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.
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.
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÷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?
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Fill the tank
TryTry it on the board and compare the result with your prediction.
The petrol bowser
Try it yourselfYour edits are saved in this browser.
Set 40 litres. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Multiply or divide
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
time=distance÷speedChoose by the units: km ÷ km/h = h.
200÷80=2.5Divide: 200 km at 80 km/h.
2.5 h=2 h 30 minInterpret: half an hour is 30 minutes.
80×2.5=200Check by going back.
Read Leo's working. Do the units come out as hours?
How long does it take to drive 270 km at 90 km/h, in hours?
Leo works out a journey time. Click the line where his working goes wrong.
Say the exchange rate is 1 Australian dollar = 0.65 US dollars. How many US dollars do you get for 200 Australian dollars?
ModifyChange the problem on purpose and see what happens.
Race, shower, money back
Need a hint?
Car B gains 15 km every hour.
Another hint
45 ÷ 15 = ?
Try it yourselfYour edits are saved in this browser.
Two cars leave together at 60 km/h and 75 km/h. When is Car B 45 km ahead?
Use the board. This ticks when you get there.
With the same example rate (1 Australian dollar = 0.65 US dollars), how many Australian dollars are 52 US dollars worth?
A shower uses 9 L per minute. How long is a shower that uses 72 L?
Use the board. This ticks when you get there.
A tap drips 0.5 L every hour. How many litres does it waste in a week?
PractiseA few on your own, to check you have it.
Using rates
Grapes cost $4.20 per kg. How much do 3.5 kg cost, in dollars?
A bus averages 85 km/h for 2.5 hours. How far does it go, in km?
A cyclist rides 45 km at 15 km/h. How many hours does it take?
Review: convert 72 km/h to metres per second.
A part-time job pays $18.50 per hour. How much is 6.5 hours of work, in dollars?
A cricket team needs 120 runs off the last 15 overs. What run rate (runs per over) does it need?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
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.
Try it yourselfYour edits are saved in this browser.
Need a hint?
360 ÷ 4.5 = ?
Another hint
The time is 4 and a half hours.
A 360 km drive must take 4.5 hours. Set the speed and time.
Use the board. This ticks when you get there.
For teachers
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.
I can check an answer with its units and its size.
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.
Running the lesson
Predict: ask what 40 ÷ 2.05 would mean.
Run: ask how much one more litre adds.
Investigate: ask Leo whether 9000 hours makes sense.
Modify: ask why the gap grows by the same amount each hour.
Make: ask what the model leaves out.
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.
Stretch and support
Stretch: Plan a trip budget overseas: look up today's exchange rate, convert three prices, and include a 3% card fee.
Support: A card: total = rate × amount; amount = total ÷ rate; units in every line.
Where it leads
This completes AC9M8M05.
Where it leads: rate models (next lessons), gradient as a rate (Linear relations), and speed in science.
What each step is for
fill the tank: Committing to a cost from a price per litre shows whether they multiply the rate by the amount.
the petrol bowser: Moving the litres along the double number line shows the cost growing by the rate for every litre.
multiply or divide: Units decide multiply or divide (Leo multiplies to find a time); currency conversion is a rate too.
race, shower, money back: Comparing two rates over time, working backwards from a total, and converting money both ways.
using rates: Totals, distances, times and a conversion, mixed; multiplying when the units say divide is the planted slip; the speed conversion review keeps units fresh.
plan a drive: Choosing a speed and time for a fixed distance uses distance = speed × time in both directions.