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# Equivalent fractions

Number Foundations · Fractions and decimals

<!-- Speaker note: Students match shaded amounts on fraction bars to find equivalent fractions, learn that multiplying or dividing the top and the bottom by the same number keeps the amount, and simplify by the highest common factor (AC9M7N04, node 1). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** fractions are equal when they shade the same amount. Multiply or divide the top and the bottom by the same number: 2/3 = 4/6 = 8/12. Dividing by the highest common factor gives the simplest form.

- I can find equivalent fractions by multiplying or dividing the top and the bottom by the same number.
- I can write a fraction in its simplest form using the highest common factor.
- I can explain with a picture why two fractions are equal.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need fractions as parts of a whole (Year 6) and factors and highest common factors (Year 7 Number). The one idea: equal fractions shade the same amount; the same multiplication or division on the top and the bottom keeps the amount. The common mistake is adding the same number to both (2/3 = 3/4); it is planted in Lily's working. Simplest-form answers are marked by form: an equal fraction that can still be simplified is told so, not marked right. -->

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## Why it matters

> Half a pizza cut into 2 slices or into 8: the same amount to eat. The numbers change; the amount does not.

<!-- Speaker note: A fraction names an amount: $\tfrac{2}{3}$ is 2 of 3 equal parts. The top number is the **numerator** (how many parts) and the bottom number the **denominator** (how many equal parts in the whole); this lesson calls them the top and the bottom. The same amount can be cut into more parts, or fewer, and named by a different fraction. Fractions that name the same amount are **equivalent**. In this lesson you find equivalent fractions on fraction bars, then write a fraction in its **simplest form**. -->

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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

**Which fraction is the same amount as $\tfrac{2}{3}$?**

- $\tfrac{4}{5}$
- $\tfrac{3}{4}$
- $\tfrac{2}{6}$
- $\tfrac{4}{6}$

<!-- Speaker note: What happens: 4/6. Split each third into 2: 6 parts, and the 2 shaded thirds become 4 shaded sixths. 3/4 (add 1 to both) is more than 2/3. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 3/4 added 1 to both. -->

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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%2Fequivalent-fractions%2Fstep-2&code=widget%3A+bars%0Aa%3A+2%0Ab%3A+3%0Aslider+c%3A+0%2C+12%2C+1%0Aslider+d%3A+1%2C+12%2C+2%0A&fresh=1}}

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

<!-- Speaker note: $d$ is the number of parts in the orange bar and $c$ the number shaded. The board says when the two shade the same amount. -->

<!-- Speaker note: Teacher: ask what happened to each third when the orange bar went to 6 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?

* 4/6. Split each third into 2: 6 parts, and the 2 shaded thirds become 4 shaded sixths. 3/4 (add 1 to both) is more than 2/3.

<!-- 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>

## Simplest form

Here is how to find equivalent fractions, and the simplest one.

A fraction is in **simplest form** (lowest terms) when the top and the bottom have no common factor but 1. Try one: simplify $\tfrac{12}{18}$.

$$\tfrac{2}{3} = \tfrac{2 \times 4}{3 \times 4} = \tfrac{8}{12}$$

1) $\tfrac{12}{18} = \tfrac{12 \div 2}{18 \div 2} = \tfrac{6}{9}$ — Join parts: divide the top and the bottom by a common factor.
2) $\tfrac{6}{9} = \tfrac{6 \div 3}{9 \div 3} = \tfrac{2}{3}$ — 6 and 9 still share 3, so divide again.
3) $\tfrac{12}{18} = \tfrac{12 \div 6}{18 \div 6} = \tfrac{2}{3}$ — Or in one step: divide by the highest common factor, 6.

<!-- 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 = 2 × 4 ÷ 3 × 4 = 8 ÷ 12 and say "Split each part into 4: multiply the top and the bottom by 4." Line 1: write 12 ÷ 18 = 12 ÷ 2 ÷ 18 ÷ 2 = 6 ÷ 9 and say "Join parts: divide the top and the bottom by a common factor." Line 2: write 6 ÷ 9 = 6 ÷ 3 ÷ 9 ÷ 3 = 2 ÷ 3 and say "6 and 9 still share 3, so divide again." Line 3: write 12 ÷ 18 = 12 ÷ 6 ÷ 18 ÷ 6 = 2 ÷ 3 and say "Or in one step: divide by the highest common factor, 6." -->

<!-- Speaker note: Teacher: model the worked example with the bars: split each part (multiply), join parts (divide). -->

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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>

## Simplest form

**Write $\tfrac{12}{18}$ in its simplest form.**

* Answer: 2/3

<!-- Speaker note: The highest common factor of 12 and 18 is 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 2/3. -->

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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

**Lily looks for a fraction equal to $\tfrac{2}{3}$. Click the line where her working goes wrong.**

```
 1  Find a fraction equal to 2/3.
 2  Do the same to the top and the bottom.
 3  Add 1 to the top and add 1 to the bottom.
 4  So 2/3 = 3/4.
```

* Answer: line 3

<!-- Speaker note: Line 3: adding changes the amount (3/4 is more than 2/3). Doing the same to the top and the bottom means multiplying or dividing: 2/3 = 4/6. -->

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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>

## Why they are equal

**Why is $\tfrac{2}{3} = \tfrac{8}{12}$?**

- 8 − 2 = 6 and 12 − 3 = 9
- 8 and 12 are both even
- Each third is split into 4 twelfths, so 2 thirds are 8 twelfths

* Answer: Each third is split into 4 twelfths, so 2 thirds are 8 twelfths

<!-- Speaker note: × 4 on the top and the bottom: the same amount, cut into 4 times as many parts. -->

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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



- **Twelfths:** Each task changes one thing.
- **A missing number:** 
- **Simplify:** 
- **Fewest parts:** 
- **Already simplest:** 

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

<!-- Speaker note: What this step is for: One change at a time: a different fraction on the board, a missing number, simplifying, simplifying on the board, then the edge case of a fraction already in lowest terms. Hints, in order: Each quarter split into 3 makes twelfths. / Make d smaller: try 6 parts, then 3. / Keep the same amount shaded. For students who need more support: Use the bars for every question; physical fraction walls help too. For early finishers: Find all the fractions equal to 3/4 with a bottom number less than 50. What do their bottoms have in common? When students get stuck: A student adds the same number to the top and the bottom (Lily's mistake). Ask: Set both on the bars. Is the same amount shaded? Fix: Same means the same multiplication or division: 2/3 = 4/6. / A student simplifies in steps and stops too early (6/9 for 12/18). Ask: Do the top and the bottom still share a factor? Fix: Keep dividing, or divide by the highest common factor once. / A student cannot find the missing number in 3/5 = ?/20. Ask: What was the bottom multiplied by? Fix: Do the same to the top. -->

<!-- Speaker note: Teacher: before the fewest-parts task, ask: how will you know you cannot join any more? -->

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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>

## Simplify

**Write $\tfrac{18}{24}$ in its simplest form.**

* Answer: 3/4

<!-- Speaker note: HCF 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4. -->

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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>

## Missing number

**Fill the gap: $\tfrac{5}{8} = \tfrac{15}{?}$.**

* Answer: 24

<!-- Speaker note: × 3: 8 × 3 = 24. -->

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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>

## Compare

**Which is bigger, $\tfrac{2}{3}$ or $\tfrac{3}{5}$? Write both with 15 parts.**

- 3/5 (it has bigger numbers)
- They are equal
- 2/3 (10/15 is more than 9/15)

* Answer: 2/3 (10/15 is more than 9/15)

<!-- Speaker note: 2/3 = 10/15 and 3/5 = 9/15, so 2/3 is bigger. -->

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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: integers

**Review: work out $-3 + 5$.**

* Answer: 2

<!-- Speaker note: Start at −3 and move 5 right: 2. -->

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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: neither in lowest terms

Then: what is the simplest form of both your fractions?

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

<!-- Speaker note: What this step is for: Building two equal fractions where neither is in lowest terms shows that one amount has many names, all multiples of its simplest one. Hints, in order: Start with a simple fraction such as 1/2 and multiply its top and bottom by 2 for one bar. / Multiply by 3 for the other bar. -->

<!-- Speaker note: Teacher: collect pairs and ask for the simplest form of each. -->

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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

**Write 20/32 in its simplest form, and give one other fraction equal to it.**

* 5/8 (÷ 4). For example 10/16 or 15/24.

<!-- 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: Fractions, decimals and percentages (node 2), then Fractions and decimals on the line (node 3). Where it leads: adding and subtracting fractions with different denominators (common denominators), and ratio. -->