfths5 twelfths8 twelfths

Mixed number × whole number

Convert the mixed number to an improper fraction first. Then use the whole-number multiplier rule and convert back if a mixed answer is useful.

Convert, multiply, convert back

$$1\frac{3}{4} = \frac{7}{4}$$

$$\frac{7}{4} \times 4 = \frac{28}{4} = 7$$

Practise with worked steps

Start here — a recipe uses $\frac{3}{5}$ L per batch; how much for 4 batches?

A recipe uses $\frac{3}{5}$ of a litre of milk for one batch. How much milk is needed for four batches?

Repeat the equal group

$$\frac{3}{5} \times 4 = \frac{12}{5}$$

$$\frac{12}{5} = 2\frac{2}{5}$$

The answer is 2 $\frac{2}{5}$ L.

Build up — find 2 $\frac{1}{3}$ + 1 $\frac{3}{4}$

Find 2 $\frac{1}{3}$ plus 1 $\frac{3}{4}$.

Use twelfths

$$\frac{1}{3} = \frac{4}{12}$$

$$\frac{3}{4} = \frac{9}{12}$$

$$2\frac{1}{3} + 1\frac{3}{4} = 3\frac{13}{12} = 4\frac{1}{12}$$

The answer is 4 $\frac{1}{12}$.

Challenge — find 4 $\frac{1}{6}$ − 2 $\frac{3}{4}$

Find 4 $\frac{1}{6}$ minus 2 $\frac{3}{4}$.

Regroup into twelfths

$$4\frac{1}{6} = 4\frac{2}{12} = 3\frac{14}{12}$$

$$2\frac{3}{4} = 2\frac{9}{12}$$

$$3\frac{14}{12} - 2\frac{9}{12} = 1\frac{5}{12}$$

The answer is 1 $\frac{5}{12}$.

Watch out — easily mixed up

Follow the fraction path

Review P4 mixed numbers and improper fractions when you need to convert between forms. Continue to Fraction and Division for equal sharing and fraction-to-decimal division, then Dividing with Fractions for the P6 reciprocal method.

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. What is 1 $\frac{1}{4}$ + 2 $\frac{2}{3}$?

   3 $\frac{5}{12}$3 $\frac{11}{12}$4 $\frac{1}{12}$

2. What is 3 $\frac{1}{4}$ − 1 $\frac{2}{3}$?

   2 $\frac{5}{12}$1 $\frac{5}{12}$1 $\frac{7}{12}$

3. What is $\frac{7}{4}$ × 3?

   5 $\frac{1}{4}$$\frac{7}{12}$4 $\frac{1}{4}$

4. A $\frac{3}{4}$-metre ribbon has $\frac{2}{3}$ of that same ribbon used. How many metres are used?

   $\frac{5}{12}$ m$\frac{1}{2}$ m$\frac{7}{8}$ m

5. What is $\frac{7}{4}$ × $\frac{5}{3}$?

   3 $\frac{1}{12}$2 $\frac{7}{12}$2 $\frac{11}{12}$

6. Four identical containers each hold 1 $\frac{3}{4}$ litres. How many litres are there?

   7 L5 L6 $\frac{3}{4}$ L

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