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# Using rates

Number Foundations · Rates

<!-- Speaker note: Students use rates both ways (total = rate × amount, amount = total ÷ rate) for petrol, speed, water and exchange rates, letting the units decide multiply or divide (AC9M8M05, node 2). About 35 minutes (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

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## Today

**We are learning to** use a rate to find a total (rate × amount), or the amount (total ÷ rate). For speed: distance = speed × time and time = distance ÷ speed. Show the units at every step, and check the answer makes sense.

- I can use a rate to find a total or an amount.
- I can use distance, speed and time in any order.
- I can check an answer with its units and its size.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need rates and unit rates (previous lesson). The one idea: a rate works both ways; the units decide whether to multiply or divide. The common mistake is multiplying distance by speed to find a time (Leo's mistake). The exchange rate in the lesson (0.65 US dollars per Australian dollar) is a round example, not today's rate. Wider: plan holiday spending with a current exchange rate (and its fees); compare two phone plans. -->

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## Why it matters

> How much will it cost to fill the car? How long is the drive to the coast? How many dollars for your holiday money? Each answer comes from one rate, used the right way round.

<!-- Speaker note: A rate links two quantities, so you can go either way: - **total = rate × amount**: 30 L of petrol at $2.05 per L costs $30 × 2.05$ = \$61.50; - **amount = total ÷ rate**: \$61.50 at $2.05 per L buys $61.50 \div 2.05 = 30$ L. For speed, **distance = speed × time** and **time = distance ÷ speed**. Write the units at each step: they tell you whether to multiply or divide (km ÷ km/h = h). Then check: is the answer about the right size? -->

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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

**Petrol costs $2.05 per litre. How much do 40 litres cost?**

- $8.20
- $19.51
- $82
- $42.05

<!-- Speaker note: What happens: $82: 40 × $2.05 = $82. (40 ÷ 2.05 = 19.51 would be how many litres $40 buys.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what 40 ÷ 2.05 would mean. -->

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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%2Fusing-rates%2Fstep-2&code=widget%3A+rate%0Atop%3A+%24%2C+cost%0Abottom%3A+L%2C+petrol%0Arate%3A+2.05%0Aslider+x%3A+0%2C+60%2C+10%2C+5%0A&fresh=1}}

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

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask how much one more litre adds. -->

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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?

* $82: 40 × $2.05 = $82. (40 ÷ 2.05 = 19.51 would be how many litres $40 buys.)

<!-- 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>

## Time

Here is the routine.

$$\text{time} = \text{distance} \div \text{speed}$$

1) $200 \div 80 = 2.5$ — Divide: 200 km at 80 km/h.
2) $2.5 \text{ h} = 2 \text{ h } 30 \text{ min}$ — Interpret: half an hour is 30 minutes.
3) $80 \times 2.5 = 200$ — Check by going back.

<!-- 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 time = distance ÷ speed and say "Choose by the units: km ÷ km/h = h." Line 1: write 200 ÷ 80 = 2.5 and say "Divide: 200 km at 80 km/h." Line 2: write 2.5 h = 2 h 30 min and say "Interpret: half an hour is 30 minutes." Line 3: write 80 × 2.5 = 200 and say "Check by going back." -->

<!-- Speaker note: Teacher: ask Leo whether 9000 hours makes sense. -->

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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>

## Time

**How long does it take to drive 270 km at 90 km/h, in hours?**

* Answer: 3

<!-- Speaker note: 270 ÷ 90 = 3 hours. -->

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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

**Leo works out a journey time. Click the line where his working goes wrong.**

```
 1  Leo drives 150 km at 60 km/h. How long does it take?
 2  Time = distance × speed = 150 × 60.
 3  So it takes 9000 hours.
```

* Answer: line 2

<!-- Speaker note: Line 2: time = distance ÷ speed. 150 ÷ 60 = 2.5 hours. Units check: km ÷ km per h = h. -->

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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>

## Exchange rates

**Say the exchange rate is 1 Australian dollar = 0.65 US dollars. How many US dollars do you get for 200 Australian dollars?**

* Answer: 130

<!-- Speaker note: 200 × 0.65 = 130 US dollars. -->

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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>

## Race, shower, money back



- **Catch up:** 
- **Money back:** 
- **A shower:** 
- **A dripping tap:** 

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

<!-- Speaker note: What this step is for: Comparing two rates over time, working backwards from a total, and converting money both ways. Hints, in order: Car B gains 15 km every hour. / 45 ÷ 15 = ? For students who need more support: A card: total = rate × amount; amount = total ÷ rate; units in every line. For early finishers: Plan a trip budget overseas: look up today's exchange rate, convert three prices, and include a 3% card fee. The hard part, on purpose: Choosing to multiply or divide. Ask students to write the units into the calculation (km ÷ km/h = h) before they press a key. When students get stuck: A student multiplies to find a time. Ask: What units does your answer have? Fix: km ÷ km/h gives hours: divide. / A student writes 2.5 hours as 2 h 50 min. Ask: What is half an hour in minutes? Fix: 0.5 h = 30 min. / A student converts money the wrong way. Ask: Should you get more or fewer US dollars than Australian dollars? Fix: Check the size: 1 AUD buys less than 1 USD here. -->

<!-- Speaker note: Teacher: ask why the gap grows by the same amount each hour. -->

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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>

## Cost

**Grapes cost $4.20 per kg. How much do 3.5 kg cost, in dollars?**

* Answer: 14.7

<!-- Speaker note: 3.5 × 4.20 = $14.70. -->

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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>

## Distance

**A bus averages 85 km/h for 2.5 hours. How far does it go, in km?**

* Answer: 212.5

<!-- Speaker note: 85 × 2.5 = 212.5 km. -->

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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>

## Time

**A cyclist rides 45 km at 15 km/h. How many hours does it take?**

* Answer: 3

<!-- Speaker note: 45 ÷ 15 = 3 hours. -->

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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: m/s

**Review: convert 72 km/h to metres per second.**

* Answer: 20

<!-- Speaker note: 72 ÷ 3.6 = 20 m/s. -->

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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>

## Pay

**A part-time job pays $18.50 per hour. How much is 6.5 hours of work, in dollars?**

* Answer: 120.25

<!-- Speaker note: 18.50 × 6.5 = $120.25. -->

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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>

## Run rate

**A cricket team needs 120 runs off the last 15 overs. What run rate (runs per over) does it need?**

* Answer: 8

<!-- Speaker note: 120 ÷ 15 = 8 runs per over. -->

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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: plan a drive

Then say whether your speed is a sensible average for a country drive with a stop, and what the model leaves out.

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

<!-- Speaker note: What this step is for: Choosing a speed and time for a fixed distance uses distance = speed × time in both directions. Hints, in order: 360 ÷ 4.5 = ? / The time is 4 and a half hours. -->

<!-- Speaker note: Teacher: ask what the model leaves out. -->

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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 plane flies 2100 km at 700 km/h. How long does it take?**

* 2100 ÷ 700 = 3 hours.

<!-- 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 completes AC9M8M05. Where it leads: rate models (next lessons), gradient as a rate (Linear relations), and speed in science. -->