my homework lesson 3 use the distributive property to multiply

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McGraw Hill My Math Grade 4 Chapter 5 Lesson 3 Answer Key Use the Distributive Property to Multiply

All the solutions provided in McGraw Hill Math Grade 4 Answer Key PDF Chapter 5 Lesson 3 Use the Distributive Property to Multiply  will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 4 Answer Key Chapter 5 Lesson 3 Use the Distributive Property to Multiply

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Question 1. Mathematical PRACTICE 7 Identify Structure How would you use the Distributive Property to find 12 × 18? Then find the product.

Answer: Using the distributive property, the product of 12 x 18 is 216. Explanation: Given that, the value is 12 x 18. Now, we will find the value using the distributive property. So, the given value is distributive as, 12 x 18 = 12 x( 10 + 8) = 12 x 10 + 12 x 8 = 120 + 96 = 216. Hence, the product of the given values is 216.

Question 2. How would you use the Distributive Property to find 14 × 17? Then find the product.

Answer: Using the distributive property, the product of 14 x 17 is 238. Explanation: Given that, the value is 14 x 17. Now, we will find the value using the distributive property. So, the given value is distributive as, 14 x 17 = 14 x( 10 + 7) = 14 x 10 + 14 x 7 = 140 + 98 = 238. Hence, the product of the given values is 238.

Practice It

Draw an area model. Then use the Distributive Property to find each product

Question 4. Find 47 × 19. 47 × 19 = 47 × (10 + 9) = (47 × ____) + (47 × ____) = ____ + ____ = ______

Question 5. Find 52 × 11. 52 × 11 = ___ × (___ + _____) = (_____ × ____) + (_____ × ____) = _____ × ____ = _______

Question 6. Find 46 × 22. 46 × 22 = ____ × (____ + ____) = (____ × ____) + (____ × ____) = ____ × ____ = ______

Use the Distributive Property to solve.

Question 7. Mathematical PRACTICE 7 Identify Structure There are 15 types of animals in each part of the zoo. The zoo has 12 parts. How many types of animals are there in all?

Answer: 180 types of animals are there in all. Explanation: There are 15 types of animals in each part of the zoo. The zoo has 12 parts. Now, we will find out how many types of animals are there in all. So, the values are 15 and 12. Using the distributive property, distribute the values. 15 x 12 = 15 x ( 10 + 2) = 15 x 10 + 15 x 2 = 150 + 30 = 180. Therefore, 180 types of animals are there in all.

Answer: Using the distributive property, the product of 12 x 17 is  204. Explanation: Given that, the value is 12 x 17. Now, we will find the value using the distributive property. So, the given value is distributive as, 12 x 17 = 12 x ( 10 + 7) = 12 x 10 + 12 x 7 = 120 + 84 = 204. Hence, the product of the given values is 204.

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McGraw Hill My Math Grade 4 Chapter 5 Lesson 3 My Homework Answer Key

Draw an area model. Then use the Distributive Property to find each product.

Question 1. 73 × 34 = ____ 73 × 34 = 73 × (30 + 4) = (73 × ___) + (73 × ____) = _____ + _____ = _______

Question 2. 82 × 22 = _____ 82 × 22 = 82 × (20 + 2) = (82 × ___) + (82 × ____) = _____ + ____ = _____

Question 3. 18 × 39 = ____ 18 × 39 = ____ ×(____ + ____) = (______ × _____) + (______ × _____) = ______ × _____ = _______

Problem Solving

Question 4. There are 48 nails in one box. How many nails are in 17 boxes? _____ nails. 17 × 48 = ____ × (____ + ____) = (______ × _____) + (______ × _____) = ______ × _____ = _______

Question 5. Each notebook has 64 pages. How many total pages are there in 33 notebooks? _____ pages 33 × 64 = ____ × (____ + ____) = (______ × _____) + (______ × _____) = ______ × _____ = _______

Question 6. Each jar contains 55 buttons. There are 16 jars on the shelf. How many buttons are there altogether? ____ buttons

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Distributive property

Here you will learn about the distributive property, including what it is, and how to use it to solve problems.

Students will first learn about the distributive property as part of operations and algebraic thinking in 3rd grade.

What is the distributive property?

The distributive property states that multiplying the sum of two or more numbers is the same as multiplying the addends separately.

For example,

When multiplying 2 \times 8, you can break 8 up into 2 + 6.

The distributive property says that you can multiply the parts separately and then add the products together.

Any way you solve the equivalent expressions, the product is the same.

For most expressions, there is more than one way to use the distributive property. 

When multiplying 2 \times 8, you can break 8 up into 5 + 3.

[FREE] Distributive Property Worksheet (Grade 1 to 3)

Use this worksheet to check your grade 1 to 3 students’ understanding of the distributive property. 15 questions with answers to identify areas of strength and support!

Common Core State Standards

How does this relate to 3rd grade math?

• Grade 3 – Operations and Algebraic Thinking (3.OA.B.5) Apply properties of operations as strategies to multiply and divide. Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication.) 3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication.) Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Distributive property.)

How to use the distributive property

In order to use the distributive property:

Identify an equation multiplying two numbers.

Show one of the numbers being multiplied as a sum of numbers.

Multiply each number in the sum.

Add the partial products together to find the final product.

Distributive property examples

Example 1: distributive property with basic facts.

Show how to solve 3 \times 5 using the distributive property.

You can use the distributive property with 3 \times 5, since it is multiplication.

2 Show one of the numbers being multiplied as a sum of numbers.

Either number can be used, but for this example let’s break up 5 into 4 + 1.

3 \times 5=3 \times(4+1)

3 Multiply each number in the sum.

\begin{aligned} & 3 \times(4+1) \\\\ & =(3 \times 4)+(3 \times 1) \\\\ & =12+3 \end{aligned}

4 Add the partial products together to find the final product.

12 + 3 = 15

3 \times 5=15 can be solved using the distributive property.

Example 2: distributive property with basic facts

Show how to solve 12 \times 9 using the distributive property.

You can use the distributive property with 12 \times 9, since it is multiplication.

Either number can be used, but for this example let’s break up 9 into 3 + 3 + 3.

12 \times 9=12 \times(3+3+3)

\begin{aligned} & 12 \times(3+3+3) \\\\ & =(12 \times 3)+(12 \times 3)+(12 \times 3) \\\\ & =36+36+36 \end{aligned}

36 + 36 + 36 = 108

12 \times 9=108 can be solved using the distributive property.

Example 3: distributive property with basic facts

Show how to solve 7 \times 6 using the distributive property.

You can use the distributive property with 7 \times 6, since it is multiplication.

Either number can be used, but for this example let’s break up 7 into 4 + 3.

7 \times 6=(4+3) \times 6

\begin{aligned} & (4+3) \times 6 \\\\ & =(4 \times 6)+(3 \times 6) \\\\ & =24+18 \end{aligned}

24 + 18 = 42

7 \times 6=42 can be solved using the distributive property.

Example 4: distributive property with basic facts

Show how to solve 4 \times 11 using the distributive property.

You can use the distributive property with 4 \times 11, since it is multiplication.

Either number can be used, but for this example let’s break up 11 into 10 + 1.

4 \times 11=4 \times(10+1)

\begin{aligned} & 4 \times(10+1) \\\\ & =(4 \times 10)+(4 \times 1) \\\\ & =40+4 \end{aligned}

40 + 4 = 44

4 \times 11=44 can be solved using the distributive property.

Example 5: distributive property with basic facts

Show how to solve 8 \times 5 using the distributive property.

You can use the distributive property with 8 \times 5, since it is multiplication.

Either number can be used, but for this example let’s break up 8 into 2 + 6.

8 \times 5=(2+6) \times 5

\begin{aligned} & (2+6) \times 5 \\\\ & =(2 \times 5)+(6 \times 5) \\\\ & =10+30 \end{aligned}

10 + 30 = 40

8 \times 5=40 can be solved using the distributive property.

Example 6: distributive property with basic facts

Show how to solve 3 \times 12 using the distributive property.

You can use the distributive property with 3 \times 12, since it is multiplication.

Either number can be used, but for this example let’s break up 12 into 1 + 1 + 10.

3 \times 12=3 \times(1+1+10)

\begin{aligned} & 3 \times(1+1+10) \\\\ & =(3 \times 1)+(3 \times 1)+(3 \times 10) \\\\ & =3+3+30 \end{aligned}

3 + 3 + 30 = 36

3 \times 12=36 can be solved using the distributive property.

Teaching tips for the distributive property

• Intentionally choose practice problems that lend themselves to being solved with the distributive property, as it is not always necessary or useful in all solving situations.
• Instead of just giving students the distributive property definition, draw attention to examples of the distributive property as they come up in daily math activities. You may even keep an anchor chart of different examples. Over time, students will start using it and recognizing it on their own and then you can introduce them to the property and its official definition through their own examples.
• Include plenty of student discourse around this topic to ensure that students understand that breaking apart a number and then multiplying it in parts does not change the total product. This could include students sharing their thinking or critiquing the thinking of others.

Easy mistakes to make

• Thinking there is only one way to use the distributive property to solve Often, there is more than one way to use the distributive property when solving. For example, \begin{aligned} & 4 \times 5 \hspace{4.65cm} 4 \times 5 \\ & =(2+2) \times 5 \hspace{3.5cm} =(1+3) \times 5 \\ & =(2 \times 5)+(2 \times 5) \hspace{1cm} \text{ OR } \hspace{1cm} =(1 \times 5)+(3 \times 5) \\ & =10+10 \hspace{3.9cm} =5+15 \\ & =20 \hspace{4.63cm} =20 \end{aligned}

Related properties of equality lessons

• Properties of equality
• Order of operations
• Associative property
• Commutative property

Practice distributive property questions

1. Which of the following equations shows 12 \times 6 using the distributive property?

The numbers are being multiplied, so the distributive property can be used.

2. Which of the following equations shows 7 \times 9 using the distributive property?

3. Which of the following equations shows 11 \times 8 using the distributive property?

4. Which of the following equations shows 3 \times 7 using the distributive property?

5. Which of the following equations is NOT a way to solve 10 \times 5 using the distributive property?

This strategy is NOT a way to solve with the distributive property.

All the other equations break 10 or 5 up into a sum and add the products of the parts, using the distributive property correctly:

6. Which of the following equations is NOT a way to solve 9 \times 8 using the distributive property?

This strategy is NOT a way to solve 9 \times 8 with the distributive property.

All the other equations break 9 or 8 up into a sum and add the products of the parts, using the distributive property correctly:

Distributive property FAQs

Yes, the distributive property can be used with integers (including negative numbers) and rational numbers (including fractions and decimals), as long as the numbers are all being multiplied. In middle and high school, students will learn how to use the distributive property with any real number and/or algebraic expression.

No, even though the associative property also uses parentheses, they are different properties. The associative property says you can change the grouping of numbers when adding or multiplying and the sum or product will be the same. This is different from the distributive property.

Yes, this is called the distributive property of multiplication over subtraction.

This is a general term and means the same as the distributive property.

No, because of the order of operations (or PEMDAS), the products will be found first and then added together. However, it is good practice to group each partial product with parentheses.

The next lessons are

• Addition and subtraction
• Multiplication and division
• Types of numbers

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7.3 Distributive Property

Learning objectives.

By the end of this section, you will be able to:

• Simplify expressions using the distributive property
• Evaluate expressions using the distributive property

Be Prepared 7.7

Before you get started, take this readiness quiz.

Multiply: 3 ( 0.25 ) . 3 ( 0.25 ) . If you missed this problem, review Example 5.15

Be Prepared 7.8

Simplify: 10 − ( −2 ) ( 3 ) . 10 − ( −2 ) ( 3 ) . If you missed this problem, review Example 3.51

Be Prepared 7.9

Combine like terms: 9 y + 17 + 3 y − 2 . 9 y + 17 + 3 y − 2 . If you missed this problem, review Example 2.22 .

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $9.25 ; $9.25 ; that is, 9 9 dollars and 1 1 quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

They need 3 3 times $9 , $9 , so $27 , $27 , and 3 3 times 1 1 quarter, so 75 75 cents. In total, they need $27.75 . $27.75 .

If you think about doing the math in this way, you are using the Distributive Property.

Distributive Property

If a , b , c a , b , c are real numbers, then

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression 3 ( x + 4 ) , 3 ( x + 4 ) , the order of operations says to work in the parentheses first. But we cannot add x x and 4 , 4 , since they are not like terms. So we use the Distributive Property, as shown in Example 7.17 .

Example 7.17

Simplify: 3 ( x + 4 ) . 3 ( x + 4 ) .

Try It 7.33

Simplify: 4 ( x + 2 ) . 4 ( x + 2 ) .

Try It 7.34

Simplify: 6 ( x + 7 ) . 6 ( x + 7 ) .

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 7.17 would look like this:

Example 7.18

Simplify: 6 ( 5 y + 1 ) . 6 ( 5 y + 1 ) .

Try It 7.35

Simplify: 9 ( 3 y + 8 ) . 9 ( 3 y + 8 ) .

Try It 7.36

Simplify: 5 ( 5 w + 9 ) . 5 ( 5 w + 9 ) .

The distributive property can be used to simplify expressions that look slightly different from a ( b + c ) . a ( b + c ) . Here are two other forms.

Other forms

Example 7.19

Simplify: 2 ( x − 3 ) . 2 ( x − 3 ) .

Try It 7.37

Simplify: 7 ( x − 6 ) . 7 ( x − 6 ) .

Try It 7.38

Simplify: 8 ( x − 5 ) . 8 ( x − 5 ) .

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Example 7.20

Simplify: 3 4 ( n + 12 ) . 3 4 ( n + 12 ) .

Try It 7.39

Simplify: 2 5 ( p + 10 ) . 2 5 ( p + 10 ) .

Try It 7.40

Simplify: 3 7 ( u + 21 ) . 3 7 ( u + 21 ) .

Example 7.21

Simplify: 8 ( 3 8 x + 1 4 ) . 8 ( 3 8 x + 1 4 ) .

Try It 7.41

Simplify: 6 ( 5 6 y + 1 2 ) . 6 ( 5 6 y + 1 2 ) .

Try It 7.42

Simplify: 12 ( 1 3 n + 3 4 ) . 12 ( 1 3 n + 3 4 ) .

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

Example 7.22

Simplify: 100 ( 0.3 + 0.25 q ) . 100 ( 0.3 + 0.25 q ) .

Try It 7.43

Simplify: 100 ( 0.7 + 0.15 p ) . 100 ( 0.7 + 0.15 p ) .

Try It 7.44

Simplify: 100 ( 0.04 + 0.35 d ) . 100 ( 0.04 + 0.35 d ) .

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

Example 7.23

Simplify: m ( n − 4 ) . m ( n − 4 ) .

Notice that we wrote m · 4 as 4 m . m · 4 as 4 m . We can do this because of the Commutative Property of Multiplication. When a term is the product of a number and a variable, we write the number first.

Try It 7.45

Simplify: r ( s − 2 ) . r ( s − 2 ) .

Try It 7.46

Simplify: y ( z − 8 ) . y ( z − 8 ) .

The next example will use the ‘backwards’ form of the Distributive Property, ( b + c ) a = b a + c a . ( b + c ) a = b a + c a .

Example 7.24

Simplify: ( x + 8 ) p . ( x + 8 ) p .

Try It 7.47

Simplify: ( x + 2 ) p . ( x + 2 ) p .

Try It 7.48

Simplify: ( y + 4 ) q . ( y + 4 ) q .

When you distribute a negative number, you need to be extra careful to get the signs correct.

Example 7.25

Simplify: −2 ( 4 y + 1 ) . −2 ( 4 y + 1 ) .

Try It 7.49

Simplify: −3 ( 6 m + 5 ) . −3 ( 6 m + 5 ) .

Try It 7.50

Simplify: −6 ( 8 n + 11 ) . −6 ( 8 n + 11 ) .

Example 7.26

Simplify: −11 ( 4 − 3 a ) . −11 ( 4 − 3 a ) .

You could also write the result as 33 a − 44 . 33 a − 44 . Do you know why?

Try It 7.51

Simplify: −5 ( 2 − 3 a ) . −5 ( 2 − 3 a ) .

Try It 7.52

Simplify: −7 ( 8 − 15 y ) . −7 ( 8 − 15 y ) .

In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, − a = −1 · a . − a = −1 · a .

Example 7.27

Simplify: − ( y + 5 ) . − ( y + 5 ) .

Try It 7.53

Simplify: − ( z − 11 ) . − ( z − 11 ) .

Try It 7.54

Simplify: − ( x − 4 ) . − ( x − 4 ) .

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Example 7.28

Simplify: 8 − 2 ( x + 3 ) . 8 − 2 ( x + 3 ) .

Try It 7.55

Simplify: 9 − 3 ( x + 2 ) . 9 − 3 ( x + 2 ) .

Try It 7.56

Simplify: 7 x − 5 ( x + 4 ) . 7 x − 5 ( x + 4 ) .

Example 7.29

Simplify: 4 ( x − 8 ) − ( x + 3 ) . 4 ( x − 8 ) − ( x + 3 ) .

Try It 7.57

Simplify: 6 ( x − 9 ) − ( x + 12 ) . 6 ( x − 9 ) − ( x + 12 ) .

Try It 7.58

Simplify: 8 ( x − 1 ) − ( x + 5 ) . 8 ( x − 1 ) − ( x + 5 ) .

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part ⓐ , we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example 7.30

When y = 10 y = 10 evaluate: ⓐ 6 ( 5 y + 1 ) 6 ( 5 y + 1 ) ⓑ 6 · 5 y + 6 · 1 . 6 · 5 y + 6 · 1 .

Notice, the answers are the same. When y = 10 , y = 10 ,

Try it yourself for a different value of y . y .

Try It 7.59

Evaluate when w = 3 : w = 3 : ⓐ 5 ( 5 w + 9 ) 5 ( 5 w + 9 ) ⓑ 5 · 5 w + 5 · 9 . 5 · 5 w + 5 · 9 .

Try It 7.60

Evaluate when y = 2 : y = 2 : ⓐ 9 ( 3 y + 8 ) 9 ( 3 y + 8 ) ⓑ 9 · 3 y + 9 · 8 . 9 · 3 y + 9 · 8 .

Example 7.31

When y = 3 , y = 3 , evaluate ⓐ −2 ( 4 y + 1 ) −2 ( 4 y + 1 ) ⓑ −2 · 4 y + ( −2 ) · 1 . −2 · 4 y + ( −2 ) · 1 .

Try It 7.61

Evaluate when n = −2 : n = −2 : ⓐ −6 ( 8 n + 11 ) −6 ( 8 n + 11 ) ⓑ −6 · 8 n + ( −6 ) · 11 . −6 · 8 n + ( −6 ) · 11 .

Try It 7.62

Evaluate when m = −1 : m = −1 : ⓐ −3 ( 6 m + 5 ) −3 ( 6 m + 5 ) ⓑ −3 · 6 m + ( −3 ) · 5 . −3 · 6 m + ( −3 ) · 5 .

Example 7.32

When y = 35 y = 35 evaluate ⓐ − ( y + 5 ) − ( y + 5 ) and ⓑ − y − 5 − y − 5 to show that − ( y + 5 ) = − y − 5 . − ( y + 5 ) = − y − 5 .

Try It 7.63

Evaluate when x = 36 : x = 36 : ⓐ − ( x − 4 ) − ( x − 4 ) ⓑ − x + 4 − x + 4 to show that − ( x − 4 ) = − x + 4 . − ( x − 4 ) = − x + 4 .

Try It 7.64

Evaluate when z = 55 : z = 55 : ⓐ − ( z − 10 ) − ( z − 10 ) ⓑ − z + 10 − z + 10 to show that − ( z − 10 ) = − z + 10 . − ( z − 10 ) = − z + 10 .

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Section 7.3 Exercises

Practice makes perfect.

In the following exercises, simplify using the distributive property.

4 ( x + 8 ) 4 ( x + 8 )

3 ( a + 9 ) 3 ( a + 9 )

8 ( 4 y + 9 ) 8 ( 4 y + 9 )

9 ( 3 w + 7 ) 9 ( 3 w + 7 )

6 ( c − 13 ) 6 ( c − 13 )

7 ( y − 13 ) 7 ( y − 13 )

7 ( 3 p − 8 ) 7 ( 3 p − 8 )

5 ( 7 u − 4 ) 5 ( 7 u − 4 )

1 2 ( n + 8 ) 1 2 ( n + 8 )

1 3 ( u + 9 ) 1 3 ( u + 9 )

1 4 ( 3 q + 12 ) 1 4 ( 3 q + 12 )

1 5 ( 4 m + 20 ) 1 5 ( 4 m + 20 )

9 ( 5 9 y − 1 3 ) 9 ( 5 9 y − 1 3 )

10 ( 3 10 x − 2 5 ) 10 ( 3 10 x − 2 5 )

12 ( 1 4 + 2 3 r ) 12 ( 1 4 + 2 3 r )

12 ( 1 6 + 3 4 s ) 12 ( 1 6 + 3 4 s )

r ( s − 18 ) r ( s − 18 )

u ( v − 10 ) u ( v − 10 )

( y + 4 ) p ( y + 4 ) p

( a + 7 ) x ( a + 7 ) x

−2 ( y + 13 ) −2 ( y + 13 )

−3 ( a + 11 ) −3 ( a + 11 )

−7 ( 4 p + 1 ) −7 ( 4 p + 1 )

−9 ( 9 a + 4 ) −9 ( 9 a + 4 )

−3 ( x − 6 ) −3 ( x − 6 )

−4 ( q − 7 ) −4 ( q − 7 )

−9 ( 3 a − 7 ) −9 ( 3 a − 7 )

−6 ( 7 x − 8 ) −6 ( 7 x − 8 )

− ( r + 7 ) − ( r + 7 )

− ( q + 11 ) − ( q + 11 )

− ( 3 x − 7 ) − ( 3 x − 7 )

− ( 5 p − 4 ) − ( 5 p − 4 )

5 + 9 ( n − 6 ) 5 + 9 ( n − 6 )

12 + 8 ( u − 1 ) 12 + 8 ( u − 1 )

16 − 3 ( y + 8 ) 16 − 3 ( y + 8 )

18 − 4 ( x + 2 ) 18 − 4 ( x + 2 )

4 − 11 ( 3 c − 2 ) 4 − 11 ( 3 c − 2 )

9 − 6 ( 7 n − 5 ) 9 − 6 ( 7 n − 5 )

22 − ( a + 3 ) 22 − ( a + 3 )

8 − ( r − 7 ) 8 − ( r − 7 )

−12 − ( u + 10 ) −12 − ( u + 10 )

−4 − ( c − 10 ) −4 − ( c − 10 )

( 5 m − 3 ) − ( m + 7 ) ( 5 m − 3 ) − ( m + 7 )

( 4 y − 1 ) − ( y − 2 ) ( 4 y − 1 ) − ( y − 2 )

5 ( 2 n + 9 ) + 12 ( n − 3 ) 5 ( 2 n + 9 ) + 12 ( n − 3 )

9 ( 5 u + 8 ) + 2 ( u − 6 ) 9 ( 5 u + 8 ) + 2 ( u − 6 )

9 ( 8 x − 3 ) − ( −2 ) 9 ( 8 x − 3 ) − ( −2 )

4 ( 6 x − 1 ) − ( −8 ) 4 ( 6 x − 1 ) − ( −8 )

14 ( c − 1 ) − 8 ( c − 6 ) 14 ( c − 1 ) − 8 ( c − 6 )

11 ( n − 7 ) − 5 ( n − 1 ) 11 ( n − 7 ) − 5 ( n − 1 )

6 ( 7 y + 8 ) − ( 30 y − 15 ) 6 ( 7 y + 8 ) − ( 30 y − 15 )

7 ( 3 n + 9 ) − ( 4 n − 13 ) 7 ( 3 n + 9 ) − ( 4 n − 13 )

In the following exercises, evaluate both expressions for the given value.

If v = −2 , v = −2 , evaluate

• ⓐ 6 ( 4 v + 7 ) 6 ( 4 v + 7 )
• ⓑ 6 · 4 v + 6 · 7 6 · 4 v + 6 · 7

If u = −1 , u = −1 , evaluate

• ⓐ 8 ( 5 u + 12 ) 8 ( 5 u + 12 )
• ⓑ 8 · 5 u + 8 · 12 8 · 5 u + 8 · 12

If n = 2 3 , n = 2 3 , evaluate

• ⓐ 3 ( n + 5 6 ) 3 ( n + 5 6 )
• ⓑ 3 · n + 3 · 5 6 3 · n + 3 · 5 6

If y = 3 4 , y = 3 4 , evaluate

• ⓐ 4 ( y + 3 8 ) 4 ( y + 3 8 )
• ⓑ 4 · y + 4 · 3 8 4 · y + 4 · 3 8

If y = 7 12 , y = 7 12 , evaluate

• ⓐ −3 ( 4 y + 15 ) −3 ( 4 y + 15 )
• ⓑ −3 · 4 y + ( −3 ) · 15 −3 · 4 y + ( −3 ) · 15

If p = 23 30 , p = 23 30 , evaluate

• ⓐ −6 ( 5 p + 11 ) −6 ( 5 p + 11 )
• ⓑ −6 · 5 p + ( −6 ) · 11 −6 · 5 p + ( −6 ) · 11

If m = 0.4 , m = 0.4 , evaluate

• ⓐ −10 ( 3 m − 0.9 ) −10 ( 3 m − 0.9 )
• ⓑ −10 · 3 m − ( −10 ) ( 0.9 ) −10 · 3 m − ( −10 ) ( 0.9 )

If n = 0.75 , n = 0.75 , evaluate

• ⓐ −100 ( 5 n + 1.5 ) −100 ( 5 n + 1.5 )
• ⓑ −100 · 5 n + ( −100 ) ( 1.5 ) −100 · 5 n + ( −100 ) ( 1.5 )

If y = −25 , y = −25 , evaluate

• ⓐ − ( y − 25 ) − ( y − 25 )
• ⓑ − y + 25 − y + 25

If w = −80 , w = −80 , evaluate

• ⓐ − ( w − 80 ) − ( w − 80 )
• ⓑ − w + 80 − w + 80

If p = 0.19 , p = 0.19 , evaluate

• ⓐ − ( p + 0.72 ) − ( p + 0.72 )
• ⓑ − p − 0.72 − p − 0.72

If q = 0.55 , q = 0.55 , evaluate

• ⓐ − ( q + 0.48 ) − ( q + 0.48 )
• ⓑ − q − 0.48 − q − 0.48

Everyday Math

Buying by the case Joe can buy his favorite ice tea at a convenience store for $1.99 $1.99 per bottle. At the grocery store, he can buy a case of 12 12 bottles for $23.88 . $23.88 .

ⓐ Use the distributive property to find the cost of 12 12 bottles bought individually at the convenience store. (Hint: notice that $1.99 $1.99 is $2 − $0.01 . $2 − $0.01 . )

ⓑ Is it a bargain to buy the iced tea at the grocery store by the case?

Multi-pack purchase Adele’s shampoo sells for $3.97 $3.97 per bottle at the drug store. At the warehouse store, the same shampoo is sold as a 3-pack 3-pack for $10.49 . $10.49 .

ⓐ Show how you can use the distributive property to find the cost of 3 3 bottles bought individually at the drug store.

ⓑ How much would Adele save by buying the 3-pack 3-pack at the warehouse store?

Writing Exercises

Simplify 8 ( x − 1 4 ) 8 ( x − 1 4 ) using the distributive property and explain each step.

Explain how you can multiply 4 ( $5.97 ) 4 ( $5.97 ) without paper or a calculator by thinking of $5.97 $5.97 as 6 − 0.03 6 − 0.03 and then using the distributive property.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Lesson 3 Hands On: Use the Distributive Property to Multiply

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Distributive Property of Multiplication

According to the distributive property of multiplication, when we multiply a number with the sum of two or more addends, we get a result that is equal to the result that is obtained when we multiply each addend separately by the number. The distributive property of multiplication applies to the sum and the difference of two more numbers.

What is the Distributive Property of Multiplication?

The distributive property of multiplication which holds true for addition and subtraction helps to distribute the given number on the operation to solve the given equation easily. In simple words, when a number is multiplied by the sum of two numbers, then the product is the same as the product that we get when the number is distributed to the two numbers inside the brackets and multiplied by each of them separately. Let us understand this with an example. When we get an expression like 6(3 + 5), we use the order of operations by first solving the brackets and then we multiply the result with the other number in the following way: 6(3 + 5) = 6 (8) = 6 × 8 = 48.

However, when we apply the distributive property of multiplication on the same expression 6(3 + 5), we distribute the number 6 to 3 and then to 5 in the following way: (6 × 3) + (6 × 5) = 48. We get the same result with both the methods. Now, the question is, why do we use the distributive property if we get the same result by both the methods. The answer is that the distributive property is used to solve expressions that have variables instead of numbers. Since different variables cannot be added or subtracted, the distributive property helps in this case.

Distributive Property of Multiplication Formula

The formula for the distributive property of multiplication is a(b + c) = ab + ac. This formula explains that we get the same product on both sides of the equation even when we multiply 'a' with the sum of 'b' and 'c' on the left-hand-side, or, when we distribute 'a' to 'b' and then to 'c' on the right-hand-side. Observe the following formula for the distributive property of multiplication. It is to be noted that this property is applicable to addition and subtraction.

Distributive Property of Multiplication Over Addition

The distributive property of multiplication over addition states that multiplying the sum of two or more addends by a number gives the same result as multiplying each addend individually by the number and then adding or the products together. This property of multiplication over addition is used when we need to multiply a number by a sum. For example, let us solve the expression 7(9 + 3). If solve it in the usual order of operations, we will solve the brackets first and then we will multiply the number with the obtained result. 7(9 + 3) = 7(12) = 84

However, according to the distributive property of multiplication over addition, we multiply 7 by each addend. This is called distributing the number 7 to 9 and 3, and then we add each product. So, let us find the product of the distributed number: 7 × 9 and 7 × 3. This gives us: 7(9) + 7(3) = 63 + 21 = 84. This shows that we get the same product.

Observe the following equation which shows the usual method on the left-hand side and the distributive property of multiplication over addition on the right-hand side. Applying the distributive property, we distribute the number 7 to 9 and 3, then we multiply the respective numbers by 7 and add the results. In each case, the result is the same.

7(9 + 3) = 7(9) + 7(3)

7(12) = 63 + 21

Distributive Property of Multiplication Over Subtraction

The distributive property of multiplication over subtraction states that the multiplication of a number by the difference of two other numbers is equal to the difference of the products of the distributed number. The formula for the distributive property of multiplication over subtraction is: a(b - c) = ab - ac. For example, let us solve: 9(20 - 10).

Using the usual order of operations, we find the difference of the numbers given in brackets and then we multiply the result by 9.

9(20 -10) = 9(10) = 90

Now, let us use the distributive property of multiplication over subtraction to solve 9(20 - 10). We multiply 9 by each value inside the bracket and then find the difference of the products.

So, let us multiply: 9 × 20 and 9 × 10. This gives us: 9(20) - 9(10) = 180 - 90 = 90. The result is the same as the above.

Observe the following equation in which the usual method is shown on the left-hand side and the distributive property of multiplication is applied on the right-hand side. Applying the distributive property of multiplication over subtraction, we distribute the number 9 to 20 and 10, then we multiply the respective numbers by 9 and subtract the products. In both the cases, we get the same answer.

9(20 - 10) = 9(20) - 9(10)

9(10) = 180 - 90

Related Articles

Check out the following articles related to the distributive property of multiplication.

• Distributive Property Formula
• Associative Property of Multiplication
• Multiplication

Distributive Property of Multiplication Examples

Example 1: Evaluate using the distributive property of multiplication: 8(10 + 2).

Solution: We will solve the expression using the distributive property of multiplication over addition.

8(10 + 2) = 8(10) + 8(2) = 80 + 16 = 96

Example 2: Solve the expression with the help of the distributive property of multiplication: 5(9 - 4).

Solution: Using the distributive property of multiplication over subtraction,

5(9 - 4) = 5(9) - 5(4) = 45 - 20 = 25

Example 3: Solve the following by using the distributive property of multiplication: 2(12 - 8).

Solution: According to the distributive property of multiplication,

2(12 - 8) = 2(12) - 2(8) = 24 - 16 = 8

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Practice Questions on Distributive Property of Multiplication

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FAQs on the Distributive Property of Multiplication

What is the distributive property of multiplication in math.

According to the distributive property of multiplication, when we multiply a number with the sum of two or more addends, we get a result that is equal to the result that is obtained when we multiply each addend separately by the number. The distributive property of multiplication applies to the sum and the difference of two more numbers . It is used to solve expressions easily by distributing a number to the numbers given in brackets. For example, if we apply the distributive property to solve the expression: 3(2 + 4), we would solve it in the following way: 3(2 + 4) = (3 × 2) + (3 × 4) = 6 + 12 = 18.

What is the Distributive Property of Multiplication Formula?

The distributive property of the multiplication formula is applied on addition and subtraction and is expressed as:

• a(b+c) = ab + bc
• a(b-c) = ab - bc

How to Solve the Distributive Property of Multiplication Over Addition?

The distributive property of multiplication over addition is used when we multiply a value by the sum of two or more numbers. For example, let us solve the expression: 5(5 + 9). This expression can be solved by multiplying 5 by both the addends. So, 5(5) + 5(9) = 25 + 45 = 70.

How to Solve Distributive Property of Multiplication Over Subtraction?

The distributive property of multiplication over subtraction is applied when we multiply a value by the difference of two numbers. For example, let us solve the expression: 3(9 - 5). The expression can be solved by multiplying 3 by each term and then find the differences of the products. So, 3(9) - 3(5) = 27 - 15 = 12.

What does Distributive Property of Multiplication Look Like?

The distributive property of multiplication can be seen with the help of its formula which is applicable to addition and subtraction in the following way:

• Distributive property of the multiplication over addition: a(b+c) = ab + bc.
• Distributive property of the multiplication over subtraction: a(b-c) = ab - bc

Give an example of the Distributive Property of Multiplication.

The distributive property of multiplication can be understood through various examples. For example, let us solve the expression 4(7 + 3). We will distribute the number 4 to 7 and 3. This will make it 4(7) + 4(3) = 28 + 12 = 40.

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Distributive Property

We can use the distributive property to expand and factor expressions.

Expanding an expression is like opening an umbrella:

We multiply each term inside the parentheses ( x + 3 y + 4 ) by the outside term 2 .

Factoring an expression is like closing an umbrella:

We divide each term in the expression 2 x + 3 y + 8 by the greatest common factor or GCF 2 . The factored expression goes inside the parentheses, and the GCF goes on the outside.

Just like how the umbrella itself doesn’t change as it opens and closes, the value of the expression remains the same whether it’s expanded or factored.

What is a term? 🙋🏿

In a mathematical expression or equation, a term can be a single number, a single variable, or numbers and variables multiplied together or divided by each other.

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How to Use the Distributive Property

Step 1. combine any like terms., step 2. expand the expression., step 2. identify the greatest common factor (gcf) of the terms in the expression., step 3. factor the expression..

Multiplication Using the Distributive Property Lesson Plan: Mt. Multiplis

This lesson plan, adaptable for grades 3-5, features the demo version of the multiplication game, Mt. Multiplis from our partner, Teachley . This interactive game uses visual models that encourage students to think about and solve problems in different ways using the properties of multiplication, particularly the distributive property.  All Teachley games are developed and supported by cognitive science research.

Lesson Plan Common Core State Standards Alignments

Students will:.

• Use effective math strategies based on mathematical properties to solve multiplication problems.
• Express their mathematical thinking using the SnapThought® tool.
• Computers or other devices with Internet access
• Interactive whiteboard

Preparation:

Preview and play Mt. Multiplis to plan how you will adapt it to your students’ needs.

For ideas on how to use SnapThought with this game, read Mt. Multiplis: SnapThought Prompts for more information and specific SnapThought prompts to provide students during game play.

• 1:1 with students and devices
• Two to three students sharing one device and swapping ideas and the device back and forth
• Station model where small groups rotate through using the devices
• The demo version that this lesson is based on includes two levels of the game.
• The first level features 5-factor problems, utilizing groups of 5.
• The second level features 4-factor problems with 4 groups of x. At this level, the game encourages students to think in terms of 2 groups of x and then double it.

Lesson Procedure:

• Play one of the recommended BrainPOP movies suggested in Preparation to build background or to make connections to prior knowledge.
• Write a simple 5-factor multiplication problem on the board, such as, “3 x 5 = _” and ask students to share how they would solve this problem. Although some may have just memorized the fact, encourage others to describe how they might solve it if they didn’t know the answer. Provide manipulatives and/or pencil and paper for students to show their strategies. After students share their ideas, model how to think think strategically about the problem by showing how 3 x 5 is the same as 2 groups of 5 plus 1 more group of 5 (10 + 5).
• Repeat step 2 but with a 4-factor problem, such as 4 x 6. Again, ask students to share how they solve the problem if they haven’t memorized the fact. Using the example from step 2, they might solve it by thinking of two groups of 6 plus another two groups (12 + 6 + 6). After students share their ideas, be sure to model the doubling strategy by showing how to double 2 groups of 6 (12 doubled) to get the answer 24.
• Click the bridge to plan it, and then again to build it.
• Drag the factor cards on the left onto the bridge area. The total is additive.
• Once you build a bridge, cross it by clicking the island you want to go to.
• Demonstrate how they can undo the last move by clicking the Undo button on the right. Or they can clear the whole bridge by clicking Reset.
• On Level 2, demonstrate how to use the Double It! button on the left side to double the planks to come up with the answer.
• Depending on your students’ needs, you can play several rounds of the game as a whole class, inviting volunteers to come up and try the different strategies.
• Now have students explore the game independently or in pairs, Allow 10-15 minutes for game play.
• If students have individual logins through My BrainPOP, encourage them to use the SnapThought® tool to take snapshots during game play, and reflect on their strategies. See Mt. Multiplis: SnapThought Prompts for suggested prompts.
• Circulate the room as students play, listening as they use or discuss their strategies, and provide help as needed.
• Bring students back to a whole class discussion and have volunteers share their strategies for game play. Ask if they used any strategies that were new for them. Ask what they liked about the strategies they used and what they didn’t like.
• Encourage students to revisit Mt. Multiplis throughout the school year to practice multiplication strategies. NOTE: The full version of the game is available at teachley.com .

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• Grade 3 Mathematics Module 1, Topic C, Lesson 10

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Distributive Property - Array Model

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Common Core Standards: 3.OA.1, 3.OA.5, 3.OA.3, 3.OA.4

New York State Common Core Math Grade 3, Module 1, Lesson 10

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NYS Math Module 1 Grade 3 Lesson 10 Concept Development

A guitar has 6 strings. How many strings are there on 3 guitars? Draw an array to represent the total number of guitar strings. 8 × 2 = _____

NYS Math Module 1 Grade 3 Lesson 10 Problem Set

• 7 × 3 = (5 × 3) + (2 × 3) = _____ (5 × 3) + (2 × 3) = 15 + __________ 15 + ______= __________

Is (3 × 4) + (2 × 4) = 12 + 8 = 20 the same as 5 × 4?

Joe planted 6 rows of carrots in his garden. There are 6 carrots in each row. How many carrots did he plant?

NYS Math Module 1 Grade 3 Lesson 10 Homework

• 6 × 3 = __________ 12 + ______ = ______ 6 × 3 = ______
• 8 × 2 = ___ ( × 2) = __ ( × 2) = ______ (4 × 2) + (4 × 2) = ______ + ______ ____ × 2 = ________
• Adriana is organizing her books on shelves. She puts 3 books in each row. a. Use the multiplication sentences on the right to draw arrays to show the books on Adriana’s top and bottom shelves. __________× 3 = 15 __________× 3 = 3

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