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# Multiplying and dividing fractions

Number Foundations · Fractions and decimals

<!-- Speaker note: 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 (Predict and Run 8 min · Investigate 15 min · Modify 10 min · Practise and Make 12 min). -->

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## Today

**We are learning to** 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.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> 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.

<!-- Speaker note: "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. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Is $\tfrac{1}{2} \times \tfrac{1}{3}$ bigger or smaller than $\tfrac{1}{2}$?**

- The same
- Bigger
- Smaller

<!-- Speaker note: What happens: Smaller: 1/2 × 1/3 = 1/6, a third of a half. Multiplying by a number less than 1 takes only part. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose bigger hold the 'multiplying makes bigger' belief. Keep the vote visible. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fmultiplying-fractions%2Fstep-2&code=widget%3A+fracarea%0Aslider+a%3A+0%2C+6%2C+1%0Aslider+b%3A+1%2C+8%2C+2%0Aslider+c%3A+0%2C+6%2C+1%0Aslider+d%3A+1%2C+8%2C+1%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/multiplying-fractions/?full=step-2#step-2)

<!-- Speaker note: $a$ of $b$ columns are blue; $c$ of $d$ rows are orange. The overlap is purple. -->

<!-- Speaker note: Teacher: ask why the overlap is smaller than both shaded parts. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* Smaller: 1/2 × 1/3 = 1/6, a third of a half. Multiplying by a number less than 1 takes only part.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Multiply across; divide by flipping

Here is how each kind works.

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

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

1) $\tfrac{3}{4} \times 20 = \tfrac{60}{4} = 15$ — A whole number is over 1: 20 = 20/1. 15 is less than 20.
2) $\tfrac{1}{2} \div 3 = \tfrac{1}{6}$ — Share a half into 3: each share is a sixth (the same as × 1/3).
3) $3 \div \tfrac{1}{4} = 3 \times 4 = 12$ — How many quarters in 3? Dividing by 1/4 is multiplying by 4.
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).
5) $0.4 \times 3 = 1.2$ — Decimals: 4 tenths × 3 = 12 tenths.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 2 ÷ 3 × 3 ÷ 4 = 2 × 3 ÷ 3 × 4 = 6 ÷ 12 = 1 ÷ 2 and say "Multiply the tops and the bottoms, then simplify." Line 1: write 3 ÷ 4 × 20 = 60 ÷ 4 = 15 and say "A whole number is over 1: 20 = 20/1. 15 is less than 20." Line 2: write 1 ÷ 2 ÷ 3 = 1 ÷ 6 and say "Share a half into 3: each share is a sixth (the same as × 1/3)." Line 3: write 3 ÷ 1 ÷ 4 = 3 × 4 = 12 and say "How many quarters in 3? Dividing by 1/4 is multiplying by 4." Line 4: write 2 ÷ 3 ÷ 1 ÷ 6 = 2 ÷ 3 × 6 = 4 and say "Divide by a fraction: multiply by it flipped (1/6 flipped is 6/1 = 6)." Line 5: write 0.4 × 3 = 1.2 and say "Decimals: 4 tenths × 3 = 12 tenths." -->

<!-- Speaker note: Teacher: contrast with adding: no common denominator needed here. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Multiply across; divide by flipping

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

* Answer: 2/5

<!-- Speaker note: (2 × 3)/(3 × 5) = 6/15 = 2/5. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

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

```
 1  Which is bigger: 3/4 × 20, or 20?
 2  3/4 × 20 means 3/4 of 20.
 3  Multiplying always makes a number bigger.
 4  So 3/4 × 20 is bigger than 20.
```

* Answer: line 3

<!-- Speaker note: Line 3: multiplying by a number less than 1 makes smaller. 3/4 of 20 is 15, less than 20. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## How many fit?

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

* Answer: 12

<!-- Speaker note: 4 quarters in each whole, so 12 in 3: 3 ÷ 1/4 = 3 × 4 = 12. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Three quarters of two fifths:** Each task changes one thing.
- **Of a whole number:** 
- **Divide by a whole number:** 
- **Divide by a fraction:** 
- **Bigger or smaller?:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/multiplying-fractions/?full=step-4#step-4)

<!-- Speaker note: What this step is for: 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. Hints, in order: Columns: 3 of 4. Rows: 2 of 5. For students who need more support: Use the board for products; strips of paper folded into quarters for division. For early finishers: Explain with the area square why 2/3 × 3/4 = 3/4 × 2/3. 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. -->

<!-- Speaker note: Teacher: for the last task, ask students to sort multipliers into 'less than 1' and 'more than 1'. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Multiply

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

* Answer: 1/2

<!-- Speaker note: 20/40 = 1/2. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Decimals

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

* Answer: 2.4

<!-- Speaker note: 0.6 × 4 = 2.4 L. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Divide in context

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

* Answer: 6

<!-- Speaker note: 3/4 = 6/8: six eighths. (3/4 ÷ 1/8 = 3/4 × 8 = 6.) -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: adding

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

* Answer: 7/10

<!-- Speaker note: 5/10 + 2/10 = 7/10. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a quarter, another way

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

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/multiplying-fractions/?full=step-6#step-6)

<!-- Speaker note: What this step is for: 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. Hints, in order: Try 1/2 of something that is 1/2 once simplified, such as 2/4. / Or 3/4 × 1/3. -->

<!-- Speaker note: Teacher: collect products equal to 1/4; ask what they all simplify to. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Work out 3/5 × 10/12, and 2 ÷ 1/3.**

* 30/60 = 1/2. 2 ÷ 1/3 = 6 (six thirds in 2).

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this 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). -->