Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 2 - Spatial algorithms and digital solutions

Spatial algorithms and digital solutions

Design, test and refine algorithms that generate spatial patterns or solve geometric problems.

Updated 2026-07-26 - 12 min read

Spatial algorithms and digital solutions 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.

Specify spatial algorithms precisely

Define initial state, coordinate or heading conventions, ordered operations, repetitions and stopping conditions. 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 specify spatial algorithms precisely 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: A reproducible algorithm must state each operation and reference convention explicitly.

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

What essential state does a turtle algorithm need before moving?

Step 1 - identify the governing idea: Define initial state, coordinate or heading conventions, ordered operations, repetitions and stopping conditions.

Step 2 - apply it to this evidence: Movement commands depend on where the turtle is and which way it faces.

Result: Starting position and heading

The relationship is visible in the working: Movement commands depend on where the turtle is and which way it faces. 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:

  • Only colour — It does not agree with the required relationship: Movement commands depend on where the turtle is and which way it faces.
  • Only final point — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A random seed only — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Which command sequence traces a regular hexagon?

Step 1 - identify the governing idea: Define initial state, coordinate or heading conventions, ordered operations, repetitions and stopping conditions.

Step 2 - apply it to this evidence: Six exterior turns of 60° total 360°.

Result: Repeat 6: move d, turn 60°

The relationship is visible in the working: Six exterior turns of 60° total 360°. 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:

  • Repeat 6: turn 120° — It does not agree with the required relationship: Six exterior turns of 60° total 360°.
  • Repeat 5: turn 60° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Repeat 6: turn 90° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which rule rotates (x,y) 90° anticlockwise about the origin?

Step 1 - identify the governing idea: Define initial state, coordinate or heading conventions, ordered operations, repetitions and stopping conditions.

Step 2 - apply it to this evidence: The x-axis unit vector maps to the y-axis unit vector.

Result: (x,y) → (−y,x)

The relationship is visible in the working: The x-axis unit vector maps to the y-axis unit vector. 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,y) → (y,x) — It does not agree with the required relationship: The x-axis unit vector maps to the y-axis unit vector.
  • (x,y) → (x,−y) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (x,y) → (−x,−y) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Test invariants and edge cases

Verify properties that should remain true and test boundaries, negative coordinates, degenerate inputs and repeated operations. 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 test invariants and edge cases 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: Testing must target likely failure modes and general invariants, not only a routine input.

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

What invariant should a rigid rotation preserve?

Step 1 - identify the governing idea: Verify properties that should remain true and test boundaries, negative coordinates, degenerate inputs and repeated operations.

Step 2 - apply it to this evidence: Rigid transformations preserve shape and size.

Result: Distances between all pairs of points

The relationship is visible in the working: Rigid transformations preserve shape and size. 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 x-coordinates — It does not agree with the required relationship: Rigid transformations preserve shape and size.
  • Distance from the x-axis — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Vertex order only — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Which test best checks a reflection algorithm?

Step 1 - identify the governing idea: Verify properties that should remain true and test boundaries, negative coordinates, degenerate inputs and repeated operations.

Step 2 - apply it to this evidence: A reflection is its own inverse.

Result: Reflect a point twice and confirm it returns

The relationship is visible in the working: A reflection is its own inverse. 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:

  • Test only the origin — It does not agree with the required relationship: A reflection is its own inverse.
  • Use one positive coordinate — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Compare colours — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Why test a polygon algorithm with zero side length?

Step 1 - identify the governing idea: Verify properties that should remain true and test boundaries, negative coordinates, degenerate inputs and repeated operations.

Step 2 - apply it to this evidence: Boundary inputs may expose unstated assumptions or division errors.

Result: It reveals degenerate-input behaviour

The relationship is visible in the working: Boundary inputs may expose unstated assumptions or division errors. 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 proves every polygon works — It does not agree with the required relationship: Boundary inputs may expose unstated assumptions or division errors.
  • Zero creates a regular polygon — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No test is needed — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Debug and refine spatial solutions

Compare expected and actual state after each step, isolate the first divergence and change one cause at a time. 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 debug and refine spatial solutions 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 faulty transition must be corrected or later inputs will reproduce the error.

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 square closes after four sides but ends facing west instead of east. What likely failed?

Step 1 - identify the governing idea: Compare expected and actual state after each step, isolate the first divergence and change one cause at a time.

Step 2 - apply it to this evidence: Position can close while heading reveals incorrect turn accumulation.

Result: The total turning was not 360°

The relationship is visible in the working: Position can close while heading reveals incorrect turn accumulation. 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 side lengths were all zero — It does not agree with the required relationship: Position can close while heading reveals incorrect turn accumulation.
  • The start point moved — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Coordinates cannot represent heading — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A translated shape has one vertex moved by a different vector. What is the defect?

Step 1 - identify the governing idea: Compare expected and actual state after each step, isolate the first divergence and change one cause at a time.

Step 2 - apply it to this evidence: Every point must receive the same displacement.

Result: The translation rule was applied inconsistently

The relationship is visible in the working: Every point must receive the same 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:

  • The original shape was too large — It does not agree with the required relationship: Every point must receive the same displacement.
  • Translations change area — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The axes are invalid — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

What is the best debugging trace?

Step 1 - identify the governing idea: Compare expected and actual state after each step, isolate the first divergence and change one cause at a time.

Step 2 - apply it to this evidence: The earliest divergence localises the faulty operation.

Result: Record state after each command and find the first mismatch

The relationship is visible in the working: The earliest divergence localises the faulty operation. 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:

  • Inspect only the final image — It does not agree with the required relationship: The earliest divergence localises the faulty operation.
  • Rewrite everything at once — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Change multiple constants together — 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. Which command sequence traces a regular hexagon?
  2. Which test best checks a reflection algorithm?
  3. A translated shape has one vertex moved by a different vector. What is the defect?

Answers

  1. Repeat 6: move d, turn 60° — Six exterior turns of 60° total 360°.
  2. Reflect a point twice and confirm it returns — A reflection is its own inverse.
  3. The translation rule was applied inconsistently — Every point must receive the same displacement.

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