Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 1 - Direct proportion, rates, ratio and scale modelling

Direct proportion, rates, ratio and scale modelling

Formulate and evaluate practical models using proportional relationships, rates, ratios and scale.

Updated 2026-07-26 - 11 min read

Direct proportion, rates, ratio and scale modelling is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Recognise direct proportion

A constant rate with a non-zero starting fee is linear but not direct proportion.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by recognise direct proportion and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Only lines through the origin with constant y/x represent direct proportion.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Is y=4x directly proportional?

Step 1 - identify the governing idea: Direct proportion has form y=kx, constant ratio y/x=k, and graph through the origin.

Step 2 - apply it to this evidence: y/x=4 and the graph passes through the origin.

Result: Yes

The relationship is visible in the working: y/x=4 and the graph passes through the origin. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • No, because 4 is added — It does not agree with the required relationship: y/x=4 and the graph passes through the origin.
  • Only for x=4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No linear equation is proportional — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Is C=3n+10 directly proportional?

Step 1 - identify the governing idea: Direct proportion has form y=kx, constant ratio y/x=k, and graph through the origin.

Step 2 - apply it to this evidence: The fixed fee 10 gives a non-zero intercept.

Result: No

The relationship is visible in the working: The fixed fee 10 gives a non-zero intercept. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Yes, because it is linear — It does not agree with the required relationship: The fixed fee 10 gives a non-zero intercept.
  • Yes, because 3 is constant — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only when n=10 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

For direct proportion, y=18 when x=6. Find k.

Step 1 - identify the governing idea: Direct proportion has form y=kx, constant ratio y/x=k, and graph through the origin.

Step 2 - apply it to this evidence: k=y/x=18/6.

Result: 3

The relationship is visible in the working: k=y/x=18/6. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 12 — It does not agree with the required relationship: k=y/x=18/6.
  • 108 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1/3 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve rate, ratio and scale problems

Writing units and the comparison direction prevents reciprocal and scale errors.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve rate, ratio and scale problems and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Length scales by k, while area scales by k².

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

A car travels 270 km in 3 h. Average rate?

Step 1 - identify the governing idea: Rates compare quantities with different units; ratios compare multiplicative parts; scales link corresponding lengths in a stated direction.

Step 2 - apply it to this evidence: 270÷3=90 with distance per time units.

Result: 90 km/h

The relationship is visible in the working: 270÷3=90 with distance per time units. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 810 km/h — It does not agree with the required relationship: 270÷3=90 with distance per time units.
  • 0.011 h/km — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 273 km/h — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Share $360 in ratio 2:3:4. Smallest share?

Step 1 - identify the governing idea: Rates compare quantities with different units; ratios compare multiplicative parts; scales link corresponding lengths in a stated direction.

Step 2 - apply it to this evidence: There are 9 parts; each is $40; 2 parts=$80.

Result: $80

The relationship is visible in the working: There are 9 parts; each is $40; 2 parts=$80. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • $120 — It does not agree with the required relationship: There are 9 parts; each is $40; 2 parts=$80.
  • $40 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • $90 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A map scale is 1:50,000. What real distance is 4 cm?

Step 1 - identify the governing idea: Rates compare quantities with different units; ratios compare multiplicative parts; scales link corresponding lengths in a stated direction.

Step 2 - apply it to this evidence: 4×50,000=200,000 cm=2 km.

Result: 2 km

The relationship is visible in the working: 4×50,000=200,000 cm=2 km. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 200 km — It does not agree with the required relationship: 4×50,000=200,000 cm=2 km.
  • 20 km — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.8 km — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Formulate and evaluate a proportional model

Unit prices may change with bulk discounts; speed may not remain constant; scale drawings may introduce measurement error. These limitations belong in evaluation.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by formulate and evaluate a proportional model and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The constant applies only while the relationship and conditions remain proportional.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Paint covers 12 m² per litre. Model litres L for area A.

Step 1 - identify the governing idea: A proportional model should define variables, justify k, solve the practical question and test whether proportionality remains plausible.

Step 2 - apply it to this evidence: Each litre covers 12 m², so required litres are area divided by 12.

Result: L=A/12

The relationship is visible in the working: Each litre covers 12 m², so required litres are area divided by 12. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • L=12A — It does not agree with the required relationship: Each litre covers 12 m², so required litres are area divided by 12.
  • L=A+12 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • L=12/A — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

The model L=A/12 gives 2.5 L. Paint is sold in 1 L tins. Practical decision?

Step 1 - identify the governing idea: A proportional model should define variables, justify k, solve the practical question and test whether proportionality remains plausible.

Step 2 - apply it to this evidence: Capacity must meet the requirement and tins are whole units.

Result: Buy 3 tins

The relationship is visible in the working: Capacity must meet the requirement and tins are whole units. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Buy 2 tins — It does not agree with the required relationship: Capacity must meet the requirement and tins are whole units.
  • Buy 2.5 tins — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Reject the coverage model — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A recipe is scaled from 4 to 10 serves. Ingredient 300 g becomes?

Step 1 - identify the governing idea: A proportional model should define variables, justify k, solve the practical question and test whether proportionality remains plausible.

Step 2 - apply it to this evidence: Scale factor is 10/4=2.5; 300×2.5=750.

Result: 750 g

The relationship is visible in the working: Scale factor is 10/4=2.5; 300×2.5=750. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 306 g — It does not agree with the required relationship: Scale factor is 10/4=2.5; 300×2.5=750.
  • 120 g — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3000 g — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Is C=3n+10 directly proportional?
  2. Share $360 in ratio 2:3:4. Smallest share?
  3. The model L=A/12 gives 2.5 L. Paint is sold in 1 L tins. Practical decision?

Answers

  1. No — The fixed fee 10 gives a non-zero intercept.
  2. $80 — There are 9 parts; each is $40; 2 parts=$80.
  3. Buy 3 tins — Capacity must meet the requirement and tins are whole units.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources