Choosing, checking and explaining proportional models | Year 7

01 / LEARN

Understand the idea.

A model is a useful simplification

A proportional model keeps a constant amount per unit. If one quantity doubles, the other doubles too. Equal bottles filled to the same level provide this model: each additional bottle contributes the same volume, and zero bottles contribute zero volume.

State what stays constant before predicting. Different bottle sizes, changing fill levels, bulk discounts or a setup fee can break the relationship. A model is useful only for the conditions and quantities it actually describes.

Test before predicting

Calculate a unit value from the known data and compare it with other pairs when available. Consistent pairs support a proportional model, but a short table alone cannot guarantee the rule for every input. Use the situation and its assumptions as well.

A fixed starting charge prevents direct proportion even when each extra hour costs the same. For a machine, distinguish working time from time including stoppages: a steady working rate predicts output only while the machine is operating at that rate.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Use a steady-rate model

A machine packs 24 boxes in 3 minutes at a steady rate. Predict boxes in 8 working minutes.

Worked solution

1
Find the working rate.
$24 \div 3 = 8 boxes per minute$
The stated steady rate allows the output to be shared equally across the three working minutes.
2
Apply the rate for eight minutes.
$8 \times 8 = 64 boxes$
Each of the eight working minutes contributes eight boxes.
3
State the condition for the prediction.
Sixty-four boxes assumes the machine continues at the same rate without stopping during those working minutes.

Answer

64 boxes, assuming uninterrupted work.

EXAMPLE 02

Reject an unsuitable model

Bike hire costs 5 dollars to start plus 3 dollars per hour. Does doubling one hour to two hours double the total cost?

Worked solution

1
Calculate the cost for one hour.
$5 + 3 \times 1 = 8 dollars$
The fixed starting charge is added once to the hourly charge.
2
Calculate the cost for two hours.
$5 + 3 \times 2 = 11 dollars$
Only the hourly part doubles; the starting charge remains five dollars.
3
Compare with double the original cost.
$2 \times 8 = 16 \neq 11$
Doubling time does not double total cost, so this cost is not directly proportional to time.

Answer

No. Two hours costs 11 dollars.

03 / YOUR TURN

Try it yourself.

Check your understanding

Three equal bags contain 18 kg. Is 10 bags containing 60 kg a reasonable prediction?

Show answer & reasoning

Yes, if every bag has the same mass.

Each equal bag contains $18 \div 3 = 6$ kilograms. Ten such bags contain $10 \times 6 = 60$ kilograms, so the prediction is reasonable provided every bag has that same mass.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

Four equal packets hold 28 stickers. Predict the stickers in 9 packets and state an assumption.
3 marks
A tap fills 15 L in 3 minutes. Find the time for 40 L at the same rate.
3 marks
A car park charges 4 dollars to enter plus 2 dollars per hour. Is the total cost directly proportional to time?
3 marks
Distance and fuel pairs are $(100, 8)$ , $(200, 16)$ and $(300, 24)$ in km and L. Predict fuel for 250 km.
3 marks
A printer makes 120 pages in 4 working minutes. During 10 clock minutes it pauses for 2 minutes. Predict pages at its steady rate.
3 marks
A gardener predicts 6 small plants need twice the water of 3 large plants. Why are the counts alone insufficient?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Four equal packets hold 28 stickers. Predict the stickers in 9 packets and state an assumption.
3 marks
One packet has $\frac{28}{4} = 7$ stickers.
Nine have $9 (7) = 63$ .
63 stickers; packets hold equal numbers.
A tap fills 15 L in 3 minutes. Find the time for 40 L at the same rate.
3 marks
The rate is $\frac{15}{3} = 5$ L per minute.
The time is $\frac{40}{5} = 8$ minutes.
8 minutes.
A car park charges 4 dollars to enter plus 2 dollars per hour. Is the total cost directly proportional to time?
3 marks
One hour costs 6 dollars and two hours cost 8 dollars.
The second amount is not twice the first because of the fixed fee.
No.
Distance and fuel pairs are $(100, 8)$ , $(200, 16)$ and $(300, 24)$ in km and L. Predict fuel for 250 km.
3 marks
250 km is $2.5 \times 100$ km.
The model predicts $2.5 (8) = 20$ L.
20 L.
A printer makes 120 pages in 4 working minutes. During 10 clock minutes it pauses for 2 minutes. Predict pages at its steady rate.
3 marks
The rate is $\frac{120}{4} = 30$ pages per working minute.
Working time is $10 - 2 = 8$ minutes.
Output is $30 (8) = 240$ pages.
240 pages.
A gardener predicts 6 small plants need twice the water of 3 large plants. Why are the counts alone insufficient?
3 marks
Direct proportion to plant count assumes equal water needs per plant.
Their different sizes make that assumption unjustified.
The plants need not use equal water.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 7 ratio comparisons and practical proportional models, including units, part-to-whole meaning and assumptions.

Official curriculum source ↗

In this lesson

Understand the idea.

A model is a useful simplification

Test before predicting

See the method in action.

Use a steady-rate model

Find the working rate.

Apply the rate for eight minutes.

State the condition for the prediction.

Reject an unsuitable model

Calculate the cost for one hour.

Calculate the cost for two hours.

Compare with double the original cost.

Try it yourself.

Check your understanding

Keep practising