X Tables
Maths Facts

Why multiplication can be done in any order

Use arrays and turning rectangles.

Quick answer

Multiplication can be done in any order because the result stays the same. This is called the commutative property of multiplication. Whether you have 3 groups of 4 or 4 groups of 3, you still get 12 altogether.

Learning outcome

By the end of this page, you will understand that multiplication works the same way whether you count 3 groups of 4 or 4 groups of 3, and you will be able to show this using arrays and real-life examples. Success signs:
  • You can draw an array to show 3 x 4 and 4 x 3 and see they make the same total
  • You can explain that 6 x 2 is the same as 2 x 6
  • You can find real-life situations where the order of multiplication doesn’t change the answer

Why this matters

Understanding that multiplication can be done in any order helps you solve problems more quickly and easily. When you know this rule, you can rearrange numbers to make calculations simpler. For example, if you are working out 7 x 8, you might find it easier to think of it as 8 x 7, or even break it into smaller parts. This rule also helps you check your answers and build confidence in your maths.

How it works

Multiplication is all about equal groups. When we say 3 x 4, we mean 3 groups of 4. But if we turn this around and say 4 x 3, we mean 4 groups of 3. The total number of items stays the same. We can prove this by drawing arrays. Step-by-step:
  1. Draw a rectangle to show 3 groups of 4
  2. Count all the squares inside
  3. Turn the rectangle and draw 4 groups of 3
  4. Count again – you should get the same number

Examples you can see

Let’s look at some arrays to show how 3 x 4 and 4 x 3 give the same answer.
3 x 4 4 x 3
3 rows of 4 = 12 4 rows of 3 = 12
Try drawing your own arrays for 5 x 2 and 2 x 5. You’ll find they both make 10 squares.

Try it as a mini adventure

The Magic Garden You are a gardener in a magical garden where flowers grow in perfect rows and columns. You have a small plot with 3 rows of 5 flowers each. You decide to change the shape of your plot and make it 5 rows of 3 flowers each. How many flowers are in your garden now? Step-by-step:
  1. Start with 3 rows of 5 flowers = 15 flowers
  2. Change to 5 rows of 3 flowers = 15 flowers
  3. Count each flower in both shapes – you’ll see they match!
This shows that no matter how you arrange your flowers, you still have the same number. The magic garden proves that 3 x 5 = 5 x 3 = 15.

Common mistakes

  • Mistake: Thinking that 3 x 4 and 4 x 3 are different because one has more rows and the other has more columns.
    Why it happens: Children sometimes get confused by the visual layout of arrays.
    Fix: Draw both arrays and count the total number of squares – they will be the same.
  • Mistake: Believing that 6 x 2 is different from 2 x 6 because one is harder to count.
    Why it happens: Children often think that larger numbers must be harder to work with.
    Fix: Use arrays or objects to show that both give the same result.

Parent and carer guide

As a parent or carer, you can help by:
  • Using everyday items like buttons, coins, or toys to make arrays
  • Letting your child draw or build arrays for different multiplication facts
  • Reinforcing the idea that 3 x 4 is the same as 4 x 3 with real examples
  • Not worrying if your child makes a mistake – it’s part of learning!
  • Encouraging your child to explain their thinking when they solve a problem

Practice questions

  1. Draw an array for 2 x 6 and 6 x 2. Do they have the same total?
  2. Fill in the blank: 4 x 7 = ___ x 4
  3. True or false: 5 x 3 = 3 x 5
  4. Draw 3 x 5 and 5 x 3. Count the total squares in each.
  5. Which of these shows the same total: 2 x 9 or 9 x 2?
Answers:
  1. Yes, both arrays have 12 squares
  2. 7
  3. True
  4. Both have 15 squares
  5. 2 x 9 and 9 x 2 both equal 18

FAQ

Why does multiplication work the same way in any order?

Because multiplication is about equal groups, and the total stays the same whether you have 3 groups of 4 or 4 groups of 3.

Does this rule work for all numbers?

Yes, it works for any numbers you multiply together – that’s why it’s called a property!

Can I use this to solve harder problems?

Yes! If 6 x 7 is hard, you can think of it as 7 x 6, or even break it into smaller parts like 7 x 5 + 7 x 2.

What if I have 0 in my multiplication?

Even with 0, the rule still works. For example, 0 x 5 = 0 and 5 x 0 = 0 – both are zero.

How can I practice this at home?

Use toys, drawings, or kitchen items to make arrays and show that 3 x 4 = 4 x 3.