Commutative Property of Multiplication Lesson

The commutative property of multiplication is one of the fundamental properties of arithmetic. This property applies to real numbers, integers, fractions, and other types of numbers. The commutative property states that the order in which the numbers are arranged in a multiplication expression does not change the product.

For example, 6 · 4 and 4 · 6 both equal 24, regardless of the order they are multiplied.

Mathematically, the commutative property of multiplication can be expressed as follows:

a · b = b · a

or

a · b · c = c · b · a

In simpler terms, this means that when you are multiplying two or more numbers, it doesn’t matter how you order or arrange them. You can arrange the numbers in any order that you would like, and then multiply from left to right per the order of operations rule. The product will be the same in both cases.

For example:

2 · 4 · 5 = 8 · 5 = 40

4 · 5 · 2 = 20 · 2 = 40

5 · 2 · 4 = 10 · 4 = 40

In all three cases, the result is 40, demonstrating the commutative property of multiplication. This property is a basic rule that influences arithmetic operations and is often taken for granted in mathematical computations.

The commutative property of multiplication applies anytime you are multiplying two or more numbers together. Let’s look at another example, with four numbers.

2 · 3 · 4 · 5 = 6 · 4 · 5 = 24 · 5 = 120

2 · 5 · 3 · 4 = 10 · 3 · 4 = 30 · 4 = 120

2 · 3 · 5 · 4 = 6 · 5 · 4 = 30 · 4 = 120

In simple terms, it does not matter the order in which we multiply, we will still get the same answer. The commutative property of multiplication is the way in which we order the numbers to multiply, and can often be very helpful when doing mental math. When doing mental math with multiplication, we can use the commutative property of multiplication to order the numbers in a way that is easier to multiply in our heads.

For example, if we had to multiply the following numbers:

5 · 7 · 4 · 2 = 35 · 4 · 2 = 140 · 2 = 280

Doing, 35 · 4, in your head might be difficult for some, especially at a younger age. However, if we use the commutative property of multiplication, we can rearrange the numbers as shown below, which are easier to multiply in our head:

5 · 2 · 7 · 4 = 10 · 7 · 4 = 70 · 4 = 280

Multiplying numbers by ten is fairly simple for most people, and therefore using the commutative property of multiplication, makes this a much easier multiplication problem to do in our heads.

Examples

Example 1: Use the commutative property of multiplication to multiply the following expression easier using mental math.

8 · 6 · 5

Solution:

8 · 6 · 5

40 · 6

240

Example 2: Use the commutative property of multiplication to fill in the blank in the problem below.

3 · 4 · 5 = 4 · 5 · ___

Solution:

Since the numbers 4 and 5 appear on both sides of the equation, we know that the missing number is 3. The 3 was moved from the beginning of the expression to the end, making the problem:

3 · 4 · 5 = 4 · 5 · 3

12 · 5 = 20 · 3

60 = 60

Example 3: Use the commutative property of multiplication to fill in the blanks below. Then find the product.

3 · ____ · 5 = 2 · 5 · ____

Solution:

Looking at both sides of the equation, we can see that 5 is used on both sides of the equation, and the other two numbers are 2 and 3.

The next step is to determine where the numbers 2 and 3 go on each corresponding side of the equation. The equation should be as follows:

3 · 2 · 5 = 2 · 5 · 3

6 · 5 = 10 · 3

30 = 30

FAQs on Associative Property of Multiplication

1. Why is the Commutative Property of Multiplication important in mathematics?

    • The commutative property simplifies mathematical operations and allows for flexibility in solving problems. It is a fundamental concept that is widely used in various branches of mathematics.

    2. What is the commutative property of multiplication?

      • The commutative property of multiplication is a fundamental mathematical rule that states the order in which you multiply two numbers does not affect the result.

      3. What is the General Rule for the Commutative Property of Multiplication?

        • The general rule for the commutative property of multiplication is written as . For any two numbers a and b. This means that the order of two numbers being multiplied does not affect the product.

        4. Can the Commutative Property be used with other mathematical operations?

          • The commutative property can also be used with addition, as it does not matter the order in which we add numbers, we will get the same answer. However, the commutative property does not work with subtraction or division.
          • One example to demonstrate why it does not work with subtraction is:

          3 – 2 ≠ 2 -3

          1 ≠ -1

          • Similarly it does not work with division, and here is an example to demonstrate this:

          24/6 ≠ 6 /24

          4 ≠ 1/4

          5. Does the Commutative Property of Multiplication apply only to whole numbers?

            • No, the commutative property of multiplication applies to all real numbers, such as decimals, fractions, negative numbers, and more.

            6. Does the Commutative Property of Multiplication only apply to two numbers?

              • No, the property is applicable to multiplication of two or more numbers. For example:

              a · b · c = c · b · a