The idea, step by step
A model is a useful simplification
A proportional model assumes that multiplying one quantity by a factor multiplies the other by the same factor. Twice as many equal bottles filled to the same level hold twice the volume. The assumption matters: different bottle sizes or fill levels would break the prediction. Explain what is being kept constant.
Test before predicting
Compare several pairs for a constant unit value and look for a fixed starting amount. Bike hire with a booking fee is not directly proportional to time, even when every extra hour has the same charge. A steady production rate may describe working time but not time including breaks. Check the practical interpretation as carefully as the arithmetic.
A common trap
Assuming any two increasing quantities are directly proportional.
Instead: A fixed starting fee or changing unit rate breaks the assumption. Check the scaling and explain the conditions for your prediction.
See the thinking
Use a steady-rate model
A machine packs 24 boxes in 3 minutes at a steady rate. Predict boxes in 8 working minutes.
- The rate is boxes per working minute.
- Eight working minutes gives boxes.
- This assumes the machine works at the same rate without stopping.
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?
- One hour costs dollars.
- Two hours costs dollars.
- Eleven is not double eight. The fixed starting fee breaks direct proportion.
Check your understanding
Three equal bags contain 18 kg. Is 10 bags containing 60 kg a reasonable prediction?
Show answer
Yes, if every bag has the same mass.
Each bag contains kg, so 10 contain kg.
Your turn
6 questions · 18 marks · Work at your own pace.
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 , and 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 stickers.
Nine have .
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 L per minute.
The time is 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 , and in km and L. Predict fuel for 250 km.3 marks
250 km is 2.5 times 100 km.
The model predicts 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 pages per working minute.
Working time is minutes.
Output is 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.
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 ↗