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Common Core State Standards

560 Unit 6 Number Systems and Algebra Concepts

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Advance PreparationReview pages 269–270 of the Student Reference Book. If students do not use TI-15 or Casio fx-55 calculators, check that their calculators are programmed for order of operations.

Teacher’s Reference Manual, Grades 4–6 p. 102

Key Concepts and Skills• Evaluate numerical expressions involving

parentheses and exponents. [Number and Numeration Goal 4]

• Add, subtract, multiply, and divide signed numbers. [Operations and Computation Goals 1 and 2]

• Apply the order of operations to evaluate expressions. [Patterns, Functions, and Algebra Goal 3]

Key ActivitiesStudents evaluate expressions according to the order of operations.

Ongoing Assessment: Informing Instruction See page 563.

Ongoing Assessment: Recognizing Student Achievement Use journal page 222. [Patterns, Functions, and Algebra Goal 3]

Key Vocabularyorder of operations � nested parentheses

MaterialsMath Journal 2, pp. 222 and 223Student Reference Book, p. 247Study Link 6 �5calculator � slate

Playing Name That NumberStudent Reference Book, p. 329Math Masters, p. 462per partnership: complete deck of number cards (the Everything Math Deck, if available)Students practice applying order of operations.

Math Boxes 6 �6Math Journal 2, p. 224 Students practice and maintain skillsthrough Math Box problems.

Study Link 6 �6Math Masters, p. 196 Students practice and maintain skillsthrough Study Link activities.

ENRICHMENTExploring a Grouping SymbolMath Masters, p. 197Students explore the use of the fraction bar as a grouping symbol.

EXTRA PRACTICE Evaluating ExpressionsStudent Reference Book, p. 247 (optional)Math Masters, p. 198Students practice using the order of operations to evaluate expressions.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Order of OperationsObjectives To review the order of operations; and to evaluateexpressions containing parentheses.e

560_EMCS_T_TLG2_G6_U06_L06_576922.indd 560 3/24/11 1:18 PM

Lesson 6�6 561

Getting Started

Math MessageEvaluate the following expressions.

1. 12 - 8 / 4 10 2. 24 / 3 + 6 143. 6 ∗ 5 + 7 37 4. 17 - 9 ∗ 3 -105. 15 + 62 / 3 27 6. 23 / 8 + 3 4

Study Link 6�5 Follow-UpAsk students to share the patterns they found in the tables. Make sure that the following patterns are mentioned:

x + y = y + x y - x is the opposite of x - y. x ∗ y = y ∗ x y ÷ x is the reciprocal of x ÷ y.

Mental Math and Reflexes Students write a number to model each situation.

In golf, par is the number of strokes a player should need to complete a hole or round. In Jeff’s first round of golf, he shot 3 over par. 3

In Jeff’s second round of golf, he shot 5 under par. -5

Karina’s math test score was 3 1 _ 2 points above the class mean. 3 1 _ 2

Anastasia’s math test score was 5.75 points below the class mean. -5.75

On Monday, the stock market dropped 63.2 points. -63.2 On Tuesday, the stock market rose 58 2 _ 7 points. 58 2 _ 7

1 Teaching the Lesson

▶ Math Message Follow-Up

WHOLE-CLASS ACTIVITY

Algebraic Thinking Record students’ answers on the board. If students disagree, record all proposed answers. Then have students use a scientific calculator to resolve discrepancies. If they do not have scientific calculators, skip ahead to the next section.

Order of operations was formally introduced in Fifth Grade Everyday Mathematics. Prior to this lesson, Everyday Mathematics has stressed the use of parentheses to evaluate expressions. If students did not use order of operations in this activity, their answers for expressions 1, 4, and 5 will differ from the answers they obtained with their calculators.

Ask students to insert parentheses in each expression to show which operation the calculator did first. Record these expressions on the board.

ExpressionNumber Expression Left to

Right Calculator Parentheses

1. 12 - 8 / 4 1 10 12 - (8 / 4)2. 24 / 3 + 6 14 14 (24 / 3) + 63. 6 ∗ 5 + 7 37 37 (6 ∗ 5) + 74. 17 - 9 ∗ 3 24 -10 17 - (9 ∗ 3)5. 15 + 62 / 3 17 27 15 + (62 / 3)6. 23 / 8 + 3 4 4 (23 / 8) + 3

For expressions 1, 4, and 5, the scientific calculator did not evaluate the expressions in the order in which the numbers and operations were entered. In each of these expressions, the calculator carried out the addition or subtraction last. A scientific calculator will follow order of operations, regardless of the order in which an expression is entered.

NOTE Scientific calculators have built-in order of operations. Many four-function calculators do not. Please check students’ calculators to be sure.

Mathematical PracticesSMP2, SMP3, SMP4, SMP5, SMP6Content Standards6.NS.5, 6.EE.1, 6.EE.2c

561-565_EMCS_T_TLG2_G6_U06_L06_576922.indd 561 3/20/12 8:26 AM

Evaluate each expression.1. 28 � 15 / 3 � 8 2. 1 � (5 * 10) / 43. 10 * 6 / 2 � 30 4. 10 * (12 / 6 � 4) / 12 � 1

Check your answers on page 423.

Algebra

Rules for the Order of Operations

1. Do the operations inside parentheses.Follow rules 2–4 when you are working inside parentheses.

2. Calculate all expressions with exponents.3. Multiply and divide in order from left to right.4. Add and subtract in order from left to right.

P E M D A SP E M D A S

Order of Operations

In many everyday situations, the order in which things are doneis important. When you bake a cake, for example, you crack theeggs before adding them to the batter. In mathematics, too,many operations should be done in a certain order.

Some people remember the order of operations by memorizingthis sentence:

Evaluate. 5 * 4 � 6 * 3 � 2 � ?

Multiply first. 5 * 4 � 6 * 3 � 2Subtract next. 20 � 18 � 2Then add. 2 � 2

45 * 4 � 6 * 3 � 2 � 4

Evaluate 52 � (3 * 4 � 2) / 5.

Clear parentheses first. 52 � (3 * 4 � 2) / 5Calculate exponents next. 52 � 10 / 5Divide. 25 � 10 / 5Then add. 25 � 2

2752 � (3 * 4 � 2) / 5 � 27

Student Reference Book, p. 247

Student Page

Adjusting the Activity

562 Unit 6 Number Systems and Algebra Concepts

▶ Reviewing Order of Operations WHOLE-CLASSDISCUSSION

(Student Reference Book, p. 247)

Algebraic Thinking Write the following sentence on the board:

Nancy fed Tom the big gray cat.

Ask students to punctuate the sentence so it is clear that Nancy is not giving Tom a cat to eat. Nancy fed Tom, the big gray cat. Explain that the use of parentheses in an expression and the order of operations make up a mathematical “punctuation system.” As a class, read and work through the examples on Student Reference Book, page 247.

Discuss why, in the absence of parentheses, it is necessary to adhere to the order of operations. Sample answer: Everyone can agree on the same value for an expression.

Use the mnemonic device Please Excuse My Dear Aunt Sally to identify the order in which operations should be performed. Point out that in expressions with only addition and subtraction or only multiplication and division, neither operation has priority over the other. They are done in order from left to right. For example, the answer to 4 - 3 + 1 is 2, not 0, and the answer to 24 / 3 ∗ 4 is 32, not 2.

Have students create original mnemonic devices to remember the order of operations, for example, Please Exile My Devious Annoying Sibling. Post students’ devices in the classroom.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

Have students evaluate the Check Your Understanding expressions and explain how they got their results. Suggest that they follow the format of the examples in the Student Reference Book and write each step on a separate line. This makes it easy to keep track of the order in which operations are done. Have students use a calculator to check their results. If the calculator is scientific, students can enter the expression from left to right. Pose additional expressions if students need more practice.

ELL

561-565_EMCS_T_TLG2_U06_L06_576922.indd 562 1/29/11 8:07 AM

Evaluate each expression. Show your work. Then compare your results to those of your partner.

1. 4 � 6 � 3 � 2. 33 � 16 / 4 �

3. 4 � 7 � (3 � 5) � 4. 24 / 6 � 4 �

5. 8 � 4 � 49 � 7 � 6. 9 � 6 � 3 � 28 �

7. 7 � 5 � 13 � 23 � 17 � 8. 100 � 50 � 2 � 4 � 5 �

9. 7 / 7 � 4 � 32 � 10. 12 � 22 � 33 � 2113

95�25

4639

1620

2927

Order of OperationsLESSON

6�6

Date Time

247

� �

�

�

�

�

�

�

Math Journal 2, p. 222

Student Page

11. 10�1 � 16 � 0.5 � 12 � 12. ((�12� � �

14�) � 3) � 6 �33 �

13. �(�8) � (�4) � 6 � (�12) / 4 �

14. �4 � (�18) / 6 � (�3 � �3 � 5) �

15. �5(�6 � (�3)) / 7.5 �

16. �(�34� � �

12�) � �

12� � (�

12� � (��

12�)) �

17. Evaluate the following expressions for x � �2.

a. x � �x � 14 / 2 � b. �x � (6 � x) � 33 / 9 �

Order of Operations continuedLESSON

6�6

Date Time

10.1 3

35

�3

2

3 5

��34�

Try This

Math Journal 2, p. 223

Student Page

Lesson 6�6 563

▶ Evaluating Expressions

INDEPENDENT ACTIVITY

(Math Journal 2, pp. 222 and 223)

Algebraic Thinking Assign journal pages 222 and 223. Have students evaluate the expressions independently and then compare results with partners. Write disagreements on the board and ask students to justify their thinking. Have students use a scientific calculator to resolve any disputes.

Ongoing Assessment: Informing InstructionWatch for students who may be confused by the omission of the multiplication symbol in algebraic expressions. Remind students that expressions such as 2 ∗ N and 2N are equivalent. Similarly, 5(4 + 3) and 5 ∗ (4 + 3) have the same meaning.

Some expressions use two sets of parentheses, one inside the other. The operation in the inner parentheses is done before the operation in the outer parentheses. Double parentheses are often referred to as nested parentheses. To support English language learners, write nested parentheses on the board and include some examples.

When most students have finished, bring the class together and discuss expressions that students found difficult.

Ongoing Assessment: JournalPage 222 �Problems 1–8Recognizing Student Achievement

Use journal page 222, Problems 1–8 to assess students’ ability to apply the order of operations to evaluate numerical expressions. Students are making adequate progress if they are able to evaluate expressions in Problems 1–8. Some students may be able to evaluate expressions containing exponents (Problems 9 and 10). [Patterns, Functions, and Algebra Goal 3]

PROBLEMBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEMMMLEBLLBLBLELLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMMOOOOOOOOOOBBBBBBLBLBBLBLBLLBLLLLPROPROPROPROPROPROPROPPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROOROROROOOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEELELEELEEEEEEEELLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBLELEEELEMMMMMMMMOOOOOOOOOBLBLBLBLBBLBBBLLOOOOROROROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGLLLLLLLLLLLLLVINVINVINVINNNNVINVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOOLOOLOLOLOO VINVINVVLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLOLOOO VVVVLLLLLLLLLLVVVVVVVVVVOOSOSOOSOSOSOSOSOSOSOOSOSOSOSOSOOOOSOSOSOSOSSOOSOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVLLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING

ELL

EM3cuG6TLG2_561-565_U06L06.indd 563 1/22/11 1:52 PM

Math Boxes LESSON

6�6

Date Time

4. Label the axes of this mystery graph and describea situation it might represent.

x-axis

y-axis

Situation

light turns green. move after the traffic

A car begins toSpeed (mph)

Time (seconds)

1. Solve. Simplify your answers.

a. � 8 � 10�23�

b. 4�12� � 1�

57� �

c. � 7�130� � 5 1�

2530�

�34�

2. Multiply or divide.

a. �10(�14.35) �

b. 4 � 3 � (�5) �

c. � �2�

840

��70�60143.5

3. Triangles JKL and MNO are congruent.

M

O

N

KL

J

MO—––Which side corresponds with JL––?

91–93

178 140

97

Sample answers:

148–153

2�58�

5. Two dice are tossed. Some possible outcomes appear in the table below.

Complete the table.

a. How many equally likely outcomes are there?

b. What is the probability of tossing a multiple of 2 on both dice?

c. What is the probability of tossing a composite number on the first die and a prime number on the second die?

36

�366�, or �

16�

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

�396�, or �

14�

Math Journal 2, p. 224

Student Page

STUDY LINK

6 6 Using Order of Operations

247

Name Date Time

Please Excuse My Dear Aunt SallyParentheses Exponents Multiplication Division Addition Subtraction

Evaluate each expression.

1. 5 6 º 3 2 2. 4 º 9 / 2 ( 4 6)

3. 12

58 º 1

2 2 4. (2.3 7.8) º 4 3

5. 42 7(3 ( 5)) 6. ((2 º 4) 3) º 6 / 2

Evaluate the following expressions for m 3.

7. mm 6 4 8. ((4 11) º 3) / 9 º ( m)

9. m2 ( (m3)) 8 10. 12 º m 5

435 1

10

28

1

33

-15

72

43.4

2021

2312

170

Find each missing number.

11. 3 gal 7 qt 4 gal qt

12. 5 gal 3 qt qt

13. 13 pt qt pt

14. 10 c qt pt

15. 18 qt gal pt441216

233

Practice

Units of Capacity

2 cups (c) 1 pint (pt)

2 pints 1 quart (qt)

4 quarts 1 gallon (gal)

Math Masters, p. 196

Study Link Master Game MasterName Date Time

Name That Number Record Sheet 132

4

Round 1

Target Number: My Cards:

My Solution (number sentence):

Number of cards used:

Round 2

Target Number: My Cards:

My Solution (number sentence):

Number of cards used:

329

Name Date Time

Name That Number Record Sheet 132

4

Round 1

Target Number: My Cards:

My Solution (number sentence):

Number of cards used:

Round 2

Target Number: My Cards:

My Solution (number sentence):

Number of cards used:

329

Math Masters, p. 462

564 Unit 6 Number Systems and Algebra Concepts

2 Ongoing Learning & Practice

▶ Playing Name That Number PARTNER ACTIVITY

(Student Reference Book, p. 329; Math Masters, p. 462)

Students use the order of operations to match a target number. For each round, students write their answers as a single number sentence on the game record sheet. Ask students to organize their number sentences to avoid using parentheses whenever possible.

▶ Math Boxes 6�6

INDEPENDENT ACTIVITY

(Math Journal 2, p. 224)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-8. The skill in Problem 5 previews Unit 7 content.

Writing/Reasoning Have students write a response to the following: Explain how you know that ∠ L and ∠O in Problem 3 are congruent. Sample answer: The two

triangles are congruent and ∠ L and ∠O are corresponding, so ∠ L and ∠O are congruent.

▶ Study Link 6�6

INDEPENDENT ACTIVITY

(Math Masters, p. 196)

Home Connection Students use order of operations to evaluate expressions.

EM3cuG6TLG2_561-565_U06L06.indd 564 1/22/11 1:52 PM

LESSON

6 � 6

Name Date Time

Another Grouping Symbol

197

Cop

yrig

ht ©

Wrig

ht G

roup

/McG

raw

-Hill

Cop

yrig

ht ©

Wrig

ht G

roup

/McG

raw

-Hill

A fraction bar (sometimes referred to as the vinculum) indicates division.

An expression such as (8 � 4) � (5 � 2) can be written as �((85

�

�

42))�.

A fraction bar also acts as a grouping symbol. Both the numerator and the denominator can be treated as if there were parentheses around them. Any operations in the numeratoror the denominator must be performed before the division.

Example: Evaluate 72 � ��249º�

37

��.

72 � ��249º�

37

�� � 72 � �(2(49

º�

37))

� � 72 � �31

62� � 72 � 3 � 24

Evaluate each expression.

1. 20 � �4 �

35

� � 2. �46((56

�

�

74))� �

3. � 4. �6[242�

72�

(79� 3)]

� � 3277(8 � 1) � (42 � 3)���(10 � 7)3

417

A fraction bar (sometimes referred to as the vinculum) indicates division.

An expression such as (8 � 4) � (5 � 2) can be written as �((85

�

�

42))�.

A fraction bar also acts as a grouping symbol. Both the numerator and the denominator can be treated as if there were parentheses around them. Any operations in the numeratoror the denominator must be performed before the division.

Example: Evaluate 72 � ��249º�

37

��.

72 � ��249º�

37

�� � 72 � �(2(49

º�

37))

� � 72 � �31

62� � 72 � 3 � 24

Evaluate each expression.

1. 20 � �4 �

35

� � 2. �46((56

�

�

74))� �

3. � 4. �6[242�

72�

(79� 3)]

� �7(8 � 1) � (42 � 3)���(10 � 7)3

LESSON

6�6

Name Date Time

Another Grouping Symbol

Math Masters, p. 197

Teaching Master

Teaching Master

LESSON

6 � 6

Name Date Time

Order of Operations

Please Excuse My Dear Aunt Sally Parentheses Exponents Multiplication Division Addition Subtraction

Evaluate each expression. Compare your result to a partner’s. If you don’t agree, discuss how you evaluated the expression to decide which result is correct.

1. 26 � 15 º 2 � 6 � 2. 18 � 5 � 102 �

3. 50 � 70 / 2 � 4. 39 � 1 � 24 / 6 �

5. 18 / 3 � (37 � 13) � 6. 10 � 28 � 14 � 5 �

7. 42 � 6 / 6 � 8 � 8. 5 � 32 º 4 / 2 � 2335

756

3685

11350

Math Masters, p. 198

Lesson 6�6 565

3 Differentiation Options

ENRICHMENT

INDEPENDENT ACTIVITY

▶ Exploring a Grouping Symbol 5–15 Min

(Math Masters, p. 197)

To further explore order of operations, students use the fraction bar as a grouping symbol. They evaluate expressions that involve fractions and nested parentheses.

EXTRA PRACTICE PARTNER ACTIVITY

▶ Evaluating Expressions 15–30 Min(Student Reference Book, p. 247; Math Masters, p. 198)

Students practice evaluating numerical expressions involving the order of operations. This page provides practice with order of operations without the distraction of difficult computations.

EM3cuG6TLG2_561-565_U06L06.indd 565 1/22/11 1:52 PM