The idea, step by step
Four facts describe one grouping
From 6 × 8 = 48, we also know 8 × 6 = 48, 48 ÷ 6 = 8 and 48 ÷ 8 = 6. These four facts form a fact family. Multiplication combines equal groups; division asks how many groups or how many in each group.
For a square fact such as 7 × 7 = 49, swapping the two factors gives the same multiplication sentence. There are only two different sentences: 7 × 7 = 49 and 49 ÷ 7 = 7. Do not invent extra numbers just to make four different facts.
| Multiplication | Matching division |
|---|---|
| 6 × 8 = 48 | 48 ÷ 6 = 8 |
| 8 × 6 = 48 | 48 ÷ 8 = 6 |
Undo and check
If a number was multiplied by nine, dividing the result by nine recovers it. If 27 × 9 = 243, then 243 ÷ 9 = 27. This is useful for checking both operations.
Zero needs care. We know 0 × 5 = 0 and 0 ÷ 5 = 0, but we cannot divide by zero. A division sentence with zero as the divisor is not a valid extra fact. For ordinary four-sentence fact families, use non-zero factors.
A common trap
Reversing a division sentence as if division could always be swapped.
Instead: 48 ÷ 6 = 8 does not mean 6 ÷ 48 = 8. Multiplication factors may swap; a division dividend and divisor have different jobs.
See the thinking
Write a full family
Write the multiplication and division fact family for 9, 12 and 108.
- Use both orders of the two factors.
- Make the product the starting number for division.
- Use each non-zero factor as a divisor once.
Check a large division
A learner writes 516 ÷ 6 = 86. Check using an inverse operation.
- Multiply the proposed quotient by six.
- 86 × 6 = 480 + 36 = 516.
- The check matches the original dividend.
Check your understanding
Use 13 × 4 = 52 to write two related division facts.
Show answer
52 ÷ 13 = 4 and 52 ÷ 4 = 13.
Start each division with the product, fifty-two.
Your turn
6 questions · 11 marks · Work at your own pace.
Read the questions online
- Write the two division facts linked to 7 × 9 = 63.1 mark
- Write a full fact family using 5, 18 and 90.2 marks
- Use multiplication to check 448 ÷ 8 = 56.2 marks
- A machine multiplies a number by six and produces 234. What inverse operation recovers the starting number?2 marks
- How many different multiplication/division sentences are in the fact family for 8, 8 and 64? Write them.2 marks
- Explain why 0 × 7 = 0 does not allow the division sentence 0 ÷ 0 = 7.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Write the two division facts linked to 7 × 9 = 63.1 mark
Start with sixty-three and divide by each factor.
63 ÷ 7 = 9 and 63 ÷ 9 = 7. - Write a full fact family using 5, 18 and 90.2 marks
The factors multiply to ninety.
5 × 18 = 90; 18 × 5 = 90; 90 ÷ 5 = 18; 90 ÷ 18 = 5. - Use multiplication to check 448 ÷ 8 = 56.2 marks
50 × 8 = 400 and 6 × 8 = 48.
56 × 8 = 448, so it is correct. - A machine multiplies a number by six and produces 234. What inverse operation recovers the starting number?2 marks
Divide by the same non-zero factor.
234 ÷ 6 = 39. - How many different multiplication/division sentences are in the fact family for 8, 8 and 64? Write them.2 marks
Swapping the equal factors repeats the same sentence.
Two: 8 × 8 = 64 and 64 ÷ 8 = 8. - Explain why 0 × 7 = 0 does not allow the division sentence 0 ÷ 0 = 7.2 marks
Any number multiplied by zero gives zero, so this cannot identify one unique quotient.
Division by zero is undefined.
Curriculum reference & lesson scope
Australian Curriculum v9.0; SA Curriculum Mathematics 2026
One-/two-digit multipliers; division strategies; contextual remainders; inverse fact families; unknown factors/dividends/divisors.
Official curriculum source ↗