3.3: Properties of Multiplication

nataliehobson; Iain Pardoe

3.3: Properties of Multiplication

Properties of Multiplication

Multiplication is one of the most powerful operations in mathematics. To understand it deeply—and to help future students build flexible thinking—it’s important to recognize the properties that govern multiplication. These properties not only make computation easier but also build the foundation for algebra and higher mathematics.

We’ll focus on three major properties:

Commutative Property of Multiplication
Associative Property of Multiplication
Distributive Property of Multiplication over Addition

The Commutative Property of Multiplication

Definition: The order in which two numbers are multiplied does not change the product.

$a \times b = b \times a$

Examples: Commutative property

$3 \times 4 = 12 and 4 \times 3 = 12$

Why does the commutative property work? With an Example!

Think of multiplication as “groups of objects.”

$3 \times 4$ $3 \times 4$ means 3 groups of 4 (🍎🍎🍎🍎, 🍎🍎🍎🍎, 🍎🍎🍎🍎).
$4 \times 3 means 4 groups of 3.$

Both end up with 12 apples.

Visual:

Both arrays contain 12 stars in total.

The Associative Property of Multiplication

Definition: When multiplying three or more numbers, the way they are grouped does not change the product.

$(a \times b) \times c = a \times (b \times c)$

Examples: Associative Property

$(2 \times 3) \times 4 = 6 \times 4 = 24$

$2 \times (3 \times 4) = 2 \times 12 = 24$

Why does the associative property work?

The product stays the same because multiplication is about repeated grouping. Whether we group early or later, we’re still finding the same total number of objects.

Visual (using blocks):

Imagine you have 2 bags, each with 3 bundles of 4 sticks:

Group as (2 × 3) first → 6 bundles of 4 = 24 sticks.
Group as (3 × 4) first → 12 sticks in each bag × 2 bags = 24 sticks.

Either way, you get 24 sticks.

It is important to create an argument for this property- not just examples.

The Distributive Property of Multiplication over Addition

Definition: Multiplication can “distribute” over addition.

$a \times (b + c) = (a \times b) + (a \times c)$

Example: Distributive Property of Multiplication over Addition

$5 \times (7 + 3) = 5 \times 10 = 50$

$(5 \times 7) + (5 \times 3) = 35 + 15 = 50$

Why does multiplication distribute over addition?

Think of $5 \times (7 + 3)$ as “5 groups of 10.” But we can also see it as “5 groups of 7” plus “5 groups of 3.” Both perspectives count the same total.

Visual (array split):

This idea becomes very powerful in algebra when we expand expressions like

$a(b+c)$ $a (b + c)$ .

Practice and Reflection

Commutative Property Practice
- Show with an array that
  
  $6 \times 2 = 2 \times 6$ $6 \times 2 = 2 \times 6$ .
- Write your own example of the commutative property using a real-life situation (e.g., rows of chairs, boxes of crayons).
Associative Property Practice
- Solve both $(4 \times 2) \times 5$ and
  
  $4 \times (2 \times 5)$ $4 \times (2 \times 5)$ . Explain why the products are the same.
- Draw a block model to represent the grouping.
Distributive Property Practice
- Use the distributive property to compute $7 \times 12$ by breaking 12 into $10 + 2$ .
- Draw an array showing how
  
  $7 \times (10 + 2)$ $7 \times (10 + 2)$ splits into two smaller rectangles.

Connections to Teaching

For many students, these properties can feel abstract. Using visual models like arrays, blocks, or area diagrams makes the properties concrete and accessible. These properties also prepare students for algebra, where rearranging, regrouping, and distributing terms is essential.

As future teachers, practice identifying these properties in different contexts so you can help students build both intuition and formal understanding.

License

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Mathematics For Elementary Education I Copyright © 2023 by Iain Pardoe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.