Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Functions, conjectures and generalisation

Functions, conjectures and generalisation

Use function notation and digital tools to test conjectures, identify invariants and construct general arguments.

Updated 2026-07-26 - 12 min read

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.

Interpret function notation

f(x) names the output of function f for input x; evaluation substitutes the complete input expression wherever x appears. 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 function notation 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: Function notation records an input-output mapping, not multiplication.

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

If f(x) = 2x² − 1, find f(3).

Step 1 - identify the governing idea: f(x) names the output of function f for input x; evaluation substitutes the complete input expression wherever x appears.

Step 2 - apply it to this evidence: 2(3²) − 1 = 18 − 1 = 17.

Result: 17

The relationship is visible in the working: 2(3²) − 1 = 18 − 1 = 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:

  • 35 — It does not agree with the required relationship: 2(3²) − 1 = 18 − 1 = 17.
  • 11 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

If g(x) = 4 − x, find g(−2).

Step 1 - identify the governing idea: f(x) names the output of function f for input x; evaluation substitutes the complete input expression wherever x appears.

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

Result: 6

The relationship is visible in the working: 4 − (−2) = 6. 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: 4 − (−2) = 6.
  • −6 — 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 1.3

If h(x) = 3x + 5 and h(x) = 20, find x.

Step 1 - identify the governing idea: f(x) names the output of function f for input x; evaluation substitutes the complete input expression wherever x appears.

Step 2 - apply it to this evidence: 3x + 5 = 20 gives x = 5.

Result: 5

The relationship is visible in the working: 3x + 5 = 20 gives x = 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:

  • 20 — It does not agree with the required relationship: 3x + 5 = 20 gives x = 5.
  • 25 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 15 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Form and test conjectures

Use patterns and digital evidence to propose a general statement, then seek boundary cases and counterexamples before accepting it. 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 form and test 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: Computational evidence supports a conjecture but cannot replace a general argument over an unlimited 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 2.1

Which value disproves 'n² > n for every real n'?

Step 1 - identify the governing idea: Use patterns and digital evidence to propose a general statement, then seek boundary cases and counterexamples before accepting it.

Step 2 - apply it to this evidence: 0.5² = 0.25, which is less than 0.5.

Result: 0.5

The relationship is visible in the working: 0.5² = 0.25, which is less than 0.5. 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 — It does not agree with the required relationship: 0.5² = 0.25, which is less than 0.5.
  • 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 2.2

A conjecture holds for n = 1 to 1000. What is justified?

Step 1 - identify the governing idea: Use patterns and digital evidence to propose a general statement, then seek boundary cases and counterexamples before accepting it.

Step 2 - apply it to this evidence: Finite checks cannot cover all integers.

Result: It has strong numerical support but still needs proof

The relationship is visible in the working: Finite checks cannot cover all integers. 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 proven for every integer — It does not agree with the required relationship: Finite checks cannot cover all integers.
  • It must fail at 1001 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No evidence exists — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Which strategy most strongly tests a conjecture?

Step 1 - identify the governing idea: Use patterns and digital evidence to propose a general statement, then seek boundary cases and counterexamples before accepting it.

Step 2 - apply it to this evidence: Potential failure points are more informative than routine confirming cases.

Result: Check boundary cases and actively search for counterexamples

The relationship is visible in the working: Potential failure points are more informative than routine confirming cases. 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 one example — It does not agree with the required relationship: Potential failure points are more informative than routine confirming cases.
  • Ignore excluded values — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Round all results — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Construct general algebraic arguments

Represent an arbitrary member of the domain algebraically and transform it using accepted properties to establish the claim. 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 construct general algebraic arguments 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 proof must show why the relationship holds for every allowed case, not one representative picture.

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

Why is the sum of consecutive integers n and n+1 odd?

Step 1 - identify the governing idea: Represent an arbitrary member of the domain algebraically and transform it using accepted properties to establish the claim.

Step 2 - apply it to this evidence: The algebra represents any pair of consecutive integers.

Result: n + (n+1) = 2n+1, one more than an even number

The relationship is visible in the working: The algebra represents any pair of consecutive integers. 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:

  • Examples 4+5 and 8+9 are odd — It does not agree with the required relationship: The algebra represents any pair of consecutive integers.
  • All integers are odd — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The sum is 2n — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Prove the square of an even integer is even. Which expression works?

Step 1 - identify the governing idea: Represent an arbitrary member of the domain algebraically and transform it using accepted properties to establish the claim.

Step 2 - apply it to this evidence: The result is 2 times an integer.

Result: (2k)² = 2(2k²)

The relationship is visible in the working: The result is 2 times 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:

  • (2k)² = 2k² — It does not agree with the required relationship: The result is 2 times an integer.
  • 2² = 4 only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • k² is always even — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Which statement establishes that x(x+1) is even for integer x?

Step 1 - identify the governing idea: Represent an arbitrary member of the domain algebraically and transform it using accepted properties to establish the claim.

Step 2 - apply it to this evidence: The product contains an even factor for every integer x.

Result: One of two consecutive integers is even

The relationship is visible in the working: The product contains an even factor for every integer x. 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 is always even — It does not agree with the required relationship: The product contains an even factor for every integer x.
  • x+1 is always odd — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The product is always zero — 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. If g(x) = 4 − x, find g(−2).
  2. A conjecture holds for n = 1 to 1000. What is justified?
  3. Prove the square of an even integer is even. Which expression works?

Answers

  1. 6 — 4 − (−2) = 6.
  2. It has strong numerical support but still needs proof — Finite checks cannot cover all integers.
  3. (2k)² = 2(2k²) — The result is 2 times an integer.

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