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# What is a ratio?

Number Foundations · Ratios

<!-- Speaker note: Students build ratios from rows of equal blocks, write part-to-part and part-to-whole ratios in the order asked, and link part to whole to a fraction (AC9M7N08, node 1). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** 2 red counters and 3 blue: the ratio of red to blue is 2 : 3 (part to part); red to all the counters is 2 : 5 (part to whole). Order matters.

- I can write a ratio of two quantities in a : b form, in the order asked.
- I can tell part to part from part to whole.
- I can write a part-to-whole ratio as a fraction.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need fractions as parts of a whole (Equivalent fractions). The one idea: a ratio compares quantities; the order in which they are named is the order of the numbers. The common mistakes are writing part to part when part to whole was asked (Pip), reversing the order, and using the whole (5) as a part. Use the same bag all lesson so students see 2 : 3 and 2 : 5 about the same objects. -->

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## Why it matters

> A fruit punch is 2 cups of juice for every 3 cups of lemonade. Swap them and it tastes completely different. How do we write a recipe like that down?

<!-- Speaker note: A **ratio** compares two (or more) quantities of the same kind. "2 red to 3 blue" is written $2 : 3$. The order matters: red first means 2 comes first. There are two kinds of comparison: - **Part to part:** red to blue is $2 : 3$. - **Part to whole:** red to *all* the counters is $2 : 5$, because there are $2 + 3 = 5$ counters in all. This is the same as the fraction $\tfrac{2}{5}$. -->

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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

**A bag has 2 red counters and 3 blue counters. What is the ratio of red to blue?**

- 5 : 2
- 2 : 5
- 2 : 3
- 3 : 2

<!-- Speaker note: What happens: 2 : 3. Red comes first, so 2 comes first. 3 : 2 is blue to red; 2 : 5 is red to all the counters. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 3 : 2 reversed the order; 2 : 5 compared to the whole. -->

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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%2Fratios%2Fstep-2&code=widget%3A+ratio%0Amode%3A+tape%0Alabels%3A+red%2C+blue%0Aslider+a%3A+1%2C+8%2C+2%2C+1%0Aslider+b%3A+1%2C+8%2C+3%2C+1%0A&fresh=1}}

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

<!-- Speaker note: The board always shows both: part to part, and part to whole. -->

<!-- Speaker note: Teacher: ask students to say both ratios aloud, e.g. 'three to five' and 'three to eight'. -->

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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?

* 2 : 3. Red comes first, so 2 comes first. 3 : 2 is blue to red; 2 : 5 is red to all the counters.

<!-- 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>

## Whole bag

**A bag has 2 red and 3 blue counters. The ratio of red to all the counters is $2 : ?$**

* Answer: 5

<!-- Speaker note: 2 + 3 = 5 counters in all, so red to all is 2 : 5. -->

<!-- Speaker note: Teacher: have students hold up the whole bag and a part to show which is which. -->

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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

**Pip finds the ratio of apples to all the fruit. Click the line where the working goes wrong.**

```
 1  A bowl has 4 apples and 3 pears. Find the ratio of apples to all the fruit.
 2  There are 4 + 3 = 7 pieces of fruit altogether.
 3  Apples to all the fruit: 4 : 3
 4  So 4 of every 7 pieces are apples.
```

* Answer: line 3

<!-- Speaker note: Line 3: 4 : 3 is apples to pears. Apples to all the fruit is 4 : 7, the 7 from line 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>

## Which way round?

**A drink has 2 cups of blue cordial and 5 cups of water. What is the ratio of water to cordial?**

- 5 : 2
- 7 : 2
- 5 : 7
- 2 : 5

* Answer: 5 : 2

<!-- Speaker note: Water first: 5, then cordial: 2. The ratio of water to cordial is 5 : 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>

## Change the bag



- **Dogs and cats:** 
- **Add the parts:** 
- **As a fraction:** 
- **Strong or weak?:** 

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

<!-- Speaker note: What this step is for: Changing one number at a time, then turning part to whole into a fraction, links ratio to the fractions students know. For students who need more support: Draw one block for each item and colour them; count before writing. For early finishers: A ratio of boys to girls is 3 : 4. What fraction of the group are girls? What if 2 more boys join, no more girls? Is the ratio still 3 : 4? The hard part, on purpose: Telling part to part from part to whole. Ask 'is the second number a part, or the whole bag?' every time before writing. When students get stuck: A student writes 4 : 3 for apples to all the fruit (Pip's slip). Ask: How many pieces of fruit are there altogether? Fix: The second number is the whole: 4 : 7. / A student writes the ratio the wrong way round. Ask: Which is named first in the question? Fix: The first thing named is the first number. / A student says 3 : 5 and 3/5 are the same. Ask: What is the whole in each? Fix: 3 : 5 is part to part, 3/8 is part to whole. -->

<!-- Speaker note: Teacher: ask which ratio matches the fraction 3/8. -->

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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>

## Pencils to pens

**A box has 5 pens and 8 pencils. The ratio of pencils to pens is $8 : ?$**

* Answer: 5

<!-- Speaker note: Pencils first: 8, then pens: 5. 8 : 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>

## Pens to all

**In the same box, the ratio of pens to all the items is $5 : ?$**

* Answer: 13

<!-- Speaker note: 5 + 8 = 13 items, so pens to all the items is 5 : 13. -->

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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>

## Wins to losses

**A team won 7 games and lost 3. The ratio of wins to losses is $7 : ?$**

* Answer: 3

<!-- Speaker note: Wins first: 7. Losses: 3. 7 : 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>

## Review: estimating

**Review: estimate $6.8 \times 31$.**

* Answer: 210

<!-- Speaker note: 6.8 ≈ 7 and 31 ≈ 30, so 7 × 30 = 210. -->

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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 three-colour mix

Then write the ratio of the first colour to the whole.

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

<!-- Speaker note: What this step is for: A ratio can have three parts, and the whole is all of them added; making 10 parts shows many ratios give the same total. Hints, in order: 3 + ? + ? = 10. / Try 3 : 2 : 5 or 3 : 4 : 3. -->

<!-- Speaker note: Teacher: collect ratios; check each adds to 10. -->

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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

**A jar has 4 black and 5 white beads. Write the ratio of black to white, and of black to all the beads.**

* 4 : 5 and 4 : 9.

<!-- 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: Equivalent ratios, scaling a ratio up and down. Where it leads: sharing in a ratio, rates, scale drawings, and proportion in Years 8 and 9. -->