Understand the idea.
Make the modelling choices explicit
A model connects a real question to a calculation. Define the input and output, including units, before choosing . Direct proportion assumes that each additional unit requires the same amount and that there is no fixed starting amount.
Calculate from an observation, then compare its prediction with other available data. Matching unit rates support the model for those conditions. State the prediction in context, together with the assumptions that make the rate usable.
| Wall area (m²) | Paint (L) |
|---|---|
| 12 | 1.5 |
| 20 | 2.5 |
| 28 | 3.5 |
A model is useful within conditions
Paint use depends on the surface, paint type, coats and wastage. A rate found for one set of conditions cannot automatically describe another. Predictions beyond the observed range need particular care because the conditions may change.
Compare predictions with measurements to judge the model: differences may reflect measurement uncertainty or a changing rate. Then apply practical constraints. A calculated quantity of paint can be fractional, but buses or cans may need rounding up to whole purchases.
See the method in action.
A paint model
Use the supplied paint data to estimate the amount for 44 m² under the same conditions.
Worked solution
- 1
Find paint use per square metre.
Paint volume is the output and surface area is the input.
- 2
Check the other supplied observations.
All three observations give the same paint use per square metre.
- 3
Predict the amount for the new area.
The direct-proportion model multiplies the area by the common unit rate.
- 4
State the condition on the estimate.
The estimate assumes unchanged surface, paint, coats and wastage; otherwise the rate may differ.
5.5 L, assuming unchanged surface, coats and wastage.
Reject an unsuitable proportional rule
A coach quote is 440 dollars for 20 students and 560 dollars for 30 students. Is a constant cost per student supported?
Worked solution
- 1
Find the cost per student for each quote.
A directly proportional total cost would have one constant cost per student.
- 2
Compare the two rates.
The differing rates rule out one direct-proportion rule fitting both quotes exactly.
- 3
Find a linear model that fits both quotes.
The change in quote gives twelve dollars per extra student and a fixed charge of two hundred dollars.
- 4
Check that model against both prices.
A fixed-charge model fits these observations, although further data would be needed to confirm it more generally.
No; a fixed-charge model fits these data better.
Try it yourself.
Check your understanding
A model predicts 2.4 identical buses. What decision is needed if all passengers must travel?
Show answer & reasoning
Hire at least 3 buses.
A bus cannot be hired as a fraction. Since two buses would provide less than the modelled capacity requirement of buses, round up and hire at least three.
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- A constant-rate model uses 3 L of cleaner for 24 m². State the rate in L/m².2 marks
- Use that model to predict cleaner for 60 m².2 marks
- State two conditions needed before applying that cleaner model to another room.2 marks
- A prediction is 7.5 L. Cleaner comes only in 4 L containers. How many are needed?2 marks
- A machine produces 30 items in 2 minutes and 75 in 5 minutes. Does a proportional model fit? Find its rule.3 marks
- The same machine needs 3 minutes to warm up before producing items. Explain why total items are not directly proportional to time measured from switch-on.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A constant-rate model uses 3 L of cleaner for 24 m². State the rate in L/m².2 marks
L/m².
L/m² - Use that model to predict cleaner for 60 m².2 marks
L.
L - State two conditions needed before applying that cleaner model to another room.2 marks
The rate may change with surface, dilution, number of applications or wastage.
For example: same surface type and same cleaning method/strength. - A prediction is 7.5 L. Cleaner comes only in 4 L containers. How many are needed?2 marks
containers.
Round up to 2; they supply 8 L.
containers - A machine produces 30 items in 2 minutes and 75 in 5 minutes. Does a proportional model fit? Find its rule.3 marks
Both rates equal items/min.
The rule gives zero output at zero production time, assuming no startup loss.
, with in minutes. - The same machine needs 3 minutes to warm up before producing items. Explain why total items are not directly proportional to time measured from switch-on.3 marks
During the first 3 minutes, output is zero.
After warm-up, for ; this is not from switch-on.
The warm-up period introduces a delay.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Year 9 proportional modelling, rates, scale, similarity and enlargement. Assumptions, interpretation and model limitations are taught alongside calculation.
Official curriculum source ↗