Times Table Methods
Commutativity in times tables
Show why 3 × 7 and 7 × 3 have the same answer using arrays and turning rectangles.
Quick answer
3 × 7 = 7 × 3 because the order of the numbers doesn’t change the answer. This is called the commutative property of multiplication.Learning outcome
By the end of this page, you will understand that the order of numbers in multiplication does not change the result, and you can use arrays to show this. Success signs:- You can draw an array for 4 × 6 and 6 × 4 and see they have the same total.
- You can explain that 8 × 3 is the same as 3 × 8.
- You can solve a multiplication problem by turning it around to make it easier.
- You can play a game where you match multiplication facts that are the same.
Why this matters
Commutativity in multiplication helps you solve problems more easily. When you know that 6 × 7 = 42, you also know that 7 × 6 = 42. This means you don’t have to learn two separate facts. It also helps you check your answers. If you're not sure about 5 × 8, you can think of 8 × 5 instead – it's the same total! In games, sports, and real life, this rule helps you count things quickly. For example, if you have 5 rows of 6 stickers, it’s the same as 6 rows of 5 stickers. In cooking, if you need 3 bags of 4 carrots, it’s the same as 4 bags of 3 carrots. This rule makes maths more flexible and fun!How it works
Multiplication is about groups. When we say 3 × 7, we mean 3 groups of 7 things. When we say 7 × 3, we mean 7 groups of 3 things. But if you count all the things in both cases, you get the same number. Let’s use arrays to show this. An array is a way to arrange things in rows and columns. Step 1: Draw 3 rows of 7 items (like stars or dots). Step 2: Count them all – that’s 21 items. Step 3: Now draw 7 rows of 3 items. Step 4: Count them again – it’s still 21 items! The total is the same, even though the rows and columns are in different orders.Examples you can see
Let’s look at some examples using arrays and real-life situations. Example 1: 2 × 5 = 5 × 2 Imagine you have 2 rows of 5 apples. That’s 10 apples in total. Now imagine you have 5 rows of 2 apples. That’s also 10 apples. Example 2: 4 × 3 = 3 × 4 You can draw 4 rows of 3 blocks. That makes 12 blocks. Or draw 3 rows of 4 blocks. That also makes 12 blocks. Example 3: 6 × 2 = 2 × 6 If you have 6 rows of 2 toys, that’s 12 toys. If you have 2 rows of 6 toys, that’s also 12 toys. These examples show that no matter how you arrange the rows and columns, the total stays the same.Try it as a mini adventure
Adventure: The Magic Array Garden You are a gardener with a magical garden. You plant 3 rows of 5 sunflowers. But your garden is magical – when you turn the garden, the rows become columns and the columns become rows! Step 1: Draw your garden as 3 rows of 5 sunflowers. Step 2: Count the sunflowers – how many are there? Step 3: Now turn your garden. It becomes 5 rows of 3 sunflowers. Step 4: Count again – how many sunflowers are there now? You’ll find that both gardens have the same number of sunflowers. This is the magic of commutativity! Mini challenge: Try this with 4 rows of 6 flowers. Then turn it into 6 rows of 4 flowers. Count each time and see if the numbers match.Common mistakes
- Mistake: Thinking that 3 × 7 and 7 × 3 are different answers. Why it happens: Children often think that changing the order changes the result. Fix: Use arrays or drawings to show that the total is the same.
- Mistake: Confusing multiplication with addition. Why it happens: When children add 3 + 7 = 10, they might think 3 × 7 = 10. Fix: Use repeated addition or arrays to show multiplication as groups.
- Mistake: Not realising that 5 × 4 and 4 × 5 are the same. Why it happens: Children might not see the connection between the two ways of writing it. Fix: Practice drawing both arrays and counting the same total.
Parent and carer guide
Parents and carers can help by:- Using everyday objects like toys, fruit, or blocks to show arrays.
- Encouraging children to draw arrays for different multiplication facts.
- Playing simple games like “Find the matching pair” – e.g., 6 × 4 and 4 × 6.
- Reinforcing that the order doesn’t matter by saying, “It’s the same number of items, just arranged differently.”
- Not worrying if a child gets confused at first – it takes time to understand this idea.
Practice questions
- What is 5 × 3? What is 3 × 5? Are they the same?
- Draw an array for 2 × 6 and 6 × 2. Do they have the same total?
- Which of these is the same as 4 × 7?
- 7 × 4
- 7 × 3
- 4 × 3
- 3 × 7
- Fill in the missing number: 9 × 2 = ___ × 9
- True or false: 6 × 8 = 8 × 6
- 5 × 3 = 15, 3 × 5 = 15. Yes, they are the same.
- Yes, both arrays have 12 items.
- 7 × 4
- 2
- True
FAQ
What is the commutative property?
The commutative property means that when you multiply two numbers, the order doesn’t matter. So 3 × 5 = 5 × 3.
Why does this work in multiplication?
Because multiplication is about equal groups. Whether you have 3 groups of 5 or 5 groups of 3, you still get the same total.
Can I use this for division?
No, division does not follow the commutative property. For example, 10 ÷ 2 = 5, but 2 ÷ 10 = 0.2, which are different.
Is this rule only for small numbers?
No, it works for all numbers – big or small. It’s a rule that always stays true in maths.
How can I help my child learn this?
Use arrays, drawings, or objects. Practice with simple examples like 2 × 4 and 4 × 2. Keep it fun and hands-on!
| Number | 0 × n | 1 × n | 2 × n | 3 × n | 4 × n | 5 × n | 6 × n | 7 × n | 8 × n | 9 × n | 10 × n | 11 × n | 12 × n |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 0 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 7 | 0 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |