Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 2 - Coordinate distance, midpoint and gradient

Coordinate distance, midpoint and gradient

Find distance, midpoint and gradient between points on the Cartesian plane and interpret each result.

Updated 2026-07-26 - 10 min read

Coordinate distance, midpoint and gradient 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.

Calculate gradient

A negative gradient records a decreasing line, not an impossible distance. A vertical line has zero run and undefined gradient.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate gradient 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: Gradient is change in y divided by change in x.

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

Find the gradient through (1,2) and (5,10).

Step 1 - identify the governing idea: Gradient is vertical change divided by horizontal change. Reversing both subtraction orders gives the same result.

Step 2 - apply it to this evidence: (10−2)/(5−1)=8/4=2.

Result: 2

The relationship is visible in the working: (10−2)/(5−1)=8/4=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:

  • 1/2 — It does not agree with the required relationship: (10−2)/(5−1)=8/4=2.
  • 3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Find the gradient through (−2,5) and (4,−1).

Step 1 - identify the governing idea: Gradient is vertical change divided by horizontal change. Reversing both subtraction orders gives the same result.

Step 2 - apply it to this evidence: (−1−5)/(4−(−2))=−6/6=−1.

Result: −1

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

Example 1.3

What is the gradient through (3,1) and (3,7)?

Step 1 - identify the governing idea: Gradient is vertical change divided by horizontal change. Reversing both subtraction orders gives the same result.

Step 2 - apply it to this evidence: The run is 3−3=0, so division by zero is undefined.

Result: Undefined

The relationship is visible in the working: The run is 3−3=0, so division by zero is undefined. 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:

  • 0 — It does not agree with the required relationship: The run is 3−3=0, so division by zero is undefined.
  • 6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1/6 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Calculate midpoint

Averaging each coordinate places the new point halfway horizontally and vertically. It should be equidistant from both endpoints.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate midpoint 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: Average x-coordinates together and y-coordinates together.

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

Find the midpoint of (2,4) and (8,10).

Step 1 - identify the governing idea: The midpoint is ((x₁+x₂)/2,(y₁+y₂)/2).

Step 2 - apply it to this evidence: Average x: 5; average y: 7.

Result: (5,7)

The relationship is visible in the working: Average x: 5; average y: 7. 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:

  • (10,14) — It does not agree with the required relationship: Average x: 5; average y: 7.
  • (3,3) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (5,14) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Find the midpoint of (−5,3) and (1,−7).

Step 1 - identify the governing idea: The midpoint is ((x₁+x₂)/2,(y₁+y₂)/2).

Step 2 - apply it to this evidence: x average is −2 and y average is −2.

Result: (−2,−2)

The relationship is visible in the working: x average is −2 and y average is −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,−4) — It does not agree with the required relationship: x average is −2 and y average is −2.
  • (2,2) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (−3,−5) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

One endpoint is (4,6), midpoint is (7,2). Find the other endpoint.

Step 1 - identify the governing idea: The midpoint is ((x₁+x₂)/2,(y₁+y₂)/2).

Step 2 - apply it to this evidence: The unknown coordinate must average with (4,6) to give (7,2).

Result: (10,−2)

The relationship is visible in the working: The unknown coordinate must average with (4,6) to give (7,2). 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:

  • (3,4) — It does not agree with the required relationship: The unknown coordinate must average with (4,6) to give (7,2).
  • (14,4) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (10,2) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Calculate coordinate distance

Coordinate differences form perpendicular horizontal and vertical legs. Squaring removes direction because distance is non-negative.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate coordinate distance 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: Use Pythagoras on the horizontal and vertical changes, then take the square root.

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

Find the distance from (1,1) to (4,5).

Step 1 - identify the governing idea: Distance is √((x₂−x₁)²+(y₂−y₁)²).

Step 2 - apply it to this evidence: Changes are 3 and 4, so distance is √(9+16)=5.

Result: 5 units

The relationship is visible in the working: Changes are 3 and 4, so distance is √(9+16)=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 units — It does not agree with the required relationship: Changes are 3 and 4, so distance is √(9+16)=5.
  • 25 units — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • √7 units — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Find the distance from (−2,3) to (4,−5).

Step 1 - identify the governing idea: Distance is √((x₂−x₁)²+(y₂−y₁)²).

Step 2 - apply it to this evidence: Changes are 6 and −8; √(36+64)=10.

Result: 10 units

The relationship is visible in the working: Changes are 6 and −8; √(36+64)=10. 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:

  • 14 units — It does not agree with the required relationship: Changes are 6 and −8; √(36+64)=10.
  • 2 units — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 100 units — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Find the exact distance from (0,0) to (5,2).

Step 1 - identify the governing idea: Distance is √((x₂−x₁)²+(y₂−y₁)²).

Step 2 - apply it to this evidence: Distance is √(5²+2²)=√29.

Result: √29 units

The relationship is visible in the working: Distance is √(5²+2²)=√29. 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 units — It does not agree with the required relationship: Distance is √(5²+2²)=√29.
  • 29 units — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • √7 units — 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. Find the gradient through (−2,5) and (4,−1).
  2. Find the midpoint of (−5,3) and (1,−7).
  3. Find the distance from (−2,3) to (4,−5).

Answers

  1. −1 — (−1−5)/(4−(−2))=−6/6=−1.
  2. (−2,−2) — x average is −2 and y average is −2.
  3. 10 units — Changes are 6 and −8; √(36+64)=10.

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