investigate • order of operations...having the order of operations, or a set of rules that...

427A Chapter 7

About the MathProfessional Development

Professional Development Videos

LESSON AT A GLANCE

Interactive Student Edition

Personal Math Trainer

Math on the Spot

MP3 Construct viable arguments and critique the reasoning of others.In this lesson, students investigate how the order in which they perform operations might change the answer, given an equation with more than one operation. Students have an opportunity to construct viable arguments, as they are encouraged to predict what will happen when they use different orders of the operations to find the answer. Following their investigation, students are encouraged to compare their result with their prediction.

From their investigation, students can see a value in having the order of operations, or a set of rules that mathematicians agree on for the order in which operations are carried out, as follows:

First, multiply and divide from left to right.

Then, add and subtract from left to right.

The order of operations allows everyone to get the same answer.

LESSON 7.11

Investigate • Order of Operations

Learning ObjectivePerform operations in order when there are no parentheses.

Language ObjectiveStudent pairs discuss and hypothesize the reason there are rules such as the order of operations.

MaterialsMathBoard

F C R Focus:Common Core State Standards3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the

unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

MATHEMATICAL PRACTICESMP2 Reason abstractly and quantitatively. MP4 Model with mathematics. MP6 Attend to precision.

F C R Coherence:Standards Across the GradesBefore2.OA.C.4

Grade 33.OA.D.8

After4.OA.A.3

F C R Rigor:Level 1: Understand Concepts....................Share and Show ( Checked Items)Level 2: Procedural Skills and Fluency.......On Your OwnLevel 3: Applications..................................Think Smarter and Go Deeper

F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 363J.

FOCUS COHERENCE RIGOR

ENGAGE1

Lesson 7.11 427B

Daily RoutinesCommon Core

Daily RoutinesCommon Core

Guess the Meaning

Word

order ofoperations

which operationyou do 1st, 2nd,3rd, and so on

My Guess

Why are there rules such as the order of

operations?

with the Interactive Student Edition

Essential QuestionWhy are there rules such as the order of operations?

Making ConnectionsInvite students to talk about rules and instructions.

Have you ever gotten a toy or item that needed to be assembled? yes Why was it important to follow the instructions that came with the item? So I knew how to put the parts together in the right order. What do you think would happen if you didn’t follow the order of the steps? I might put the item together wrong.

Learning ActivityWhat is the problem the students are trying to solve? Connect the story to the problem. Ask the following questions.

• Describe in your own words what is meant by a two-step problem. a problem that involves two operations to solve

• If you are adding three numbers, does the order matter? no

• If you have to do two different operations to solve a problem, do you think the order you perform the operations matters? yes

Literacy and MathematicsChoose one or more of the following activities:

• Have students write directions for a simple task, like making a sandwich. Have them discuss in pairs what might happen if the directions were in a different order.

• Have students write a story about someone who attempts to follow directions in the wrong order.

Vocabulary BuilderMaterials Guess the Meaning (see eTeacher Resources)

Guess the Meaning Have students complete the Guess the Meaning chart for the term order of operations.

Problem of the Day 7.11There are 40 students competing in a team race. Each team of runners will have 10 students. Write an expression that could be used to find the number of teams that all 40 students would make. __________

Vocabulary order of operations

• Interactive Student Edition• Multimedia Glossary e

40 ÷ 10

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EXPLORE2

InvestigateInvestigate

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Chapter 7 427

connect You can use what you know about acting out a two-step problem to write one equation to describe and solve a two-step problem.

• If you solved a two-step problem in a different order, what do you think might happen?

Use different orders to find 4 + 16 ÷ 2.

A. Make a list of all the possible orders you can use to find the answer to 4 + 16 ÷ 2.

B. Use each order in your list to find the answer. Show the steps you used.

Order of OperationsEssential Question Why are there rules such as the order of operations?

1. Did following different orders change the answer? ______

2. MATHEMATICALPRACTICE 8 Draw Conclusions If a problem has more than one

type of operation, how does the order in which you perform the operations affect the answer?

3. Explain the need for setting an order of operations that everyone follows.

Draw ConclusionsDraw Conclusions

Lesson 7.11

Operations and Algebraic Thinking—3.OA.D.8

MATHEMATICAL PRACTICESMP4, MP7, MP8

add, divide or divide, add

You might not get the same answer.

4 + 16 = 20; 20 ÷ 2 = 10

16 ÷ 2 = 8; 4 + 8 = 12

Yes

If you follow a different order, it changes the answer.

Possible explanation: so everyone will know what operation to do rst

and will nd the correct answer

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Name

Order of Operations

First: Multiply and divide from left to right.

Then: Add and subtract from left to right.

}}

Order of Operations

Danny buys a marker for $4. He also buys 5 pens for $2 each. How much money does he spend?

You can write $4 1 5 3 $2 5 c to describe and solve the problem.

Find $4 1 5 3 $2 5 c.

When there is more than one type of operation in a problem, use the order of operations, or the set of rules for the order in which to do operations.

Step 1 Multiply from left to right.

$4 1 5 3 $2 5 c

multiply

$4 1 $10 5 c

Step 2 Next, add from left to right.

$4 1 $10 5 c

add

$14 5 c

So, Danny spends $14 .

Write correct if the operations are listed in the correct order. If not correct, write the correct order of operations.

1. 5 1 6 3 3 add, multiply

2. 20 4 4 2 3 divide, subtract

Follow the order of operations to find the unknown number.

3. 9 2 7 1 2 5 k

k 5 _

4. 8 1 2 3 5 5 m

m 5 _

5. 7 3 8 2 6 5 g

g 5 _ 6. 16 1 4 4 2 5 s

s 5 _

7. 12 2 6 4 2 5 y

y 5 _

8. 36 4 6 1 13 5 f

f 5 _

Lesson 7.11Reteach

19

50

9

18

18

4

correctmultiply, add

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7-25 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

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Name

1. 2 1 3 3 5 20

2. 12 4 2 2 5 2

3. 4 3 6 2 5 12

4. 1 6 3 3 5 20

5. 5 1 4 6 5 8

6. 20 2 4 8 5 18

7. 14 2 2 3 5 0

8. 30 4 6 3 5 45

9. 9 3 2 20 5 16

10. 1 2 3 5 5 45

11. 1 24 4 4 5 7

12. 9 1 7 2 5 11

7. Explain how you found the unknown number in Exercise 9.

Lesson 7.11Enrich

Order of Operations

Find the unknown number that makes the equation true. Follow the order of operations.

Possible explanation: � rst, I looked for a number

that, when you subtract 20, gives a difference of 16.

That number is 36. Since 9 is one of the factors, the

unknown number must be 4, because 9 3 4 5 36.

1

4

7

18

12

6

5

35

9

16

2

3

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1

2

3 DifferentiatedInstruction

427 Chapter 7

Enrich 7.11Reteach 7.11

LESSON 7.11

InvestigateMATHEMATICAL PRACTICES

MP1 Make sense of problems and persevere in solving them. First, read the problem together. Then, tell students they will first investigate how the order in which they perform operations might change the answer.• What steps did you use to find the answer

when doing the addition part first? Possible answer: add 4 + 16 = 20, divide 20 by 2 = 10

• What steps did you use to find the answer when doing the division part first? Possible answer: divide 16 by 2 = 8; add 4 + 8 = 12

MP6 Attend to precision. Display the expression 14 − 5 × 2.• Try performing the operations in order

from left to right. What do you get? 14 − 5 = 9; 9 × 2 = 18.

• Now try multiplying first and then subtracting. What do you get? 5 × 2 = 10; 14 − 10 = 4.

• How did the different orders change your answer? I got two different answers.

Draw ConclusionsMP8 Look for and express regularity in repeated reasoning. Help students conclude the benefit of using the order of operations, or a set of rules for which operations should be done in which order.• Why do you think setting an order of

operations is needed? Possible answer: so that everyone can do it in the same way to get the same answer

ELL Strategy: Model Language

Remind students that operation means to add, subtract, multiply or divide. Draw the symbols for each operation. Order means the sequence; what to do first, then second, and so on.•Post a problem from the lesson.

•Point and say, First, multiply and divide from left to right. Then, add and subtract from left to right.

•Have students identify the part of the equation that should be done first, and circle it or draw arrows and label it “first.” Have students label the part that should be done next.

3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.


Meeting Individual Needs

DifferentiatedInstruction

Share and ShShare and ShShare and Show MATHBOARDMATHBOARD

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1. 4 + 5 × 2 multiply, add 2. 8 ÷ 4 × 2 multiply, divide


So, Meghan has _ left.

• Does your answer make sense? Explain.

STEP 1

Multiply from left to right.

STEP 2

Subtract from left to right.

When solving problems with more than one type of operation, you need to know which operation to do first. A special set of rules, called the order of operations, gives the order in which calculations are done in a problem.

First, multiply and divide from left to right.

Then, add and subtract from left to right.

Meghan buys 2 books for $4 each. She pays with a $10 bill. How much money does she have left?

You can write $10 − 2 × $4 = c to describe and solve the problem.

Use the order of operations to solve $10 − 2 × $4 = c.

$10 − 2 × $4 = c$10 − $8 = c

$10 − $8 = c $2 = c

3. 12 + 16 ÷ 4 add, divide 4. 9 + 2 × 3 add, multiply

5. 4 + 6 ÷ 3 divide, add 6. 36 − 7 × 3 multiply, subtract

Make ConnectionsMake Connections

MathTalk MATHEMATICAL PRACTICES 7

Identify Relationships What operation should you do first to find: 12 − 6 ÷ 2 and 12 ÷ 6 − 2? What is the answer to each problem?

yes; possible explanation: if I subtracted fi rst, my

answer would be $32 and that is more money than

I started with.

correct

$2

divide, multiply

correct correct

divide, add multiply, add

In both problems, you

divide fi rst; 9; 0.

EXPLAIN3

COMMON ERRORS Advanced Learners

Lesson 7.11 428

Share and Show MATHBOARDMATHBOARDMBMMMBBBMATHABOARDMMMAAATHATHTHHAAAAAAAAATTAAAABOARDBOARDBOARD

The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.

Make ConnectionsHelp students make connections between solving equations that involve more than one operation and using the order of operations.Focus students on the opening problem. Help them connect the equations to the words and events in the word problem.• Which part of the equation stands for the

2 books that Meghan buys for $4 each? Explain. 2 × 4; Possible explanation: it means 2 groups of 4. Meghan buys 2 books for $4 each.

• Which operations are done first? multiplication and division

• If an addition symbol is to the left of a division symbol, which operation do you do first? division

Error Students may make errors in determining the correct order of operations.

Example Students indicate “add, divide” is the correct order for 12 + 16 ÷ 4.Springboard to Learning Remind students that the order of operations is a set of rules. Have students first circle a multiplication or division part of the problem, so that they know what part to do first.

Logical / MathematicalPartners

Materials Spinners (3-section) (see eTeacher Resources)

• Have students work in pairs. Have them label one spinner with the numbers 1 to 3, another spinner with the numbers 4 to 6, and a third spinner with the numbers 7 to 9.

• Partners take turns spinning each of the three pointers. Students should write an equation with two different operations that form the greatest answer, for example: for 2, 6, 7: 2 + 7 × 6 = x. Students should continue to use the order of operations for completing the equation.

• Each partner shares his or her equation. The player who has the equation with the greater answer wins.

• Students can play again. The player who has the equation with the lesser answer wins.

Problem Solving • ApplicationsProblem Solving • Applications

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Chapter 7 • Lesson 11 429

13. 2, 6, and 5

_ + _ × _ = 16

14. 4, 12, and 18

_ − _ ÷ _ = 15

7. 63 ÷ 9 − 2 = f

f = _

8. 7 − 5 + 8 = y

y = _

9. 3 × 6 − 2 = h

h = _

Follow the order of operations to find the unknown number. Use your MathBoard.

10. 80 − 64 ÷ 8 = n

n = _

11. 3 × 4 + 6 = a

a = _

12. 2 × 7 ÷ 7 = c

c = _

MATHEMATICALPRACTICE 4 Write an Equation Algebra Use the numbers listed to

make the equation true.

15. 8, 9, and 7

_ × _ − _ = 47

16. 2, 4, and 9

_ ÷ _ + _ = 11

17. WRITE Math Pose a Problem Write a word problem that can be solved by using 2 × 5 ÷ 5. Solve your problem.

18. SMARTER Is 4 + 8 × 3 equal to 4 + 3 × 8? Explain how you know without finding the answers.

5 10 16

72 18 2

or 6, 5, 2

or 8, 7, 9

Possible problem: Juanita has 2 pages of stickers with

5 stickers on each page. How many stickers can she give

Possible explanation: by the Commutative Property of

her 5 friends if each friend gets the same number of stickers?

Multiplication, 8 × 3 = 3 × 8. So, the answers are equal since

2 stickers

you multiply � rst to � nd the answer for each.

6

7 4

18 122

8 2

5

9 9

4

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Quick Check

If

Rt I 1

2

3

Quick Check

If

Rt I 1

2

3

Then

ELABORATE4

Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.

429 Chapter 7

MP4 Model with mathematics. Exercises 13–16 require students to use higher order thinking skills as they position numbers in an equation to make the equation true. Tell students to continue to apply the order of operations as they complete the equations.Exercise 17 requires students to analyze an expression with two operations to write a matching word problem. Tell students they need to apply the order of operations when writing situations for their word problem.

SMARTER

a student misses the checked exercises

Differentiate Instruction with • Reteach7.11

• PersonalMathTrainer3.OA.D.8

• RtITier1Activity(online)

Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

MP5 Use appropriate tools strategically. Some calculators are programmed to perform operations using the order of operations. If you have such a calculator available, model how to press keys in sequence for a problem that has two operations, for example:

+ × • How do you think this calculator

is programmed to use the order of operations? Possibleanswer:aruleisusedbythecalculatorasitreadsthesymbolsfortheoperationsthatareputin.

Math on the Spot Video TutorUsethisvideotohelpstudentsmodelandsolvethistypeofThink Smarterproblem.


Problem Solving • ThinkingProblem Solving • Applications

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES

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Picture Book Art

The Eric Carle Museum of Picture Book Art in Amherst, Massachusetts, is the first museum in the United States that is devoted to picture book art. Picture books introduce literature to young readers.

The museum has 3 galleries, a reading library, a café, an art studio, an auditorium, and a museum shop. The exhibits change every 3 to 6 months, depending on the length of time the picture art is on loan and how fragile it is.

The table shows prices for some souvenirs in the bookstore in the museum.

20. Kallon bought 3 Caterpillar note cards and 1 Caterpillar pen. How much did he spend on souvenirs?

21. DEEPER Raya and 4 friends bought their teacher 1 Firefly picture frame. They shared the cost equally. Then Raya bought an Exhibition poster. How much money did Raya spend in all? Explain.

19. SMARTER For numbers 19a–19d, select True or False for each equation.

19a. 24 ÷ 3 + 5 = 13 True False

19b. 5 + 2 × 3 = 21 True False

19c. 15 − 3 ÷ 3 = 14 True False

19d. 18 ÷ 3 × 2 = 12 True False

Souvenir PricesSouvenir Price

Firefl y Picture Frame $25

Exhibition Posters $10

Caterpillar Note Cards $8

Caterpillar Pens $4

Sun Note Pads $3

$28

$15; Possible explanation: since there are 5 friends, I divided

the cost of the frame by 5. Then I added the cost of the

poster to that number.

EVALUATE5 Formative Assessment

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

Lesson 7.11 430

Students practice division facts to place counters on the gameboard.

GamesDivision Cover-Up

Students read the book and use division facts to find how much wood they need to build a fence.

LiteratureThe Garden Fence

Students complete purple Activity Card 19 by relating division facts to known multiplication facts.

ActivitiesDivision Dilemmas

Essential QuestionUsing the Language ObjectiveReflect Have student pairs discuss and hypothesize to answer the Essential Question. Why are there rules such as the order of operations? Possible answer: the rules help make sure that the answer is the same no matter who completes the equation.

Math Journal WRITE MathGive a description of the rules for the order of operations in your own words.

SMARTER

Successful completion of this item indicates a good understanding of the order of operations. Students who choose True for part b or False for part c need to remember that multiplication and division are done before adding and subtracting. Students who choose False for part d may need a reminder that they should work from left to right when the equation is all division and multiplication or all addition and subtraction.

Connect to Social StudiesInvite volunteers to read aloud the information about the museum that is devoted to picture book art.• What kind of art do you think the museum

displays? Possible answer: drawings and paintings found in picture books

For Exercises 20–21, students need to use the information in the table to solve the problems. For each problem, encourage students to write an equation with more than one operation.

DEEPER

For Exercise 21, have students write a complete answer that includes an explanation of how they solved the problem. Encourage students to describe the operations they used in their explanation.

Extend the Math Activity


Problem SolvingProblem Solving

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Order of Operations


Follow the order of operations to find the unknown number.

1. 45 − 3 × 5 subtract, multiply 2. 3 × 4 ÷ 2 divide, multiply

3. 5 + 12 ÷ 2 divide, add 4. 7 × 10 + 3 add, multiply

8. Shelley bought 3 kites for $6 each. She gave the clerk $20. How much change should Shelley get?

9. Tim has 5 apples and 3 bags with 8 apples in each bag. How many apples does Tim have in all?

Chapter 7 431

5. 6 + 4 × 3 = n

n = _

6. 8 − 3 + 2 = k

k = _

7. 24 ÷ 3 + 5 = p

p = _

multiply, subtract

Lesson 7.11Practice and Homework

COMMON CORE STANDARD—3.OA.D.8 Solve problems involving the four operations, and identify and explain patterns in arithmetic.

10. WRITE Math Give a description of the rules for the order of operations in your own words.

multiply, divide

multiply, addcorrect

$2 29 apples

18 7 13

Check students’ work.

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431 Chapter 7

Practice and HomeworkUse the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers.

Parentheses with the Order of Operations

Investigate Students write parentheses to indicate which part of equations are completed first with the order of operations.

• Discuss with students the concept that they can use parentheses in equations to show which part of the equation needs to be completed first before other parts of the equation. Parentheses are used when there is a change in the rules, or the order of operations. Provide examples, such as:

(18 2 6) 4 2 5 x (3 1 2) 3 5 5 y

• Ask students how the last equation would have a different answer if the parentheses were not there. Possible answer: the equation would have a value of 13 because I would multiply first, and then add.

• Have students write an equation with two operations. Then have them place parentheses to change the order of the operations. Have students write word problems to match their equations.

Summarize Have students write explanations for how the parentheses in their equations change the meanings and answers.


Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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Lesson Check (3.OA.D.8)

Spiral Review (3.OA.C.7, 3.OA.D.9, 3.NBT.A.3)

Tables 1 2 3 4

Chairs 4 8 12 16

5. For decorations, Meg cut out 8 groups of 7 snowflakes each. How many snowflakes did Meg cut out in all?

6. A small van can hold 6 students. How many small vans are needed to take 36 students on a field trip to the music museum?

3. Each story in Will’s apartment building is 9 feet tall. There are 10 stories in the building. How tall is the apartment building?

4. Describe a pattern in the table.

1. Natalie is making doll costumes. Each costume has 4 buttons that cost 3¢ each and a zipper that costs 7¢. How much does she spend on buttons and a zipper for each costume?

2. Leonardo’s mother gave him 5 bags with 6 flower bulbs in each bag to plant. He has planted all except 3 bulbs. How many flower bulbs has Leonardo planted?

19¢

90 feet

56 snow� akes

27 � ower bulbs

table; multiply the number of tables by 4.

Possible answer: add 4 chairs for each

6 small vans

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Lesson 7.11 432

Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section.


investigate • order of operations...having the order of operations, or a set of rules that...

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