# The Distributive Property

This is part of a collection of definitions about math properties. These downloadable images can easily be incorporated into a lesson plan.

### To see the complete collection of these definitions, click on this link.

The next section provides background information on math properties. Refer to this section as you review the math properties definitions.

## The Distributive Property

Learn about the distributive property through this video. This is part of a series of videos on mathematical properties. Go to Media4Math.com to see thousand...

The video was uploaded on 9/10/2022.

You can view the video here.

The video lasts for 4 minutes and 10 seconds.

Video Transcript

The distributive property allows you to multiply a factor across multiple terms of an expression. In this video we will look at the case where the constant terms a and b are positive.

What is the area of this rectangle? Use the distributive property to express the area. What is the area when x = 4?

The area of a rectangle is the product of the length and width. With this rectangle, the length is a variable expression.

The area of the rectangle is the product of these two expressions, 6 and the quantity x + 5. Using the distributive property with an expression of this form has the a-term distributed to each of the terms in the variable expression.

So, we take the 6 and distribute it to the x and the 5, as shown here. The result is 6x + 30.

We evaluate this expression for x = 4, as shown here. We get 24 + 30, which is equal to 54.

Let’s look at another example.

What is the area of this parallelogram? Use the distributive property to express the area. What is the area when x = 10?

The area of a parallelogram is the product of the base and the height. With this parallelogram, the height is a variable expression.

The area of the parallelogram is the product of these two expressions, 5 and the quantity x + 7. Using the distributive property with an expression of this form has the a-term distributed to each of the terms in the variable expression.

So, we take the 5 and distribute it to the x and the 7, as shown here. The result is 5x + 35.

We evaluate this expression for x = 10, as shown here. We get 50 + 35, which is equal to 85

Let’s look at a final example.

A number is increased by 12. The sum is multiplied by 15. Use the distributive property to write the expression. Evaluate the expression for x = 20.

This is an example of converting words into an algebraic expression. Let’s start with this part, “a number is increased by 12.” Since we don’t know which number, we designate it as x. This number is increased by 12, so we write x + 12.

This sum, or the entire expression is multiplied by 15, so we enclose the expression in parentheses and multiply the whole expression by 15, as shown here. This is the mathematical equivalent of the verbal expression.

We now distribute the 15 to both terms, as shown here. We get 15x + 180.

We evaluate this expression for x = 20, as shown here. We get 300 + 180, which adds up to 480.

## Math Properties

As you work with numbers, it's important to know properties of operations and equations. Learn these and you'll have an easier time combining certain expressions.

Let's start by looking at the properties relating to addition.

Commutative Property.  When adding two numbers, it doesn't matter in what order you add them.  The Commutative Property of Addition means that for any two numbers a and b,

### a + b = b + a

Why is this an important property to use? Take a look at this expression:

### 7 + 25 + 3

Use the Commutative Property to rewrite the expression this way

### 7 + 3 + 25

Do you see how the 7 and 3 easily combine to form a 10, making the sum much easier to find:

### 7 + 3 + 25 = 10 + 25 = 35

Look for opportunities to use the Commutative Property to simplify addition.

Associative Property.  When adding three or more numbers, you can choose any pair of numbers to add first. The Associative Property of Addition means that you can add any pair of numbers first.

### (a + b) + c = a + (b + c)

Here's how to use the Associative Property to simplify addition:

### (25 + 3) + 7 = 25 + (3 + 7) = 25 + 10 = 35

In this case using the Associative Property simplifies the addition.

### Multiplication Properties

Commutative Property.  When multiplying two numbers, it doesn't matter in what order you multiply them.  The Commutative Property of Multiplication means that for any two numbers a and b,

### a • b = b • a

Why is this an important property to use? Take a look at this expression:

### 25 • 27 • 4

Use the Commutative Property to rewrite the expression this way

### 27 • 25 • 4

Do you see how the product of 25 and 4 is 100, making the product much easier to find:

### 27 • 25 • 4 = 27 • 100 = 2700

Look for opportunities to use the Commutative Property to simplify multiplication.

Associative Property.  When multiplying three or more numbers, you can choose any pair of numbers to multiply first. The Associative Property of Multiplication means that you can multiply any pair of numbers first.

### (a • b) • c = a • (b • c)

Here's how to use the Associative Property to simplify addition:

### (37 • 20) • 5 = 37 • (20 • 5) = 37 • 100 = 3700

In this case using the Associative Property simplifies the multiplication.

Distributive Property.  When multiplying a numerical expression by a common factor, this factor can be distributed, as shown below.

### Equation Properties

As you solve an equation, use the properties shown above in the solution process. Also, take advantage of the following equation properties

Addition Property of Equality.  When solving an equation, you may have to remove a number from one side of the equation in order to isolate the variable. Use the Addition Property of Equality, which states that whatever you add to one side of an equation, you also have to add to the other side of the equation.

Here is an example of using the Addition Property of Equality to solve an equation:

Notice how 2 is added to both sides in order to get x by itself to solve the equation.

Subtraction Property of Equality.  When solving an equation, you may have to remove a number from one side of the equation in order to isolate the variable. Use the Subtraction Property of Equality, which states that whatever you subtract from one side of an equation, you also have to subtract from the other side of the equation.

Here is an example of using the Subtraction Property of Equality to solve an equation:

Notice how 2 is subtracted from both sides in order to get x by itself to solve the equation.

Multiplication Property of Equality.  When solving an equation, you may have to remove a number from one side of the equation in order to isolate the variable. Use the Multiplication Property of Equality, which states that whatever you multiply by on one side of an equation, you also have to multiply by on the other side of the equation.

Here is an example of using the Multiplication Property of Equality to solve an equation:

Division Property of Equality.  When solving an equation, you may have to remove a number from one side of the equation in order to isolate the variable. Use the Division Property of Equality, which states that whatever you divide by on one side of an equation, you also have to divide by on the other side of the equation. Note: You cannot divide by zero.

Here is an example of using the Division Property of Equality to solve an equation:

Notice how both sides of the equation are divided by 2 in order to get x by itself to solve the equation.