Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 3 - Enlargements and invariants

Enlargements and invariants

Apply enlargement transformations and identify what changes and what remains invariant.

Updated 2026-07-26 - 11 min read

Enlargements and invariants 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.

Construct enlargements

The centre anchors the transformation. Negative k places the image on the opposite ray; 0<k<1 produces a reduction.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by construct enlargements 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: Enlargement includes reductions and negative factors; every image point depends on the centre.

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

From origin, point (2,−3) enlarged by factor 2 maps to?

Step 1 - identify the governing idea: For centre O and scale factor k, image point P′ lies on line OP with directed distance OP′=k·OP.

Step 2 - apply it to this evidence: Multiply both coordinate displacements by 2.

Result: (4,−6)

The relationship is visible in the working: Multiply both coordinate displacements by 2. 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:

  • (4,−3) — It does not agree with the required relationship: Multiply both coordinate displacements by 2.
  • (2,−6) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (1,−1.5) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

From origin, (6,4) enlarged by factor 1/2 maps to?

Step 1 - identify the governing idea: For centre O and scale factor k, image point P′ lies on line OP with directed distance OP′=k·OP.

Step 2 - apply it to this evidence: A factor below 1 reduces each displacement.

Result: (3,2)

The relationship is visible in the working: A factor below 1 reduces each displacement. 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:

  • (12,8) — It does not agree with the required relationship: A factor below 1 reduces each displacement.
  • (5.5,3.5) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (6,2) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

From origin, (3,1) with factor −2 maps to?

Step 1 - identify the governing idea: For centre O and scale factor k, image point P′ lies on line OP with directed distance OP′=k·OP.

Step 2 - apply it to this evidence: Negative factor reverses direction and doubles distance.

Result: (−6,−2)

The relationship is visible in the working: Negative factor reverses direction and doubles distance. 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:

  • (6,2) — It does not agree with the required relationship: Negative factor reverses direction and doubles distance.
  • (−3,−1) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (1.5,0.5) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Identify invariant properties

Orientation is preserved for positive factors and reversed by a negative factor combined with central inversion, while similarity remains.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by identify invariant properties 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 is two-dimensional, so it scales by the square of the length factor.

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 factor 3. Area scale factor?

Step 1 - identify the governing idea: Under enlargement, corresponding angles, parallelism and shape are invariant; lengths and perimeter scale by |k|, and area by k².

Step 2 - apply it to this evidence: Area scales by 3².

Result: 9

The relationship is visible in the working: Area scales by 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 scales by 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 triangle angle is 47°. After factor 4 enlargement?

Step 1 - identify the governing idea: Under enlargement, corresponding angles, parallelism and shape are invariant; lengths and perimeter scale by |k|, and area by k².

Step 2 - apply it to this evidence: Corresponding angles are invariant.

Result: 47°

The relationship is visible in the working: Corresponding angles are invariant. 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:

  • 188° — It does not agree with the required relationship: Corresponding angles are invariant.
  • 11.75° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 51° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Perimeter 18 cm enlarged by factor 0.5 becomes?

Step 1 - identify the governing idea: Under enlargement, corresponding angles, parallelism and shape are invariant; lengths and perimeter scale by |k|, and area by k².

Step 2 - apply it to this evidence: Every length, hence total perimeter, halves.

Result: 9 cm

The relationship is visible in the working: Every length, hence total perimeter, halves. 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:

  • 4.5 cm — It does not agree with the required relationship: Every length, hence total perimeter, halves.
  • 18 cm — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 36 cm — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Interpret digital enlargements

Visual similarity alone does not prove one transformation. Coordinates and invariants provide checkable evidence.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by interpret digital enlargements 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: Verify a common centre and constant scale ratio.

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

Corresponding distances from centre are 4→10 and 6→15. Scale factor?

Step 1 - identify the governing idea: A valid enlargement has a common centre where lines through corresponding points meet and a constant directed distance ratio.

Step 2 - apply it to this evidence: 10/4=15/6=2.5.

Result: 2.5

The relationship is visible in the working: 10/4=15/6=2.5. 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:

  • 6 — It does not agree with the required relationship: 10/4=15/6=2.5.
  • 1.5 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

One side doubles but another triples. Is it one enlargement?

Step 1 - identify the governing idea: A valid enlargement has a common centre where lines through corresponding points meet and a constant directed distance ratio.

Step 2 - apply it to this evidence: A single enlargement requires one common scale factor.

Result: No

The relationship is visible in the working: A single enlargement requires one common scale factor. 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 both grew — It does not agree with the required relationship: A single enlargement requires one common scale factor.
  • Only angles matter — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Yes with factor 5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Lines joining three corresponding vertex pairs do not meet at one point. Best conclusion?

Step 1 - identify the governing idea: A valid enlargement has a common centre where lines through corresponding points meet and a constant directed distance ratio.

Step 2 - apply it to this evidence: All corresponding rays must pass through the centre.

Result: The mapping is not a single enlargement with one centre

The relationship is visible in the working: All corresponding rays must pass through the centre. 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:

  • The scale factor is zero — It does not agree with the required relationship: All corresponding rays must pass through the centre.
  • It must be factor 1 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Area alone proves it — 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. From origin, (6,4) enlarged by factor 1/2 maps to?
  2. A triangle angle is 47°. After factor 4 enlargement?
  3. One side doubles but another triples. Is it one enlargement?

Answers

  1. (3,2) — A factor below 1 reduces each displacement.
  2. 47° — Corresponding angles are invariant.
  3. No — A single enlargement requires one common scale factor.

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