Meet the Maths Opposites: How Multiplication & Division Are Best Friends!

Hello Maths Detectives! Did you know that multiplication and division are like a super-powered team? They are inverse operations, which is a fancy way of saying they are opposites that can “undo” each other, just like addition and subtraction! If you know one, you can often figure out the other. They belong to the same fact family!

What Does “Inverse Operations” Mean for Multiplication and Division?

It means they have a special connection:

• Multiplication is about putting equal groups together to find a total.
• Division is about splitting a total into equal groups OR finding out how many equal groups you can make.

Because they are opposites:

• If you multiply two numbers to get an answer, you can use division with that answer and one of the original numbers to find the other original number.
• If you divide a number by another, you can use multiplication with the answer and the number you divided by to get back to your starting number.

The Amazing Fact Family! (e.g., 3, 4, and 12)

Let’s look at the numbers 3, 4, and 12. These three numbers are part of a fact family.

If we know the multiplication fact: 3 × 4 = 12 (This also means 4 × 3 = 12 because multiplication is commutative – you can swap the numbers around!)

This one multiplication fact automatically tells us TWO division facts:

• 12 ÷ 4 = 3 (If you have 12 and make 4 equal groups, there will be 3 in each group. Or, how many 4s fit into 12? Three.)
• 12 ÷ 3 = 4 (If you have 12 and make 3 equal groups, there will be 4 in each group. Or, how many 3s fit into 12? Four.)

So, the numbers 3, 4, and 12 give us these related facts:

• 3 × 4 = 12
• 4 × 3 = 12
• 12 ÷ 3 = 4
• 12 ÷ 4 = 3 That’s a whole family of facts from just knowing one!

How Understanding This Relationship Helps You:

• Learn Division Facts Easily: If you know your multiplication tables, you already know most of your division facts! For 24 ÷ 3, just think “What times 3 makes 24?” (It’s 8!).
• Check Your Answers:
  • If you calculate 35 ÷ 5 = 7, you can check it with multiplication: Does 7 × 5 = 35? Yes!
  • If you calculate 6 × 8 = 48, you can check it with division: Does 48 ÷ 8 = 6? Yes!
• Solve Missing Number Problems: For example, if you have 7 × ? = 21, you can use division (21 ÷ 7 = 3) to find the missing number!

Explore the Multiplication & Division Connection! (18 Puzzles)

Ready to see how these operations work together? For each puzzle below, you might need to write the related facts for a multiplication or division sum, check answers using the inverse operation, or find a missing number!

(Your web app with the 18 questions will go here. Questions should focus on writing related facts from a given multiplication/division sum, identifying members of a fact family, or using the inverse to find a missing number.)

Why is Knowing This Relationship a Maths Superpower?

• Makes Learning Easier: You learn two operations (multiplication and division) almost at the same time by seeing how they connect.
• Boosts Recall: Understanding the link helps facts stick in your memory better.
• Powerful Problem Solving: You can approach problems from different angles if you know how these operations relate.
• Foundation for Future Maths: This understanding is vital for fractions, algebra, and much more!

Tips for Grown-Ups: Building the Multiplication-Division Link

Helping children see multiplication and division as inverse operations, and understand fact families, is crucial for their mathematical fluency and conceptual understanding.

• Use Arrays and Grouping: Show an array (e.g., 3 rows of 4 dots for 3×4=12). Then show how that same array can be seen as 12 dots split into 3 groups of 4 (12÷3=4) or 4 groups of 3 (12÷4=3).
• Fact Family Triangles: Write three related numbers (e.g., 5, 6, 30) on the corners of a triangle. Cover one number and ask for the multiplication or division facts that use the other two to find it.
• “What’s the Opposite Question?”: If they solve 7 × 3 = 21, ask, “So, what would 21 divided by 3 be?”
• Real-World Scenarios: “I have 18 sweets to share equally among 3 friends (division). Each friend gets 6 sweets. So, 3 friends with 6 sweets each means there are 18 sweets in total (multiplication).”
• Write All Four Facts: When learning a new multiplication fact, encourage them to write down the two related division facts (and the commuted multiplication fact) straight away.