Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 2 - Linear expressions and algebraic properties

Linear expressions and algebraic properties

Expand, factorise and simplify linear expressions using the distributive, associative and commutative properties.

Updated 2026-07-26 - 12 min read

Linear expressions and algebraic properties 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.

Collect like terms

Only terms with identical variable parts and exponents can be combined by adding or subtracting their coefficients. 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 collect like terms 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 complete variable part must match; for example, x and x² are not like terms.

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

Simplify 5x + 3x − 4.

Step 1 - identify the governing idea: Only terms with identical variable parts and exponents can be combined by adding or subtracting their coefficients.

Step 2 - apply it to this evidence: The x-terms combine to 8x while the constant remains separate.

Result: 8x − 4

The relationship is visible in the working: The x-terms combine to 8x while the constant remains separate. 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:

  • 4x — It does not agree with the required relationship: The x-terms combine to 8x while the constant remains separate.
  • 8x − 1 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4x − 4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Simplify 7a − 2 + 4 − 3a.

Step 1 - identify the governing idea: Only terms with identical variable parts and exponents can be combined by adding or subtracting their coefficients.

Step 2 - apply it to this evidence: 7a − 3a = 4a and −2 + 4 = 2.

Result: 4a + 2

The relationship is visible in the working: 7a − 3a = 4a and −2 + 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:

  • 4a − 6 — It does not agree with the required relationship: 7a − 3a = 4a and −2 + 4 = 2.
  • 10a + 2 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8a — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Simplify 2m² + 5m − m² + 3m.

Step 1 - identify the governing idea: Only terms with identical variable parts and exponents can be combined by adding or subtracting their coefficients.

Step 2 - apply it to this evidence: The squared terms combine separately from the linear terms.

Result: m² + 8m

The relationship is visible in the working: The squared terms combine separately from the linear terms. 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:

  • 9m² — It does not agree with the required relationship: The squared terms combine separately from the linear terms.
  • m² + 2m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8m³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Expand using the distributive property

Multiply the factor outside brackets by every term inside the brackets, including each sign. 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 expand using the distributive property 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 bracket represents a grouped sum, so distribution gives 3x + 12.

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

Expand 4(2x − 3).

Step 1 - identify the governing idea: Multiply the factor outside brackets by every term inside the brackets, including each sign.

Step 2 - apply it to this evidence: Multiplying both terms by 4 gives 8x and −12.

Result: 8x − 12

The relationship is visible in the working: Multiplying both terms by 4 gives 8x and −12. 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:

  • 8x − 3 — It does not agree with the required relationship: Multiplying both terms by 4 gives 8x and −12.
  • 6x − 7 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8x + 12 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Expand −2(3y + 5).

Step 1 - identify the governing idea: Multiply the factor outside brackets by every term inside the brackets, including each sign.

Step 2 - apply it to this evidence: The negative factor changes the sign of both products.

Result: −6y − 10

The relationship is visible in the working: The negative factor changes the sign of both products. 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:

  • −6y + 5 — It does not agree with the required relationship: The negative factor changes the sign of both products.
  • 6y − 10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • −5y − 7 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Simplify 3(x + 2) + 2(x − 1).

Step 1 - identify the governing idea: Multiply the factor outside brackets by every term inside the brackets, including each sign.

Step 2 - apply it to this evidence: Expansion gives 3x + 6 + 2x − 2, which simplifies to 5x + 4.

Result: 5x + 4

The relationship is visible in the working: Expansion gives 3x + 6 + 2x − 2, which simplifies to 5x + 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:

  • 5x + 1 — It does not agree with the required relationship: Expansion gives 3x + 6 + 2x − 2, which simplifies to 5x + 4.
  • 6x + 4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5x + 8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Factorise by taking out a common factor

Factorising reverses expansion: divide every term by the greatest useful common factor and place the quotients in brackets. 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 factorise by taking out a common factor 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 factor must multiply back to the original expression; 6x + 9 = 3(2x + 3).

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

Factorise 8x + 12.

Step 1 - identify the governing idea: Factorising reverses expansion: divide every term by the greatest useful common factor and place the quotients in brackets.

Step 2 - apply it to this evidence: The greatest common factor is 4; dividing both terms gives 2x and 3.

Result: 4(2x + 3)

The relationship is visible in the working: The greatest common factor is 4; dividing both terms gives 2x and 3. 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(4x + 12) — It does not agree with the required relationship: The greatest common factor is 4; dividing both terms gives 2x and 3.
  • 4(8x + 12) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8(x + 12) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Factorise 15a − 20.

Step 1 - identify the governing idea: Factorising reverses expansion: divide every term by the greatest useful common factor and place the quotients in brackets.

Step 2 - apply it to this evidence: Both coefficients share factor 5 and the negative sign remains inside.

Result: 5(3a − 4)

The relationship is visible in the working: Both coefficients share factor 5 and the negative sign remains inside. 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:

  • 5(3a + 4) — It does not agree with the required relationship: Both coefficients share factor 5 and the negative sign remains inside.
  • 15(a − 20) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3(5a − 20) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Which expression is equivalent to 6x + 18 and fully factorised?

Step 1 - identify the governing idea: Factorising reverses expansion: divide every term by the greatest useful common factor and place the quotients in brackets.

Step 2 - apply it to this evidence: Expanding 6(x + 3) returns 6x + 18.

Result: 6(x + 3)

The relationship is visible in the working: Expanding 6(x + 3) returns 6x + 18. 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(2x + 18) — It does not agree with the required relationship: Expanding 6(x + 3) returns 6x + 18.
  • 6(x + 18) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2(3x + 18) — 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. Simplify 7a − 2 + 4 − 3a.
  2. Expand −2(3y + 5).
  3. Factorise 15a − 20.

Answers

  1. 4a + 2 — 7a − 3a = 4a and −2 + 4 = 2.
  2. −6y − 10 — The negative factor changes the sign of both products.
  3. 5(3a − 4) — Both coefficients share factor 5 and the negative sign remains inside.

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