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e

Texas Essential Knowledge and Skills

Algebraic Reasoning—5.4.F Simplify numerical expressions that do not involve exponents, including up to two levels of grouping

MATHEMATICAL PROCESSES5.1.B Use a problem-solving model5.1.F Analyze mathematical relationships 5.1.G Display, explain, and justify mathematical ideas and arguments

Are You Ready?Access Prior KnowledgeUse the Are You Ready? 7.4 in the Assessment Guide to assess students’ understanding of the prerequisite skills for this lesson.

Vocabularysimplify, order of operations

Go to Multimedia eGlossary at thinkcentral.com

7.4 PROBLEM SOLVING • Simplify Numerical Expressions

In what order must operations be simplified to find the solution to a problem?

Essential Question?

Lesson OpenerMaking ConnectionsInvite students to tell you what they know about doing things in the correct order.

What activities or projects do you do that have steps that must be done in a specific order? (Possible answers: putting together a model airplane, baking, playing certain games) What happens when you do not do the steps in the right order? Do all activities or projects have steps that need to be done in a specific order? Why or why not?

Using the Digital LessonYou may wish to ask the students to model the stated problem using counters. Have students brainstorm different ways to break down the problem into smaller steps.

Learning TaskWhat is the problem the students are trying to solve? Connect the story to the problem.

• What are you trying to find? (the total number of nails used)

• What do the numbers 40, 16, and 4 represent in the problem? (the number of nails in the roof, the number of nails in each side, and the number of sides)

• Why do we multiply 4 and 16? (There are 4 sides and each has 16 nails).

Literacy and MathematicsChoose one or more of the following activities.

• Ask students how the expression shown would change if each side had 20 nails instead of 16. Have them write their expressions and compare with a partner.

• Elicit the meaning of operations as it is used in this context. Ask students to name mathematical operations they have learned.

In what order must operations be simpli f ied to f ind the solution to a

problem?

Lesson 7.4 309A

Essential Question?

Unlock the ProblemUnlock the Problem

Name______

18

3 x 4 + 3 x 2

12 + 6

First, I added.

Then, I multiplied.

42

Heather

3 x 4 + 3 x 2

3 x 7 x 2

7.4Name

Connect Remember that a numerical expression is a

mathematical phrase that uses only numbers

and operation symbols.

(5 − 2) × 7 72 ÷ 9 + 16 (24 − 15) + 32

To simplify, or find the value of, a numerical expression

with more than one operation, you must follow rules called

the order of operations. The order of operations tells

you in what order you should simplify an expression.

In what order must operations be simplified to find the solution to a problem?

Algebraic Reasoning—5.4.F

MATHEMATICAL PROCESSES5.1.B, 5.1.F, 5.1.G

So, _ cups of flour and sugar

are needed.

Explain why Heather’s answer is not correct.

A Heather did not follow the order ofoperations correctly.

B Follow the order of operations by multiplying first and then adding.

A cake recipe calls for 4 cups of flour and 2 cups of sugar. To triple

the recipe, how many cups of flour and sugar are needed in all?

Simplify 3 × 4 + 3 × 2 to find the total number of cups.

Order of Operations

1. Perform operations in parentheses.

2. Multiply and divide from left to right.

3. Add and subtract from left to right.

Simplify Numerical Expressions

18

Heather added before she multiplied.

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Language SupportELLEnglish Language LearnersELPS 2.C.2, 4.C.3, 4.F.10

Strategy: Model Concepts• Write the following words on the board, and ask students to identify

which operations are associated with the words. less than, more than, difference subtractionin all, more than, sum, plus additionby, times, product multiplicationshare equally, equal parts, divide, quotient division

• Have students write a sentence or word problem that uses one of the words (or a related word) for each operation. Possible answer: Maria has 6 more CDs than Molly, who has 20.

• Invite students to share their sentences and word problems.

VisualSmall Group

Have students note that each of the expressions near the top of the page contains more than one operation.

• Why do we need to follow the order of operations whenever we simplify an expression that has more than one operation? More than one answer is possible if the operations are not performed in the correct order.

Unlock the ProblemBefore you discuss Heather’s solution, make sure students understand what each part of the expression 3 × 4 + 3 × 2 represents.

• The problem tells us that the expression 3 × 4 + 3 × 2 is used to find the total number of cups. What do 3 × 4 and 3 × 2 represent? 3 × 4 represents tripling the amount of flour, and 3 × 2 represents tripling the amount of sugar.

Have students focus their attention on the numerical expressions at the top of this page and explain or write how the order of operations would be used to simplify each expression.

(5 – 2) × 7 = 3 × 7 = 21

72 ÷ 9 + 16 = 8 + 16 = 24

(24 – 15) + 32 = 9 + 32 = 41

Leveled Activities ELPS

Beginning: Activity 8 2.I.4, 3.G.1, 3.H.3

Intermediate: Activity 40 4.F.6, 4.G.2, 4.G.4

Advanced: Activity 14 4.C.4, 4.F.9, 4.G.2

Advanced High: Activity 18 4.C.4, 4.E, 4.F.7

Go to thinkcentral.com for the ELL Activity Guide containing these leveled activities.

309 Module 7

Share and ShowShare and Show

Mathematical ProcessesMath Talk

Problem SolvingProblem Solving

So, Lena will use _ cups of flour, milk, and sugar in all.

5 × ( 3 + 1 + 2 )

5 × ( _ )

_

Write the expression.

First, perform the operations in parentheses.

Then multiply.

Simplify Expressions with Parentheses To simplify an expression

with parentheses, follow the order of operations. Perform the operations

in parentheses first. Multiply from left to right. Then add and subtract

from left to right.

ExampleEach batch of cupcakes Lena makes uses 3 cups of flour, 1 cup of milk, and 2 cups

of sugar. Lena wants to make 5 batches of cupcakes. How many cups of flour, milk,

and sugar will she need in all?

1. 10 + 36 ÷ 9

Think: I need to divide first.

_

2. 10 + (25 − 10) ÷ 5

_

3. 9 − (3 × 2) + 8

_

Simplify the numerical expression.

4. Use Symbols Write 12 + 17 – 3 × 2 with parentheses

so it has a value of 23.

5. Multi-Step The value of 100 – 30 ÷ 5 with parentheses

can have a value of 14 or 94. Explain.

Raina simplified the expression 5 × 2 + 2 by adding first and

then multiplying. Will her answer be correct? Explain.

6

30

30

When the parentheses are around 100 – 30, the value is 14. When the parentheses are

around 30 ÷ 5, the value is 94.

No. Possible explanation: She needs to follow the order of operations and multiply fi rst and then add. The correct answer is 12.

12 + 17 − (3 × 2)

111314

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Enrich

4 + (4 × (4 ÷ 4))

4 + (4 × 1)

4 + 4

8

Go to Go to thinkcentral.com for additional enrichmentactivities in the Enrich Activity Guide.

1

2

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a student misses the checked exercises

Quick Check

IF

THENDifferentiate Instruction withRtI Tier 1 Lesson 55

Simplify Expressions with Parentheses Point out to students that regardless of other operations in an expression, they should always perform the operations in parentheses first. So, even if an expression involves multiplication outside of the parentheses and addition inside the parentheses, you perform the addition first.

ExampleWrite the expression 5 × (3 + 1 + 2) on the board or overhead, and discuss what the expression represents.

• What does (3 + 1 + 2) represent? the number of cups of ingredients Lena needs to make one batch of cupcakes

• Why do you multiply the expression (3 + 1 + 2) by 5? because Lena wants to make 5 batches of cupcakes

Share and ShowBefore students simplify each expression, ask them to identify the operations and the order in which they will be performed.

Use the checked exercises for Quick Check.

Problem Solving Go Deeper• In Exercise 4, how can you place parentheses

to increase the value of the expression? Possible answer: I can place parentheses around 17 – 3 to increase the value.

VisualIndividual / Partners

• Consider the following expression: 4 + (4 × (4 ÷ 4))

• Since the expression has parentheses inside parentheses, you must perform the operations in the innermost set of parentheses first.

• After you perform operations in the innermost set of parentheses, you can perform the operations in the next set of parentheses.

• Students may challenge each other by using the classic Four Fours puzzle. Students use 4 fours, the basic operation symbols, as well as grouping symbols to form expressions for the numbers 0–10.

• You may want to have students compile a list of expressions, such as, 4 × 4 – (4 × 4); 4 _ 4 + 4 – 4; 4 × 4 ÷ (4 + 4), and so on.

Lesson 7.4 310

Unlock the ProblemUnlock the Problem

back

middle side

side

Name

7. Multi-Step In the wild, an adult giant panda eats about 30 pounds of

food each day. Write and simplify a numerical expression that shows

how many pounds of food 6 pandas eat in 3 days.

8. Connect Write and simplify two equivalent numerical

expressions that show the Distributive Property of Multiplication.

6. Multi-Step A movie theater has 4 groups of seats. The largest group

of seats, in the middle, has 20 rows, with 20 seats in each row. There

are 2 smaller groups of seats on the sides, each with 20 rows and

6 seats in each row. A group of seats in the back has 5 rows, with

30 seats in each row. How many seats are in the movie theater?

a. What do you need to know?

b. What operation can you use to find the number of seats in the back

group of seats? Write the expression.

c. What operation can you use to find the number of seats in both

groups of side seats? Write the expression.

d. What operation can you use to find the number of seats in the

middle group? Write the expression.

e. Write an expression to represent the total

number of seats in the theater.

f. How many seats are in the theater? Show the

steps you use to solve the problem.

multiplication; 5 × 30

3 × (30 × 6), 540 pounds

Possible answer: 4 × (9 + 6) and (4 × 9) + (4 × 6); 60

multiplication; Possible expression: (20 × 20)

Possible expression:

(5 × 30) + 2 × (20 × 6) + (20 × 20)

Check students’ work.

790 seats

multiplication; possible expression: 2 × (20 × 6)

I need to fi nd the number of seats that are in each

section of the movie theater.

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COMMON ERRORSError Expressions are simplified from left to right without consideration of the type of operation.

Example 12 + 8 ÷ 2 = 10

Springboard to Learning Have students work cooperatively to develop an acronym, mnemonic device, or another method that can be used to remember the steps to follow when using the order of operations.

CE

Unlock the ProblemHave students read Problem 6 and discuss what they need to know. This is a multi-step problem in which they first must determine the number of seats in each group.

Math on the Spot Video Tutor

Through the Math on the Spot Video Tutor, students will be guided through an interactive solving of this type of H.O.T. problem. Use this video to also help students solve the H.O.T. problem in the Interactive Student Edition. With these videos and H.O.T. problems, students will build skills needed in the TEXAS assessment.

MV

Math on the Spot videos are in theInteractive Student Edition and atthinkcentral.com.

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A numerical expression is a mathematical phrase that includes only numbers and operation symbols.

You simplify the expression when you perform all the computations to find its value.

To simplify an expression, use the order of operations.

Simplify the expression (10 + 6 × 6) - 4 × 10.

Step 1 Start with computations inside the parentheses.

Step 2 Perform the order of operations inside the parentheses.

Step 3 Rewrite the expression with the parentheses simplified.

Step 4 Multiply and divide from left to right.

Step 5 Add and subtract from left to right.

So, (10 + 6 × 6) - 4 × 10 = 6.

Simplify the numerical expression.

1. 8 - (7 × 1) 2. 5 - 2 + 12 ÷ 4 3. 8 × (16 ÷ 2)

4. 4 × (28 - 20 ÷ 2) 5. (30 - 9 ÷ 3) ÷ 9 6. (6 × 6 - 9) - 9 ÷ 3

7. 11 ÷ (8 + 9 ÷ 3) 8. 13 × 4 - 65 ÷ 13 9. 9 + 4 × 6 - 75 ÷ 15

5.4.FSimplify Numerical ExpressionsOBJECTIVE Use the order of operations to simplify numerical expressions.

LESSON 55

Multiply and divide from left to right.

10 + 6 × 6 = 10 +

Add and subtract from left to right.

10 + 36 =

46 - 4 × 10

46 - 4 × 10 = 46 -

46 - 40 =

36

10 + 6 × 6

Order of Operations 1. Parentheses2. Multiply and Divide3. Add and Subtract

46

40

6

1

72

1

6

3

47

64

24

28

Algebraic Reasoning 109

FINISH

START

Enrich© Houghton Mifflin Harcourt Publishing Company

E46

Name Enrich 46

Order of Operations Game

Three players are playing a board game. Complete the exercises

below, and move each player’s piece the same number of spaces as

the answer for the unknown value. Circle the player who wins the game.

Each black space counts as one space.

Player 1 Player 2 Player 3

(50 2 2) 4 4 5 5 1 10 4 5 5 108 4 (27 2 9) 5

(343 2 5 ) 4 26 2 11 5 (7 3 7) 4 (3 1 4) 5 6 1 3 2 7 5

(55 2 1) 4 9 5 (16 3 3) 4 (4 3 6)

5

(64 4 16) 3 (11 2 6)

5

(15 2 36 4 4) 1 (9 3 2)

5

2 3 (3 1 51 4 17)

5

144 2 (10 1 4 3 5 3 5 )

5

(64 1 6) 4 ( 3 5) 5 2 81 4 ( 4 4) 5 9 (4 3 ) 2 (1 1 8 3 2) 5 3

6. Stretch Your Thinking A fourth player joins the game and is

given an expression that moves the game piece directly to the second

black space on the board. The expression has a division, a multiplication,

and a subtraction operation. Write a possible expression.

12 7 6

2 7 2

62 20

24 12 34

367 5

Possible answer: (159 2 3 3 3) 4 3 5 50

1.

2.

3.

4.

5.

1

2

3

RtI Tier 1 Lesson 55 Enrich 46

311 Module 7

Daily Assessment TaskDaily Assessment Task

Mathematical Processes

TEXAS Test Prep12. Which expression has a value of 6?

A (6 ÷ 3) × 4 + 8

B 27 − 9 ÷ 3 × (4 + 1)

C (18 + 12) × 6 − 4

D 71 − 5 × (9 + 4)

Fill in the bubble completely to show your answer.

9. The books in an unusual library are labeled with numerical expressions.

Katrina needs to find the book with a numerical expression that has a

value of 12. Which one should she choose?

A (6 + 26) – (4 × 5)

B 2 × (15 – 7)

C 10 + 8 – 36 ÷ 9

D (25 – 4) ÷ 7

10. Franco needs to simplify 3 × (8 – 4) + 7. What should be his first step?

A Add 7 to 4.

B Subtract 4 from 8.

C Multiply 3 by 7.

D Add 3 to 7.

11. Multi-Step Sarah simplified each of the expressions shown. What is

the difference between the values of the two expressions?

3 × 12 – 9 + 10 3 × (12 – 9) + 10

A 9 C 12

B 18 D 22

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Games

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Daily Assessment Task 1

2

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Differentiated Centers Kit

• Soar to Success MathWarm-Up 14.40

ActivitiesMultiplication RelayStudents complete purple Activity Card 11 by working in teams to solve multiplication problems.

LiteratureA Drive Through HistoryStudents read about how the Alvarez family uses multiplication and division to plan their vacation.

GamesWhat’s Left?Students write and solve division problems to make a specific remainder.

TEXAS Test Prep CoachIn the Test Prep exercise, if students selected:

A They read the question as 16 instead of 6.

B They ignored the numbers and operation in parentheses.

C They did not follow the order of operations and read the question as 60 instead of 6.

Essential Question? WriteMathWriteMath

In what order must operations be simplified to find the solution to a problem? 1: perform operations in parentheses; 2: multiply and divide from left to right; 3: add and subtract from left to right.

• Enrich 46

• Homework and Practice Lesson 7.4

Can students simplify operations in the correct order to find the solution to a problem?

Lesson 7.4 312

TEXAS Test PrepLesson CheckLesson Check

15. Mrs. Harrison writes a numerical expression

on the board that has a value of 16. Which

expression does Mrs. Harrison write?

A (15 + 3) − 14 ÷ 2

B 20 − 4 ÷ 2

C 3 × (5 + 1)

D 18 + 13 − (5 × 3)

16. Omar wants to simplify the expression below.

What should be his first step?

6 × (3 + 9) − 5

A Multiply 6 by 3.

B Subtract 5 from 9.

C Add 9 to 3.

D Subtract 5 from 6.

17. Which is the value of the numerical

expression below?

30 ÷ (3 + 2) × 8

A 96

B 48

C 26

D 40

18. Becka says that when she simplifies the

expression below, the value is 6 less than her

age. What is Becka’s age?

(7 + 11) − 8 ÷ 2

A 20

B 14

C 26

D 11

19. Multi-Step Kalani and Porter each simplified

one of the expressions below. If they add their

results together, what will be their sum?

Kalani: 4 × 11 − 5 + 12

Porter: 4 × 11 − (5 + 12)

A 78

B 44

C 102

D 54

20. Multi-Step Nicole simplified the expression

below. Then she multiplied the value of the

expression by 5. What is Nicole’s product?

36 + 6 ÷ 3 + 2

A 16

B 80

C 40

D 200

Fill in the bubble completely to show your answer.

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Problem Solving

Problem SolvingProblem Solving

Homeworkand Practice

Name

Simplify Numerical Expressions7.4

Algebraic Reasoning—5.4.FMATHEMATICAL PROCESSES 5.1.B, 5.1.F, 5.1.G

Simplify the numerical expression.

Rewrite the expression with parentheses to equal the given value.

1. 10 − (2 × 4) + 12

4. 25 − 4 × 5

2. 14 + 35 ÷ 7

5. 3 × (6 + 3) − 15

3. 11 + (31 − 4) ÷ 9

6. 30 − 14 ÷ 2

7. 12 + 6 ÷ 2 + 4; value: 13

10. 60 − 3 + 2 × 5; value: 35

8. 42 − 24 ÷ 6; value: 3

11. 18 + 9 ÷ 3; value: 21

9. 9 + 16 − 2 × 4; value: 17

12. 5 × 2 + 4 + 3; value: 45

13. Julian paid his library fines. He paid $2 each for

4 overdue books and $16 for a lost book. Write

and simplify a numerical expression that shows

how much Julian paid in library fines.

14. For 5 days in a row last week, Addison read

9 pages of a poetry book and 16 pages of her

favorite novel. Write and simplify a numerical

expression that shows how many pages Addison

read last week.

14

12 + 6 ÷ (2 + 4)

5

60 − (3 + 2) × 5

($2 × 4) + $16; $24 5 × (9 + 16); 125

12

18 + (9 ÷ 3)

19

(42 − 24) ÷ 6

14

9 + 16 − (2 × 4)

23

5 × (2 + 4 + 3)

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Module 7 • Lesson 4 313

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Homework and PracticeUse the Homework and Practice pages to provide students with more practice on the concepts and skills of this lesson.

313-314 Module 7