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McGraw Hill My Math Grade 3 Chapter 9 Lesson 4 Answer Key The Associative Property

All the solutions provided in  McGraw Hill My Math Grade 3 Answer Key PDF Chapter 9 Lesson 4 The Associative Property will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 3 Answer Key Chapter 9 Lesson 4 The Associative Property

The Associative Property of Multiplication states that the grouping of factors does not change the product.

Math in My World

Helpful Hint The Associative Property also allows you to group the easier factors.

Either way 2 × 3 × 4 = _______________. The _______________ Property shows that grouping does not change the product. Answer: The Associative Property Property shows that grouping does not change the product.

Explanation: For example, 7 ×(2 × 3) = (7 × 2) × 3 = 42. So, The associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product.

Guided Practice 

Use parentheses to group two factors. Then find each product.

Explanation: Given, 2 × 4 × 6 ( 2 × 4 ) ×6 Multiply the factors inside the parentheses first. 8 × 6  = 48 So, 2 × ( 4 × 6 ) = 48.

Explanation: Given, 2 × 4 × 3 ( 2 × 4 ) × 3 Multiply the factors inside the parentheses first. 8 × 3  = 24 So, 2 × ( 4 × 3 ) = 24.

Independent Practice

Explanation: Given,4 × 1 × 3 (4 × 1 ) × 3 Multiply the factors inside the parentheses first. 4 × 3  = 12 So, 4 × ( 1 × 3 ) = 12.

Explanation: Given, 2 × 3 × 3 ( 2 × 3 ) × 3 Multiply the factors inside the parentheses first. 6 × 3  = 18 So, 2 × ( 3 × 3 ) = 18.

Question 6. 6 × 2 × 2 = _____________ Answer: 6 × 2 × 2 = 24

Explanation: Given, 6 × 2 × 2 ( 6 × 2 ) × 2 Multiply the factors inside the parentheses first. 12 × 2  = 24 So, 6 × ( 2 × 2 ) = 24.

Question 7. 2 × 3 × 2 = ______________ Answer: 2 × 3 × 2 =12

Explanation: Given, 2 × 3 × 2 ( 2 × 3 ) × 2 Multiply the factors inside the parentheses first. 6 × 2  = 12 So, 2 × (3 × 2 ) = 12.

Algebra Find each missing factor.

Explanation: Given, figure value is 2 Then, (6 × 1) × 2 Multiply the factors inside the parentheses first. 6 × 2  = 12 So, 6 × ( 1 ×2 ) = 12.

Explanation: Given, figure value is 3 Then, ( 4 × 3 ) × 2 Multiply the factors inside the parentheses first. 12 × 3  = 24 So, 4 × ( 3 × 2 ) = 24.

Explanation: Given, figure value is 4 and 2 Then, ( 4 × 2 ) × 5 Multiply the factors inside the parentheses first. 8 × 5  = 40 So, 4 × ( 2 × 5 ) = 40.

Explanation: Given, figure value is 2 Then, ( 6 × 2 ) × 3 Multiply the factors inside the parentheses first. 12 × 3  = 36 So, 6 × ( 2 × 3 ) = 36.

Explanation: Given, figure value is 3 and 4 Then, ( 3 × 3 ) ×4 Multiply the factors inside the parentheses first. 9 × 4  = 36 So, 3 × ( 3 × 4 ) = 36.

Explanation: Given, figure value is 2 and 3 Then, ( 5 × 2 ) ×3 Multiply the factors inside the parentheses first. 10 × 3  = 30 So, 5 × ( 2 × 3 ) = 36.

Problem Solving

Question 18. Mathematical PRACTICE Make a Plan There are 5 apples. Troy cuts each apple into 2 pieces. Beth cuts each piece into 4 slices. What is the total number of apple slices? Answer: There are 40 apple slices

Explanation: Given, There are 5 apples. Troy cuts each apple into 2 pieces. Beth cuts each piece into 4 slices. That makes, 5 × ( 2 × 4 ) Multiply the factors inside the parentheses first. 10 × 4  = 40 So, 5 × ( 2 × 4 ) = 40 Hence, There are 40 apple slices.

Question 19. Troy and Beth each cut 2 bananas into 4 pieces. What is the total number of banana pieces? Answer: There are 16 banana pieces

Explanation: Given, Troy and Beth each cut 2 bananas into 4 pieces. That makes, 2 × 2 × 4 Multiply the factors inside the parentheses first. 4 ×4  = 16 So, 2 × ( 2 × 4 ) = 16.

Question 20. A clerk unpacked 2 boxes of nails. Each box held 4 cartons with 10 packages of nails. How many packages of nails did the clerk unpack? Answer: Clerk unpacked 80 packages of nails

Explanation: Given, A clerk unpacked 2 boxes of nails. Each box held 4 cartons with 10 packages of nails. That makes, 2 × ( 4 × 10 ) Multiply the factors inside the parentheses first. 8 × 10  = 80 So, 2 × ( 4 × 10 ) = 80. Hence, Clerk unpacked 80 packages of nails.

HOT Problems

Question 21. Mathematical PRACTICE Find the Error From the following, circle the number sentence that is not true. Explain. (2 × 3) × 3 = 2 × (3 × 3) 3 × (1 × 5) = (3 × 1) × 5 4 × (4 × 2) = (3 × 4) × 4 6 × (4 × 2) = (6 × 4) × 2 Answer: 4 × (4 × 2) = (3 × 4) × 4 is not true

Explanation: From the given question, 4 × (4 × 2) = (3 × 4) × 4 is not true, because Here, not only the factors are grouped together but also the numbers are changed as 2 as 3 So, 4 × (4 × 2) = (3 × 4) × 4 is not true

Question 22. Building on the Essential Question Explain why the grouping of the factors does not matter when finding (3 × 4) × 2. Answer: 24, Either way the result of the product is same.

Explanation: Given, (3 × 4) × 2. (3 × 4) × 2. Multiply the factors inside the parentheses first. 12 × 2 = 24 or 3 × (4 × 2) Group the same factors another way. 3 × 8 = 24 Either way the result of the product is same.

McGraw Hill My Math Grade 3 Chapter 9 Lesson 4 My Homework Answer Key

Question 1. 2 × 3 × 6 = ______________ Answer: 2 × 3 × 6 = 36

Explanation: Given, 2 × 3 × 6 ( 2 × 3 ) ×6 Multiply the factors inside the parentheses first. 6 × 6  = 36 So, 2 × ( 3 × 6 ) = 36.

Question 2. 5 × 2 × 2 = ________________ Answer: 5 × 2 × 2 = 20

Explanation: Given, 5 × 2 × 2 ( 5 × 2 ) × 2 Multiply the factors inside the parentheses first. 10 × 2  = 20 So, 5 × ( 2 × 2 ) = 20.

Question 7. Mathematical PRACTICE Use Number Sense Mariette bought 4 packs of sparkling water. There were 6 bottles in each pack. If each bottle cost $2, how much did Mariette spend on sparkling water? Answer: Mariette spend $48 on sparkling water.

Explanation: Given, Mariette bought 4 packs of sparkling water. There were 6 bottles in each pack. If each bottle cost $2, that makes, 4 × ( 6 × 2 ) Multiply the factors inside the parentheses first. 4 × 12  = 48 So, ( 4 × 6 ) × 2 =48. Hence, Mariette spend $48 on sparkling water.

Question 8. Jamal and Brianna each bought 3 oranges. They sliced each orange into 6 pieces. How many orange slices did Jamal and Brianna have altogether? Answer: Jamal and Brianna have altogether 36 orange slices.

Explanation: Given, Jamal and Brianna each bought 3 oranges. They sliced each orange into 6 pieces. That makes, 2 × ( 3 × 6 ) Multiply the factors inside the parentheses first. 6 × 6  = 36 So, ( 2 × 3 ) × 6 = 36. Hence, Jamal and Brianna have altogether 36 orange slices.

Question 9. Mr. and Mrs. Perry packed their lunch 5 days in a row. Each of them packed 3 oatmeal cookies for dessert every day. What is the total number of cookies they packed for lunch that week? Answer: The total number of cookies they packed for lunch that week is 30.

Explanation: Given, Mr. and Mrs. Perry packed their lunch 5 days in a row. Each of them packed 3 oatmeal cookies for dessert every day. That makes, 2 × ( 5 × 3 ) Multiply the factors inside the parentheses first. 2 × 15  = 30 So, ( 2 × 5 ) × 3 = 30. Hence, The total number of cookies they packed for lunch that week is 30.

Vocabulary Check

Question 10. Write a definition for the Associative Property of Multiplication. Answer: The associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product.

Explanation: For Example, ( 5 × 4 ) × 2 or  5 × ( 4 × 2 ).

Test Practice

Explanation: Given, (3 × 3) × 7 Multiply the factors inside the parentheses first. 9 × 7  = 63 So, 3 × ( 3 × 7 ) = 63. Hence, option D is correct.

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Associative property

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Grouping in multiplication

The associative property of multiplication tells us that the way in which factors are grouped in a multiplication equation does not change the answer.  3 x (4 x 5) is the same as (3 x 4) x 5 . 

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Lesson Plan

Associative property of multiplication, view aligned standards, learning objectives.

Students will be able to apply the associative property to multiply single-digit factors.

Introduction

• Ask students what the word associate means. Use it in a sentence. For example, "I associate with Matthew during recess," or "We often associate the color blue with sadness." Give students a moment to discuss the word with peers.
• Call on a few students to give a definition for the word associate and then develop a meaning with the class (i.e. joined or connected)
• On the board, draw a quick picture to illustrate the word (i.e. draw two people holding hands)
• Explain: Today we are going to explore the associative property of multiplication .
• Write the name of the property on the board and underline the word associative .
• Have students share their ideas with a supportive partner or a peer with the same home language (L1), if possible.
• Allow students to share their ideas in their L1.

Intermediate

• Ask students to restate the definition of associate in their own words to a partner and then to the whole group.

Associative Property

With our Associative Property lesson plan, students learn what the associative property is in mathematics for both addition and multiplication and how to use it. Students practice identifying problems that use the associative property as a part of this lesson.

Included with this lesson are some adjustments or additions that you can make if you’d like, found in the “Options for Lesson” section of the Classroom Procedure page. One of the optional additions to this lesson is to have students physically group themselves to demonstrate the property in front of the class.

Description

Additional information, what our associative property lesson plan includes.

Lesson Objectives and Overview: Associative Property explains the associative property, discusses both addition and multiplication, and provides several examples for teachers to do with students. Students learn that associative property allows you to add or multiply regardless of how the numbers are grouped. At the end of the lesson, students will be able to understand the associative property of addition and multiplication. This lesson is for students in 2nd grade, 3rd grade, and 4th grade.

Classroom Procedure

Every lesson plan provides you with a classroom procedure page that outlines a step-by-step guide to follow. You do not have to follow the guide exactly. The guide helps you organize the lesson and details when to hand out worksheets. It also lists information in the blue box that you might find useful. You will find the lesson objectives, state standards, and number of class sessions the lesson should take to complete in this area. In addition, it describes the supplies you will need as well as what and how you need to prepare beforehand. The supplies you will need for this lesson include scissors, glue, and extra paper.

Options for Lesson

Included with this lesson is an “Options for Lesson” section that lists a number of suggestions for activities to add to the lesson or substitutions for the ones already in the lesson. One optional addition to this lesson is to h ave students begin to incorporate the commutative and associative property. If you have more advanced students, you can begin to incorporate more than three numbers,. Finally, you could have students physically group themselves to demonstrate the property in front of the class.

Teacher Notes

The teacher notes page includes lines that you can use to add your own notes as you’re preparing for this lesson.

ASSOCIATIVE PROPERTY LESSON PLAN CONTENT PAGES

The Associative Property lesson plan includes one content page. The associative property shows us that we can add or multiply numbers regardless of the way the numbers are grouped. We add or multiply grouped numbers (numbers in parenthesis) first. When we add three or more numbers, the sum is the same no matter which numbers you add or multiply first. It doesn’t matter which order you add or multiply.

The lesson includes an example of the associative property of addition. (4 + 5) + 2 is the same as 4 + (5 + 2). 9 + 2 = 4+ 7. 11 = 11.

It also includes an example of the associative property of multiplication. (3 x 4) x 1 is the same as 3 x (4 x 1). 12 x 1 = 3 x 4. 12 = 12.

ASSOCIATIVE PROPERTY LESSON PLAN WORKSHEETS

The Associative Property lesson plan includes four worksheets: an activity worksheet, a practice worksheet, a homework assignment, and a quiz. You can refer to the guide on the classroom procedure page to determine when to hand out each worksheet.

MATCHING ACTIVITY WORKSHEET

The activity worksheet asks students to cut out the cards on the worksheet, match them, and glue them on a separate page.

EQUAL OR NOT PRACTICE WORKSHEET

For the practice worksheet, students will use the associative property to determine if the equations are equal or not equal.

ASSOCIATIVE PROPERTY HOMEWORK ASSIGNMENT

The homework assignment asks students to fill in the missing numbers in different equations using the associative property.

This lesson includes a quiz that you can use to test students’ understanding of the lesson material. For the quiz, students will draw a line between the matching equations.

Worksheet Answer Keys

This lesson plan includes answer keys for the practice worksheet, the homework assignment, and the quiz.  If you choose to administer the lesson pages to your students via PDF, you will need to save a new file that omits these pages. Otherwise, you can simply print out the applicable pages and keep these as reference for yourself when grading assignments.

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Associative Property Lesson Plan: Expressions and Equations

*Click to open and customize your own copy of the  Associative Property Lesson Plan .

This lesson accompanies the BrainPOP topic Associative Property , and supports the standard of applying the properties of operations to generate equivalent expressions. Students demonstrate understanding through a variety of projects.

Step 1: ACTIVATE PRIOR KNOWLEDGE

Display this expression: 

Ask students:

• How could you rewrite this expression to help you to solve it? 

Step 2: BUILD KNOWLEDGE

• Read the description on the Associative Property topic page .
• Play the Movie , pausing to check for understanding. 
• Have students read one of the following Related Reading articles: “Quirky Stuff” or “Plus/Minus.” Partner them with someone who read a different article to share what they learned with each other.

Step 3: APPLY and ASSESS 

Students take the Associative Property Quiz , applying essential literacy skills while demonstrating what they learned about this topic.

Step 4: DEEPEN and EXTEND

Students express what they learned about the associative property while practicing essential literacy skills with one or more of the following activities. Differentiate by assigning ones that meet individual student needs.

• Make-a-Movie : Produce a tutorial explaining how to use the associative property to solve a multiplication problem. 
• Make-a-Map : Create a concept map that uses the associative property to identify expressions equivalent to 32 x 15.
• Creative Coding : Code a math problem that challenges the solver to use the associative property. 

More to Explore 

Monster School Bus :  Combine numbers in different ways in this interactive game. 

Related BrainPOP Topics : Deepen understanding of the properties of operations with these topics: Commutative Property and Distributive Property .

Teacher Support Resources:

• Pause Point Overview : Video tutorial showing how Pause Points actively engage students to stop, think, and express ideas.  
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• Learning Activities Support : Resources for best practices using BrainPOP.

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Understanding multiplication

Associative property

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

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

What is the associative property?

The associative property , or the associative law in maths, says that when you add or multiply numbers, the grouping of numbers can be different and the correct answer will still be the same.

It can also be referred to as the associative property of addition and the associative property of multiplication.

For example,

When adding 5 + 6 + 1, you can group the numbers in different ways:

Notice that even with different groupings, the sum of the numbers is the same.

This is also true when multiplying a set of numbers.

When multiplying 2 \times 5 \times 3, you can group the numbers in different ways:

Notice that even with different groupings, the product of the numbers is the same.

The associative property can be used to find friendly numbers when solving. Friendly numbers are numbers that are easy to add or multiply mentally – like multiples of 10.

The associative property lets us regroup and create friendlier numbers. 43 + 60 is easier to solve mentally than 44 + 59.

The associative property lets us regroup and create friendlier numbers. 7 \times 30 is easier to solve mentally than (7 \times 5) \times 6=35 \times 6.

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 associative property

In order to use the associative property:

Check to see that the operation is addition or multiplication.

Change the grouping of the numbers and solve.

[FREE] Associative Property Worksheet (Grade 3)

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

Associative property examples

Example 1: simple associative property with addition.

Use the associative property to solve 19 + 4 + 26.

All the numbers are being added, so the associative property can be used.

2 Change the grouping of the numbers and solve.

Example 2: simple associative property with multiplication

Use the associative property to solve 7 \times 4 \times 5.

All the numbers are being multiplied, so the associative property can be used.

Example 3: associative property with multiplication

Use the associative property to solve 2 \times 4 \times 5 \times 2.

Example 4: associative property – addition with friendly numbers

Use the associative property with friendly numbers to solve 22 + 49.

Example 5: associative property – addition with friendly numbers

Use the associative property with friendly numbers to solve 78 + 15.

Example 6: associative property – multiplication with friendly numbers

Use the associative property with friendly numbers to solve 5 \times 12.

Teaching tips for the associative property

• Intentionally choose practice problems that lend themselves to being solved with the associative property, as it is not always necessary or useful in all solving situations.
• Instead of just giving students the associative property definition, draw attention to examples of the associative 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 regrouping the numbers when adding or multiplying does not change the sum or product. This could include students sharing their thinking or critiquing the thinking of others.

Easy mistakes to make

• Using the associative property for subtraction or division The associative property only works when grouping the numbers differently doesn’t change the answer. This does not work for subtraction or division. For example, \begin{aligned} & 10-5-2 \hspace{1cm} 10-(5-2)\\ & =5-2 \hspace{1.4cm} =10-3 \\ & =3 \hspace{1.95cm} =7 \\ \end{aligned} Changing the grouping of the numbers, changes the answer.
• Thinking there is only one way to use the associative property to solve with friendly numbers Often, there is more than one way to use the associative property when solving. For example, \begin{aligned} & 45+48 \hspace{2.8cm} 45+48 \\ & =(43+2)+48 \hspace{1.7cm} =45+(5+43) \\ & =43+(2+48) \hspace{0.6cm} \text{OR} \hspace{0.6cm} =(45+5)+43 \\ & =43+50 \hspace{2.5cm} =50+43 \\ & =93 \hspace{3.23cm} =93 \\ \end{aligned}
• Confusing the order of operations Equations are always solved starting from the left-hand side and moving to the right-hand side. While students do not need to be introduced formally to the order of all operations, it is important that they read and understand equations in this way. Otherwise, the different groupings or use of parentheses may not mean anything to them.

Related properties of equality lessons

This associative property topic guide is part of our series on properties of equality. You may find it helpful to start with the main properties of equality topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

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

Practice associative property questions

1. Which of the following equations shows an alternative way to solve 5 + 11 + 9 using the associative property?

2. Which of the following equations showsan alternative way to solve 5 \times 5 \times 8 using the associative property?

3. Which equation shows the associative property?

When solving 6 + 7 + 13, the associative property says you can group the numbers differently and still get the same answer, so (6 + 7) + 13 = 6 + (7 + 13).

4. Which equation shows the associative property?

When solving 7 \times 5 \times 10 the associative property says you can group the numbers differently and still get the same answer, so (7 \times 5) \times 10=7 \times(5 \times 10).

5. Which shows how to use the associative property AND friendly numbers to solve 26 + 45?

Friendly numbers are numbers that are easy to add mentally – like multiples of 10.

Change the grouping of the numbers and use friendly numbers to solve.

6. Which shows how to use the associative property AND friendly numbers to solve 8 \times 5?

Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

Associative property FAQs

Yes, the associative property can be used with fractions, decimals, negative numbers and rational numbers, as long as they are all being added or multiplied.

The associative property changes the grouping of the numbers but not their location within the equation. The commutative property changes the order, or the location, of numbers within an equation. They both affect the order of operations , which can make them easy to confuse.

The next lessons are

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

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Home > Math Worksheets > Algebra Worksheets > Associative Property

This property of operations comes into play when ever you have a string of numbers that are either being multiplied or added. It tells use that the order in which you group or approach the problem does not really matter. For example: 3 + 6 +8 +2 has the same result as if you added the numbers in this form: 6 + 3 +2 + 8. The same applies to multiplication. If we were to multiply 7 with 5 and 4, we could arrange it as 7 x 5 x 4 or 5 x 4 x 7. The final product would be the same regardless of how we arranged it. This solid and detailed section of worksheets will help you to learn how to use the associative property to regroup algebraic expressions, both in adding and in multiplying. This set of worksheets introduces your students to the concept of the associative property, provides examples, short practice sets, longer sets of questions, and quizzes. The associative property helps students transition into early algebra concepts and starts them thinking up that alley. The following worksheets will help your students to understand how to manipulate equations using the associative property.

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Print associative property worksheets, click the buttons to print each worksheet and associated answer key., associative property lesson.

Learn how to rewrite equations and still have the outcome be the same. This makes for a nice introduction to algebra. Example: (4 + 5) + 3

Worksheet 1

Rewrite these equations to help make them more manageable to work with. This is a skill that will go a long way for you. Example: a + (m + d)

Worksheet 2

We focus on a mix of numbers and variables used on this sheet. Example: 8 + (2 + 5)

Review Sheet

Practice rewriting the following 6 equations by using what you have learned so far. Example: w(gs)

Rewrite these 10 equations that need your love and care to organize. Example: T x (a x h)

Do Now Worksheet

Complete these 3 problems dealing with associative properties and put your answers in the "My Answer" box. This can be done with the whole class as a reminder or quick refresher.

Addition and Multiplication Lesson

This worksheet explains how to rewrite equations using the associative properties of addition and multiplication. A sample problem is solved.

Lesson and Practice

The associative property of addition says that when we add more than two numbers the grouping of the addends does not change the sum. For multiplication it says that when we multiply more than two numbers the grouping of the factors does not change the product.

Rewrite Using Associative Property Worksheet

Students will rewrite equations using the skills that they have learned. Ten problems are provided.

We will get even more work on these skills. Ten problems are provided.

Skill Drill

A great way to get these skills amped up as an entire class. Three problems are provided.

What is the Associative Property?

The associative property, in mathematics, states that the sum or product of three or more numbers will always be the same regardless of their grouping. In mathematics, grouping refers to using parentheses () or brackets []. This property can only be used in addition or multiplication, not subtraction or division.

The formula that use differs based on the operation that you are using:

For Addition: (a + b) + c = a + (b + c)

For Multiplication: (a x b) x c = a x (b x c)

The formula shows the outcome will remain the same regardless of the grouping.

Associative Property of Addition

The sum of three or more numbers also follows the associative property. The sum of these numbers will remain the same no matter how the numbers are arranged. Let’s see some examples so we can better understand the use of this when working with addition:

Example: What if we let a=2, b=5, and c=8. The equation then becomes (2+5) + 8 = 2 + (5+8). Solving the right-hand side will get us 2 + 13 = 15. Solving the left-hand side will get us 7 + 8 = 15. Hence, right hand side = left hand side.

Associative Property of Multiplication

The product of three or more numbers also follows the associative property. The product of these numbers will remain the same no matter how the numbers are arranged. Let’s see some examples to better understand how to use this with multiplication operations:

Example: In the formula for the associative property of multiplication, let a=5, b=3, and c=7. The equation then becomes (5 x 3) x 7 = 5 x (3 x 7). Solving the right-hand side will get us 5 x 21 = 105. Solving the left-hand side will get us 15 x 7 = 105. Hence, right-hand side = left-hand side.

Associative Property of Rational Numbers

The sum or product of three or more rational numbers also follows the associative property. The sum or product of these numbers remains the same no matter how the numbers are arranged. Let’s see some examples to better understand how this works.

Example 1: In the formula of associative property of rational numbers in addition, let a/b=1/2, c/d= 3/4, e/f= 5/6. The equation then becomes (1/2 + 3/4) + 5/6 = 1/2 + (3/4 + 5/6). Solving the right-hand side will get us 1/2 + 19/12 = 25/12. Solving the left-hand side will get us 5/4 + 5/6 = 25/12. Hence, right hand side = left hand side.

Example 2: In the formula of associative property of rational numbers in multiplication, let a/b=1/2, c/d= 3/4, e/f= 5/6. The equation then becomes (1/2 x 3/4) x 5/6 = 1/2 x (3/4 x 5/6). Solving the right-hand side will get us 1/2 x 3/8 = 3/16. Solving the left-hand side will get us 3/8 x 5/6 = 3/16. Hence, right hand side = left hand side.

Note that this does not apply to division or subtraction. As we move into algebra we run into unknown variables, but the operations are the same and this property has the same place. If add or multiply variables, it does not matter how we assemble them. For example, a + b + c = c + a + b. We make heavy use of this property to help us reorganize equations into forms that are easier to work with.

How to Rewrite Equations Using the Associative Property

We use parentheses to group a different pair of numbers together. Meanwhile, we use this property when we are rewriting any number in an expression. Associative Property - With respect to addition: If you are changing the order, the answer will not change. If you have real numbers a, b, c, then, the equation will be like; ( a + b ) + c = a + ( b + c). With respect to multiplication: The result will also be the same while changing the order of the multiplicative numbers. We will suppose the same thing while finding the following expression that is; 5 × 1/3 × 3. Here, we will get the same result by changing the order of the expression. For example; (5 × 1/3) × 3 = (5/3) × 3 = 5 | 5 × ( 1/3 × 3 ) = 5 × 1 = 5. If you have a, b, c as a real number while multiplying the expressions, the equation will be; ( a × b ) × c = a × ( b × c). Rewriting the Equations - With respect to addition: (3+ 0.6) + 0.4. Change the grouping or order -> 3 + ( 0.6 + 0.4 ) = 1. With respect to multiplication: ( - 4 × 2/5 ) × 15 = .Change the order or grouping -> - 4 × ( 2/5 × 15) = 6.

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9.3.1: Associative, Commutative, and Distributive Properties

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Learning Objectives

• Identify and use the commutative properties for addition and multiplication.
• Identify and use the associative properties for addition and multiplication.
• Identify and use the distributive property.

Introduction

There are many times in algebra when you need to simplify an expression. The properties of real numbers provide tools to help you take a complicated expression and simplify it.

The associative, commutative, and distributive properties of algebra are the properties most often used to simplify algebraic expressions. You will want to have a good understanding of these properties to make the problems in algebra easier to solve.

The Commutative Properties of Addition and Multiplication

You may encounter daily routines in which the order of tasks can be switched without changing the outcome. For example, think of pouring a cup of coffee in the morning. You would end up with the same tasty cup of coffee whether you added the ingredients in either of the following ways:

• Pour 12 ounces of coffee into mug, then add splash of milk.
• Add a splash of milk to mug, then add 12 ounces of coffee.

The order that you add ingredients does not matter. In the same way, it does not matter whether you put on your left shoe or right shoe first before heading out to work. As long as you are wearing both shoes when you leave your house, you are on the right track!

In mathematics, we say that these situations are commutative—the outcome will be the same (the coffee is prepared to your liking; you leave the house with both shoes on) no matter the order in which the tasks are done.

Likewise, the commutative property of addition states that when two numbers are being added, their order can be changed without affecting the sum. For example, \(\ 30+25\) has the same sum as \(\ 25+30\).

\(\ 30+25=55\)

\(\ 25+30=55\)

Multiplication behaves in a similar way. The commutative property of multiplication states that when two numbers are being multiplied, their order can be changed without affecting the product. For example, \(\ 7 \cdot 12\) has the same product as \(\ 12 \cdot 7\).

\(\ 7 \cdot 12=84\)

\(\ 12 \cdot 7=84\)

These properties apply to all real numbers. Let’s take a look at a few addition examples.

Commutative Property of Addition

For any real numbers \(\ a\) and \(\ b\), \(\ a+b=b+a\).

Subtraction is not commutative. For example, \(\ 4-7\) does not have the same difference as \(\ 7-4\). The \(\ -\) sign here means subtraction.

However, recall that \(\ 4-7\) can be rewritten as \(\ 4+(-7)\), since subtracting a number is the same as adding its opposite. Applying the commutative property for addition here, you can say that \(\ 4+(-7)\) is the same as \(\ (-7)+4\). Notice how this expression is very different than \(\ 7-4\).

Now look at some multiplication examples.

Commutative Property of Multiplication

For any real numbers \(\ a\) and \(\ b\), \(\ a \cdot b=b \cdot a\).

Order does not matter as long as the two quantities are being multiplied together. This property works for real numbers and for variables that represent real numbers.

Just as subtraction is not commutative, neither is division commutative. \(\ 4 \div 2\) does not have the same quotient as \(\ 2 \div 4\).

Write the expression \(\ (-15.5)+35.5\) in a different way, using the commutative property of addition, and show that both expressions result in the same answer.

\(\ (-15.5)+35.5=20\) and \(\ 35.5+(-15.5)=20\)

Rewrite \(\ 52 \cdot y\) in a different way, using the commutative property of multiplication. Note that \(\ y\) represents a real number.

• \(\ 5 y \cdot 2\)
• \(\ 26 \cdot 2 \cdot y\)
• \(\ y \cdot 52\)
• Incorrect. You cannot switch one digit from 52 and attach it to the variable \(\ y\). The correct answer is \(\ y \cdot 52\).
• Incorrect. This is another way to rewrite \(\ 52 \cdot y\), but the commutative property has not been used. The correct answer is \(\ y \cdot 52\).
• Incorrect. You do not need to factor 52 into \(\ 26 \cdot 2\). The correct answer is \(\ y \cdot 52\).
• Correct. The order of factors is reversed.

The Associative Properties of Addition and Multiplication

The associative property of addition states that numbers in an addition expression can be grouped in different ways without changing the sum. You can remember the meaning of the associative property by remembering that when you associate with family members, friends, and co-workers, you end up forming groups with them.

Below are two ways of simplifying the same addition problem. In the first example, 4 is grouped with 5, and \(\ 4+5=9\).

\(\ 4+5+6=9+6=15\)

Here, the same problem is worked by grouping 5 and 6 first, \(\ 5+6=11\).

\(\ 4+5+6=4+11=15\)

In both cases, the sum is the same. This illustrates that changing the grouping of numbers when adding yields the same sum.

Mathematicians often use parentheses to indicate which operation should be done first in an algebraic equation. The addition problems from above are rewritten here, this time using parentheses to indicate the associative grouping.

\(\ (4+5)+6=9+6=15\)

\(\ 4+(5+6)=4+11=15\)

It is clear that the parentheses do not affect the sum; the sum is the same regardless of where the parentheses are placed.

Associative Property of Addition

For any real numbers \(\ a\), \(\ b\), and \(\ c\),

\(\ (a+b)+c=a+(b+c)\).

The example below shows how the associative property can be used to simplify expressions with real numbers.

Rewrite \(\ 7+2+8.5-3.5\) in two different ways using the associative property of addition. Show that the expressions yield the same answer.

\(\ (7+2)+8.5-3.5=14\) and \(\ 7+2+(8.5+(-3.5))=14\)

Multiplication has an associative property that works exactly the same as the one for addition. The associative property of multiplication states that numbers in a multiplication expression can be regrouped using parentheses. For example, the expression below can be rewritten in two different ways using the associative property.

Original expression: \(\ -\frac{5}{2} \cdot 6 \cdot 4\)

Expression 1: \(\ \left(-\frac{5}{2} \cdot 6\right) \cdot 4=\left(-\frac{30}{2}\right) \cdot 4=-15 \cdot 4=-60\)

Expression 2: \(\ -\frac{5}{2} \cdot(6 \cdot 4)=-\frac{5}{2} \cdot 24=-\frac{120}{2}=-60\)

The parentheses do not affect the product. The product is the same regardless of where the parentheses are.

Associative Property of Multiplication

For any real numbers \(\ a\), \(\ b\), and \(\ c\), \(\ (a \cdot b) \cdot c=a \cdot(b \cdot c)\).

Rewrite \(\ \frac{1}{2} \cdot\left(\frac{5}{6} \cdot 6\right)\) using only the associative property.

• \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\)
• \(\ \left(\frac{5}{6} \cdot 6\right) \cdot \frac{1}{2}\)
• \(\ 6 \cdot\left(\frac{5}{6} \cdot \frac{1}{2}\right)\)
• \(\ \frac{1}{2} \cdot 5\)
• Correct. Here, the numbers are regrouped. Now \(\ \frac{1}{2}\) and \(\ \frac{5}{6}\) are grouped in parentheses instead of \(\ \frac{5}{6}\) and \(\ 6\).
• Incorrect. The order of numbers is not changed when you are rewriting the expression using the associative property of multiplication. How they are grouped should change. The correct answer is \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\).
• Incorrect. The order of numbers is not changed when you are rewriting the expression using the associative property of multiplication. Only how they are grouped should change. The correct answer is \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\).
• Incorrect. Multiplying within the parentheses is not an application of the property. The correct answer is \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\).

Using the Associative and Commutative Properties

You will find that the associative and commutative properties are helpful tools in algebra, especially when you evaluate expressions. Using the commutative and associative properties, you can reorder terms in an expression so that compatible numbers are next to each other and grouped together. Compatible numbers are numbers that are easy for you to compute, such as \(\ 5+5\), or \(\ 3 \cdot 10\), or \(\ 12-2\), or \(\ 100 \div 20\). (The main criteria for compatible numbers is that they “work well” together.) The two examples below show how this is done.

Evaluate the expression \(\ 4 \cdot(x \cdot 27)\) when \(\ x=-\frac{3}{4}\).

\(\ 4 \cdot(x \cdot 27)=-81\) when \(\ x=\left(-\frac{3}{4}\right)\)

Simplify: \(\ 4+12+3+4-8\)

\(\ 4+12+3+4-8=15\)

Simplify the expression: \(\ -5+25-15+2+8\)

• Incorrect. When you use the commutative property to rearrange the addends, make sure that negative addends carry their negative signs. The correct answer is 15.
• Correct. Use the commutative property to rearrange the expression so that compatible numbers are next to each other, and then use the associative property to group them.
• Incorrect. Check your addition and subtraction, and think about the order in which you are adding these numbers. Use the commutative property to rearrange the addends so that compatible numbers are next to each other. The correct answer is 15.
• Incorrect. It looks like you ignored the negative signs here. When you use the commutative property to rearrange the addends, make sure that negative addends carry their negative signs. The correct answer is 15.

The Distributive Property

The distributive property of multiplication is a very useful property that lets you rewrite expressions in which you are multiplying a number by a sum or difference. The property states that the product of a sum or difference, such as \(\ 6(5-2)\), is equal to the sum or difference of products, in this case, \(\ 6(5)-6(2)\).

\(\ \begin{array}{l} 6(5-2)=6(3)=18 \\ 6(5)-6(2)=30-12=18 \end{array}\)

The distributive property of multiplication can be used when you multiply a number by a sum. For example, suppose you want to multiply 3 by the sum of \(\ 10+2\).

\(\ 3(10+2)=?\)

According to this property, you can add the numbers 10 and 2 first and then multiply by 3, as shown here: \(\ 3(10+2)=3(12)=36\). Alternatively, you can first multiply each addend by the 3 (this is called distributing the 3), and then you can add the products. This process is shown here.

\(\ \begin{array}{l} 3(10+2)=3(12)=36 \\ 3(10)+3(2)=30+6=36 \end{array}\)

The products are the same.

Since multiplication is commutative, you can use the distributive property regardless of the order of the factors.

The Distributive Properties

For any real numbers \(\ a\), \(\ b\), and \(\ c\):

Multiplication distributes over addition:

\(\ a(b+c)=a b+a c\)

Multiplication distributes over subtraction:

\(\ a(b-c)=a b-a c\)

Rewrite the expression \(\ 10(9-6)\) using the distributive property.

• \(\ 10(6)-10(9)\)
• \(\ 10(3)\)
• \(\ 10(6-9)\)
• \(\ 10(9)-10(6)\)
• Incorrect. Since subtraction isn’t commutative, you can’t change the order. The correct answer is \(\ 10(9)-10(6)\).
• Incorrect. This is a correct way to find the answer. But the question asked you to rewrite the problem using the distributive property. The correct answer is \(\ 10(9)-10(6)\)
• Incorrect. You changed the order of the 6 and the 9. Note that subtraction is not commutative and you did not use the distributive property. The correct answer is \(\ 10(9)-10(6)\).
• Correct. The 10 is correctly distributed so that it is used to multiply the 9 and the 6 separately.

Distributing with Variables

As long as variables represent real numbers, the distributive property can be used with variables. The distributive property is important in algebra, and you will often see expressions like this: \(\ 3(x-5)\). If you are asked to expand this expression, you can apply the distributive property just as you would if you were working with integers.

Remember, when you multiply a number and a variable, you can just write them side by side to express the multiplied quantity. So, the expression “three times the variable \(\ x\)” can be written in a number of ways: \(\ 3 x\), \(\ 3(x)\), or \(\ 3 \cdot x\).

Use the distributive property to expand the expression \(\ 9(4+x)\).

\(\ 9(4+x)=36+9 x\)

Use the distributive property to evaluate the expression \(\ 5(2 x-3)\) when \(\ x=2\).

When \(\ x=2,5(2 x-3)=5\).

In the example above, what do you think would happen if you substituted \(\ x=2\) before distributing the 5? Would you get the same answer of 5? The example below shows what would happen.

Combining Like Terms

The distributive property can also help you understand a fundamental idea in algebra: that quantities such as \(\ 3x\) and \(\ 12x\) can be added and subtracted in the same way as the numbers 3 and 12. Let’s look at one example and see how it can be done.

Add: \(\ 3 x+12 x\)

\(\ 3 x+12 x=15 x\)

Do you see what happened? By thinking of the \(\ x\) as a distributed quantity, you can see that \(\ 3x+12x=15x\). (If you’re not sure about this, try substituting any number for in this expression…you will find that it holds true!)

Groups of terms that consist of a coefficient multiplied by the same variable are called “like terms”. The table below shows some different groups of like terms:

Whenever you see like terms in an algebraic expression or equation, you can add or subtract them just like you would add or subtract real numbers. So, for example,

\(\ 10 y+12 y=22 y\), and \(\ 8 x-3 x-2 x=3 x\).

Be careful not to combine terms that do not have the same variable: \(\ 4 x+2 y\) is not \(\ 6 x y\)!

Simplify: \(\ 10 y+5 y+9 x-6 x-x\).

\(\ 10 y+5 y+9 x-6 x-x=15 y+2 x\)

Simplify: \(\ 12 x-x+2 x-8 x\).

• Incorrect. It looks like you added all of the terms. Notice that \(\ -x\) and \(\ -8 x\) are negative. The correct answer is \(\ 5 x\).
• Incorrect. You combined the integers correctly, but remember to include the variable too! The correct answer is \(\ 5x\).
• Correct. When you combine these like terms, you end up with a sum of \(\ 5x\)
• Incorrect. It looks like you subtracted all of the terms from \(\ 12x\). Notice that \(\ -x\) and \(\ -8 x\) are negative, but that \(\ 2 x\) is positive. The correct answer is \(\ 5 x\).

The commutative, associative, and distributive properties help you rewrite a complicated algebraic expression into one that is easier to deal with. When you rewrite an expression by a commutative property, you change the order of the numbers being added or multiplied. When you rewrite an expression using an associative property, you group a different pair of numbers together using parentheses. You can use the commutative and associative properties to regroup and reorder any number in an expression as long as the expression is made up entirely of addends or factors (and not a combination of them). The distributive property can be used to rewrite expressions for a variety of purposes. When you are multiplying a number by a sum, you can add and then multiply. You can also multiply each addend first and then add the products together. The same principle applies if you are multiplying a number by a difference.

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Division and the Associative Property

Related Topics: Lesson Plans and Worksheets for Grade 4 Lesson Plans and Worksheets for all Grades More Lessons for Grade 4 Common Core For Grade 4

Examples, solutions, and videos to help Grade 4 students learn how to use division and the associative property to test for factors and observe patterns.

Common Core Standards: 4.OA.4

New York State Common Core Math Grade 4, Module 3, Lesson 23

Worksheets for Grade 4

NYS Math Module 3 Grade 4 Lesson 23 Concept Development

Problem 1: Use division to find factors of larger numbers. Find the unknown factor: 28 = 7 × ___. Problem 2: Use the associative property to find additional factors of larger numbers. Problem 3: Use division or the associative property to find factors of larger numbers.

NYS Math Module 3 Grade 4 Lesson 23 Problem Set

• Explain your thinking or use division to answer the following. a. Is 2 a factor of 84? b. Is 2 a factor of 83?
• Use the associative property to find more factors of 24 and 36. a. 24 = 12 × 2 = ( ___ × 3) × 2 = ___ × (3 × 2) = ___ × 6 = ___ b. 36 = ____ × 4 = ( ____ × 3) × 4 = ____ × (3 × 4) = ____ × 12 = ____
• In class, we used the associative property to show that when 6 is a factor, then 2 and 3 are factors, because 6 = 2 × 3. Use the fact that 8 = 4 × 2 to show that 2 and 4 are factors of 56, 72, and 80. 56 = 8 × 7 72 = 8 × 9 80 = 8 × 10

NYS Math Module 3 Grade 4 Lesson 23 Homework

• Explain your thinking or use division to answer the following. a. Is 2 a factor of 72? c. Is 3 a factor of 72? e. Is 6 a factor of 72? f. Is 4 a factor of 60? h. Is 8 a factor of 60?
• Use the associative property to find more factors of 12. a. 12 = 6 × 2 = ___ × (3 × 2) = ___ × 6 = ___
• In class, we used the associative property to show that when 6 is a factor, then 2 and 3 are factors, because 6 = 2 × 3. Use the fact that 10 = 5 × 2 to show that 2 and 5 are factors of 70.
• The first statement is false. The second statement is true. Explain why using words, pictures, or numbers. If a number has 2 and 6 as factors, then it has 12 as a factor. If a number has 12 as a factor, then both 2 and 6 are factors.

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Use Division And The Associative Property To Test For Factors And Observe Patterns

Description.

Objective:  Use division and the associative property to test for factors and observe patterns.

In Lesson 23, students use division to examine numbers to 100 for factors and make observations about patterns they observe, for example, “When 2 is a factor, the numbers are even.”

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• Grade 4 Mathematics Module 3, Topic F, Lesson 23

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