Year 5 Mathematics · AC9M5A01

Recognise and explain the connection between multiplication and division as inverse operations and use this to develop families of number facts

Students use one multiplicative relationship to generate related multiplication and division equations, solve unknowns and extend facts by scaling factors and products

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Learning goalsSay it simply

Students use one multiplicative relationship to generate related multiplication and division equations, solve unknowns and extend facts by scaling factors and products.

Multiplication combines equal factors and division undoes that relationship by finding a missing factor or number of groups.

The product is the whole and the factors are related parts of the multiplicative structure. Division checks multiplication and solves missing-factor problems.

Scaling one factor by 10 scales the product by 10. The related division facts change consistently because the relationship is preserved.

Learning routine: Represent → Reason → Calculate → Interpret → Verify

Success looks like

  • Represent accurately
  • Explain the relationship
  • Choose a suitable method or unit
  • Apply in context
  • Check and justify
Key conceptTeach from the board

Build a fact family from 8 × 14 = 112

relationshipequationmultiply8 × 14 = 112commute14 × 8 = 112divide by first factor112 ÷ 8 = 14divide by second factor112 ÷ 14 = 8

The product is the whole and the factors are related parts of the multiplicative structure. Division checks multiplication and solves missing-factor problems.

  1. Read every label and identify the quantities, parts or evidence.
  2. Explain the relationship shown—not just the final answer.
  3. Check the conclusion against the original question and units.

Scale a fact family while preserving structure

8 × 14 = 112→80 × 14 = 1 120→1 120 ÷ 80 = 14→1 120 ÷ 14 = 80

Scaling one factor by 10 scales the product by 10. The related division facts change consistently because the relationship is preserved.

Now transfer the same relationship to a new situation and justify the result with precise vocabulary.

Clean visual examplesOne-page board

Clean one-page examples

AC9M5A01 - Recognise and explain the connection between multiplication and division as inverse operations and use this to develop families of number facts
Example 1

relationship equation multiply 8 × 14 = 112 commute 14 × 8 = 112 divide by first factor 112 ÷ 8 = 14 divide by second factor 112 ÷ 14 = 8

Example 2

8 × 14 = 112 → 80 × 14 = 1 120 → 1 120 ÷ 80 = 14 → 1 120 ÷ 14 = 80

Example 3

relationship equation multiply 8 × 14 = 112 commute 14 × 8 = 112 divide by first factor 112 ÷ 8 = 14 divide by second factor 112 ÷ 14 = 8

Example 4

8 × 14 = 112 → 80 × 14 = 1 120 → 1 120 ÷ 80 = 14 → 1 120 ÷ 14 = 80

Curriculum examplesCopied content

Content description: recognise and explain the connection between multiplication and division as inverse operations and use this to develop families of number facts.

  • E1: using materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication
  • E2: using arrays, multiplication tables, and physical and virtual materials to develop families of facts; for example, 3 \times 4 = 12, 4 \times 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3
  • E3: demonstrating multiplicative partitioning using materials, diagrams or arrays and recording 2 multiplication and 2 division facts for each grouping; 4 \times 6 = 24, 6 \times 4 = 24, 24 ÷ 4 = 6 and 24 ÷ 6 = 4; explaining how each is different from and connected to groups in the materials, diagrams or arrays
  • E4: using materials, diagrams or arrays to recognise and explain the inverse relationship between multiplication and division; for example, solving 240 ÷ 20 = \square by thinking 20\times\square = 240; using the inverse to make calculations easier; for example, solving 17\times\square= 221 using division, \square = 221 ÷ 17
Questions and answersWith answers

Core idea: Multiplication combines equal factors and division undoes that relationship by finding a missing factor or number of groups.

Remember

  • Represent accurately
  • Explain the relationship
  • Choose a suitable method or unit
  • Apply in context
  • Check and justify

Important questions

  • Write four facts for 8, 14 and 112. Explain using the model or evidence above.
  • Use inverse to check 936 ÷ 8. Explain using the model or evidence above.
  • Scale 7 × 6 = 42 by 10. Explain using the model or evidence above.
  • Solve 15 × □ = 180. Explain using the model or evidence above.
  • Explain why division is not commutative. Explain using the model or evidence above.
Practice and reviewReady for practice
  • Division fact uses a different product — Related facts must preserve the same whole.
  • Every order change is allowed — Multiplication is commutative; division generally is not.
  • Scaling factor but not product — If one factor changes, the product changes proportionally.
  • Fact recalled without explanation — Connect the equation to groups, arrays or missing factors.

Learn from the Topic Guide and fixed Teacher Slides, complete the Practice Sheet, use Practice for supported feedback, then take the Test when ready.

Curriculum alignmentStart here

Students use one multiplicative relationship to generate related multiplication and division equations, solve unknowns and extend facts by scaling factors and products.

Multiplication combines equal factors and division undoes that relationship by finding a missing factor or number of groups.

The product is the whole and the factors are related parts of the multiplicative structure. Division checks multiplication and solves missing-factor problems.

Scaling one factor by 10 scales the product by 10. The related division facts change consistently because the relationship is preserved.

Learning routine: Represent → Reason → Calculate → Interpret → Verify

Success looks like

  • Represent accurately
  • Explain the relationship
  • Choose a suitable method or unit
  • Apply in context
  • Check and justify
Related resourcesKeep teaching

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Classroom sequence

Teach: multiplication and division undo each other. Model: 11 × 5 = 55, so 55 ÷ 11 = 5. Ask: what does each number represent? Guided try: build a fact family. Misconception check: do not reverse the divisor and quotient without changing the meaning. Independent try: explain a new family.