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# Multiplication Property – An Essential Building Block in Mathematics

## Introduction

The Multiplication Property is an essential aspect of arithmetic that helps us understand and solve mathematical problems involving multiplication. This article will thoroughly explore the Multiplication Property, giving you the knowledge and resources, you need to understand this essential mathematical topic.

This concept is typically introduced in 3rd grade and further developed in subsequent grades as students advance their mathematical understanding.

## Math Domain

Multiplication Property falls under Arithmetic and Number Theory in elementary mathematics.

## Applicable Common Core Standards

The concept of Multiplication Property is covered under the following Common Core Standards:

3.OA.A.1: Interpret products of whole numbers

3.OA.B.5: Apply properties of operations as strategies to multiply and divide

4.NBT.B.5: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers using strategies based on place value and the properties of operations.

## Definition

Multiplication is the process of finding the product of two or more numbers called the factors (multiplicand and multiplier). It has different properties: commutative, associative, identity, and distributive. These properties of multiplication show us the different ways of finding the product.

## Key Concepts

Commutative Property of Multiplication: This property states that when two numbers are multiplied, the order in which they are multiplied does not affect the product. In other words, a × b = b × a.

Associative Property of Multiplication: This property states that grouping numbers does not affect the product when multiplying three or more numbers. In other words, (a × b) × c = a × (b × c).

Distributive Property of Multiplication: This property states that multiplication can be distributed over addition and subtraction. This property helps in the solution of problems with brackets. Additionally, it speeds up our mental computations.

Identity Property of Multiplication: The identity property of multiplication says that when a number is multiplied by 1, the product will still be the same number.  One (1) is called the identity element in multiplication.

## Discussion with Illustrative Examples

Commutative Property of Multiplication

Example: 3 × 5 = 5 × 3

This example demonstrates that the order in which the numbers are multiplied does not affect the result. In both cases, the product is 15.

Associative Property of Multiplication

Example: (2 × 3) × 4 = 2 × (3 × 4)

Grouping the numbers does not affect the product. In both cases, the product is 24.

Distributive Property of Multiplication

Example: 2×3+4  = (2 × 3) +(2× 4)

Multiplication is distributive since we obtain the same result in both situations.

Identity Property of Multiplication

Examples:

In all cases, the answer will always be the same when multiplying a number by 1.

## Examples with Solutions

Example 1

Find the missing value.

a. 2×9=__×2

b. 4×6×__=4×__×8

c. 5×3+7=(5×__)+(__×7)

d. 10×1=___

Solution

a. 2×9=9×2  Commutative Property of Multiplication

b. 4×6×8=4×6×8  Associative Property of Multiplication

c. 5×3+7=(5×3)+(5×7) Distributive Property of Multiplication

d. 10×1=10  Identity Property of Multiplication

Example 2

Find the product of 4 × 7 using the Commutative Property of Multiplication.

Solution

4 × 7 = 7 × 4 = 28

Example 3

Solve for the product of (2 × 6) × 7 using the Associative Property of Multiplication.

Solution

(2 × 6) × 7 = 12×7 = 84

2 × (6 × 7) = 2 42 = 84

## Real-life Application with Solution

Problem 1

Johnny has three bags of oranges, and each bag contains five oranges. How many oranges does Johnny have in total?

Solution

If we use the Commutative Property of Multiplication to solve this problem, we have,

3 bags × 5 oranges per bag = 5 oranges per bag × 3 bags = 15 oranges

Johnny has a total of 15 oranges.

Problem 2

The cost of buying a pink flower is $2, while the cost of buying a white flower is$1. Write a numerical expression of buying five pink flowers and five white flowers. Use the distributive property to find the total.

Solution

Since we want to get the cost of buying five flowers of each color, pink and white, where pink costs $2 each and white costs$1 each, we may have the numerical expression, 5×(2+1)

Using the distributive property, we have,

5×(2+1)=(5×2)+(5×1)

=10+5

=15

Therefore, the total cost of buying five pink and five white flowers is \$15.

## Practice Test

1. 6 × 9 = ____ × 6
2. (5 × 2) × 3 = 5 × (____ × 3)
3. 7 × 4 = ____ × ____
4. (10 × 2) × 5 = 10 × (2 × ____)
5. 8 × (3 × 4) = (8 × ____) × 4
6. 12 × 2 = ____ × 12
7. (9 × 3) × 4 = 9 × (3 × ____)
8. What is an example of the commutative property of multiplication?

a. 3 x 3 = 2 x 3

b. 3 x 2 = 2 x 3

1. Which is an example of the associative property of multiplication?

a. (2 x 1) x 2 = 2 x (1 x 2)

b. (2 x 1) + 2 = (2 x 2) + (2 x 1)Which is an example of distributive property?

1. Which is an example of the distributive property of multiplication?

a. (2 x 1) + 1 = 1 + (2 x 1)

b. 3 x (2 + 1) = (3 x 2) + (3 x 1)

1. Romeo went to his grandmother’s garden with planted pineapples. There were two sections of pineapples. Each section has the same number of pineapples. He saw four rows of pineapples, and each row had five pineapples. How many pineapples did Romeo see in his grandmother’s garden?

### What distinguishes the Associative Property of Multiplication from the Commutative Property of Multiplication?

The main difference between the two properties is that the Commutative Property of Multiplication deals with the order of the numbers being multiplied (a × b = b × a), while the Associative Property of Multiplication deals with the grouping of the numbers being multiplied ((a × b) × c = a × (b × c)).

### Do the Commutative and Associative Properties also apply to addition?

Yes, both the Commutative and Associative Properties apply to addition as well. The Commutative Property of Addition implies that a + b equals b + a, while the Associative Property of Addition says that (a + b) + c = a + (b + c).

### Do the Associative and Commutative Properties also apply to subtraction and division?

No, these properties do not apply to subtraction and division. Subtraction and division are not commutative nor associative operations.

### Why are the Commutative and Associative Properties important in mathematics?

These properties are essential because they allow us to manipulate and rearrange mathematical expressions and equations more easily, which can simplify problem-solving and help us better understand the underlying concepts.

### What is the importance of multiplication property?

Specific rules are applied while multiplying numbers based on the properties of multiplication. All types of mathematical expressions, whether algebraic expressions, fractions, or integers, can be solved using these properties since they make it simple to simplify expressions.