---
marp: true
theme: default
html: true
---

# Ratio and rate models

Number Foundations · Modelling

<!-- Speaker note: Students model practical problems as a ratio or a rate (paint coverage, cordial, fuel, sharing, filling), solve them and interpret the answer in the situation (AC9M8M07, node 1). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 10 min · Practise and Make 12 min). -->

---

## Today

**We are learning to** model a practical problem: say what you need, decide whether it is a ratio (parts of the same kind, like cordial to water) or a rate (two different units, like m² per litre), set it up, solve, then interpret the answer in the situation, such as rounding up to whole tins.

- I can decide whether a problem needs a ratio or a rate.
- I can set up and solve a ratio or rate model.
- I can interpret the answer in the situation, such as rounding up to whole tins.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need ratios (Year 7) and rates (previous lessons). The one idea: same kind of quantity, ratio; different units, rate; then interpret the answer in the situation. The common mistake is rounding the wrong way for the context (Ana's mistake). Wider: ratios in First Nations string and cordage making (fibre lengths and twisting), taught with local knowledge where you can; radiocarbon dating as a ratio of isotopes. -->

---

## Why it matters

> Painting a room, mixing cordial for a party, filling a pool, sharing the cost of a gift: the maths is easy once you choose the right model and remember what the answer means.

<!-- Speaker note: To **model** a practical problem: 1. **Say what you need to find**, with its units. 2. **Choose the model.** Quantities of the **same kind** (litres of cordial to litres of water) make a **ratio**. Quantities with **different units** (square metres per litre, litres per minute) make a **rate**. 3. **Set it up and solve.** 4. **Interpret** the answer in the situation: paint comes in whole tins, people come in whole numbers, money is to the cent. -->

---

<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 litre of paint covers 12 m². How much paint do you need for a 30 m² wall?**

- 0.4 L
- 2.5 L
- 42 L
- 360 L

<!-- Speaker note: What happens: 2.5 L: 30 ÷ 12 = 2.5. (360 multiplies the area by the rate; that would be m² × m² per L, which makes no sense.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what units 30 ÷ 12 gives. -->

---

<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%2Frate-models%2Fstep-2&code=widget%3A+rate%0Atop%3A+m%C2%B2%2C+area%0Abottom%3A+L%2C+paint%0Arate%3A+12%0Aslider+x%3A+0%2C+8%2C+1%2C+0.5%0A&fresh=1}}

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

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask how many tins, and why. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 25px; }</style>

---

<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?

* 2.5 L: 30 ÷ 12 = 2.5. (360 multiplies the area by the rate; that would be m² × m² per L, which makes no sense.)

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

---

<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>

## Ratio or rate

Here is the routine.

$$\text{need: litres of paint for 20 m}^2$$

1) $\text{12 m}^2 \text{ per L: a rate}$ — Different units, so a rate.
2) $20 \div 12 \approx 1.7 \text{ L}$ — m² ÷ (m² per L) = L.
3) $\text{buy 2 one-litre tins}$ — Interpret: tins are whole, and 1 would not be enough.

<!-- 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 need: litres of paint for 20 m^2 and say "Say what you are finding, with units." Line 1: write 12 m^2 per L: a rate and say "Different units, so a rate." Line 2: write 20 ÷ 12 1.7 L and say "m² ÷ (m² per L) = L." Line 3: write buy 2 one-litre tins and say "Interpret: tins are whole, and 1 would not be enough." -->

<!-- Speaker note: Teacher: ask Ana whether 2 tins are enough. -->

---

<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>

## Ratio or rate

**Cordial is mixed 1 part cordial to 4 parts water. Is this a ratio or a rate model?**

- A rate: litres per litre
- A percentage
- A ratio: both parts are litres of liquid
- Neither

* Answer: A ratio: both parts are litres of liquid

<!-- Speaker note: Same kind of quantity, same units: a ratio, 1 : 4. -->

---

<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 buys paint. Click the line where her answer stops making sense.**

```
 1  Ana needs 2.4 L of paint for her wall.
 2  The paint is sold in 1 L tins.
 3  2.4 rounds to 2, so she buys 2 tins.
```

* Answer: line 3

<!-- Speaker note: Line 3: 2.4 does round to 2, but 2 tins hold only 2 L, not enough. Here she must round up: 3 tins. -->

---

<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>

## Fuel use

**A car uses 7 L of petrol per 100 km. How many litres does a 350 km trip use?**

* Answer: 24.5

<!-- Speaker note: 350 km is 3.5 lots of 100 km: 3.5 × 7 = 24.5 L. -->

---

<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>

## Share, scale, fill



- **Share a gift:** 
- **Scale a recipe:** 
- **Which model:** 
- **Fill a tank:** 

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

<!-- Speaker note: What this step is for: A ratio share, a recipe scaled by a rate, choosing the model, and a fill time, each interpreted in context. Hints, in order: The ratio is 3 : 5, so 8 parts. / Set the total to $240. For students who need more support: A model card: what do I need (units)? Ratio or rate? Calculation. Does the answer fit the situation? For early finishers: Plan the paint for your classroom: measure the walls, subtract the windows and door, choose a paint and tin size, and cost it. The hard part, on purpose: Choosing the model. Ask students to name the two quantities and their units before they calculate: same units, ratio; different units, rate. When students get stuck: A student multiplies area by coverage. Ask: What units would that give? Fix: m² ÷ (m² per L) = L: divide. / A student uses 4 parts for a 1 : 4 ratio. Ask: How many parts altogether? Fix: 1 + 4 = 5 parts. / A student rounds to the nearest. Ask: Is that enough for the job? Fix: Round to suit the situation: up for tins and buses. -->

<!-- Speaker note: Teacher: ask which of the problems were ratios. -->

---

<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>

## Paint

**5 L of paint at 12 m² per litre covers how many square metres?**

* Answer: 60

<!-- Speaker note: 5 × 12 = 60 m². -->

---

<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>

## Cordial

**Cordial is mixed 1 : 4 with water. How many litres of cordial are in 3 L of drink?**

* Answer: 0.6

<!-- Speaker note: 5 parts make 3 L, so one part is 0.6 L of cordial. -->

---

<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>

## Buses

**A bus seats 45 people. How many buses are needed for 200 people?**

* Answer: 5

<!-- Speaker note: 200 ÷ 45 = 4.4…, and a part bus is still a bus: 5. -->

---

<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: percentage increase

**Review: increase $50 by 10%.**

* Answer: 55

<!-- Speaker note: 50 × 1.1 = $55. -->

---

<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: paint a fence

Then say how many 4 L tins to buy, and why.

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

<!-- Speaker note: What this step is for: A two-step model (area doubled for two coats, then the rate) with an answer to interpret. Hints, in order: Two coats on 42 m² is 84 m² of painting. / 84 ÷ 12 = ? -->

<!-- Speaker note: Teacher: ask how many 4 L tins, and what is left over. -->

---

<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 recipe uses 2 cups of rice to 3 cups of water. How much water for 5 cups of rice? Is this a ratio or a rate?**

* A ratio (cups to cups): 5 × 1.5 = 7.5 cups of water.

<!-- 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: This starts AC9M8M07. Where it leads: checking models and their assumptions (next lesson), and linear models (Linear relations). -->