Practical proportional models and limitations | Year 9

01 / LEARN

The idea, step by step

Make the modelling choices explicit

Start with the question: what must be predicted? Name the input, output and units. A direct-proportion model $y = k x$ is reasonable only when the unit rate stays constant and there is no fixed initial amount.

Obtain $k$ from suitable data, use the rule, then check a second observation if one is available. Report the result in context rather than stopping at an unexplained number.

Paint used on the same type of prepared wall
Wall area (m²)	Paint (L)
12	1.5
20	2.5
28	3.5

A model is useful within conditions

A constant paint-coverage model assumes the same surface, paint, number of coats and wastage. Extending it to rough brick or a second coat without adjustment would be unjustified.

Compare a prediction with a new measurement. A small discrepancy might be rounding or measurement error; a consistent change in unit rates suggests changing the model. Also consider practical constraints: paint cans and buses may need whole-number purchases even when the model gives a fraction.

A common trap

A proportional prediction remains reliable for any input size.

Instead: The constant rate and practical conditions must remain suitable. Check restrictions, available data and whole-number decisions before trusting an extrapolation.

02 / WORKED EXAMPLES

See the thinking

Example 1

A paint model

Use the supplied paint data to estimate the amount for 44 m² under the same conditions.

$k = 1.5 / 12 = 0.125$ L/m².
Check: $2.5 / 20 = 0.125$ and $3.5 / 28 = 0.125$ .
Predicted paint is $44 \times 0.125 = 5.5$ L.

5.5 L, assuming unchanged surface, coats and wastage.

Example 2

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?

Unit costs are $440 / 20 = 22$ and $560 / 30 \approx 18.67$ dollars/student.
They differ, so one direct-proportion cost rule is not supported.
Both quotes fit a fixed 200 dollars charge plus 12 dollars per student: $200 + 12 n$ .

No; a fixed-charge model fits these data better.

Check your understanding

A model predicts 2.4 identical buses. What decision is needed if all passengers must travel?

Show answer

Hire at least 3 buses.

Buses are whole objects; rounding down would leave passengers without seats.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

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
$3 / 24 = 0.125$ L/m².
0.125 L/m²
Use that model to predict cleaner for 60 m².
2 marks
$60 \times 0.125 = 7.5$ L.
7.5 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
$7.5 / 4 = 1.875$ containers.
Round up to 2; they supply 8 L.
2 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 $15$ items/min.
The rule gives zero output at zero production time, assuming no startup loss.
$N = 15 t$ , with t 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, $N = 15 (t - 3)$ for $t \geq 3$ ; this is not $N = k t$ from switch-on.
The warm-up period introduces a delay.

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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 ↗

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