Inverse fact families | Year 5

01 / LEARN

Understand the idea.

Four facts describe one grouping

A multiplication and division fact family describes the same total in related ways. From $6 \times 8 = 48$ , swapping the factors gives $8 \times 6 = 48$ . Dividing the total by either non-zero factor gives the other factor.

Four different sentences are possible when the two non-zero factors differ. For a square fact such as $7 \times 7 = 49$ , swapping factors changes nothing. Its two different family facts are the multiplication and $49 \div 7 = 7$ .

The 6, 8 and 48 fact family
Multiplication	Matching division
6 × 8 = 48	48 ÷ 6 = 8
8 × 6 = 48	48 ÷ 8 = 6

Undo and check

Multiplication and division are inverse operations. If multiplying $27$ by $9$ makes $243$ , then dividing $243$ by $9$ returns $27$ . This lets you check a division using a multiplication you can calculate independently.

Keep the divisor non-zero. Although $0 \times 5 = 0$ and $0 \div 5 = 0$ are valid, division by zero is undefined. Also keep the product at the start of each division fact; swapping dividend and divisor is not the same as swapping multiplication factors.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Write a full family

Write the multiplication and division fact family for 9, 12 and 108.

Worked solution

1
Write the multiplication using the two factors.
$9 \times 12 = 108$
9 and 12 combine to make the product 108.
2
Swap the factors.
$12 \times 9 = 108$
The order of factors does not change their product.
3
Divide the product by the first factor.
$108 \div 9 = 12$
Undoing multiplication by 9 recovers the factor 12.
4
Divide the product by the other factor.
$108 \div 12 = 9$
Undoing multiplication by 12 completes the four-fact family.

Answer

9 × 12 = 108; 12 × 9 = 108; 108 ÷ 9 = 12; 108 ÷ 12 = 9.

EXAMPLE 02

Check a large division

A learner writes 516 ÷ 6 = 86. Check using an inverse operation.

Worked solution

1
Use multiplication to check the proposed quotient.
$86 \times 6 = (80 \times 6) + (6 \times 6)$
A correct quotient multiplied by its divisor must reproduce 516.
2
Calculate the partial products.
$480 + 36 = 516$
Both parts of 86 have been multiplied by 6.
3
Compare with the original dividend.
$516 \div 6 = 86$
The multiplication check matches 516, so the proposed division is correct.

Answer

The division is correct.

03 / YOUR TURN

Try it yourself.

Check your understanding

Use 13 × 4 = 52 to write two related division facts.

Show answer & reasoning

52 ÷ 13 = 4 and 52 ÷ 4 = 13.

Keep the product $52$ as the dividend. Dividing by each factor gives $52 \div 13 = 4$ and $52 \div 4 = 13$ .

Keep practising

6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

Write the two division facts linked to $7 \times 9 = 63$ .
1 mark
Write a full fact family using 5, 18 and 90.
2 marks
Use multiplication to check $448 \div 8 = 56$ .
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 $0 \times 7 = 0$ does not allow the division sentence $0 \div 0 = 7$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write the two division facts linked to $7 \times 9 = 63$ .
1 mark
Start with sixty-three and divide by each factor.
$63 \div 7 = 9$ and $63 \div 9 = 7$ .
Write a full fact family using 5, 18 and 90.
2 marks
The factors multiply to ninety.
$5 \times 18 = 90$ ; $18 \times 5 = 90$ ; $90 \div 5 = 18$ ; $90 \div 18 = 5$ .
Use multiplication to check $448 \div 8 = 56$ .
2 marks
$50 \times 8 = 400$ and $6 \times 8 = 48$ .
$56 \times 8 = 448$ , 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.
$234 \div 6 = 39$ .
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: $8 \times 8 = 64$ and $64 \div 8 = 8$ .
Explain why $0 \times 7 = 0$ does not allow the division sentence $0 \div 0 = 7$ .
2 marks
Any number multiplied by zero gives zero, so this cannot identify one unique quotient.
Division by zero is undefined.

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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 ↗

In this lesson

Understand the idea.

Four facts describe one grouping

Undo and check

See the method in action.

Write a full family

Write the multiplication using the two factors.

Swap the factors.

Divide the product by the first factor.

Divide the product by the other factor.

Check a large division

Use multiplication to check the proposed quotient.

Calculate the partial products.

Compare with the original dividend.

Try it yourself.

Check your understanding

Keep practising