Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 3 - Proportion and scaling models

Proportion and scaling models

Model direct and inverse proportional relationships and solve unfamiliar scaling problems across length, area and volume.

Updated 2026-07-26 - 11 min read

Proportion and scaling models is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

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.

Model direct and inverse proportion

Direct proportion has y = kx; inverse proportion has y = k/x for non-zero x, with constant product xy. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by model direct and inverse 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: Direct proportion passes through the origin with constant ratio; inverse proportion has constant product.

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

If y is directly proportional to x and y=18 when x=6, find k.

Step 1 - identify the governing idea: Direct proportion has y = kx; inverse proportion has y = k/x for non-zero x, with constant product xy.

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:

  • 108 — It does not agree with the required relationship: k = y/x = 18/6.
  • 12 — 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.

Example 1.2

If y is inversely proportional to x and y=4 when x=3, find y when x=6.

Step 1 - identify the governing idea: Direct proportion has y = kx; inverse proportion has y = k/x for non-zero x, with constant product xy.

Step 2 - apply it to this evidence: The constant product is 12, so y=12/6.

Result: 2

The relationship is visible in the working: The constant product is 12, so y=12/6. 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:

  • 8 — It does not agree with the required relationship: The constant product is 12, so y=12/6.
  • 7 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 24 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which table shows direct proportion?

Step 1 - identify the governing idea: Direct proportion has y = kx; inverse proportion has y = k/x for non-zero x, with constant product xy.

Step 2 - apply it to this evidence: The ratio y/x is constantly 5.

Result: x: 1,2,3; y: 5,10,15

The relationship is visible in the working: The ratio y/x is constantly 5. 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:

  • x: 1,2,3; y: 5,6,7 — It does not agree with the required relationship: The ratio y/x is constantly 5.
  • x: 1,2,3; y: 6,3,2 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x: 0,1,2; y: 5,10,15 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Scale length, area and volume

A linear scale factor k multiplies lengths by k, areas by k² and volumes by k³. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by scale length, area and volume 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: Area combines two length dimensions and volume three, so their scale factors are squared and cubed.

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 shape is enlarged by linear factor 3. Find area factor.

Step 1 - identify the governing idea: A linear scale factor k multiplies lengths by k, areas by k² and volumes by k³.

Step 2 - apply it to this evidence: Area factor is 3².

Result: 9

The relationship is visible in the working: Area factor is 3². 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:

  • 3 — It does not agree with the required relationship: Area factor is 3².
  • 6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 27 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A solid is enlarged by factor 2.5. Find volume factor.

Step 1 - identify the governing idea: A linear scale factor k multiplies lengths by k, areas by k² and volumes by k³.

Step 2 - apply it to this evidence: Volume factor is 2.5³ = 15.625.

Result: 15.625

The relationship is visible in the working: Volume factor is 2.5³ = 15.625. 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:

  • 6.25 — It does not agree with the required relationship: Volume factor is 2.5³ = 15.625.
  • 7.5 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Two similar figures have area ratio 49:81. Find corresponding length ratio.

Step 1 - identify the governing idea: A linear scale factor k multiplies lengths by k, areas by k² and volumes by k³.

Step 2 - apply it to this evidence: Length ratio is the positive square root of the area ratio.

Result: 7:9

The relationship is visible in the working: Length ratio is the positive square root of the area ratio. 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:

  • 49:81 — It does not agree with the required relationship: Length ratio is the positive square root of the area ratio.
  • 14:18 only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2401:6561 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve unfamiliar scaling problems

Annotate givens, identify the proportional dimension, convert units and verify the result with scale and context. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve unfamiliar scaling 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: The exponent depends on whether the quantity measures length, area or volume.

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

A model tank at 1:20 scale holds 0.5 L. Estimate full tank capacity.

Step 1 - identify the governing idea: Annotate givens, identify the proportional dimension, convert units and verify the result with scale and context.

Step 2 - apply it to this evidence: Volume factor is 20³=8000; 0.5×8000=4000 L.

Result: 4000 L

The relationship is visible in the working: Volume factor is 20³=8000; 0.5×8000=4000 L. 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:

  • 10 L — It does not agree with the required relationship: Volume factor is 20³=8000; 0.5×8000=4000 L.
  • 200 L — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8000 L — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A map area is 6 cm² at scale 1:500. What real area is represented?

Step 1 - identify the governing idea: Annotate givens, identify the proportional dimension, convert units and verify the result with scale and context.

Step 2 - apply it to this evidence: Area factor is 500²; 6×250000=1,500,000 cm²=150 m².

Result: 150 m²

The relationship is visible in the working: Area factor is 500²; 6×250000=1,500,000 cm²=150 m². 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:

  • 1500 m² — It does not agree with the required relationship: Area factor is 500²; 6×250000=1,500,000 cm²=150 m².
  • 3000 m² — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3 m² — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A cube's volume becomes 343 cm³ after enlargement by factor 7. What was its original volume?

Step 1 - identify the governing idea: Annotate givens, identify the proportional dimension, convert units and verify the result with scale and context.

Step 2 - apply it to this evidence: Volume factor is 7³=343, so original volume is 343/343=1 cm³.

Result: 1 cm³

The relationship is visible in the working: Volume factor is 7³=343, so original volume is 343/343=1 cm³. 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:

  • 49 cm³ — It does not agree with the required relationship: Volume factor is 7³=343, so original volume is 343/343=1 cm³.
  • 7 cm³ — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2401 cm³ — 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. If y is inversely proportional to x and y=4 when x=3, find y when x=6.
  2. A solid is enlarged by factor 2.5. Find volume factor.
  3. A map area is 6 cm² at scale 1:500. What real area is represented?

Answers

  1. 2 — The constant product is 12, so y=12/6.
  2. 15.625 — Volume factor is 2.5³ = 15.625.
  3. 150 m² — Area factor is 500²; 6×250000=1,500,000 cm²=150 m².

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