Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 1 - Integer and rational-number operations

Integer and rational-number operations

Choose and apply efficient operations with integers, fractions and decimals, including brackets and powers.

Updated 2026-07-26 - 12 min read

Integer and rational-number operations 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.

Operate with signed numbers

Signs depend on the operation: subtracting a negative adds its opposite, while multiplying or dividing equal signs gives a positive result. 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 operate with signed numbers 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 equal-sign rule applies to multiplication and division; subtraction must first be rewritten as adding the opposite.

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

Evaluate −7 − (−12).

Step 1 - identify the governing idea: Signs depend on the operation: subtracting a negative adds its opposite, while multiplying or dividing equal signs gives a positive result.

Step 2 - apply it to this evidence: Subtracting −12 is equivalent to adding 12: −7 + 12 = 5.

Result: 5

The relationship is visible in the working: Subtracting −12 is equivalent to adding 12: −7 + 12 = 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:

  • −19 — It does not agree with the required relationship: Subtracting −12 is equivalent to adding 12: −7 + 12 = 5.
  • 19 — 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

Evaluate (−6)(−4) ÷ 3.

Step 1 - identify the governing idea: Signs depend on the operation: subtracting a negative adds its opposite, while multiplying or dividing equal signs gives a positive result.

Step 2 - apply it to this evidence: The product of two negative factors is 24, and 24 ÷ 3 = 8.

Result: 8

The relationship is visible in the working: The product of two negative factors is 24, and 24 ÷ 3 = 8. 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:

  • −8 — It does not agree with the required relationship: The product of two negative factors is 24, and 24 ÷ 3 = 8.
  • 2 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 72 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A temperature changes from 3°C to −5°C. What is the change?

Step 1 - identify the governing idea: Signs depend on the operation: subtracting a negative adds its opposite, while multiplying or dividing equal signs gives a positive result.

Step 2 - apply it to this evidence: Final minus initial is −5 − 3 = −8°C.

Result: −8°C

The relationship is visible in the working: Final minus initial is −5 − 3 = −8°C. 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:

  • 8°C — It does not agree with the required relationship: Final minus initial is −5 − 3 = −8°C.
  • −2°C — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2°C — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Operate with positive and negative fractions

Add or subtract fractions using a common denominator; multiply numerators and denominators; divide by multiplying by the reciprocal. 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 operate with positive and negative fractions 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: Denominators name the size of the parts, so addition first requires equivalent fractions with the same part size.

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

Evaluate 3/4 − 5/6.

Step 1 - identify the governing idea: Add or subtract fractions using a common denominator; multiply numerators and denominators; divide by multiplying by the reciprocal.

Step 2 - apply it to this evidence: Using denominator 12 gives 9/12 − 10/12 = −1/12.

Result: −1/12

The relationship is visible in the working: Using denominator 12 gives 9/12 − 10/12 = −1/12. 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:

  • −2/2 — It does not agree with the required relationship: Using denominator 12 gives 9/12 − 10/12 = −1/12.
  • 8/10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1/12 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Evaluate (−2/3) × (9/10).

Step 1 - identify the governing idea: Add or subtract fractions using a common denominator; multiply numerators and denominators; divide by multiplying by the reciprocal.

Step 2 - apply it to this evidence: Cancel 3 with 9 and 2 with 10, then retain the single negative sign.

Result: −3/5

The relationship is visible in the working: Cancel 3 with 9 and 2 with 10, then retain the single negative sign. 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:

  • 3/5 — It does not agree with the required relationship: Cancel 3 with 9 and 2 with 10, then retain the single negative sign.
  • −18/13 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • −11/30 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Evaluate 5/8 ÷ (−15/4).

Step 1 - identify the governing idea: Add or subtract fractions using a common denominator; multiply numerators and denominators; divide by multiplying by the reciprocal.

Step 2 - apply it to this evidence: Multiply 5/8 by the reciprocal −4/15 and simplify to −1/6.

Result: −1/6

The relationship is visible in the working: Multiply 5/8 by the reciprocal −4/15 and simplify to −1/6. 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:

  • −75/32 — It does not agree with the required relationship: Multiply 5/8 by the reciprocal −4/15 and simplify to −1/6.
  • 1/6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • −10/23 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Apply order of operations to rational numbers

Evaluate grouping symbols and powers first, then multiplication or division, then addition or subtraction at the same level from left to right. 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 apply order of operations to rational numbers 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: Operations have agreed precedence; left-to-right applies only among operations at the same level.

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

Evaluate 6 − 2(−3 + 5).

Step 1 - identify the governing idea: Evaluate grouping symbols and powers first, then multiplication or division, then addition or subtraction at the same level from left to right.

Step 2 - apply it to this evidence: The bracket is 2, then 2 × 2 = 4, and 6 − 4 = 2.

Result: 2

The relationship is visible in the working: The bracket is 2, then 2 × 2 = 4, and 6 − 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:

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

Example 3.2

Evaluate −3² + 10.

Step 1 - identify the governing idea: Evaluate grouping symbols and powers first, then multiplication or division, then addition or subtraction at the same level from left to right.

Step 2 - apply it to this evidence: The exponent applies to 3 before the leading negative: −(3²) + 10 = −9 + 10 = 1.

Result: 1

The relationship is visible in the working: The exponent applies to 3 before the leading negative: −(3²) + 10 = −9 + 10 = 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:

  • 19 — It does not agree with the required relationship: The exponent applies to 3 before the leading negative: −(3²) + 10 = −9 + 10 = 1.
  • −19 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 7 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Evaluate 1.5 + 3/4 × 2.

Step 1 - identify the governing idea: Evaluate grouping symbols and powers first, then multiplication or division, then addition or subtraction at the same level from left to right.

Step 2 - apply it to this evidence: Multiply first: 3/4 × 2 = 1.5, then add to obtain 3.

Result: 3

The relationship is visible in the working: Multiply first: 3/4 × 2 = 1.5, then add to obtain 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:

  • 4.5 — It does not agree with the required relationship: Multiply first: 3/4 × 2 = 1.5, then add to obtain 3.
  • 2.25 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1.875 — 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. Evaluate (−6)(−4) ÷ 3.
  2. Evaluate (−2/3) × (9/10).
  3. Evaluate −3² + 10.

Answers

  1. 8 — The product of two negative factors is 24, and 24 ÷ 3 = 8.
  2. −3/5 — Cancel 3 with 9 and 2 with 10, then retain the single negative sign.
  3. 1 — The exponent applies to 3 before the leading negative: −(3²) + 10 = −9 + 10 = 1.

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