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# Equivalent and simplest ratios

Number Foundations · Ratios

<!-- Speaker note: Students make copies of a ratio on the board to see that multiplying both terms keeps the ratio, find missing terms, and simplify by dividing both terms by a common factor (AC9M7N08, node 2). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 8 min). -->

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## Today

**We are learning to** 2 : 3 = 4 : 6 = 6 : 9: multiply or divide both numbers by the same number. 12 : 18 simplifies to 2 : 3 by dividing both by 6.

- I can find an equivalent ratio by multiplying or dividing both terms by the same number.
- I can write a ratio in simplest form.
- I can scale a recipe up using equivalent ratios.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need equivalent fractions (the same multiplying and dividing) and highest common factors. The one idea: equivalent ratios are copies of the pattern: both terms multiplied (or divided) by the same number. The common mistake is additive thinking (Ana: 2 → 6 is +4, so 3 → 7). This lesson is the hinge for sharing in a ratio: students who cannot scale and simplify will struggle next. Link to equivalent fractions: 2 : 3 behaves like 2/3 for scaling, but a ratio is not a fraction (part to part). -->

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## Why it matters

> A recipe for 4 uses 2 cups of flour and 3 of milk. How much milk do you need for 10 cups of flour?

<!-- Speaker note: Doubling a recipe keeps the taste the same. Two ratios are **equivalent** when one is the other with *both* numbers multiplied (or both divided) by the same number: $$2 : 3 = 4 : 6 = 6 : 9 \qquad (\times 2,\ \times 3)$$ A ratio is in **simplest form** when no number greater than 1 divides both terms. To simplify, divide both by their highest common factor: $12 : 18$, divide by 6, gives $2 : 3$. It is the same idea as equivalent fractions. Adding the same number to both terms does **not** keep the ratio: $2 : 3$ is not $4 : 5$. -->

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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 ratio is equivalent to $2 : 3$?**

- 4 : 5
- 3 : 4
- 5 : 6
- 4 : 6

<!-- Speaker note: What happens: 4 : 6. Both terms are doubled. 4 : 5 and 3 : 4 each add the same number to both terms, which changes the ratio. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 4 : 5 added 2 to both terms. -->

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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-ratios%2Fstep-2&code=widget%3A+ratio%0Amode%3A+tape%0Alabels%3A+flour%2C+milk%0Aa%3A+2%0Ab%3A+3%0Aslider+k%3A+1%2C+8%2C+1%2C+1%0A&fresh=1}}

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

<!-- Speaker note: The groups of blocks repeat, so the ratio stays the same. -->

<!-- Speaker note: Teacher: ask how many blocks of each colour with 3 copies, then how the numbers are related to 2 and 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>

## What happened?

* 4 : 6. Both terms are doubled. 4 : 5 and 3 : 4 each add the same number to both terms, which changes the ratio.

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

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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>

## Fill in the blank

**Fill in the blank: $2 : 3 = 10 : ?$**

* Answer: 15

<!-- Speaker note: 2 → 10 is × 5, so 3 → 3 × 5 = 15. -->

<!-- Speaker note: Teacher: ask students to draw Ana's recipe as blocks. -->

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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

**Ana scales a recipe. Click the line where her working goes wrong.**

```
 1  A recipe uses 2 cups of flour for every 3 cups of milk.
 2  For 6 cups of flour, how much milk?
 3  Flour went from 2 to 6: that is + 4.
 4  So milk goes from 3 to 7: 3 + 4.
 5  Answer: 7 cups of milk.
```

* Answer: line 4

<!-- Speaker note: Line 4: adding 4 to the milk too is wrong. Flour went × 3 (2 to 6), so milk goes × 3: 3 × 3 = 9 cups. -->

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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

**Simplify $12 : 18$. It is $2 : ?$**

* Answer: 3

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

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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>

## Scale and simplify



- **Simplest form:** 
- **Divide by ten:** 
- **Scale the recipe:** 
- **Spot the impostor:** 

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

<!-- Speaker note: What this step is for: Simplifying on the board, simplifying by hand, scaling a recipe and spotting an impostor use the same rule from several angles. For students who need more support: Keep the copy blocks on the board and count the groups. For early finishers: A ratio is 2 : 3. Find all ratios equivalent to it with both terms less than 30, and say how you know you found all of them. The hard part, on purpose: Multiplying both terms, not adding the same amount to both. Ask 'how many copies is that?' before any arithmetic. When students get stuck: A student adds the same number to both terms (Ana's slip). Ask: Draw 2 flour and 3 milk as blocks. What happens to the blocks when you make 3 copies? Fix: The number of blocks is multiplied, not added to. / A student simplifies one term only. Ask: Did you divide both terms by the same number? Fix: Divide both by their highest common factor. / A student cannot find the multiplier. Ask: What do you multiply 2 by to get 10? Fix: Find the multiplier from the term you know, then use it on the other. -->

<!-- Speaker note: Teacher: for the impostor, ask what the multiplier would have to be. -->

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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 by five

**Simplify $15 : 25$. It is $3 : ?$**

* Answer: 5

<!-- Speaker note: Divide both by 5: 3 : 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>

## Multiply

**Fill in the blank: $5 : 2 = ? : 8$**

* Answer: 20

<!-- Speaker note: 2 → 8 is × 4, so 5 × 4 = 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>

## Pink paint

**Pink paint is 3 parts red to 1 part white. How many parts of white go with 12 parts of red?**

* Answer: 4

<!-- Speaker note: 3 → 12 is × 4, so white is 1 × 4 = 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>

## Review: part to whole

**Review: the ratio of blue to yellow is $3 : 4$. Blue to the whole is $3 : ?$**

* Answer: 7

<!-- Speaker note: 3 + 4 = 7, so blue to the whole is 3 : 7. -->

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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: big copies of 3 : 5

Then explain how you know 24 : 40 is the same ratio as 3 : 5.

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

<!-- Speaker note: What this step is for: Finding copies of 3 : 5 with both terms over 20 shows the multiplier is what keeps the ratio, and many copies work. Hints, in order: Set a = 3 and b = 5 first. / Try 7 copies. -->

<!-- Speaker note: Teacher: collect the multipliers used: all give the same ratio. -->

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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

**Simplify 24 : 36, and find the missing term: 3 : 5 = 12 : ?**

* 24 : 36 = 2 : 3. 3 → 12 is × 4, so 5 × 4 = 20.

<!-- 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: Sharing in a ratio, which uses equivalent ratios to split a total. Where it leads: rates and unit rates, scale drawings, similar shapes, and proportion in Years 8 and 9. -->