Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 2 - Linear functions, conjectures and generalisation

Linear functions, conjectures and generalisation

Represent linear functions in tables, rules and graphs, test conjectures and generalise patterns.

Updated 2026-07-26 - 12 min read

Linear functions, conjectures and generalisation 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.

Identify linear patterns and constant change

A linear pattern has a constant first difference when inputs increase by equal amounts. 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 identify linear patterns and constant change 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: Linear growth requires equal additive change for equal input steps, not merely an upward trend.

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

The outputs are 5, 8, 11, 14 for inputs 1, 2, 3, 4. Is the relation linear?

Step 1 - identify the governing idea: A linear pattern has a constant first difference when inputs increase by equal amounts.

Step 2 - apply it to this evidence: Each unit increase in input adds 3 to the output.

Result: Yes, the first difference is 3

The relationship is visible in the working: Each unit increase in input adds 3 to the output. 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:

  • No, because the outputs are not multiples — It does not agree with the required relationship: Each unit increase in input adds 3 to the output.
  • Yes, because every pattern is linear — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No, because the values are odd and even — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Which sequence is nonlinear?

Step 1 - identify the governing idea: A linear pattern has a constant first difference when inputs increase by equal amounts.

Step 2 - apply it to this evidence: The differences 2, 4 and 8 are not constant; the pattern doubles.

Result: 2, 4, 8, 16

The relationship is visible in the working: The differences 2, 4 and 8 are not constant; the pattern doubles. 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, 7, 11, 15 — It does not agree with the required relationship: The differences 2, 4 and 8 are not constant; the pattern doubles.
  • 10, 8, 6, 4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1, 1, 1, 1 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A pattern has rule y = 4x − 1. What is the change in y when x increases by 1?

Step 1 - identify the governing idea: A linear pattern has a constant first difference when inputs increase by equal amounts.

Step 2 - apply it to this evidence: The coefficient of x gives the constant additive change.

Result: 4

The relationship is visible in the working: The coefficient of x gives the constant additive change. 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: The coefficient of x gives the constant additive change.
  • 3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4x — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Connect tables, rules and graphs

For y = mx + b, m is the constant rate of change and b is the output when x = 0; the graph is a straight line. 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 connect tables, rules and graphs 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 coefficient m controls repeated change; b is the starting value or vertical intercept.

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

For y = 3x + 2, find y when x = 5.

Step 1 - identify the governing idea: For y = mx + b, m is the constant rate of change and b is the output when x = 0; the graph is a straight line.

Step 2 - apply it to this evidence: Substitution gives 3(5) + 2 = 17.

Result: 17

The relationship is visible in the working: Substitution gives 3(5) + 2 = 17. 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:

  • 25 — It does not agree with the required relationship: Substitution gives 3(5) + 2 = 17.
  • 10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 32 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A table has y-values 7, 11, 15 for x = 0, 1, 2. Which rule fits?

Step 1 - identify the governing idea: For y = mx + b, m is the constant rate of change and b is the output when x = 0; the graph is a straight line.

Step 2 - apply it to this evidence: The starting value is 7 and the constant increase is 4.

Result: y = 4x + 7

The relationship is visible in the working: The starting value is 7 and the constant increase is 4. 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:

  • y = 7x + 4 — It does not agree with the required relationship: The starting value is 7 and the constant increase is 4.
  • y = 4x — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • y = 11x − 4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Which point lies on y = 2x − 3?

Step 1 - identify the governing idea: For y = mx + b, m is the constant rate of change and b is the output when x = 0; the graph is a straight line.

Step 2 - apply it to this evidence: Substituting x = 4 gives y = 8 − 3 = 5.

Result: (4, 5)

The relationship is visible in the working: Substituting x = 4 gives y = 8 − 3 = 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:

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

Test and generalise conjectures

Use multiple strategically chosen cases to form a conjecture, express the general rule algebraically and distinguish evidence from proof. 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 and generalise conjectures 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: Examples can support or disprove a conjecture, but a general justification is needed to establish why it always holds.

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 learner claims every multiple of 4 is even. Which justification is general?

Step 1 - identify the governing idea: Use multiple strategically chosen cases to form a conjecture, express the general rule algebraically and distinguish evidence from proof.

Step 2 - apply it to this evidence: Writing an arbitrary multiple as 4n proves it is twice an integer.

Result: 4n = 2(2n), so every multiple of 4 has factor 2

The relationship is visible in the working: Writing an arbitrary multiple as 4n proves it is twice an integer. 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, 8 and 12 are even — It does not agree with the required relationship: Writing an arbitrary multiple as 4n proves it is twice an integer.
  • The claim sounds true — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Most multiples of 4 are even — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Which value disproves the conjecture n² > n for every integer n?

Step 1 - identify the governing idea: Use multiple strategically chosen cases to form a conjecture, express the general rule algebraically and distinguish evidence from proof.

Step 2 - apply it to this evidence: For n = 0, n² = n, so the strict inequality fails.

Result: 0

The relationship is visible in the working: For n = 0, n² = n, so the strict inequality fails. 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 — It does not agree with the required relationship: For n = 0, n² = n, so the strict inequality fails.
  • 3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • −2 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

The perimeter of a row of n joined unit squares is 2n + 2. What is the perimeter for 20 squares?

Step 1 - identify the governing idea: Use multiple strategically chosen cases to form a conjecture, express the general rule algebraically and distinguish evidence from proof.

Step 2 - apply it to this evidence: Substitution gives 2(20) + 2 = 42.

Result: 42 units

The relationship is visible in the working: Substitution gives 2(20) + 2 = 42. 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:

  • 40 units — It does not agree with the required relationship: Substitution gives 2(20) + 2 = 42.
  • 22 units — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 80 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. Which sequence is nonlinear?
  2. A table has y-values 7, 11, 15 for x = 0, 1, 2. Which rule fits?
  3. Which value disproves the conjecture n² > n for every integer n?

Answers

  1. 2, 4, 8, 16 — The differences 2, 4 and 8 are not constant; the pattern doubles.
  2. y = 4x + 7 — The starting value is 7 and the constant increase is 4.
  3. 0 — For n = 0, n² = n, so the strict inequality fails.

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