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# Rates

Number Foundations · Rates

<!-- Speaker note: Students match rates to distances and times on a double number line, tell rates from ratios, find unit rates and convert between rate units such as km/h and m/s (AC9M8M05, node 1). 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** a rate compares two quantities with different units: 60 km per hour, $4 per kg, 80 beats per minute. The unit rate is how much for 1 of the second quantity: divide the first by the second. A ratio compares quantities in the same units. Speeds convert with 1 km/h = 1000 m ÷ 3600 s.

- I can write a rate with its two units.
- I can find a unit rate by dividing in the right order.
- I can convert a speed between km/h and m/s, and tell a rate from a ratio.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need ratios (Number Foundations, Ratios). The one idea: a rate is "this much per one of that", with two different units; divide the first by the second. The common mistake is dividing the wrong way round (Mia's mistake). The board's double number line lines up the two quantities; the dashed line marks the unit rate, and the notes give km/h in m/s. Wider: compare several units for the same speed (km/h, m/s, knots); rates in First Nations land management, such as how fast a cool burn spreads in different conditions (teach with local knowledge where you can). -->

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## Why it matters

> Speed limits, the price of petrol, your heart rate and how fast your phone downloads: all rates. Each one says 'this much per one of that'.

<!-- Speaker note: A **rate** compares two quantities measured in **different units**: 60 km per hour (km/h), $4.50 per kilogram ($/kg), 80 beats per minute. A **ratio** compares quantities in the **same** units: 2 cups of flour to 3 cups of milk, 2 : 3. The **unit rate** is the amount for **one** of the second quantity. Divide the first by the second, and keep the units: $$150 \text{ km in } 2.5 \text{ h} \;\Rightarrow\; 150 \div 2.5 = 60 \text{ km per h}$$ Rates convert like units. For speed: 1 km = 1000 m and 1 h = 3600 s, so $54 \text{ km/h} = 54 \times 1000 \div 3600 = 15 \text{ m/s}$ (divide by 3.6). -->

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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 car travels 150 km in 2 hours at a steady speed. How far does it go in 1 hour?**

- 2/150 km
- 75 km
- 152 km
- 300 km

<!-- Speaker note: What happens: 75 km: 150 ÷ 2 = 75. The car's speed is 75 km per hour. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what '1 hour's worth' is. -->

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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%2Frates%2Fstep-2&code=widget%3A+rate%0Atop%3A+km%2C+distance%0Abottom%3A+h%2C+time%0Aslider+r%3A+20%2C+120%2C+50%2C+5%0Aslider+x%3A+0%2C+6%2C+1%2C+0.5%0A&fresh=1}}

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

<!-- Speaker note: The top line is the distance, the bottom line the time. The dashed line marks 1 hour. -->

<!-- Speaker note: Teacher: ask what happens to the distance when the time doubles. -->

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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?

* 75 km: 150 ÷ 2 = 75. The car's speed is 75 km per hour.

<!-- 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>

## Rate or ratio

Here is the routine.

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

1) $150 \div 2.5 = 60$ — 150 km in 2.5 h.
2) $60 \text{ km/h}$ — Keep both units in the answer.
3) $60 \div 3.6 \approx 16.7 \text{ m/s}$ — To convert km/h to m/s, × 1000 ÷ 3600, which is ÷ 3.6.

<!-- 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 distance ÷ time and say "Divide what you want "per" by the "one" quantity." Line 1: write 150 ÷ 2.5 = 60 and say "150 km in 2.5 h." Line 2: write 60 km/h and say "Keep both units in the answer." Line 3: write 60 ÷ 3.6 16.7 m/s and say "To convert km/h to m/s, × 1000 ÷ 3600, which is ÷ 3.6." -->

<!-- Speaker note: Teacher: ask Mia to say her rate in words. -->

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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>

## Rate or ratio

**Rate or ratio?**

Groups: Rate · Ratio

60 km per hour · $3.50 per kg · 2 : 3 cordial to water · 80 beats per minute · 3 boys to 4 girls · 7 L per 100 km

* Answer: 60 km per hour → Rate; $3.50 per kg → Rate; 2 : 3 cordial to water → Ratio; 80 beats per minute → Rate; 3 boys to 4 girls → Ratio; 7 L per 100 km → Rate

<!-- Speaker note: A rate has two different units; a ratio compares the same kind of thing. -->

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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

**Mia works out the price of apples per kilogram. Click the line where her working goes wrong.**

```
 1  Mia buys 3 kg of apples for $12.
 2  To find the price in dollars per kilogram, she divides 3 by 12: 0.25.
 3  So the apples cost $0.25 per kg.
```

* Answer: line 2

<!-- Speaker note: Line 2: dollars per kilogram is dollars ÷ kilograms: 12 ÷ 3 = $4 per kg. 3 ÷ 12 = 0.25 kg per dollar, a different rate. -->

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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>

## Convert a speed

**Convert 90 km/h to metres per second.**

* Answer: 25

<!-- Speaker note: 90 × 1000 = 90 000 m in 3600 s: 90 000 ÷ 3600 = 25 m/s. -->

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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>

## Convert and compare



- **m/s to km/h:** 
- **Price per kilogram:** 
- **Best buy:** 
- **Heart rate:** 

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

<!-- Speaker note: What this step is for: Converting speeds and prices and comparing two rates with the same unit shows unit rates make comparisons fair. Hints, in order: The board shows m/s in its notes. / 25 m/s × 3.6 = 90 km/h. For students who need more support: A card: 'rate = first quantity ÷ second quantity, units: first per second'. For early finishers: Express the same speed in km/h, m/s and knots (1 knot = 1.852 km/h). Which unit do ships use, and why? The hard part, on purpose: Dividing in the right order. Ask students to say the rate in words ('dollars per kilogram') before they divide: the 'per' quantity goes second. When students get stuck: A student divides the wrong way. Ask: What is it per? That goes second. Fix: Dollars per kilogram = dollars ÷ kilograms. / A student drops the units. Ask: 60 what, per what? Fix: A rate always has two units. / A student multiplies by 3.6 to get m/s. Ask: Is a metre per second faster or slower than a kilometre per hour? Fix: m/s is the smaller number: divide by 3.6. -->

<!-- Speaker note: Teacher: ask why the cheaper bag is not the better buy. -->

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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>

## Speed

**A train travels 240 km in 3 hours. What is its speed in km per hour?**

* Answer: 80

<!-- Speaker note: 240 ÷ 3 = 80 km/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>

## Convert

**Convert 36 km/h to metres per second.**

* Answer: 10

<!-- Speaker note: 36 ÷ 3.6 = 10 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>

## Price per kg

**A 2 kg bag of flour costs $3.40. What is the price in dollars per kilogram?**

* Answer: 1.7

<!-- Speaker note: 3.40 ÷ 2 = $1.70 per kg. -->

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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: ratios

**Review: what is the ratio 12 : 18 in simplest form?**

- 2 : 3
- 3 : 2
- 6 : 9
- 12 : 18 is already simplest

* Answer: 2 : 3

<!-- Speaker note: Divide both by 6: 2 : 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>

## Make: a long drive

Then give the speed in metres per second, to 1 decimal place.

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

<!-- Speaker note: What this step is for: Choosing the rate and the time to match a distance makes students use the unit rate both ways. Hints, in order: 280 ÷ 3.5 = ? / The time is 3 and a half hours. -->

<!-- Speaker note: Teacher: ask for the speed both ways. -->

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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 cyclist rides 45 km in 1.5 hours. Find her speed in km/h and m/s.**

* 45 ÷ 1.5 = 30 km/h; 30 ÷ 3.6 ≈ 8.3 m/s.

<!-- 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 AC9M8M05. Where it leads: using rates to solve problems (next lesson), rate models, and gradient as a rate (Linear relations). -->