Model a practical problem: say what you need, decide whether it is a ratio (parts of the same kind, like cordial to water) or a rate (two different units, like m² per litre), set it up, solve, then interpret the answer in the situation, such as rounding up to whole tins.
Painting a room, mixing cordial for a party, filling a pool, sharing the cost of a gift: the maths is easy once you choose the right model and remember what the answer means.
To model a practical problem:
Say what you need to find, with its units.
Choose the model. Quantities of the same kind (litres of cordial to litres of water) make a ratio. Quantities with different units (square metres per litre, litres per minute) make a rate.
Set it up and solve.
Interpret the answer in the situation: paint comes in whole tins, people come in whole numbers, money is to the cent.
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.
Paint a wall
TryTry it on the board and compare the result with your prediction.
Coverage
Try it yourselfYour edits are saved in this browser.
Find how many litres cover the 30 m² wall. Was your prediction right? How many 1 L tins would you buy?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Choosing the model
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
need: litres of paint for 20 m2Say what you are finding, with units.
12 m2 per L: a rateDifferent units, so a rate.
20÷12≈1.7 Lm² ÷ (m² per L) = L.
buy 2 one-litre tinsInterpret: tins are whole, and 1 would not be enough.
Read Ana's working. Will 2 tins be enough?
Ana buys paint. Click the line where her answer stops making sense.
A car uses 7 L of petrol per 100 km. How many litres does a 350 km trip use?
ModifyChange the problem on purpose and see what happens.
Share, scale, fill
Need a hint?
The ratio is 3 : 5, so 8 parts.
Another hint
Set the total to $240.
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Two friends share the cost of a $240 gift in the ratio 3 : 5. Set the board. How much does each pay?
Use the board. This ticks when you get there.
A pasta recipe uses 300 g of pasta for 4 serves. How many grams for 10 serves?
A hose fills a 1200 L tank at 15 L per minute. How many minutes does it take?
PractiseA few on your own, to check you have it.
Models
5 L of paint at 12 m² per litre covers how many square metres?
Cordial is mixed 1 : 4 with water. How many litres of cordial are in 3 L of drink?
A bus seats 45 people. How many buses are needed for 200 people?
Review: increase $50 by 10%.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Paint a fence
Then say how many 4 L tins to buy, and why.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Two coats on 42 m² is 84 m² of painting.
Another hint
84 ÷ 12 = ?
A 42 m² fence needs two coats. Paint covers 12 m² per litre. Find how many litres.
Use the board. This ticks when you get there.
For teachers
Students model practical problems as a ratio or a rate (paint coverage, cordial, fuel, sharing, filling), solve them and interpret the answer in the situation (AC9M8M07, node 1). About 40 minutes.
I can decide whether a problem needs a ratio or a rate.
I can set up and solve a ratio or rate model.
I can interpret the answer in the situation, such as rounding up to whole tins.
Before class
Do the Run board yourself first. Students need ratios (Year 7) and rates (previous lessons).
The one idea: same kind of quantity, ratio; different units, rate; then interpret the answer in the situation. The common mistake is rounding the wrong way for the context (Ana's mistake).
Wider: ratios in First Nations string and cordage making (fibre lengths and twisting), taught with local knowledge where you can; radiocarbon dating as a ratio of isotopes.
Running the lesson
Predict: ask what units 30 ÷ 12 gives.
Run: ask how many tins, and why.
Investigate: ask Ana whether 2 tins are enough.
Modify: ask which of the problems were ratios.
Make: ask how many 4 L tins, and what is left over.
The hard part (on purpose)
Choosing the model. Ask students to name the two quantities and their units before they calculate: same units, ratio; different units, rate.
When students get stuck
A student multiplies area by coverage. Ask: What units would that give? Fix: m² ÷ (m² per L) = L: divide.
A student uses 4 parts for a 1 : 4 ratio. Ask: How many parts altogether? Fix: 1 + 4 = 5 parts.
A student rounds to the nearest. Ask: Is that enough for the job? Fix: Round to suit the situation: up for tins and buses.
Stretch and support
Stretch: Plan the paint for your classroom: measure the walls, subtract the windows and door, choose a paint and tin size, and cost it.
Support: A model card: what do I need (units)? Ratio or rate? Calculation. Does the answer fit the situation?
Where it leads
This starts AC9M8M07.
Where it leads: checking models and their assumptions (next lesson), and linear models (Linear relations).
What each step is for
paint a wall: Committing to an amount of paint shows whether they divide the area by the coverage rate, or multiply.
coverage: Moving the litres until the area reaches the wall shows the rate model, then the answer has to be interpreted for tins.
choosing the model: Same units point to a ratio, different units to a rate; the answer must suit the situation (Ana rounds tins down).
share, scale, fill: A ratio share, a recipe scaled by a rate, choosing the model, and a fill time, each interpreted in context.
models: Rate and ratio models with interpreting, mixed; rounding the wrong way for the situation is the planted slip; the percentage review keeps multipliers fresh.
paint a fence: A two-step model (area doubled for two coats, then the rate) with an answer to interpret.
Now you can
Decide whether a problem needs a ratio or a rate.
Set up and solve a ratio or rate model.
Interpret the answer in the situation, such as rounding up to whole tins.