Multiplying and dividing fractions

a/b × c/d is c/d of a/b: multiply the tops and multiply the bottoms. Multiplying by a fraction less than 1 makes a number smaller. Dividing asks how many fit: 3 ÷ 1/4 = 12.

Half of a pizza is left, and you eat a third of what is left. How much of the whole pizza did you eat? Fractions of fractions are multiplications.

"Of" means multiply: $\tfrac{1}{3}$ of 12 is $\tfrac{1}{3} \times 12 = 4$ . A fraction of a fraction works the same way: $\tfrac{3}{4}$ of $\tfrac{2}{3}$ is $\tfrac{2}{3} \times \tfrac{3}{4}$ .

To multiply fractions, multiply the tops and multiply the bottoms: no common denominator needed. A square cut into columns and rows shows why: the parts multiply, and so do the shaded ones.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

Explain it to me

A fraction of a fraction · Sound on for the voiceover

To multiply fractions, multiply the tops and multiply the bottoms, then simplify: $\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}$ . It is $\tfrac{3}{4}$ of $\tfrac{2}{3}$ : in a square, 3 of 4 rows of 2 of 3 columns is 6 of 12 parts.

Multiplying by a fraction less than 1 makes a number smaller ( $\tfrac{3}{4} \times 20 = 15$ ). Dividing asks how many fit: $3 \div \tfrac{1}{4} = 12$ , the same as $3 \times 4$ . To divide by a fraction, multiply by it flipped (top and bottom swapped: $\tfrac{1}{6}$ becomes $\tfrac{6}{1} = 6$ ).

$\tfrac{3}{4} of \tfrac{2}{3}: the purple overlap is 6 of the 12 small parts, so \tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}.$

\tfrac{3}{4}

\tfrac{2}{3}

: the purple overlap is 6 of the 12 small parts, so

\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Half of a third

Think of it as $\tfrac{1}{3}$ of $\tfrac{1}{2}$ .

\tfrac{1}{2} \times \tfrac{1}{3}

bigger or smaller than

\tfrac{1}{2}

?The sameBiggerSmaller

TryTry it on the board and compare the result with your prediction.

A third of a half

$a$ of $b$ columns are blue; $c$ of $d$ rows are orange. The overlap is purple.

Try it yourselfYour edits are saved in this browser.

The board shows $\tfrac{1}{2}$ of the square. Change $c$ and $d$ to show $\tfrac{1}{2} \times \tfrac{1}{3}$ . Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Multiply, and divide

Here is how each kind works.

Worked exampleBefore each step, say what you think comes next.

$\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{2 \times 3}{3 \times 4} = \tfrac{6}{12} = \tfrac{1}{2}$ Multiply the tops and the bottoms, then simplify.
$\tfrac{3}{4} \times 20 = \tfrac{60}{4} = 15$ A whole number is over 1: 20 = 20/1. 15 is less than 20.
$\tfrac{1}{2} \div 3 = \tfrac{1}{6}$ Share a half into 3: each share is a sixth (the same as × 1/3).
$3 \div \tfrac{1}{4} = 3 \times 4 = 12$ How many quarters in 3? Dividing by 1/4 is multiplying by 4.
$\tfrac{2}{3} \div \tfrac{1}{6} = \tfrac{2}{3} \times 6 = 4$ Divide by a fraction: multiply by it flipped (1/6 flipped is 6/1 = 6).
$0.4 \times 3 = 1.2$ Decimals: 4 tenths × 3 = 12 tenths.

Now try one: $\tfrac{2}{3} \times \tfrac{3}{5}$ .

Read Jack's reasoning. Work out $\tfrac{3}{4}$ of 20 first.

Work out $\tfrac{2}{3} \times \tfrac{3}{5}$ in simplest form.

Jack compares $\tfrac{3}{4} \times 20$ with 20. Click the line where his reasoning goes wrong.

How many quarters are there in 3? Work out $3 \div \tfrac{1}{4}$ .

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

Need a hint?

Columns: 3 of 4. Rows: 2 of 5.

Try it yourselfYour edits are saved in this browser.

Show $\tfrac{3}{4} \times \tfrac{2}{5}$ on the board.

Use the board. This ticks when you get there.

Work out $\tfrac{2}{5}$ of 30.

Work out $\tfrac{3}{4} \div 3$ in simplest form.

Work out $\tfrac{2}{3} \div \tfrac{1}{6}$ .

Edge case: which of these is bigger than 12?

12 \times \tfrac{1}{2}

12 \times \tfrac{5}{4}

12 \div 2

12 \times \tfrac{3}{4}

PractiseA few on your own, to check you have it.

Multiply and divide

Work out $\tfrac{4}{5} \times \tfrac{5}{8}$ in simplest form.

A bottle holds 0.6 L. How many litres do 4 bottles hold?

A ribbon is $\tfrac{3}{4}$ m long. How many pieces $\tfrac{1}{8}$ m long can be cut from it?

Review: work out $\tfrac{1}{2} + \tfrac{1}{5}$ .

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A quarter, another way

Then find one where neither fraction has a 1 on top.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Try 1/2 of something that is 1/2 once simplified, such as 2/4.

Another hint

Or 3/4 × 1/3.

Make the overlap exactly $\tfrac{1}{4}$ of the square, without using $\tfrac{1}{2} \times \tfrac{1}{2}$ .

Use the board. This ticks when you get there.

For teachers

Students use a fraction area square to see a fraction of a fraction, multiply tops and bottoms, learn that multiplying by a fraction less than 1 makes smaller, and divide by counting how many fit (AC9M7N06, node 2). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can multiply two fractions and simplify the answer.
I can divide a fraction by a whole number and by a fraction.
I can say whether a product will be bigger or smaller than the number I started with.

Before class

Do the Run board yourself first. Students need equivalent fractions and simplifying, and adding fractions (previous lesson) to contrast.

The one idea: a/b × c/d is c/d of a/b; the parts multiply, so the tops and the bottoms multiply. The hinge misconception is that multiplying always makes bigger (Jack's mistake); it matters for percentages next.

Division is introduced as "how many fit", which justifies multiplying by the flipped fraction without a rule to memorise.

Running the lesson

Predict: students who chose bigger hold the 'multiplying makes bigger' belief. Keep the vote visible.
Run: ask why the overlap is smaller than both shaded parts.
Investigate: contrast with adding: no common denominator needed here.
Modify: for the last task, ask students to sort multipliers into 'less than 1' and 'more than 1'.
Make: collect products equal to 1/4; ask what they all simplify to.

The hard part (on purpose)

Letting go of the belief that multiplying makes bigger. Ask for an estimate (bigger or smaller than the starting number?) before every calculation.

When students get stuck

A student says multiplying makes bigger (Jack's mistake). Ask: What is half of 20? Is that bigger than 20? Fix: A multiplier less than 1 takes part: the answer is smaller.
A student finds a common denominator before multiplying. Ask: Does the area picture need the same parts? Fix: Multiply the tops and the bottoms; a common denominator is for adding.
A student multiplies when dividing by a fraction (3 ÷ 1/4 = 3/4). Ask: How many quarters fit into 1? Into 3? Fix: Dividing counts how many fit: multiply by the flipped fraction.

Stretch and support

Stretch: Explain with the area square why 2/3 × 3/4 = 3/4 × 2/3.

Support: Use the board for products; strips of paper folded into quarters for division.

Where it leads

Next: Percentages (node 3), where taking a percentage is multiplying by a fraction of 100.

Where it leads: rates and ratio, and algebraic fractions (Year 10).

What each step is for

half of a third: Committing to whether 1/2 × 1/3 is bigger or smaller than 1/2 surfaces the belief that multiplying always makes bigger.
a third of a half: Shading 1/2 of the columns and 1/3 of the rows shows the overlap is 1/6: smaller than both.
multiply, and divide: Tops times tops and bottoms times bottoms; Jack's belief that multiplying makes bigger fails for fractions less than 1. Dividing counts how many fit, so dividing by a fraction less than 1 makes bigger.
one change at a time: One change at a time: a new product on the board, a fraction of a whole number, dividing by a whole number, dividing by a fraction, then which product is bigger than the number.
multiply and divide: Fractions, a decimal and a division, mixed; 'multiplying makes bigger' is the planted slip; the review keeps adding fractions fresh.
a quarter, another way: Building a product equal to 1/4 other than 1/2 × 1/2 shows many products share a value, and that simplifying is part of multiplying.

Now you can

Multiply two fractions and simplify the answer.
Divide a fraction by a whole number and by a fraction.
Say whether a product will be bigger or smaller than the number I started with.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson14. Percentages of amounts15% of $80 is 0.15 × 80 = $12. Take it off for a discount ($68, or 80 × 0.85 in one step), add it on for an increase ($92, or 80 × 1.15). And 18 out of 24 is 18 ÷ 24 = 75%.

Source: content/maths/number/multiplying-fractions/lesson.md