Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 1: Properties of Arithmetic

Lesson 3: Multiplication

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra (AMC 8), students explore the core properties of multiplication, including the commutative property, associative property, and the distributive property over addition. Learners also practice factoring out common factors and applying the order of operations to simplify expressions. The lesson builds fluency with mental math strategies by using these properties to compute products efficiently.

Section 1

Commutative Property of Multiplication

Property

Changing the order of the factors does not change their product.

a \cdot b = b \cdot a

Examples

We know that $6 \cdot 8 = 48$ . Because of the Commutative Property, we also know that $8 \cdot 6 = 48$ .
To find the product of 15 and 3, you can calculate $15 \cdot 3 = 45$ or $3 \cdot 15 = 45$ . Both give the same result.
A floor plan shows a room that is 10 feet wide and 12 feet long. Its area is $10 \times 12 = 120$ square feet, which is the same as $12 \times 10 = 120$ .

Explanation

Just like with addition, you can swap the numbers you're multiplying and still get the same answer. For example, having 3 rows with 5 apples each is the same total amount as having 5 rows with 3 apples each.

Section 2

Associative Property of Multiplication

Property

Associative Property of Multiplication: If $a$ , $b$ , and $c$ are real numbers, then

(a \cdot b) \cdot c = a \cdot (b \cdot c)

When multiplying three numbers, changing the grouping of the numbers does not change the result.

Section 3

Identity Property of Multiplication

Property

The product of any number and 1 is the number. 1 is called the multiplicative identity.

1 \cdot a = a

a \cdot 1 = a

Examples

To calculate $1 \cdot 63$ , remember that the product of any number and one is the number itself. So, $1 \cdot 63 = 63$ .
To find $(250)1$ , multiplying by one does not change the value. The result is $250$ .
The product $1 \times 7,654$ is simply $7,654$ because 1 is the multiplicative identity.

Explanation

Multiplying a number by 1 means you have exactly one group of that number, so its value (or identity) doesn't change. It's like the number is looking in a mirror; it stays the same.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Commutative Property of Multiplication

Property

Changing the order of the factors does not change their product.

a \cdot b = b \cdot a

Examples

We know that $6 \cdot 8 = 48$ . Because of the Commutative Property, we also know that $8 \cdot 6 = 48$ .
To find the product of 15 and 3, you can calculate $15 \cdot 3 = 45$ or $3 \cdot 15 = 45$ . Both give the same result.
A floor plan shows a room that is 10 feet wide and 12 feet long. Its area is $10 \times 12 = 120$ square feet, which is the same as $12 \times 10 = 120$ .

Explanation

Section 2

Associative Property of Multiplication

Property

Associative Property of Multiplication: If $a$ , $b$ , and $c$ are real numbers, then

(a \cdot b) \cdot c = a \cdot (b \cdot c)

When multiplying three numbers, changing the grouping of the numbers does not change the result.

Section 3

Identity Property of Multiplication

Property

The product of any number and 1 is the number. 1 is called the multiplicative identity.

1 \cdot a = a

a \cdot 1 = a

Examples

To calculate $1 \cdot 63$ , remember that the product of any number and one is the number itself. So, $1 \cdot 63 = 63$ .
To find $(250)1$ , multiplying by one does not change the value. The result is $250$ .
The product $1 \times 7,654$ is simply $7,654$ because 1 is the multiplicative identity.

Explanation

Multiplying a number by 1 means you have exactly one group of that number, so its value (or identity) doesn't change. It's like the number is looking in a mirror; it stays the same.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Commutative Property of Multiplication

Property

Changing the order of the factors does not change their product.

a \cdot b = b \cdot a

Examples

We know that $6 \cdot 8 = 48$ . Because of the Commutative Property, we also know that $8 \cdot 6 = 48$ .
To find the product of 15 and 3, you can calculate $15 \cdot 3 = 45$ or $3 \cdot 15 = 45$ . Both give the same result.
A floor plan shows a room that is 10 feet wide and 12 feet long. Its area is $10 \times 12 = 120$ square feet, which is the same as $12 \times 10 = 120$ .

Explanation

Section 2

Associative Property of Multiplication

Property

Associative Property of Multiplication: If $a$ , $b$ , and $c$ are real numbers, then

(a \cdot b) \cdot c = a \cdot (b \cdot c)

When multiplying three numbers, changing the grouping of the numbers does not change the result.

Section 3

Identity Property of Multiplication

Property

The product of any number and 1 is the number. 1 is called the multiplicative identity.

1 \cdot a = a

a \cdot 1 = a

Examples

To calculate $1 \cdot 63$ , remember that the product of any number and one is the number itself. So, $1 \cdot 63 = 63$ .
To find $(250)1$ , multiplying by one does not change the value. The result is $250$ .
The product $1 \times 7,654$ is simply $7,654$ because 1 is the multiplicative identity.

Explanation

Multiplying a number by 1 means you have exactly one group of that number, so its value (or identity) doesn't change. It's like the number is looking in a mirror; it stays the same.