Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 3 - Three-dimensional location and geometric algorithms

Three-dimensional location and geometric algorithms

Represent locations in three dimensions and design or follow algorithms that generate geometric patterns and transformations.

Updated 2026-07-26 - 12 min read

Three-dimensional location and geometric algorithms 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.

Interpret three-dimensional coordinates

An ordered triple (x, y, z) locates a point by three perpendicular coordinates whose order and reference origin must remain fixed. 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 interpret three-dimensional coordinates 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: Each position has a different axis meaning, so swapping coordinates usually locates a different point.

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

Which point has x = 2, y = −1 and z = 4?

Step 1 - identify the governing idea: An ordered triple (x, y, z) locates a point by three perpendicular coordinates whose order and reference origin must remain fixed.

Step 2 - apply it to this evidence: The ordered triple records x, then y, then z.

Result: (2, −1, 4)

The relationship is visible in the working: The ordered triple records x, then y, then z. 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:

  • (−1, 2, 4) — It does not agree with the required relationship: The ordered triple records x, then y, then z.
  • (4, −1, 2) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (2, 4, −1) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

From (3, 2, 1), move 4 units in the positive z-direction. Where do you arrive?

Step 1 - identify the governing idea: An ordered triple (x, y, z) locates a point by three perpendicular coordinates whose order and reference origin must remain fixed.

Step 2 - apply it to this evidence: Only the z-coordinate changes: 1 + 4 = 5.

Result: (3, 2, 5)

The relationship is visible in the working: Only the z-coordinate changes: 1 + 4 = 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:

  • (7, 2, 1) — It does not agree with the required relationship: Only the z-coordinate changes: 1 + 4 = 5.
  • (3, 6, 1) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (3, 2, −3) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

What displacement moves point (1, 5, 2) to (4, 3, 8)?

Step 1 - identify the governing idea: An ordered triple (x, y, z) locates a point by three perpendicular coordinates whose order and reference origin must remain fixed.

Step 2 - apply it to this evidence: Subtract start from finish in each coordinate.

Result: (+3, −2, +6)

The relationship is visible in the working: Subtract start from finish in each coordinate. 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, +2, −6) — It does not agree with the required relationship: Subtract start from finish in each coordinate.
  • (+5, +8, +10) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (+3, +2, +6) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Follow geometric algorithms

Execute each instruction in order while maintaining the current position, direction, length and angle state. 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 follow geometric 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: Transformations and movements are generally non-commutative, so changing the order changes the result.

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

Start at (0,0), move 3 right then 2 up. Where do you finish?

Step 1 - identify the governing idea: Execute each instruction in order while maintaining the current position, direction, length and angle state.

Step 2 - apply it to this evidence: The x-coordinate increases by 3 and y increases by 2.

Result: (3, 2)

The relationship is visible in the working: The x-coordinate increases by 3 and y increases 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:

  • (2, 3) — It does not agree with the required relationship: The x-coordinate increases by 3 and y increases by 2.
  • (3, −2) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (5, 0) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A turtle faces east, moves 5, turns left 90° and moves 2. Which final direction is it facing?

Step 1 - identify the governing idea: Execute each instruction in order while maintaining the current position, direction, length and angle state.

Step 2 - apply it to this evidence: A left turn from east faces north; the final movement does not change direction.

Result: North

The relationship is visible in the working: A left turn from east faces north; the final movement does not change direction. 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:

  • South — It does not agree with the required relationship: A left turn from east faces north; the final movement does not change direction.
  • East — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • West — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Which repeated command traces a square?

Step 1 - identify the governing idea: Execute each instruction in order while maintaining the current position, direction, length and angle state.

Step 2 - apply it to this evidence: Four equal sides and four quarter turns return to the start.

Result: Repeat 4 times: move d, turn 90°

The relationship is visible in the working: Four equal sides and four quarter turns return to the start. 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 3 times: move d, turn 90° — It does not agree with the required relationship: Four equal sides and four quarter turns return to the start.
  • Repeat 4 times: move d, turn 60° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Repeat 2 times: move d, turn 180° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Design and debug geometric algorithms

Specify initial state, exact movement and rotation commands, repetition count and a test that verifies the intended invariant or final position. 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 design and debug geometric 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: Correctness requires reproducible instructions and checks such as closure, equal side lengths or preserved distances.

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 algorithm uses turn 80° four times. What is the error?

Step 1 - identify the governing idea: Specify initial state, exact movement and rotation commands, repetition count and a test that verifies the intended invariant or final position.

Step 2 - apply it to this evidence: Four exterior turns must total 360°, so each is 90°.

Result: The turn must be 90°

The relationship is visible in the working: Four exterior turns must total 360°, so each is 90°. 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 needs five repeats — It does not agree with the required relationship: Four exterior turns must total 360°, so each is 90°.
  • The side lengths must differ — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The starting point is wrong — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

An algorithm should reflect (x,y) across the y-axis. Which rule is correct?

Step 1 - identify the governing idea: Specify initial state, exact movement and rotation commands, repetition count and a test that verifies the intended invariant or final position.

Step 2 - apply it to this evidence: Reflection across the y-axis changes horizontal sign and preserves vertical position.

Result: (x, y) → (−x, y)

The relationship is visible in the working: Reflection across the y-axis changes horizontal sign and preserves vertical position. 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) → (x, −y) — It does not agree with the required relationship: Reflection across the y-axis changes horizontal sign and preserves vertical position.
  • (x, y) → (y, x) — 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.

Example 3.3

Which check verifies a translation preserved a shape?

Step 1 - identify the governing idea: Specify initial state, exact movement and rotation commands, repetition count and a test that verifies the intended invariant or final position.

Step 2 - apply it to this evidence: A translation applies one identical displacement to every point.

Result: All vertices moved by the same displacement vector

The relationship is visible in the working: A translation applies one identical displacement to every point. 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 shape is closer to the origin — It does not agree with the required relationship: A translation applies one identical displacement to every point.
  • All coordinates are positive — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The area appears larger — 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 (3, 2, 1), move 4 units in the positive z-direction. Where do you arrive?
  2. A turtle faces east, moves 5, turns left 90° and moves 2. Which final direction is it facing?
  3. An algorithm should reflect (x,y) across the y-axis. Which rule is correct?

Answers

  1. (3, 2, 5) — Only the z-coordinate changes: 1 + 4 = 5.
  2. North — A left turn from east faces north; the final movement does not change direction.
  3. (x, y) → (−x, y) — Reflection across the y-axis changes horizontal sign and preserves vertical position.

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