Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 3 - Geometric algorithms and proof decisions

Geometric algorithms and proof decisions

Design, test and refine algorithms based on constructions and geometric theorems.

Updated 2026-07-26 - 11 min read

Geometric algorithms and proof decisions 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.

Write unambiguous geometric steps

Each step should identify the object, operation and dependency. Words such as 'roughly' or 'make it look equal' are not reproducible instructions.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by write unambiguous geometric steps 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 steps and construction constraints are required so the result can be reproduced.

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

Best first step for perpendicular bisector of AB?

Step 1 - identify the governing idea: A geometric algorithm must be ordered, unambiguous and executable by another person or digital tool.

Step 2 - apply it to this evidence: Equal radii create intersection points equidistant from A and B.

Result: Draw equal-radius arcs centred at A and B with radius greater than half AB

The relationship is visible in the working: Equal radii create intersection points equidistant from A and B. 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:

  • Guess the midpoint by eye — It does not agree with the required relationship: Equal radii create intersection points equidistant from A and B.
  • Draw any line through A — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Measure one random angle — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Why must construction steps be ordered?

Step 1 - identify the governing idea: A geometric algorithm must be ordered, unambiguous and executable by another person or digital tool.

Step 2 - apply it to this evidence: Dependencies make a different order invalid or impossible.

Result: Later objects depend on earlier constructed points or lines

The relationship is visible in the working: Dependencies make a different order invalid or impossible. 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:

  • Order is only for neatness — It does not agree with the required relationship: Dependencies make a different order invalid or impossible.
  • Geometry has no dependencies — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Alphabetical order proves accuracy — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which instruction is unambiguous?

Step 1 - identify the governing idea: A geometric algorithm must be ordered, unambiguous and executable by another person or digital tool.

Step 2 - apply it to this evidence: Centre and radius-defining point are explicit.

Result: Draw the circle centred at P through Q

The relationship is visible in the working: Centre and radius-defining point are explicit. 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:

  • Draw a medium circle — It does not agree with the required relationship: Centre and radius-defining point are explicit.
  • Make a nice arc — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Put a circle near Q — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Use theorem conditions

For example, equal alternate angles imply parallel lines only with a transversal configuration; Pythagoras requires a right triangle.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by use theorem conditions 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 the conditions before invoking the theorem.

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

An algorithm uses a²+b²=c². What condition must it check?

Step 1 - identify the governing idea: A theorem is an if–then statement: the conclusion is valid only when its conditions are established.

Step 2 - apply it to this evidence: Pythagoras requires a right triangle.

Result: The triangle is right-angled with c as hypotenuse

The relationship is visible in the working: Pythagoras requires a right triangle. 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:

  • All sides are equal — It does not agree with the required relationship: Pythagoras requires a right triangle.
  • c is the shortest side — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The triangle is drawn large — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Equal corresponding angles can justify what conclusion?

Step 1 - identify the governing idea: A theorem is an if–then statement: the conclusion is valid only when its conditions are established.

Step 2 - apply it to this evidence: The converse corresponding-angles theorem applies in a transversal configuration.

Result: The lines are parallel

The relationship is visible in the working: The converse corresponding-angles theorem applies in a transversal configuration. 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 lines are perpendicular — It does not agree with the required relationship: The converse corresponding-angles theorem applies in a transversal configuration.
  • All lengths are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The triangle is equilateral — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A point lies on a perpendicular bisector of AB. What follows?

Step 1 - identify the governing idea: A theorem is an if–then statement: the conclusion is valid only when its conditions are established.

Step 2 - apply it to this evidence: That is the perpendicular-bisector locus theorem.

Result: It is equidistant from A and B

The relationship is visible in the working: That is the perpendicular-bisector locus theorem. 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:

  • It is the midpoint of AB — It does not agree with the required relationship: That is the perpendicular-bisector locus theorem.
  • It lies on AB — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Angle APB is always 90° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Test and refine algorithms

Testing collinear points, equal values, vertical lines or degenerate shapes can expose missing decisions. Refinement adds conditions or branches.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by test and refine algorithms 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: Test varied cases and justify why the steps cover the whole stated domain.

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 midpoint algorithm divides by x₂−x₁ and fails for vertical segments. Best refinement?

Step 1 - identify the governing idea: One successful example shows possibility, not correctness for all allowed inputs.

Step 2 - apply it to this evidence: Averaging works for vertical and non-vertical segments.

Result: Use coordinate averaging instead of gradient division

The relationship is visible in the working: Averaging works for vertical and non-vertical segments. 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:

  • Ban vertical segments silently — It does not agree with the required relationship: Averaging works for vertical and non-vertical segments.
  • Rotate the screen — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Round x₂−x₁ to 1 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Best test set for a triangle-classification algorithm?

Step 1 - identify the governing idea: One successful example shows possibility, not correctness for all allowed inputs.

Step 2 - apply it to this evidence: Varied and edge cases test branches and assumptions.

Result: Acute, right, obtuse, isosceles and boundary/invalid cases

The relationship is visible in the working: Varied and edge cases test branches and assumptions. 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:

  • Three acute triangles only — It does not agree with the required relationship: Varied and edge cases test branches and assumptions.
  • One neat diagram — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only equilateral triangles — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

An algorithm returns different results when vertex labels rotate. What issue?

Step 1 - identify the governing idea: One successful example shows possibility, not correctness for all allowed inputs.

Step 2 - apply it to this evidence: A robust procedure should define or remove ordering dependence.

Result: It depends on label order rather than geometric relationships

The relationship is visible in the working: A robust procedure should define or remove ordering dependence. 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 theorem changed — It does not agree with the required relationship: A robust procedure should define or remove ordering dependence.
  • Rotation changes side lengths — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • All algorithms need fixed letters — 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. Why must construction steps be ordered?
  2. Equal corresponding angles can justify what conclusion?
  3. Best test set for a triangle-classification algorithm?

Answers

  1. Later objects depend on earlier constructed points or lines — Dependencies make a different order invalid or impossible.
  2. The lines are parallel — The converse corresponding-angles theorem applies in a transversal configuration.
  3. Acute, right, obtuse, isosceles and boundary/invalid cases — Varied and edge cases test branches and assumptions.

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