Open middle problems

Quick answer

Open middle problems are puzzles where you have to fill in the middle part of a problem with numbers or symbols, and there are often many ways to get the right answer.

Learning outcome

By the end of this page, you will understand how to solve open middle problems by using your thinking skills to try different numbers and find more than one solution.

• You can identify the middle part of a problem that needs to be filled in.
• You can try different numbers to make the problem work.
• You can explain why more than one answer might be correct.
• You can check your answer to make sure it fits the rules.

Why this matters

Open middle problems help you think more deeply about maths. They are not just about getting one right answer – they are about exploring different paths and understanding how numbers work together. These problems help you become a better problem solver in school and in real life. You might use them when sharing snacks, planning a party, or building with blocks.

How it works

Open middle problems usually have a beginning and an end, but the middle part is missing. You need to find numbers or symbols to fill in the middle. Here’s how to solve them:

1. Read the problem carefully. Look for the parts that are already filled in and the part that is missing.
2. Think about what numbers could fit. What numbers might make the problem work?
3. Try one number and see if it works. If not, try another.
4. Check your answer. Make sure it fits all the rules.
5. Try more than one answer. Often, there are many correct solutions.

Examples you can see

Let’s look at some open middle problems you can try:

Example 1: Fill in the blanks to make this addition problem true:

__ + __ = 15

There are many ways to solve this. For example:

• 5 + 10 = 15
• 7 + 8 = 15
• 3 + 12 = 15
• 1 + 14 = 15

All of these are correct answers because they all add up to 15.

Example 2: Fill in the blanks to make this multiplication problem true:

__ × __ = 24

Here are some possible answers:

• 3 × 8 = 24
• 4 × 6 = 24
• 2 × 12 = 24
• 1 × 24 = 24

Each one is correct because the multiplication works out to 24.

Example 3: Fill in the blanks to make this subtraction problem true:

__ - __ = 7

Possible answers:

• 10 - 3 = 7
• 15 - 8 = 7
• 20 - 13 = 7

All these work because the subtraction gives you 7.

Try it as a mini adventure

Adventure: The Treasure Hunt

You are going on a treasure hunt! You have a map that says:

“Find the treasure by solving this puzzle: __ + __ = 12. The first number is less than 6, and the second number is more than 5.”

Let’s solve it together:

1st time: Try 4 + 8 = 12. 4 is less than 6, and 8 is more than 5. ✔️

2nd time: Try 5 + 7 = 12. 5 is less than 6, and 7 is more than 5. ✔️

3rd time: Try 3 + 9 = 12. 3 is less than 6, and 9 is more than 5. ✔️

4th time: Try 6 + 6 = 12. 6 is not less than 6. ❌

5th time: Try 2 + 10 = 12. 2 is less than 6, and 10 is more than 5. ✔️

There are many correct answers! You can find the treasure by choosing any pair that works.

Common mistakes

• Mistake: Thinking that only one answer is correct.
• Why it happens: Some people think maths always has just one right answer.
• Fix: Remember, open middle problems often have more than one correct answer.
• Mistake: Not checking if the numbers fit the rules.
• Why it happens: It’s easy to rush and forget to double-check.
• Fix: Always read the problem again and check if your numbers fit all the rules.
• Mistake: Choosing numbers that don’t work even if they seem right.
• Why it happens: Sometimes it's hard to think of numbers that fit all conditions.
• Fix: Try different combinations and be patient. It’s okay to try again.

Parent and carer guide

Helping your child with open middle problems is a fun way to build their thinking skills. Here’s how:

• Ask your child to explain their thinking. “How did you choose those numbers?”
• Encourage them to try more than one solution. “Can you find another way?”
• Let them make mistakes and learn from them. It’s part of the process.
• Use everyday examples like sharing snacks or counting toys to make it real.
• Don’t worry if they don’t get it right away. It’s about exploring and learning.

Try this at home: Ask your child to find two numbers that add up to 10. Then ask them to find another pair. You can make it fun by using coins, blocks, or fruit.

Practice questions

1. Fill in the blanks to make this true: __ + __ = 10
2. Fill in the blanks to make this true: __ × __ = 18
3. Fill in the blanks to make this true: __ - __ = 5
4. Fill in the blanks to make this true: __ + __ = 14
5. Fill in the blanks to make this true: __ × __ = 20

Answers:

1. 3 + 7 = 10
2. 3 × 6 = 18
3. 10 - 5 = 5
4. 6 + 8 = 14
5. 4 × 5 = 20

FAQ

What is an open middle problem?

An open middle problem is a maths puzzle where the beginning and end are given, but the middle part is missing. You have to find numbers or symbols to fill in the middle.

Why are open middle problems useful?

They help you think more deeply about numbers and problem-solving. They show that there can be more than one way to solve a problem.

Can I use open middle problems in real life?

Yes! You can use them when sharing food, planning a party, or even when playing games. They help you think carefully about how to use numbers.

Are open middle problems hard?

They can be tricky at first, but they get easier with practice. They are meant to challenge your thinking and make you use your brain!

How can I get better at solving them?

Try different numbers, check your answers, and don’t be afraid to try again. The more you practice, the better you’ll get.