Using the Distributive Property

The distributive property is an important concept in math, especially in algebra. It helps students simplify expressions and solve equations. In grade 5 math, using the distributive property becomes a foundational skill that will be used in many areas of math as students continue their studies.

What is distributive property?

The distributive property is a rule that shows how multiplication interacts with addition or subtraction. It means that you can "distribute" or spread out the multiplication over each term within a set of parentheses. It's usually written as:

a × (b + c) = (a × b) + (a × c)

This rule means that if you want to multiply a number inside brackets by both 'b' and 'c', you can do it in two steps:

• Firstly multiply 'a' by 'b'.
• Then, multiply 'a' by 'c'.
• Finally, add the two results together.

Using this property we can simplify expressions and make problems easier to solve.

Breaking the distributive property

Let's look at a concrete example to understand how the distributive property works in practice:

Example 1

Consider the expression:

3 × (4 + 5)

Using the distributive property, we can simplify this step-by-step:

1. First, we multiply 3 by 4:

   3 × 4 = 12

2. Now we multiply 3 by 5:

   3 × 5 = 15

3. Now add the two results together:

   12 + 15 = 27

So 3 × (4 + 5) simplifies to 27.

Visual example

This image shows the process of using the distributive property.

Working with large numbers

We can use the distributive property not only with small numbers but also with large numbers to simplify calculations:

Example 2

Consider the expression:

6 × (7 + 9)

Let's distribute the 6:

1. First, do the calculation:

   6 × 7 = 42

2. Then calculate:

   6 × 9 = 54

3. Finally, add up both results:

   42 + 54 = 96

The result of 6 × (7 + 9) is 96.

Using the distributive property to simplify equations

The distributive property is also used to simplify equations. Here's how:

Example 3

Simplify the expression:

2 × (x + 4)

Use the distributive property to expand this:

1. Divide 2 into the following terms:

   (2 × x) + (2 × 4)

2. Perform the multiplication:

   2x + 8

This expression 2 × (x + 4) simplifies to 2x + 8.

Example 4

Now consider subtraction inside parentheses:

5 × (10 - 3)

Distributing the 5's, we get:

1. First, calculate:

   5 × 10 = 50

2. Then calculate:

   5 × 3 = 15

3. Subtract the second result from the first:

   50 - 15 = 35

The expression 5 × (10 - 3) simplifies to 35.

Why is the distributive property important?

The distributive property is important because it provides flexibility in the way we solve mathematical problems. It helps us break down problems into smaller, easier to handle parts. This property is not only useful in arithmetic, but also forms the basis of more complex algebraic procedures.

Here is a summary of why learning the distributive property is beneficial:

• It simplifies arithmetic calculations by breaking them into smaller parts.
• This lays a foundation for understanding algebra and solving equations.
• It improves problem-solving skills by providing alternative ways to deal with problems.
• It enables mental math by making calculations more manageable.

Practice problems

Practice using the distributive property with these problems:

Problem 1

Simplify the expression:

7 × (8 + 12)

Find the simplified result using the distributive property:

Problem 2

Simplify the expression:

4 × (15 - 6)

What is the simplified result?

Problem 3

Expand and simplify the expression:

9 × (x + 11)

Express the result in the form ax + b.

Answers to exercise problems

Problem 1 answer

First, apply the distributive property:

(7 × 8) + (7 × 12)

Calculate individual products:

56 + 84 = 140

Hence the expression simplifies to 140.

Problem 2 answer

Apply the distributive property:

(4 × 15) - (4 × 6)

Calculate individual products:

60 - 24 = 36

The expression simplifies to 36.

Problem 3 answer

Use the distributive property to expand:

(9 × x) + (9 × 11)

It becomes:

9x + 99

Thus, the simplified expression is 9x + 99.

Conclusion

The distributive property is a powerful tool in algebraic thinking, providing many avenues for simplifying and solving problems. As students continue to use this property, they will build a solid foundation for more complex mathematical concepts in the future. By practicing these skills, students will become more confident and competent in their math abilities, opening the door to deeper understanding and problem-solving skills.

Remember, the key to mastering the distributive property is practice and application in different scenarios. Keep practicing with more problems, and soon, using this property will become second nature.

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