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Maths Facts

Why subtraction cannot always be swapped

Compare 8 - 3 and 3 - 8 on a number line.

Quick answer

Subtraction cannot always be swapped because changing the order of the numbers changes the answer. For example, 8 - 3 = 5, but 3 - 8 = -5. The order matters in subtraction, unlike addition where swapping the numbers doesn't change the total.

Learning outcome

By the end of this page, you will understand that subtraction is not commutative – which means you can't swap the numbers around and get the same answer. You'll also be able to show this using number lines, objects, and real-life examples. Success signs:
  • You can explain why 8 - 3 is not the same as 3 - 8
  • You can use a number line to show subtraction in both directions
  • You can find real-life situations where swapping subtraction makes a difference
  • You can solve simple subtraction problems correctly

Why this matters

Understanding that subtraction cannot be swapped is important because it helps you avoid mistakes in maths. If you think 6 - 2 = 2 - 6, you'll get the wrong answer. This idea also helps when you're solving word problems or doing puzzles where order matters. Knowing that swapping numbers changes the result is a key part of being good at maths and thinking clearly about numbers.

How it works

Subtraction means taking away one number from another. When you do 8 - 3, you start with 8 and take away 3. When you do 3 - 8, you start with 3 and take away 8. This is very different! Let's look at how it works step by step:
  1. First, you need to know what subtraction means – it's taking away or finding the difference
  2. Next, you look at what happens when you swap the numbers
  3. Then you see how it shows on a number line
  4. Finally, you try real-life examples to see why order matters

Examples you can see

Let's look at some examples using a number line to help us understand. Example 1: 8 - 3 Start at 8 on the number line. Then move 3 spaces to the left (because we are subtracting). You land on 5.
  1. Start at 8
  2. Move 3 spaces left
  3. Land on 5
Example 2: 3 - 8 Start at 3 on the number line. Then move 8 spaces to the left (because we are subtracting). You land on -5.
  1. Start at 3
  2. Move 8 spaces left
  3. Land on -5
Notice the difference! 8 - 3 = 5, but 3 - 8 = -5. These are completely different answers. Let's try another example with apples:
  • You have 6 apples. You eat 2 apples. How many do you have left? 6 - 2 = 4 apples
  • You have 2 apples. You eat 6 apples. How many do you have left? 2 - 6 = -4 apples
The second situation doesn't make sense in the real world because you can't have a negative number of apples. But mathematically, it's correct – it just shows that the order changes the result.

Try it as a mini adventure

The Number Line Treasure Hunt You are on a treasure hunt! You start at a number on the number line. You must follow clues to find the treasure, but you need to be careful about which way you move. Clue 1: Start at 10. Move 4 spaces to the left. What number do you land on?
  1. Start at 10
  2. Move 4 spaces left
  3. Land on 6
Clue 2: Now, start at 4. Move 10 spaces to the left. What number do you land on?
  1. Start at 4
  2. Move 10 spaces left
  3. Land on -6
What do you notice? Clue 1 gave you 10 - 4 = 6. Clue 2 gave you 4 - 10 = -6. These are completely different answers! That's why we can't swap subtraction. Let's try one more:
  • Clue 3: Start at 7. Move 3 spaces to the left. What number do you land on? 7 - 3 = 4
  • Clue 4: Start at 3. Move 7 spaces to the left. What number do you land on? 3 - 7 = -4

Common mistakes

  • Mistake: Thinking 10 - 5 = 5 and 5 - 10 = 5 Why it happens: Some children think subtraction is like addition and that swapping numbers doesn't matter. Fix: Use a number line or count objects to show that 5 - 10 = -5, not 5.
  • Mistake: Believing that 12 - 7 = 7 - 12 Why it happens: Children might confuse subtraction with addition, where swapping does not change the result. Fix: Show that 12 - 7 = 5 and 7 - 12 = -5. These are different!
  • Mistake: Thinking that 9 - 9 = 0 and 9 - 9 = 9 Why it happens: Confusion about what happens when you subtract a number from itself. Fix: Explain that any number minus itself equals zero, and that swapping doesn't change this fact.

Parent and carer guide

As a parent or carer, you can help your child understand why subtraction cannot be swapped by:
  1. Using a number line together to show how moving left changes the result
  2. Playing simple subtraction games where you switch numbers and see different answers
  3. Using everyday examples like sharing cookies or counting toys
  4. Encouraging your child to explain their thinking when solving subtraction problems
  5. Not worrying if they make mistakes at first – it's normal to think about swapping at first
What to say:
  • "Let's try both ways and see what happens"
  • "It's okay to make mistakes – that's how we learn"
  • "Can you show me how you think about this?"
What not to worry about:
  • Your child getting confused at first – this is normal
  • They not understanding immediately – it takes time to get used to new ideas
  • They swapping numbers accidentally – this shows they're thinking!

Practice questions

  1. What is 9 - 4?
  2. What is 4 - 9?
  3. What is 15 - 8?
  4. What is 8 - 15?
  5. What is 12 - 6?
  6. What is 6 - 12?
  7. What is 20 - 13?
  8. What is 13 - 20?
  9. What is 7 - 11?
  10. What is 11 - 7?
Answers:
  1. 9 - 4 = 5
  2. 4 - 9 = -5
  3. 15 - 8 = 7
  4. 8 - 15 = -7
  5. 12 - 6 = 6
  6. 6 - 12 = -6
  7. 20 - 13 = 7
  8. 13 - 20 = -7
  9. 7 - 11 = -4
  10. 11 - 7 = 4

FAQ

Why can't we swap subtraction like we can with addition?

Because subtraction means taking away, so changing the order changes what you're taking away. For example, 5 - 2 means taking 2 away from 5, while 2 - 5 means taking 5 away from 2. These are completely different actions.

Does this mean subtraction is harder than addition?

No! Subtraction is just different from addition. Both are important and useful. The key is understanding that they work differently, which helps you solve problems more accurately.

Can we ever swap subtraction?

Not in normal subtraction. But in some special cases, like when you're adding the same number to both sides of an equation, swapping can be helpful. But that's a more advanced idea.

How can I practice this at home?

You can use number lines, count objects, or play games where you do subtraction in different orders. Try asking questions like "What's 8 - 3? What's 3 - 8?" and see how the answers are different.

What if my child still gets confused?

That's okay! It takes time to understand. Use lots of visual examples, games, and real-life situations to help them see why order matters in subtraction.