At a glance. One lesson of about 60 minutes. Students find unit rates by dividing, say the unit of a rate, compare rates fairly and add rates that share a unit of time.
Calculator: no calculator. Every division here comes out whole or to one decimal place; the skill is choosing what to divide by, not the division.
What students will learn
Know
- A rate compares two quantities in different units.
- The unit rate is the amount for one of the second quantity, and its unit has “per” in it.
- Rates can be compared, or added, when they are per the same thing.
Understand
- A bigger total is not a bigger rate: the time (or the amount) has to be accounted for.
- The same rate can be written per minute or per hour by multiplying or dividing by 60.
- A rate is a ratio of unlike quantities, so a unit goes with the number.
Do
- Find a unit rate by dividing, and say its unit.
- Compare two rates, converting to the same “per” first.
- Use a rate to find an amount over a longer time.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding and using unit rates (questions 1 to 7).
- Criterion B. Investigating patterns: the first figure. Ask which settings give columns of the same height (12 litres in 3 minutes and 24 litres in 6 minutes both give 4); listen for “the same amount in every minute”.
- Criterion C. Communicating: writing a rate with its unit and justifying a comparison in a sentence (question 9).
- Criterion D. Applying mathematics in real-life contexts: hosepipes, typing, fuel, sewing (the hook and questions 3 to 8).
The one thing to do with the figure
Ask for a prediction before anyone moves a slider. “A bucket hose fills 10 litres in 2 minutes and a drum hose fills 200 litres in 25 minutes. Which is faster?” Take both answers from the room, then ask what you would need to work out. The figure shows the idea with one tap; the worked example settles the hose question.
Then use the second figure to separate the total from the rate: find a setting where tap B delivers more litres and is slower, and one where it delivers fewer and is faster. Ask students to say which number they looked at.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the two hosepipes, with predictions. The first figure on the board. |
| 10 min | Key idea and the second figure; students find the three kinds of setting (faster, equal, slower). |
| 10 min | Both worked examples; students finish the faded sprinkler example in pairs. |
| 5 min | Question 1, the check. If many choose B, redraw it as litres per kilometre against kilometres per litre. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 3 : 5; 12; n = 7. |
|---|---|
| Worked example 1 | 5 and 8 litres per minute; the drum hose is faster. |
| Worked example 2 | 25 pages per minute, 1500 pages per hour. |
| Faded example | 6 m² per minute, 360 m² per hour. |
| 1. Fuel use as a rate | A, 8 km per litre. B has the units the wrong way round; C subtracts; D multiplies. |
| 2. Unit rate | 14. 84 ÷ 6 = 14 loaves per hour. |
| 3. Price per kilogram | 110. 275 ÷ 2.5 = 110 rupees per kilogram. |
| 4. Greatest rate | B. The rates are 24, 26, 25 and 13 words per minute. C has the most words but is not fastest. |
| 5. Changing the per | 15. 7.5 × 2 = 15 km per hour. |
| 6. Adding rates | 11. 42 ÷ 7 = 6, and 5 + 6 = 11 litres per minute. |
| 7. Comparing across units | 6. 50 m per minute and 300 m per minute; 300 ÷ 50 = 6. |
| 8. A rate over a long time | 72. 480 ÷ 20 = 24 periods, and 24 × 3 = 72 shirts. |
| 9. More is not better | B. A: uses the litres alone. D: the right bus for the wrong reason (distance alone). C: not checked by dividing. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1; 8 litres per km | Wrote 8 litres per km: the 8 is km for one litre, not litres for one km. | Ask “for one what?” and have them say the sentence: in one litre, the van travels 8 km. |
| 78 in question 2; 272.5 in question 3; 47 in question 6 | Subtracted, or added, two quantities in different units. | Ask whether litres can be taken from minutes. Different units cannot be added or subtracted; they can be divided. |
| 504 in question 2; 687.5 in question 3; 225 in question 5 | Multiplied the two numbers instead of dividing. | Check by multiplying back: unit rate times the second quantity should give the first. |
| 0.25 in question 5 | Stopped at the kilometres in one minute and did not change the time unit. | Underline the unit the question asks for. |
| 1 in question 6 | Subtracted the rates, thinking of a comparison. | Ask: is one tap emptying the tank? They add because both fill it. |
| 250 or 0.0075 in question 7 | Found a difference, or divided kilometres by metres without converting. | Write both as metres per minute first, then divide. |
| 24 or 1440 in question 8 | Stopped at the number of periods, or used 3 per minute. | Ask: how long does 3 shirts take, and how many of those fit? |
| A, C or D in question 9 | Used the litres or the distance alone, not the kilometres for one litre. | Ask how far each bus goes on one litre. |
Questions worth asking
- Is 9 a rate? No: 9 litres per minute is. Listen for a student insisting on the unit.
- Why can two taps have different totals and the same rate? Because the times differ. Listen for “same amount in each minute”.
- How is a rate different from a ratio? A ratio compares two amounts in the same unit; a rate compares different units. Listen for an example of each.
- Can you change 25 pages per minute to pages per hour without a calculator? Multiply by 60. Ask which of 25 × 6 and 25 × 60 is easier and why.
Differentiation
A way in for anyone stuck
- Give a small table with the headings amount, time, amount for one, and fill the first row together.
- Keep the numbers to those that divide exactly (12 ÷ 3, 40 ÷ 4) until the idea is secure.
- Say the unit aloud every time: “nine litres per minute”, never “nine”.
Stretch
- Question 6 and the what-if: add 5 litres per minute and 30 litres per hour in each unit. (5.5 litres per minute, 330 litres per hour.)
- A tap fills a 90-litre tank in 15 minutes. A second tap is also opened and the tank then fills in 9 minutes. What is the second tap's rate alone? (6 and 10 litres per minute, so 4 litres per minute.)
Students learning in English as an additional language
“Per” means “for each one”. Say “litres per minute” as “litres for each minute” the first few times.
| Word | Plain meaning | Example |
|---|---|---|
| rate | a comparison of two quantities in different units | 36 litres in 4 minutes |
| unit rate | the amount for one of the second quantity | 9 litres per minute |
| per | for each one | rupees per kilogram |
| steady | the same amount in each equal time | a steady 9 litres a minute |
| compare | put two things side by side to see which is greater | 7 > 3 |
Sentence starters for explaining aloud or in writing:
- The rate is … per …, because … ÷ … = …
- … is faster than … because in one minute it …
- I cannot compare these yet, because …
Two things not to say
“Just divide the big number by the small number.” Sometimes the small number is the quantity (7.5 km in 30 minutes), and dividing the larger by the smaller gives the wrong rate. Ask “for one what?”.
“Faster means more.” More litres, more kilometres or more words says nothing until the time is accounted for.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Unit rateA cyclist rides 45 km in 3 hours. Write her speed as a rate.
15 km per hour. - PriceA shop sells 8 kg of flour for 360 rupees. What is the price per kilogram?
45 rupees per kilogram.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- PayTomás is paid 4200 pesos for 6 hours of work and Lucía 3640 pesos for 5 hours. Who earns more per hour, and by how much?
Tomás: 700 pesos per hour. Lucía: 728 pesos per hour. Lucía earns 28 pesos more each hour. - ReadingA student reads 36 pages in 40 minutes. At that rate, how many pages in 2 hours?
108 pages: 36 pages every 40 minutes is 3 lots of 40 minutes in 2 hours, and 3 × 36 = 108. - Earlier work (Year 8, Unit 1)Simplify the ratio 24 : 40.
3 : 5. The highest common factor is 8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningOne machine prints 120 leaflets in 8 minutes and another prints 100 leaflets in 5 minutes. Working together, how long do they take to print 420 leaflets?
15 and 20 leaflets per minute, so 35 per minute together, and 420 ÷ 35 = 12 minutes. - Choosing a unitA snail moves 3 metres in an hour and a beetle 4 cm in a second. Which is faster, and by how many times?
The snail goes 3 m = 300 cm in an hour. The beetle goes 4 × 3600 = 14400 cm in an hour. The beetle is faster: 14400 ÷ 300 = 48 times as fast.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All four sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Pour water or sand into equal-width columns drawn on the board, or use squared paper and shade equal columns.