Fact families explained for parents (Australia)

A homework page of little triangles, each holding three numbers, says write the fact family. A fact family is a set of three numbers that belong together, plus the number sentences you can make from them. Take 3, 5 and 8: 3 + 5 = 8, 5 + 3 = 8, 8 − 3 = 5 and 8 − 5 = 3. Multiplication and division have families too: 4, 6 and 24 give 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 4 = 6 and 24 ÷ 6 = 4. The question that unlocks it at home: "Which number is the whole, and which two are the parts?"

The real lesson underneath is that addition and subtraction undo each other, and so do multiplication and division. Teachers call these inverse operations. This guide covers that link and how families make it visible. It builds on number bonds (pairs that make a total) and part-part-whole thinking, and connects to multiplication and division. It doesn't re-teach those topics, times tables or written methods.

The big idea: one whole, two parts

Every addition and subtraction family has the same shape: two parts that join to make a whole.

• Adding joins the parts: 3 + 5 = 8.
• Subtracting takes one part away from the whole and leaves the other: 8 − 5 = 3.

Times and divide work the same way, using equal groups:

• Multiplying puts equal groups together: 4 groups of 6 make 24.
• Dividing splits the total back into those groups: 24 shared into 4 groups gives 6 each, or 24 split into groups of 6 makes 4 groups.

So in a family, the whole (or total) is the biggest number. Children who spot the whole first rarely write 3 − 8 = 5 by mistake.

Two kinds of family, side by side

	Addition and subtraction	Multiplication and division
Three numbers	7, 6, 13	3, 8, 24
Joining facts	7 + 6 = 13, 6 + 7 = 13	3 × 8 = 24, 8 × 3 = 24
Undoing facts	13 − 6 = 7, 13 − 7 = 6	24 ÷ 8 = 3, 24 ÷ 3 = 8
The big number	13 is the whole	24 is the total of the groups
Turnarounds work for	+ only	× only

The two kinds never mix: 2 + 4 = 6 and 2 × 3 = 6 both reach 6, but they belong to different families.

Turnarounds: why there are usually four facts

Swapping the parts doesn't change the answer for adding (7 + 6 = 6 + 7) or multiplying (3 × 8 = 8 × 3). Australian classrooms often call these turnarounds; older children may hear commutative. Turnarounds give two joining facts, and the two undoing facts make four.

The swap does not work for subtracting or dividing. Flip 13 − 6 = 7 into 6 − 13 and you no longer get 7; likewise 24 ÷ 8 = 3, but 8 ÷ 24 is not 3. A simple rule for children: in a take-away or sharing fact, the big number goes first. An array shows the multiplication turnaround beautifully: turn a 3-by-8 grid of dots on its side and it becomes 8 by 3, with the same 24 dots.

When a family has only two facts

Some families are smaller, which surprises children who expect four every time.

• Doubles: 6, 6 and 12 give only 6 + 6 = 12 and 12 − 6 = 6. Swapping identical parts creates nothing new (see doubling and halving).
• Square numbers: 5, 5 and 25 give 5 × 5 = 25 and 25 ÷ 5 = 5 (square numbers).

Zero is worth a gentle warning. With addition it behaves: 0, 7 and 7 give 0 + 7 = 7, 7 + 0 = 7, 7 − 0 = 7 and 7 − 7 = 0. With multiplication it breaks: 7 × 0 = 0 and 0 ÷ 7 = 0 work, but 0 ÷ 0 has no sensible answer, because you can't divide by zero. Most worksheets leave zero out of times-table families, and so can you.

Fact triangles and other pictures

Many classes use a fact triangle: the whole sits at the top corner, the two parts at the bottom. For a times-table triangle the total goes at the top, the two factors below.

How to use one at home:

1. Cover the top number. Your child joins the bottom two: add, or multiply.
2. Cover a bottom number. Your child works out the missing part from the other two: subtract, or divide.
3. Say all the facts aloud. Four sentences (or two, for doubles) from one triangle.

The same idea appears as a bar model (one long bar for the whole, split into two shorter bars) or a part-part-whole diagram (part-part-whole). Different pictures, same relationship; use whichever the teacher prefers.

Hands-on at the kitchen table

1. Two-colour buttons. Put out 5 red and 3 blue buttons. "How many altogether?" (8). Cover the blue: "8 take away 3 is…?" (5). Cover the red: "8 take away 5 is…?" (3). Write the four facts.
2. Egg cartons. A carton with 2 rows of 6 shows 2 × 6 = 12 and 6 × 2 = 12. Share 12 eggs into 2 rows (12 ÷ 2 = 6), then into rows of 6 (12 ÷ 6 = 2); see equal groups.
3. Number line jumps. Jump forward 6 from 7 to land on 13, then jump back 6 to return to 7. Forward and back undo each other (empty number line).
4. Make your own triangle. Your child chooses three numbers that belong together, draws the triangle and quizzes you. Slip in a deliberate mistake for them to catch.

Ten minutes is plenty; if tempers fray, homework without tears can help.

Using fact families to solve problems

"Think addition" for subtraction. For 13 − 8, ask "What do I add to 8 to make 13?" Counting up 2 to reach 10, then 3 more, gives 5 (make ten).

"Think multiplication" for division. For 42 ÷ 6, ask "Six times what makes 42?" If your child knows 6 × 7 = 42, the answer is 7. Division facts come almost free once times-table facts are secure (times tables without tears).

Missing numbers. For 27 + □ = 63, the family says the missing part is 63 − 27, which is 36. For □ × 4 = 28, the missing factor is 28 ÷ 4, which is 7. Boxes and blanks like these are an early step towards algebra.

Checking answers. After 52 − 18 = 34, add back: 34 + 18 = 52, so it's right. After 56 ÷ 7 = 8, multiply: 8 × 7 = 56. This habit catches slips in subtraction and division, including column subtraction.

Word problems. "Mia had some stickers, gave away 15 and has 9 left. How many did she start with?" The two parts are 15 and 9, so the whole is 15 + 9 = 24. Finding the whole and parts tames a wordy story (word problems).

Questions to ask tonight

• "Which number is the whole?" → the biggest one.
• "Can you swap these two and still be right?" → yes for + and ×, no for − and ÷.
• "How many facts can this family make?" → usually four; doubles and squares make two.
• "Can you check that subtraction with an addition?" → add the answer to the number taken away.
• "Which times-table fact helps with this division?" → find the multiplication that makes the total.

Where it sits in the Australian Curriculum

Version 9 of the Australian Curriculum: Mathematics brings the pieces in steps:

• Year 1: add and subtract within 20, using part-part-whole knowledge to 10.
• Year 2: the curriculum's examples include describing how addition and subtraction relate; children recall multiplication facts for twos and use doubling and halving to find the matching division facts.
• Year 3: explain addition and subtraction as inverse operations and use that to find unknown values in number sentences; recall multiplication facts for 3, 4, 5 and 10, then extend them to division.
• Year 4: recall multiplication facts up to 10 × 10 and the related division facts.
• Year 5: explain multiplication and division as inverse operations and use this to develop families of number facts.

Victorian schools follow the Victorian Curriculum 2.0, which is closely based on version 9. Classes don't all use the phrase fact family, and timing varies, so check with your child's teacher.

What it looks like by year level

Prep–Year 2: joining and separating small collections; number bonds to 10; first add–subtract families with objects. Year maps: Prep maths, Year 1, Year 2.

Years 3–4: add–subtract families with bigger numbers; missing-number sentences; times-table families grow towards 10 × 10. Year maps: Year 3, Year 4.

Years 5–6: inverse operations used confidently to check work and solve problems with larger numbers, including checking a division with remainders: 23 ÷ 4 = 5 r 3, because 5 × 4 + 3 = 23. Year maps: Year 5, Year 6.

Common slips and how to steer them

Putting the small number first in subtraction. "3 − 8 = 5." Ask which number is the whole, then start there.

Writing a turnaround for division. "24 ÷ 6 = 4, so 6 ÷ 24 = 4." Share 6 lollies between 24 people: does each person get 4?

Expecting four facts every time. Doubles and square numbers give two — correct, not unfinished.

Learning facts as separate lists. If 7 × 8 = 56 is known but 56 ÷ 8 feels new, practise the pair on one triangle.

When number facts feel like one part of a larger concern, start with when to worry about maths, then maths anxiety in children or Is my child behind in maths?.

Related guides

• Number bonds / part-part-whole — the parts-and-whole groundwork
• Addition / subtraction — the operations in the first kind of family
• Multiplication / division — the operations in the second
• Arrays / equal groups — seeing times-table families
• Maths help hub — every maths guide in one place

Worth asking at the next parent–teacher chat: "Does the class use fact triangles or bar models — and are they on add–subtract or times-table families this term?"

FAQ

What is a fact family in maths?

Three numbers linked by a pair of opposite operations, plus the number sentences they make. For 2, 9 and 11: 2 + 9 = 11, 9 + 2 = 11, 11 − 2 = 9 and 11 − 9 = 2.

Why do schools teach fact families?

They show that each operation has a partner that undoes it, which makes subtraction and division easier and gives children a built-in way to check answers.

How many facts are in a fact family?

Usually four. Families with two equal parts, such as 6 + 6 = 12 or 5 × 5 = 25, have only two.

Can a fact family include addition and multiplication?

No. Addition pairs with subtraction; multiplication pairs with division. The two kinds stay separate.

Is a fact family the same as a number bond?

They are close cousins. A number bond shows the parts that make a whole; the fact family writes out every addition and subtraction sentence that bond gives you.

A calm next step

Choose three numbers tonight, say 4, 9 and 13, and ask "Which is the whole — and how many facts can you make?" Use buttons or a triangle, say each fact aloud, then stop. For a broader picture of your child's maths, try the free check: a one-off adaptive diagnostic that takes about 10 minutes and shows partial results only, with no account or card required. The full placement opens after signup. This is not NAPLAN practice, nor the whole app: Ziado itself offers adaptive maths (Foundation–Year 6) and phonics reading (Foundation–Year 4). Ziado maths shows how practice is pitched to your child's level. Should you continue, Individual and Family plans open with a 14-day free trial of the complete app; pricing lists current plans.