Zero groups versus zero in each group

Quick answer

0 groups of 5 and 5 groups of 0 both equal 0. This is because multiplication shows equal groups, and if there are no groups, there are no items to count.

Learning outcome

By the end of this page, you will understand that 0 groups of any number is 0, and that any number of groups with 0 items in each group is also 0.

• You can show 0 groups of 5 with no items at all
• You can show 5 groups of 0 with 5 empty groups
• You can explain why both answers are 0
• You can solve problems involving 0 groups or 0 in each group

Why this matters

Understanding zero in multiplication helps you see how maths works in real life. It helps you solve problems where there are no items, or where you have empty groups. This is useful in cooking, sharing, games, and even in your head when you're counting!

How it works

Let's look at two ways to think about zero:

1. 0 groups of 5 means: no groups, and each group would have 5 items if there were any groups.
2. 5 groups of 0 means: 5 groups, but each group has 0 items.
3. Both of these give the same answer: 0.

Examples you can see

Let's imagine we're at a party and we're setting up plates:

0 groups of 5

Imagine you have no plates at all. There are no groups of plates, so there are no items. This is 0 groups of 5, which equals 0.

5 groups of 0

Imagine you have 5 plates, but each plate has 0 items on it. You have 5 groups, but each group has nothing. This is 5 groups of 0, which also equals 0.

Visual examples

0 groups of 5: Draw 0 circles. There are 0 items. This is 0 × 5 = 0.

5 groups of 0: Draw 5 circles, but each circle has 0 items inside. This is 5 × 0 = 0.

Try it as a mini adventure

Adventure: The Empty Toy Box

You have a toy box. You are going to count your toys, but first, you need to understand how to count in two different ways.

1. First, you look at your toy box and find there are 0 toy groups in it. You have 0 groups of 3 toys. How many toys are there?
2. Next, you look at 3 toy boxes. Each toy box has 0 toys in it. How many toys are there in total?
3. What do you notice about both answers?

Both answers are 0. You can see this by drawing pictures or using your hands to show empty groups.

Common mistakes

• Mistake: Thinking that 0 groups of 5 is 5.
  Why it happens: Confusing "no groups" with "no items in groups".
  Fix: Remember: 0 groups means no groups at all, so there are no items.
• Mistake: Thinking that 5 groups of 0 is 5.
  Why it happens: Counting the number of groups instead of the items.
  Fix: Remember: 5 groups of 0 means 5 groups, each with 0 items, so 0 items total.
• Mistake: Thinking that 0 × 5 = 5 and 5 × 0 = 5.
  Why it happens: Mixing up the multiplication facts.
  Fix: Remember: any number times 0 is 0. Any number times 1 is itself.

Parent and carer guide

Help your child understand zero in multiplication by:

1. Using physical items like toys, crayons, or counters to show empty groups.
2. Repeating the idea that "no groups" means no items, even if the number of items in each group is large.
3. Repeating the idea that "groups with nothing in them" also means no items.
4. Encouraging your child to draw pictures or use fingers to show the groups.
5. Practising with real-life examples like: "How many apples are in 0 baskets?" or "How many crayons are in 5 boxes if each box has 0 crayons?"

Remember: this is a tricky concept, and it's okay if your child needs time to understand it. Keep it fun and visual!

Practice questions

1. 0 groups of 7 = ?
2. 7 groups of 0 = ?
3. 0 groups of 10 = ?
4. 10 groups of 0 = ?
5. 0 groups of 1 = ?
6. 1 group of 0 = ?
7. 0 groups of 12 = ?
8. 12 groups of 0 = ?
9. 0 groups of 0 = ?
10. 0 groups of 5 = ?

Answers:

1. 0
2. 0
3. 0
4. 0
5. 0
6. 0
7. 0
8. 0
9. 0
10. 0

FAQ

Why does 0 groups of 5 equal 0?

Because there are no groups, so there are no items to count. Even if each group would have 5 items, there are 0 groups, so the total is 0.

Why does 5 groups of 0 equal 0?

Because you have 5 groups, but each group has 0 items. So you have 5 groups of nothing, which equals 0.

Can 0 groups of 5 be the same as 5 groups of 0?

Yes! Both equal 0. This is because multiplication is commutative, and zero times any number is zero.

Is zero a number in multiplication?

Yes! Zero is a number, and multiplying by zero always gives zero.

What if I have 0 groups of 0?

That’s also 0. You have no groups, and each group has no items, so the total is 0.

Times Table Facts: Zero Times Any Number