Distributive Property

Arithmetic
principle

Also known as: distribution, distributive law, expanding brackets

Grade 3-5

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Multiplication distributes over addition: a(b + c) = ab + ac, linking two operations together. Foundation of algebra; enables factoring and expanding expressions.

This concept is covered in depth in our how the distributive property works, with worked examples, practice problems, and common mistakes.

Definition

Multiplication distributes over addition: a(b + c) = ab + ac, linking two operations together.

💡 Intuition

Three packs of (2 red + 4 blue) = (3 \times 2 red) + (3 \times 4 blue) = 6 red + 12 blue.

🎯 Core Idea

Distribution connects multiplication and addition—the bridge between operations.

Example

5(10 + 2) = 5 \times 10 + 5 \times 2 = 50 + 10 = 60 Easier than 5 \times 12 directly.

Formula

a(b + c) = ab + ac

Notation

a(b + c) is shorthand for a \times (b + c); the factor distributes to each term inside

🌟 Why It Matters

Foundation of algebra; enables factoring and expanding expressions.

💭 Hint When Stuck

Draw a box split into two parts to visualize how the outer factor multiplies each piece inside the parentheses.

Formal View

\forall a, b, c \in \mathbb{R}: a(b + c) = ab + ac \text{ and } (a + b)c = ac + bc

🚧 Common Stuck Point

Works forwards (distributing) and backwards (factoring): ab + ac = a(b+c) is equally valid.

⚠️ Common Mistakes

  • Only distributing to the first term inside the parentheses: 3(x + 4) = 3x + 4 instead of 3x + 12
  • Forgetting to distribute the sign: -(a + b) = -a + b instead of -a - b
  • Trying to distribute multiplication over multiplication: a(bc) \neq (ab)(ac)

Frequently Asked Questions

What is Distributive Property in Math?

Multiplication distributes over addition: a(b + c) = ab + ac, linking two operations together.

Why is Distributive Property important?

Foundation of algebra; enables factoring and expanding expressions.

What do students usually get wrong about Distributive Property?

Works forwards (distributing) and backwards (factoring): ab + ac = a(b+c) is equally valid.

What should I learn before Distributive Property?

Before studying Distributive Property, you should understand: multiplication, addition.

Next Steps

factoring

How Distributive Property Connects to Other Ideas

To understand distributive property, you should first be comfortable with multiplication and addition. Once you have a solid grasp of distributive property, you can move on to factoring.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Commutative, Associative, and Distributive Properties →
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Visualization

Static

Visual representation of Distributive Property