Rates: a week of ready-to-teach maths
Five lessons for Year 8 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a distance-time graph, twelve rate-or-ratio cards, twelve supermarket shelf cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets, one copy per pair: the rate-or-ratio cards are for Day 1, the shelf cards for Day 3.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
A rate compares two quantities measured in different units, such as kilometres and hours, or dollars and kilograms, so a rate always carries two units: 80 km/h, $4.50/kg. (A ratio compares quantities in the same unit, such as 3 cups to 2 cups.) A unit rate is the amount for one of the second quantity, found by dividing, and once it is known, multiplying scales it to any amount. Unit rates make fair comparisons: two packs of different sizes are compared by their price per kilogram. To change the units of a rate, change both parts: 10 m/s is 36 000 m in an hour, which is 36 km/h.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.
Writing rates in this pack
Rates are written with a slash for “per” (km/h, L/min, $/kg, words/min) and said with the word per. Money is rounded to the cent at the end of a calculation, and unit prices for small packs are given per 100 grams, as many supermarket shelf labels show them.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | What is a rate? | Sort statements into rates and ratios, write rates with both units, and say what a rate tells us | Everyday rates; rate-or-ratio cards |
| 2 | Unit rates | Divide so the second quantity is 1, use a unit rate to scale up, and read a distance-time graph | Finding a unit rate; scaling a unit rate over time |
| 3 | Best buys and fair comparisons | Compare prices per kilogram or per 100 grams, fuel use, typing speed and scoring rates | Comparing best value; shelf cards |
| 4 | Changing the units of a rate | Convert both parts of a rate, m/s to km/h and L/min to mL/s, and compare speeds given in different units | Converting the units |
| 5 | Rate problems | Use speed, distance and time in all three arrangements, catch the average-speed trap, add rates, and play the quiz | Finding a unit rate |
How the week builds
Day 1 tells a rate from a ratio; Day 2 finds unit rates and uses them; Day 3 compares with them; Day 4 changes their units; and Day 5 solves longer problems, including speed, distance and time. It builds on ratios in Year 7 and leads straight into the modelling unit later in Year 8 that uses ratios and rates in financial contexts, and into linear relations, where a rate is the slope of a straight-line graph.
Materials for the week (one trip)
- From the classroom: pencils, rulers, calculators, scissors, this pack printed.
- Cut-out sheet 1, one per pair: 12 rate-or-ratio cards (Day 1).
- Cut-out sheet 2, one per pair: 12 shelf cards (Day 3).
- Optional: a few real supermarket labels or catalogues with unit prices for Day 3.
Words for the word wall
rate, ratio, per, unit rate, speed, average speed, distance, time, unit price, best buy, value, per 100 grams, convert, km/h, m/s, L/min, mL/s, distance-time graph, steady, scale up
Dear families
This week in maths, Year 8 is working with rates: comparisons of two different kinds of quantity, such as kilometres per hour or dollars per kilogram. They find unit rates, compare supermarket prices fairly, change the units of a rate, and solve speed, distance and time problems.
Try this at home
- At the supermarket or in a catalogue, find the unit price on a shelf label (per kg or per 100 g) and compare two sizes.
- Ask: a car travels 180 km in 2 hours. What is its average speed? (90 km/h.)
- Ask: which is the better buy, 2 L of milk for $3.20 or 3 L for $4.50? ($1.60/L and $1.50/L: the 3 L.)
- Count your heartbeats for 15 seconds, then work out your heart rate in beats per minute.
My rates this week
Fill one row a day. Write a rate you noticed, with both units, and its unit rate.
| Day | The rate I noticed | As a unit rate |
|---|---|---|
| Monday | ||
| Tuesday | ||
| Wednesday | ||
| Thursday | ||
| Friday |
Printed from the free seegongsik Rates teaching pack · seegongsik.com/au/y8/measurement/AC9M8M05/pack
What is a rate?
A rate compares two quantities measured in different units, so it always carries two units. Students sort statements into rates and ratios, write rates with both units, and say what a rate tells us.
We are learning to
- tell a rate (two different units) from a ratio (the same unit),
- write a rate with both of its units, using per or a slash,
- say in words what a rate tells us, such as how far in each hour.
Success criteria
- I can explain why 60 km in an hour is a rate but 3 cups to 2 cups is a ratio.
- I can write 90 beats in 1.5 minutes as a rate per minute.
You need
The worksheet and a calculator each; the rate-or-ratio cards from cut-out sheet 1, one set per pair, cut out.
Lesson flow (about 60 minutes)
| 10 min | Rates all around us Show the board (see the box), pressing each chip. Ask: “Why does 3 : 2 say “to” while the others say “per”?” |
| 20 min | Rate or ratio? Pairs sort the twelve cards into “rate” and “ratio”, then record them in Part A. Ask: ““3 goals in 4 games”: rate or ratio? What are its two units?” |
| 30 min | Write it as a rate Parts B and C; early finishers do Part D. Ask: “80 km/h and 0.0125 h/km describe the same car. Why do we usually use the first?” |
Two shorter sessions instead? Split after the card sort; start the second with Part B.
Watch for these ideas
- Writing a rate with one unit only, such as “speed 60”.
- Calling any comparison a rate: a rate needs two different units.
- Dividing the wrong way round, so the rate is per the wrong quantity.
Answers
- Part A: rates 1, 3, 4, 7, 9, 10, 12; ratios 2, 5, 6, 8, 11. Card 12 is a rate: goals per game.
- Part B: 45 L in 3 min = 15 L/min; 960 words in 12 min = 80 words/min; 84 dollars in 6 hours = 14 $/h; 1500 m in 300 s = 5 m/s.
- Part C: 80 km/h and 0.0125 h/km: kilometres in each hour; hours for each kilometre; $4.50/kg and 0.22 kg/$: dollars for each kilogram; kilograms for each dollar.
- Part D: answers vary, for example kilometres per hour, goals per game, points per minute, metres per second.
What is a rate?
A. Rate or ratio?
Write each card number in its pile. For two cards, say what the two quantities are.
| Rate (different units) | Ratio (the same unit) |
|---|---|
B. Write it as a rate
Divide so that the second quantity is 1. Give the rate with both units.
| 45 L in 3 min | |
| 960 words in 12 min | |
| 84 dollars in 6 hours | |
| 1500 m in 300 s |
C. Which way round?
Each pair describes the same thing. Say in words what each one tells you.
| 80 km/h and 0.0125 h/km | |
| $4.50/kg and 0.22 kg/$ |
D. Stretch: a sports report
Write four rates that a sports report could use, each with its two units. Say which sport each one is from.
Unit rates
A unit rate is the amount for one of the second quantity: one hour, one kilogram, one minute. Students find unit rates by dividing, use them to scale up by multiplying, and read a distance-time graph where a steady speed is a straight line.
We are learning to
- find a unit rate by dividing both quantities so the second is 1,
- use a unit rate to find an amount for any number of units,
- read a steady speed from a distance-time graph, and see that a steeper line is faster.
Success criteria
- I can find the speed of a car that travels 175 km in 2.5 hours.
- I can say which line on a distance-time graph is faster, and why.
You need
The worksheet, a ruler and a calculator each.
Lesson flow (about 60 minutes)
| 10 min | Make the bottom 1 Show the board (see the box): the unit rate, then the graph. Ask: “Why do we divide both 240 and 3 by 3, and not just one of them?” |
| 25 min | Find it, then use it Students complete Parts A and B, writing each unit rate with both units before using it. Ask: “Once we know 80 km in one hour, how do we get 4.5 hours?” |
| 25 min | Reading the graph Part C; early finishers do Part D. Ask: “The car’s line is steeper than the cyclist’s. What does steeper mean here?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Dividing the wrong way: 3 ÷ 240 gives hours per kilometre.
- Forgetting the units, or writing km for a speed.
- Reading a graph from the wrong axis, or misreading the scale.
Answers
- Part A: 240 ÷ 3 = 80 km/h; 18 ÷ 4 = $4.50/kg; 150 ÷ 2.5 = 60 words/min; 36 ÷ 8 = 4.5 L/min.
- Part B: 80 × 4.5 = 360 km; 4.2 × 2.5 = $10.50; 12 × 15 = 180 L.
- Part C: after 3.5 hours the car has gone 280 km and the cyclist 70 km; the car reaches 300 km after 3.75 hours; the speeds are 80 and 20 km/h, from any point on each line.
- Part D: in 4 hours the cyclist rides 80 km; the car needs 1 hour for that distance (where the car’s line reaches the same height).
Unit rates
A. Find the unit rate
| 240 km in 3 h | |
| $18 for 4 kg | |
| 150 words in 2.5 min | |
| 36 L in 8 min |
B. Use the unit rate
| A car at a steady 80 km/h. How far in 4.5 hours? | |
| Apples at $4.20/kg. What do 2.5 kg cost? | |
| A tap runs at 12 L/min. How much water in 15 minutes? |
C. A distance-time graph
- How far has each gone after 3.5 hours?
- When does the car reach 300 km?
- Use one point on each line to find each speed.
D. Stretch: catching up
The cyclist rides for 4 hours. How long would the car take to drive the same distance? Use the unit rates, then check with the graph.
Best buys and fair comparisons
A bigger pack is not always cheaper per kilogram. Students compare prices per kilogram, per litre and per 100 grams on a supermarket shelf, then compare fuel use, typing speed and scoring rates fairly.
We are learning to
- find a unit price by dividing the price by the amount,
- choose the best buy and explain it with unit prices,
- compare other rates by bringing them to the same “per”, and decide whether higher or lower is better.
Success criteria
- I can say which is better value, 600 g for $4.20 or 1 kg for $6.80, using the price per 100 g.
- I can explain why a bigger pack is not always cheaper per kilogram.
You need
The worksheet and a calculator each; the shelf cards from cut-out sheet 2, one set per pair, cut out.
Lesson flow (about 60 minutes)
| 10 min | Price per kilogram Show the board (see the box). Ask: “Option B costs more money. Why is it still the better value?” |
| 25 min | The shelf Pairs lay the twelve shelf cards out by product, find every unit price, and fill Part A. Ask: “For rice, the biggest bag is not the best buy. How could you spot that without a calculator?” |
| 25 min | Other fair comparisons Parts B and C; early finishers do Part D. Ask: “For fuel use in litres per 100 km, is a bigger number better or worse?” |
Two shorter sessions instead? Split after the shelf; start the second with Part B.
Watch for these ideas
- Comparing the prices on the labels instead of the unit prices.
- Dividing the amount by the price, then picking the smaller number.
- Mixing grams and kilograms in one comparison.
Answers
- Part A: Rice $2.70, $2.80, $2.60 per kg, best 2 kg; Milk $1.50, $1.75, $1.55 per L, best 3 L; Yoghurt $0.78, $0.82, $0.85 per 100 g, best 500 g; Oats $0.40, $0.36, $0.44 per 100 g, best 1500 g.
- Part B: juice $2.40 and $2.30 per L, so 2 L; cheese $1.40 and $1.28 per 100 g, so 1000 g; batteries $1.30 and $1.20 per battery, so 10.
- Part C: 8 against 7.5 L/100 km, so B; 45 against 47.5 words/min, so B; 0.73 against 0.67 goals/game, so A.
- Part D: 20% off makes $8.00 for 1.25 kg, $6.40/kg, against $7.20/kg: the discounted bag.
Best buys and fair comparisons
A. The shelf
Find the unit price of every pack on the shelf cards, then circle the best buy for each product.
| Product | Pack 1 | Pack 2 | Pack 3 |
|---|---|---|---|
| Rice (per kg) | |||
| Milk (per L) | |||
| Yoghurt (per 100 g) | |||
| Oats (per 100 g) |
B. Which is the better buy?
| Juice: 1.5 L for $3.60, or 2 L for $4.60 | |
| Cheese: 250 g for $3.50, or 1000 g for $12.80 | |
| Batteries: 4 for $5.20, or 10 for $12.00 |
C. Other fair comparisons
| Which car uses less fuel? A: 36 L in 450 km. B: 30 L in 400 km. | |
| Who types faster? A: 270 words in 6 min. B: 380 words in 8 min. | |
| Who scores more often? A: 11 goals in 15 games. B: 8 goals in 12 games. |
D. Stretch: a discount
A 1.25 kg bag is $10.00 with 20% off this week. A 1 kg bag is $7.20. Which is cheaper per kilogram?
Changing the units of a rate
To change the units of a rate, change both parts. Students turn m/s into km/h and back, L/min into mL/s and $/kg into cents per gram, then compare speeds given in different units.
We are learning to
- change the top and the bottom of a rate into new units, one at a time,
- use × 3.6 to change m/s to km/h, and ÷ 3.6 to change back,
- put rates into the same units before comparing them.
Success criteria
- I can change 20 m/s to km/h and explain where the 3.6 comes from.
- I can say which is faster, 30 m/s or 100 km/h.
You need
The worksheet and a calculator each.
Lesson flow (about 60 minutes)
| 10 min | Top and bottom Show the board (see the box). Write 10 m/s as 10 m in 1 s, 36 000 m in 3600 s, 36 km in 1 h. Ask: “Why do we multiply by 3600 and then divide by 1000?” |
| 25 min | Convert Students complete Parts A and B, writing each step with its units. Ask: “The cheetah’s speed looks smaller than the car’s. Is it slower?” |
| 25 min | Taps, walkers and hearts Part C; early finishers do Part D. Ask: “How many seconds are in a minute, and how does that help with mL/s?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Changing only one part of the rate, such as the metres but not the seconds.
- Multiplying by 3.6 to go from km/h to m/s: the m/s number is the smaller one.
- Comparing speeds in different units by their numbers alone.
Answers
- Part A: 15 m/s × 3600 s/h ÷ 1000 m/km = × 3.6 = 54 km/h; 72 km/h ÷ 3.6 = 20 m/s; 9 L/min × 1000 mL/L ÷ 60 s/min = 150 mL/s; $2.50/kg × 100 c/$ ÷ 1000 g/kg = 0.25 c/g.
- Part B: a cheetah at full sprint 108 km/h, a car on a country road 100 km/h, a racing pigeon 90 km/h, a sprinter in a 100 m race 36 km/h; fastest to slowest: cheetah at full sprint, car on a country road, racing pigeon, sprinter in a 100 m race.
- Part C: 12 L/min = 200 mL/s; 4.8 km/h = 80 m/min, so 1 km takes 12.5 min.
- Part D: 72 × 60 = 4320 beats an hour, × 24 = 103 680 a day, about 38 million a year.
Changing the units of a rate
1 km = 1000 m · 1 h = 60 min = 3600 s · 1 L = 1000 mL · $1 = 100 c
A. Convert
Change both parts of each rate. Show each step with its units.
| 15 m/s to km/h | |
| 72 km/h to m/s | |
| 9 L/min to mL/s | |
| $2.50/kg to c/g |
B. Who is fastest?
Change every speed to km/h, then order them from fastest to slowest.
| A cheetah at full sprint: 30 m/s | |
| A car on a country road: 100 km/h | |
| A racing pigeon: 25 m/s | |
| A sprinter in a 100 m race: 36 km/h |
Fastest to slowest: __________________________________________
C. Taps and walkers
- A tap runs at 12 L/min. How many millilitres is that each second?
- A person walks at 4.8 km/h. How many metres is that each minute? How long does 1 km take?
D. Stretch: a heart rate
A resting heart beats 72 times a minute. About how many times does it beat in an hour? In a day? In a year?
Rate problems
Speed, distance and time are one rate used three ways round. Students solve speed and cost problems, catch the average-speed trap, add two rates working together, and play the self-check quiz as a class game.
We are learning to
- use speed = distance ÷ time, distance = speed × time and time = distance ÷ speed,
- find an average speed as the total distance ÷ the total time,
- add two rates that work together, such as two taps filling one tank.
Success criteria
- I can find how long a 150 km trip takes at 60 km/h, in hours and minutes.
- I can explain why the average of two speeds is not always the average speed.
You need
The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.
Lesson flow (about 60 minutes)
| 10 min | Three ways round Show the board (see the box). Then write 80 = 240 ÷ 3, 240 = 80 × 3 and 3 = 240 ÷ 80 under each other. Ask: “If you know the speed and the distance, which operation gives the time?” |
| 30 min | Problems Parts A to C; early finishers do Part D. Ask: “Mia averaged the two speeds. What did Mia forget about the time spent on each leg?” |
| 20 min | Self-check quiz as a class game Teams of four hold up one, two or three fingers for their option, then reveal and read the explanation aloud. A team that was wrong says what fooled them. Ask: “Before we reveal it: which answer would someone choose if they compared total prices?” |
Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.
Watch for these ideas
- Dividing the wrong way round, such as 3 ÷ 240.
- Writing 2.25 hours as 2 hours 25 minutes: it is 2 hours 15 minutes.
- Averaging two speeds instead of dividing the total distance by the total time.
Answers
- Part A: 1. 315 ÷ 3.5 = 90 km/h; 2. 24 × 2.25 = 54 km (2 h 15 min = 2.25 h); 3. 2160 ÷ 720 = 3 h.
- Part B: $18.50 × 12 = $222.00; $4.50 × 7 = $31.50.
- Part C: 60 km at 60 km/h takes 1 h and 60 km at 30 km/h takes 2 h, so 120 ÷ 3 = 40 km/h, not 45: more time is spent going slowly.
- Part D: together 12 + 8 = 20 L/min, so 200 ÷ 20 = 10 minutes.
Rate problems
speed = distance ÷ time · distance = speed × time · time = distance ÷ speed
A. Speed, distance and time
| 1. A train travels 315 km in 3.5 hours. What is its average speed? | |
| 2. A cyclist rides at 24 km/h for 2 hours 15 minutes. How far does the cyclist go? | |
| 3. A plane flies 2160 km at 720 km/h. How long does the flight take? |
B. Costs from rates
| 1. A part-time job pays $18.50 an hour. What is the pay for 12 hours? | |
| 2. Extra phone data costs $4.50 a gigabyte. What do 7 gigabytes cost? |
C. Find the mistake
Mia drives 60 km at 60 km/h, then another 60 km at 30 km/h. Mia says the average speed is 45 km/h. Is Mia right? Find the true average speed.
D. Stretch: two taps
One tap fills a tank at 12 L/min and another at 8 L/min. How long do both together take to fill a 200 L tank?
Rate or ratio? Cards
For Day 1, one set per pair: twelve statements
Sort the cards: does each compare two different units (a rate) or two amounts in the same unit (a ratio)?
Teacher note: rate: 1 (kilometres per hour), 3 (dollars per litre), 4 (beats per minute), 7 (millilitres per second), 9 (dollars per hour), 10 (litres per 100 kilometres), 12 (goals per game). ratio: 2 (cups to cups), 5 (people to people), 6 (length to length), 8 (volume to volume), 11 (length to length).
Supermarket shelf cards
For Day 3, one set per pair: four products, three packs each
Lay out each product’s three packs. Work out every unit price, then find the best buy for each product.
Teacher note: the unit prices and best buys are in the Day 3 answers.
What we know: Rates
Work on your own with a calculator. Give every rate with both units.
- Which is a rate and which is a ratio: “150 km in 2 hours” or “3 boys to 5 girls”? Why? ______________________
- Apples cost $28.50 for 6 kg. What is the price per kilogram? ______________________
- At a steady 75 km/h, how far do you travel in 3.5 hours? ______________________
- How long does a 210 km trip take at 70 km/h? ______________________
- Which is the better buy: 400 g for $3.20, or 1 kg for $7.50? Show the unit prices. ______________________
- Change 35 m/s to km/h. ______________________
- A hose runs at 18 L/min. How many millilitres is that each second? ______________________
- Sam rides 30 km at 30 km/h, then 30 km at 15 km/h. Sam says the average speed is 22.5 km/h. Is Sam right? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- 150 km in 2 hours is a rate (kilometres and hours); 3 boys to 5 girls is a ratio (people to people).
- $28.50 ÷ 6 = $4.75/kg.
- 75 × 3.5 = 262.5 km.
- 210 ÷ 70 = 3 hours.
- $0.80 and $0.75 per 100 g, so the 1 kg pack.
- 35 × 3.6 = 126 km/h.
- 18 × 1000 ÷ 60 = 300 mL/s.
- No: 1 h + 2 h = 3 h for 60 km, so 20 km/h.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Rates and unit rates (Q1, Q2) | calls any comparison a rate, or leaves out a unit | tells a rate from a ratio and finds a unit rate with both units | explains which quantity belongs on the bottom of a rate |
| Using rates (Q3, Q4, Q8) | divides the wrong way, or averages two speeds | uses speed, distance and time in all three arrangements, with hours and minutes | finds an average speed from the total distance and the total time |
| Best buys (Q5) | compares the prices on the labels | compares unit prices per 100 g or per kg and chooses the best buy | explains why a bigger pack is not always better value |
| Converting rates (Q6, Q7) | changes only one part of a rate | changes both parts, m/s to km/h and L/min to mL/s | explains the 3.6 from 3600 seconds and 1000 metres |
Eight questions, four ideas. A student at standard answers most questions with working, and gives every rate both units.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Rate or ratio | Unit rates | Best buys | Converting | Problems |
|---|---|---|---|---|---|
The five columns are the five days: rate or ratio, unit rates, best buys, converting rates and rate problems.