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Common Core State Standards

566 Unit 6 Number Systems and Algebra Concepts

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Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 284–294

Key Concepts and Skills• Translate word sentences into

number sentences. [Patterns, Functions, and Algebra Goal 1]

• Identify relation symbols in number sentences. [Patterns, Functions, and Algebra Goal 2]

• Determine whether equalities and inequalities are true or false. [Patterns, Functions, and Algebra Goal 2]

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

Key ActivitiesStudents review the terms, ideas, and word translations of number sentences. They determine whether a number sentence is true or false.

Ongoing Assessment: Recognizing Student Achievement Use Mental Math and Reflexes. [Patterns, Functions, and Algebra Goal 3]

Key Vocabularyrelation symbol � equation � inequality �

operation symbol

MaterialsMath Journal 2, pp. 226 and 227Student Reference Book, p. 241Study Link 6 �6

Solving Challenging Area ProblemsMath Journal 2, p. 227calculatorStudents solve challenging problems involving areas of rectangles.

Math Boxes 6 �7Math Journal 2, p. 225 Students practice and maintain skillsthrough Math Box problems.

Study Link 6 �7Math Masters, p. 199 Students practice and maintain skillsthrough Study Link activities.

READINESS

Ordering OperationsMath Masters, p. 200Students review and apply the order of operations.

ENRICHMENTTranslating Algebraic ExpressionsMath Masters, p. 201Students list and then use various word phrases that can be used to refer to an operation.

EXTRA PRACTICE Solving Custom-Made Math BoxesMath Masters, p. 405Students complete teacher-generated Math Boxes.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Review:Number Sentences

Objectives To review number sentences; and to translate wordsentences into number sentences.s

566_EMCS_T_TLG2_G6_U06_L07_576922.indd 566 3/24/11 1:18 PM

Algebra

Mathematical SymbolsDigits Variables Operation Symbols Relation Symbols Grouping Symbols

0, 1, 2, 3, 4, n x y z � plus � is equal to ( ) parentheses5, 6, 7, 8, 9 a b c d � minus � is not equal to [ ] brackets

C M P ?ppp � or * times � is less than⁄ or � divided by is greater than

is less than or equal to� is greater than or equal to

Equations 3 � 3 � 8 False 3 � 3 � 6, not 8(24 � 3) / 9 � 3 True 27 / 9 � 3100 � 92 � 9 False 92 � 9 � 90, and 90 is not equal to 100

Inequalities �27 * 4 42 False �27 * 4 � �108, which is not greater than 42.�45

� � �23

� � �12

� True �45

� � �23

� � �125� , and �

125� is less than �

12

�.27 � 72 True 27 is not equal to 72.19 � 19 False 19 is not less than itself.16 * 4 � 80 � 3 True 64 is greater than or equal to 26�

23

�.

Number Sentences

Number sentences are made up of mathematical symbols.

A number sentence must contain numbers (or variables) and arelation symbol. It may or may not contain operation symbolsand grouping symbols.

Number sentences that contain the � symbol are calledequations. Number sentences that contain any one of thesymbols �, �, , , or � are called inequalities.

If a number sentence does not contain variables, then it isalways possible to tell whether it is true or false.

True or false?

1. 32 � 14 � 18 2. 4 * 7 � 30 3. 0 � �55�

4. 25 � 5 5 * 6 5. 50 � 12 � 7 * 22 6. 84 � 84Check your answers on page 423.

Student Reference Book, p. 241

Student Page

Lesson 6�7 567

Getting Started

Math MessageIdentify the following symbols.

1. ≠ not equal to 5. < less than3. > greater than 4. = equal5. ≤ less than or equal to 6. ≥ greater than or equal to

Study Link 6�6 Follow-UpBriefly go over the answers with the class.

Mental Math and Reflexes �Students evaluate expressions. Suggestions:

9(15 ÷ 3) 45 84 ÷ 3 ÷ 4 7

14 + 7 ∗ 8 70 6 ∗ 15 + 9 99

45 ÷ 3 - 4 ∗ 2 7 9 ∗ 4 ÷ 12 ∗ 7 - 18 3

Ongoing Assessment: Mental Math and Reflexes �Recognizing Student Achievement

Use Mental Math and Reflexes to assess students’ ability to apply the order of operations in evaluating numerical expressions. Students are making adequate progress if they are able to solve the suggested problems. [Patterns, Functions, and Algebra Goal 3]

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Student Reference Book, p. 241)

Algebraic Thinking Students of Everyday Mathematics have worked with the relation symbols =, <, and > since first grade. They are probably less familiar with the symbols ≠, ≤, and ≥. Present several examples to clarify the meanings of these symbols. Suggestions:

6 + 4 ≠ 7 is a true inequality because the sum of 6 and 4 is 10, not 7.

2 ∗ 5 ≠ 10 is a false inequality because the product of 2 and 5 is 10.

7 ≤ 14 is a true inequality because 7 is less than 14.

7 + 7 ≤ 14 is a true inequality because 7 + 7 is equal to 14.

8 ≥ 2 + 10 is a false inequality because 2 + 10 is not less than 8 nor is it equal to 8.

Mathematical PracticesSMP1, SMP2, SMP3, SMP4, SMP6, SMP8Content Standards6.NS.1, 6.EE.2c, 6.G.1

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Date Time

Number SentencesLESSON

6�7Translate the word sentences below into number sentences. Study the first one.

1. Three times five is equal to fifteen.

2. Nine increased by seven is less than twenty-nine.

3. Thirteen is not equal to nine more than twenty.

4. The product of eight and six is less than or equal to the sum of twenty and thirty.

5. Thirty-seven increased by twelve is greater than fifty decreased by ten.

6. Nineteen is less than or equal to nineteen.

Tell whether each number sentence is true or false.

7. 3 � 21 � 63 8. (3 � 4) � 7 � 19

9. 42 � 12 / 6 5 10. 8 � 7 � 1

11. 24 / 4 � 2 � 8 12. 9 / (8 � 5) 3

13. 21 (7 � 3) � 5 14. 8 � 7 72

15. 63 / 7 � 8 16. 35 � 5 � 8 � 320

Insert parentheses so each number sentence is true.

17. 5 � 8 � 4 � 2 � 42 18. 7 � 9 � 6 � 21

19. 10 � 2 � 6 � 24 20. 9 � 7 / 7 � 8

21. 33 � 24 / 3 � 25 22. 36 / 7 � 2 � 3 � 12

23. 3 � 4 � 3 5 � 3 � 3 24. 48 / 8 � 4 � 100 / 10

3 � 5 � 15

9 � 7 � 2913 � 20 � 9

8 � 6 20 � 30

37 � 12 50 �1019 19

true truetruetruetruefalse

truetruefalsetrue

240 241

( ) ( )( )( )

( ) ( )

( )( )

( )( ( ))

Math Journal 2, p. 226

Student Page

25. Write three true and three false number sentences. Trade journals with yourpartner and determine which sentences are true and which are false.

Number Sentence True or false?

26. The word HOPE is printed in shadedblock letters inside a 15 ft by 5 ftrectangular billboard. What is the area ofthe unshaded portion of the billboard?

27. Square corners, 6 centimeters on a side,are removed from a 36 cm by 42 cmpiece of cardboard. The cardboard is thenfolded to form an open box. What is thesurface area of the inside of the box?

28. Pennies tossed onto the gameboard at the right have an equal chance of landinganywhere on the board. If 60% of thepennies land inside the smaller square,what is the length of a side s of thesmaller square to the nearest inch?

Date Time

Number Sentences continuedLESSON

6�7

240–241

Answers vary.

29 ft2

1,368 cm2

7 in.

6 cm

36 cm

6 cm

42 cm

3' 3' 3' 3'

5'

1'

1'2'

Try This

9 in.s

s

Try the Penny Toss!

9 in.

Math Journal 2, p. 227

Student Page

Adjusting the Activity

Adjusting the Activity

568 Unit 6 Number Systems and Algebra Concepts

Discuss and write on the board the following terms and ideas:

� A number sentence must contain a relation symbol (=, ≠, <, >, ≤, or ≥). A number sentence that contains an equal sign (=) is called an equation. A number sentence that contains any of the other relation symbols is called an inequality. Point out that an expression, such as 15 + 7, is not a number sentence because it does not contain a relation symbol. Ask students to write an equation and an inequality. Select some volunteers to write examples on the board.

� A number sentence may contain one or more operation symbols (+, -, ×, ∗, ÷, or /). Some number sentences do not, for example, 14 ≠ 20, 17 < 22, and x = 5.

� If a number sentence contains only numbers (no variables), it is always possible to tell whether the sentence is true or false.

� A common misconception is that a number sentence must be true—that if it is false, the statement is not a number sentence. Explain that number sentences, like word sentences, may be true or false, but they are still sentences.

Record an example of each symbol, idea, and term on the board. Leave these examples posted so students can refer to them throughout the lesson.

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

Ask students to solve the Check Your Understanding problems on page 241 of the Student Reference Book. Briefly review answers.

▶ Solving Problems Involving PARTNER ACTIVITY

Number Sentences(Math Journal 2, pp. 226 and 227)

Algebraic Thinking Assign Problems 1–25 on journal pages 226 and 227. Remind students to apply the order of operations.

When most students have completed the problems, bring the class together to review the answers. Have volunteers share their translated number sentences.

Have students highlight or underline the inequality symbols before working on Problems 7–16. Ask students to write word sentences for number sentences involving division (Problems 9, 11, 12, 15).

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

ELL

PROBLEMBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEMMMBLLELBLEBLLLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMMOOOOOOOOOOBBBBBBBBLBLBLBLBBLBLBLLLLLPROPROPROPROPROPROPROPROPROPROPROPRPPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROOROROROROOPPPPPPP MMMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEELELEELEEEEEEEELLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBB ELEELEMMMMMMMMMOOOOOOOOOBLBLBBLBLBLBLBBLBLBLROOOROROROROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRGGGLLLLLLLLLLLLLVVINVINVINVINNNVINVINNVINVINVINVINVINV GGGGGGGGGGGOLOOOLOOLOOLOO VINVINVVLLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINV NGGGGGGGGGGOLOLOLOLOLOLOLOOOLO VVVVVVLLLLLLLLLLVVVVVVVVVOSOSOOSOSOSOSOSOSOSOSOSOSOSOOOOOSOOSOSOSOSOSOSOSOSOSOSOOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVVLLLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING

EM3cuG6TLG2_567-570_U06L07.indd 568 1/22/11 1:54 PM

5. Which fraction is equivalent to 2.015?Choose the best answer.

�120,0,01050�

�240,0,03000�

�420030�

�21,00105

�

Date Time

Math Boxes LESSON

6�7

6. You draw one card at random from aregular deck of 52 playing cards (nojokers). What is the chance of drawing:

a. a 4?

b. a card with a prime number?

c. a face card (jack, queen, or king)?

d. an even-numbered

black card?

1. Simplify.

a. �47� of 84

b. �210� of 35

c. �83� of 9�

34�

d. �13� of �

5618�

26

48

3. Give a ballpark estimate for each quotient.

a. 137.8 � 15

b. 248.19 / 12

c. 4,507.08 � 89.76

d. 0.6 / 14.7 0.0450 2010

4. Complete each sentence using an algebraic expression.

a. If each bag of potatoes weighs at least p pounds, then 6 bags weigh at least

pounds.

b. Jack is 6 inches taller than Michael. If Jack is h inches tall, then Michael is

inches tall.h � 6

6p

261 240

59 60 148–153

2. Divide. Simplify if possible.

a. �172� � �

25

� �

b. 2�58

� � 3 �

c. 3 � 6�14

� �

d. 4�12

� � 2�170� �

�1225�

1�12

14�

Sample estimates:

�143�

�113�

�133�

�256�

1�34�

�14�

�78�

1�23�

87–89 93

Math Journal 2, p. 225

Student Page

STUDY LINK

6 � 7 Number Sentences

241–243

Name Date Time

1. a. Draw a circle around each number sentence.

17 � 27 3 º 15 � 100 56 / 8

(5 � 4) º 20 � 20 (4 � 23) / 9 12 � 12

b. Choose one item that you did not circle. Explain why it is not a number sentence.Sample answer: A number sentence must contain a relation symbol. 56 / 8 does not include one.

2. Tell whether each number sentence is true or false.

a. 9 � (6 � 2) 0.5 b. 94 � 49 � 2 º 2

c. �264� � 33 / 11 d. 70 � 25 � 45

3. Insert parentheses to make each number sentence true.

a.(28 � 6)� 9 � 31 b. 20 �(40 � 9)� 11

c.(36 / 6)/ 2 � 12 d. 4 º(8 � 4)� 16

4. Write a number sentence for each word sentence. Tell whether the numbersentence is true or false.

Word sentence Number sentence True or false?

a. If 14 is subtracted from 60,the result is 50.

b. 90 is 3 times as much as 30.

c. 21 increased by 7 is less than 40.

d. The square root of 36 is greater than half of 10. true�36� �

12� º 10

true21 � 7 � 40true90 � 3 º 30false60 � 14 � 50

truefalsefalsetrue

Practice

5. 1.867 � 0.947 � 6. 6 � 2.49 � 7. 256.3 � 4.785 � 251.5153.510.92

Math Masters, p. 199

Study Link Master

Lesson 6�7 569

2 Ongoing Learning & Practice

▶ Solving Challenging Area Problems PARTNER ACTIVITY

(Math Journal 2, p. 227)

Algebraic Thinking Assign Problems 26–28 on journal page 227 to pairs of students. While some students may not be able to solve all the problems, encourage everyone to try. Have students share solution strategies.

Problem 26: Some students may calculate the area of the word HOPE and subtract it from the area of the rectangle. 75 ft2 - 46 ft2, or 29 ft2 Others may add the areas that are not shaded. (1 ∗ 2) + (1 ∗ 2) + (1 ∗ 5) + (1 ∗ 1) + (1 ∗ 5) + (1 ∗ 1) + (3 ∗ 2) + (1 ∗ 3) + (2 ∗ 1) + (2 ∗ 1), or 29 ft2

Problem 27: The surface area of the open box is the same as the area of the original rectangular piece of cardboard (36 cm ∗ 42 cm, or 1,512 cm2) minus the area of the four square corners that are removed (4 ∗ (6 cm ∗ 6 cm), or 144 cm2). 1,512 cm2 - 144 cm2 = 1,368 cm2

Problem 28: The area of the entire gameboard is 81 in2. The area of the smaller square is 60 percent of the area of the entire board, or 48.6 in2. This is almost 49 in2. Therefore, the length of a side of the smaller square is almost 7 in., because 7 ∗ 7 = 49. Some students may notice that they can find the solution quickly by using the square root key on their calculators.

▶ Math Boxes 6�7

INDEPENDENT ACTIVITY

(Math Journal 2, p. 225)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 6-4a and 6-5. The skills in Problems 5 and 6 preview Unit 7 content.

Writing/Reasoning Have students write a response to the following: Write a number story that can be modeled by the number sentence in Problem 2b. Sample answer:

Vanessa uses yarn to make bracelets. She has 2 5 _ 8 feet of yarn and wants to make 3 bracelets. How much yarn can she use for each bracelet?

▶ Study Link 6�7

INDEPENDENT ACTIVITY

(Math Masters, p. 199)

Home Connection Students practice identifying number sentences, inserting parentheses to make true number sentences, identifying true and false number sentences, and translating word sentences into number sentences.

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2 1

The order of operations is shown in the diagram below.

in order, left to right

Use the diagram to help you label which operation you should perform first, second, third, and so on when evaluating an expression.

Example: Label the order in which you should perform the operations to evaluate theexpression 9 / (8 � 5). Then evaluate the expression.

Do the operation inside the parentheses. (8 � 5) � 3

Divide and multiply in order from left to right. 9 / 3 � 312 º 4 � 48

Add and subtract in order from left to right. 3 � 48 � 5151 � 11 � 40

9 / (8 � 5) � 12 º 4 � 11 � 40

For each expression, label the operation you would perform first, second, third, and so on. Then evaluate the expression.

1. 7 º 23 � 2. 6 � 0.3 º 10 �

3. 6 � 4 º 42 � 4. (9 � 1) / 2 º 32 �

5. 14 � 28 / 7 º 2 � 6. 1 º 7 � 5 / 1 � 126

4570

956

4�5

2�3

1

12 4 3 59 / (8 � 5) � 12 º 4 � 11

� or �º or /an( )

13 2

3 2 1

LESSON

6�7

Name Date Time

Ordering Operations

1 3 4 2

1 3 2

2 1

Math Masters, p. 200

Teaching Master

LESSON

6 � 7 Name-Collection Boxes

pyg

gp

Name

Date

Name

Date

Name

Date

Name

Date

Math Masters, p. 201

Teaching Master

570 Unit 6 Number Systems and Algebra Concepts

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Ordering Operations 5–15 Min

(Math Masters, p. 200)

To provide experience applying the order of operations, have students label the order in which operations should be performed to evaluate numeric expressions involving positive numbers.

ENRICHMENT

SMALL-GROUP ACTIVITY

▶ Translating 5–15 Min

Algebraic Expressions(Math Masters, p. 201)

To further explore words used to describe operations, make a table on the board similar to the one shown. Have students list phrases that can be used to suggest addition, subtraction, multiplication, or division.

Operation Word Phrase Operation Word Phrase

Addition

the sum of

Subtraction

the difference ofplus minusadded to subtracted fromincreased by decreased bya total of less than

Multiplication

the product of

Division

the quotient oftimes shared amongmultiplied by divided bydoubled halvedtripled split evenly

Then have students use name-collection boxes (Math Masters, p. 201) to generate word phrases for various algebraic expressions that you assign. (See margin.) Suggestions:

4 + n d - 12 5s w _ 5

EXTRA PRACTICE

INDEPENDENT ACTIVITY

▶ Solving Custom-Made 5–15 Min

Math Boxes(Math Masters, p. 405)

Use Math Masters, page 405 to generate Math Box questions that focus on a particular concept or skill for which students need extra practice.

d – 12

Name-Collection Box for d - 12

The difference ofd and 12

12 subtracted from d

A number decreased by 12

12 less than a number

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