DONE BY: GAYSAN Real MODULE Numbers -...

Real-World Video

ESSENTIAL QUESTION

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How can you use real numbers to solve real-world problems

Real Numbers 1

Get immediate feedback and help as

you work through practice sets

Personal Math Trainer

Interactively explore key concepts to see

how math works

Animated Math

Go digital with your write-in student

edition accessible on any device

Scan with your smart phone to jump directly to the online edition

video tutor and more

Math On the Spot

MODULE

Living creatures can be classified into groups The sea otter belongs to the kingdom Animalia and class Mammalia Numbers can also be classified into groups such as rational numbers and integers

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LESSON 11

Rational and Irrational Numbers

8NS1 8NS2 8EE2

LESSON 12

Sets of Real Numbers8NS1

LESSON 13

Ordering Real Numbers8NS2

Since every rational and irrational number is a real number any real-world problem that can be modeled and solved with rational or irrational numbers can be modeled and solved with real numbers

3

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8_MCAAESE206984_U1MO01indd 3 220513 109 AM

3 Module 1

DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A

DONE BY GAYSAN

YOUAre ReadyPersonal

Math Trainer

Online Practice and Helpmyhrwcom

Complete these exercises to review skills you will need for this module

Find the Square of a NumberEXAMPLE Find the square of 2 _ 3

2 _ 3 times 2 _ 3 = 2 timesthinsp2 ____ 3 timesthinsp3

= 4 _ 9

Find the square of each number

1 7 2 21 3 -3 4 4 _ 5

5 27 6 thinsp- 1 _ 4 7 thinsp-57 8 1 2 _ 5

ExponentsEXAMPLE 5 3 = 5 times 5 times 5

thinsp = 25 times 5 thinsp = 125

Simplify each exponential expression

9 9 2 10 2 4 11 ( 1 _ 3 ) 2 12 (-7) 2

13 4 3 14 (-1) 5 15 45 2 16 10 5

Write a Mixed Number as an Improper FractionEXAMPLE 2 2 _ 5 = 2 + 2 _ 5

thinsp = 10 __ 5 + 2 _ 5

thinsp = 12 __ 5

Write each mixed number as an improper fraction

17 3 1 _ 3 18 1 5 _ 8 19 2 3 _ 7 20 5 5 _ 6

Write the mixed number as a sum of a whole number and a fractionWrite the whole number as an equivalent fraction with the same denominator as the fraction in the mixed numberAdd the numerators

Use the base 5 as a factor 3 timesMultiply from left to right

Multiply the number by itself

Simplify

49 441 9

729

81

64 -1

16

2025 100000

49

16 __ 25

1 _ 9

10 __ 3 13 __ 8 17 __ 7 35 __ 6

1 __ 16 1 24 __ 25 or 1963249

Unit 14

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8_MCAAESE206984_U1MO01indd 4 230513 448 PM

Math TrainerOnline Assessment

and Intervention

Personal

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1

2

3 Response toIntervention

Professional Development

PROFESSIONAL DEVELOPMENT VIDEO

Are You ReadyAssess ReadinessUse the assessment on this page to determine if students need intensive or strategic intervention for the modulersquos prerequisite skills

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Interactive WhiteboardsEngage students with interactive whiteboard-ready lessons and activities

Personal Math Trainer Online Assessment and InterventionAssign automatically graded homework quizzes tests and intervention activitiesPrepare your students with updated practice tests aligned with Common Core

Online Teacher EditionAccess a full suite of teaching resources onlinemdashplan present and manage classes and assignments

ePlannerEasily plan your classes and access all your resources online

Interactive Answers and SolutionsCustomize answer keys to print or display in the classroom Choose to include answers only or full solutions to all lesson exercises

Intervention Enrichment

Access Are You Ready assessment online and receive instant scoring feedback and customized intervention or enrichment

Online and Print Resources

Skills Intervention worksheets

bull Skill 11 Find the Square of a Number

bull Skill 12 Exponents

bull Skill 22 Write a Mixed Number as an Improper Fraction

Differentiated Instruction

bull Challenge worksheets PRE-AP

Extend the Math PRE-AP Lesson Activities in TE

Real-World Video Viewing GuideAfter students have watched the video discuss the following bull What are some different ways that biologists classify animals bull What are some classifications of numbers mentioned in the video natural numbers integers rational numbers

Author Juli Dixon models successful teaching practices as she explores the concept of real numbers in an actual eighth-grade classroom

Real Numbers 4


Reading Start-Up

Active ReadingLayered Book Before beginning the lessons in this module create a layered book to help you learn the concepts in this module Label the flaps ldquoRational Numbersrdquo ldquoIrrational Numbersrdquo ldquoSquare Rootsrdquo and ldquoReal Numbersrdquo As you study each lesson write important ideas such as vocabulary models and sample problems under the appropriate flap

VocabularyReview Words integers (enteros) negative numbers

(nuacutemeros negativos)positive numbers

(nuacutemeros positivos)whole number (nuacutemero

entero)

Preview Words cube root (raiz cuacutebica) irrational numbers (nuacutemero

irracional) perfect cube (cubo

perfecto) perfect square (cuadrado

perfecto) principal square root (raiacutez

cuadrada principal) rational number (nuacutemero

racional) real numbers (nuacutemero real) repeating decimal (decimal

perioacutedico) square root (raiacutez cuadrada) terminating decimal

(decimal finito)

Visualize VocabularyUse the words to complete the graphic You can put more than one word in each section of the triangle

Understand VocabularyComplete the sentences using the preview words

1 One of the two equal factors of a number is a

2 A has integers as its square roots

3 The is the nonnegative square root of a number

Integers

0 83 308

1 45 192

-21 -78 -93

square root

perfect square

principal square root

whole numbers

negative numbers

positive numberswhole numbers

5Module 1

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Reading Start-Up Have students complete the activities on this page by working alone or with others

Strategies for English LearnersEach lesson in the TE contains specific strategies to help English Learners of all levels succeedEmerging Students at this level typically progress very quickly learning to use English for immediate needs as well as beginning to understand and use academic vocabulary and other features of academic language Expanding Students at this level are challenged to increase their English skills in more contexts and learn a greater variety of vocabulary and linguistic structures applying their growing language skills in more sophisticated ways appropriate to their age and grade level Bridging Students at this level continue to learn and apply a range of high-level English language skills in a wide variety of contexts includ-ing comprehension and production of highly technical texts

Active ReadingIntegrating Language ArtsStudents can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary

Additional ResourcesDifferentiated Instruction

bull Reading Strategies EL

EL

After

Students will connect that bull the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat

bull non-rational numbers are called irrational numbers

In this moduleStudents will learn how to bull express a rational number as a decimal bull approximate the value of an irrational number bull describe the relationship between sets of real numbers bull order a set of real numbers arising from mathematical and real-world contexts

Before

Students understand bull write rational numbers as decimals bull describe relationships between sets and subsets of rational numbers

bull compare rational numbers

Tracking Your Learning Progression

Focus | Coherence | Rigor

5 Module 1


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What It Means to YouYou will learn to estimate the values of irrational numbers

What It Means to YouYou will recognize a number as rational or irrational by looking at its fraction or decimal form

Estimate the value of radic_

8

8 is between the perfect squares 4 and 9So radic

_ 8 is between radic

_ 4 and radic

_ 9

radic_

8 is between 2 and 3

8 is closer to 9 so radic_

8 is closer to 3 28 2 = 784 29 2 = 841 radic

_ 8 is between 28 and 29

A good estimate for radic_

8 is 285

Classify each number as rational or irrational

0 _

3 = 1 _ 3 025 = 1 _ 4

These numbers are rational because they can be written as ratios of integers or as repeating or terminating decimals

π asymp 3141592654hellip radic_ 5 asymp 2236067977hellip

These numbers are irrational because they cannot be written as ratios of integers or as repeating or terminating decimals

Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module

Real NumbersGETTING READY FOR

Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number

Key Vocabularyrational number (nuacutemero

racional) A number that can be expressed as a ratio of two integers

irrational number (nuacutemero irracional)A number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal

Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)

EXAMPLE 8NS1

EXAMPLE 8NS2

8NS2

8NS1

Visit myhrwcom to see all CA Common Core Standards explained

8 is not a perfect square Find the two perfect squares closest to 8

Unit 16

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8_MCABESE206984_U1MO01indd 6 102913 1123 PM

GETTING READY FOR

Real NumbersUse the examples on the page to help students know exactly what they are expected to learn in this module

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California Common Core Standards Lesson 11

Lesson 12

Lesson 13

8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number

8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)

8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic

_ 2 is irrational

Go online to see a complete unpacking of the CA Common Core Standards

CA Common Core Standards

Content Areas

The Number Systemmdash8NS

Cluster Know that there are numbers that are not rational and approximate them by rational numbers

Real Numbers 6

DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B

Lesson Support Content Objective Students will learn to rewrite rational numbers and decimals take square roots and

cube roots and approximate irrational numbers

Language Objective Students will show and explain how to rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers

LESSON 11 Rational and Irrational Numbers

Building BackgroundEliciting Prior Knowledge Have students work with a partner to review the relationship between fractions and decimals Ask students to provide an example of writing a fraction or mixed number as a decimal and vice versa Discuss how students chose and wrote their examples

Learning ProgressionsIn this lesson students work with positive rational and irrational numbers They make connections among the real numbers by converting fractions and decimals and approximating irrational numbers Important understandings for students include the following

bull Understand that every number has a decimal expansion bull Convert a repeating decimal to a rational number bull Evaluate square roots of perfect squares and cube roots of perfect cubes

bull Estimate an irrational number

Work with the real number system will continue in this unit as students extend the positive rational and irrational numbers to include negative numbers and compare and order real numbers

Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students ldquoA square garden has an area of 20 square feetrdquo

Have students explain why the side length cannot be rational Then have them approximate the length of each side of the garden to the nearest tenth and hundredth Sample answer The length is the solution to s 2 = 20 radic

_ 20 which is not a rational

number 45 ft 447 ft The length is between 4 and 5 feet 20 is closer to 45 2 than to 44 2 or 46 2 It is also closer to 447 2 than to 446 2 or 448 2

3 _ 4

= 075 1 2 _ 3

= 1 _

6

7 _ 10

= 07 45 = 4 1 _ 2

20 ft 2

California Common Core Standards


8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )


_ 2 is irrational

MP6 Attend to precision

7A


Math Talk

Language Support EL

PROFESSIONAL DEVELOPMENT

Linguistic Support EL

AcademicContent Vocabulary

square ndash In this lesson the word square has multiple meanings which can cause confusion For example to square as in to take the square root of a number is a verb It is different from the nouns square or square of a number The text also refers to perfect square and principal square root of a number and the square root symbol is used These different usages of square as a mathematical term need to be clarified Sentence frames can be used to help define the meaning

To square a number means to _______The perfect square of a number means _______

Background Knowledge

suffixes ndash When added to a root word the suffix -th is used in math to indicate one of a specified number of parts such as tenth hundredth or thousandth Remind students that the suffix -th also indicates place value Note that Spanish Vietnamese Mandarin and other languages do not have the ending th sound so teachers need to enunciate carefully

cognates ndash The words terminating and terminal used in this lesson are cognates in Spanish terminar meaning ldquoto endrdquo or ldquoto finishrdquo A Spanish cognate for approximate is aproximar

Leveled Strategies for English Learners

Emerging Use cards with root words ten hundred and thousand and a card with the -th suffix Have students place them together to show place value Then complete a sentence Use the same procedure to identify decimals

Expanding Support students at this level of English proficiency by providing sentence frames for them to use to describe their mathematical reasoning

To write the fraction _______ as a decimal I _______

Bridging Have students identify different meanings of the term square by matching examples of math problems with a written out sentence frame that defines the usage of the term square to square a number perfect square square root Use this procedure also with the term cube

Be sure to clarify the different uses of the term square when referring to square roots perfect squares and so on

EL

California ELD Standards

Emerging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in basic ways

Expanding 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a growing number of ways to manipulate language

Bridging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a variety of ways to manipulate language

Rational and Irrational Numbers 7B


11L E S S O N


EngageESSENTIAL QUESTION

How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares

Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why

ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text

ExplainEXAMPLE 1

Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10

Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo

YOUR TURNTalk About ItCheck for Understanding

Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125

EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction

bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number

Interactive Whiteboard Interactive example available online

ADDITIONAL EXAMPLE 1Write each fraction as a decimal

A 2 _ 5

04 B 5 _ 9

0 _

5

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ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form

A 0355 71 ___ 200

B 0 _

43 43 __ 99

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CA Common CoreStandards

The student is expected to

The Number Systemmdash8NS1



Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )

Expressions and Equationsmdash8EE2

Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic

_ 2 is irrational

Mathematical Practices

MP6 Precision


the value of expressions (eg

7 Lesson 11


My Notes

Math On the Spotmyhrwcom

Math TrainerOnline Practice

and Help

Personal

myhrwcom

Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers

Write each decimal as a fraction in simplest form

0825

The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction

825 ____ 1000

Then simplify the fraction

825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _

37 has 2 repeating digits so multiply each side of the equation x = 0

_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _

37 you can subtract x from one side and 0 _

37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37

Now solve the equation for x Simplify if necessary

99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B

Write each fraction as a decimal

YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1

To write ldquo825 thousandthsrdquo put 825 over 1000

Divide both the numerator and the denominator by 25

100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37

Divide both sides of the equation by 99

0 _

45 0125 2 _

3

Unit 18

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8_MCAAESE206984_U1M01L1indd 8 120413 838 PM

My Notes

Math On the Spot

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= 033333333333331mdash3

ESSENTIAL QUESTION

Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05

6 can be written as 6 _ 1 05 can be written as 1 _ 2

Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely


1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B

0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1

025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N

11Rational and Irrational Numbers

How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers

8NS1

Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2

8NS1

Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator

Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed

Divide until the remainder is zero or until the digits in the quotient begin to repeat

Add zeros after the decimal point in the dividend as needed

When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)

7Lesson 11

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8_MCABESE206984_U1M01L1indd 7 11113 128 AM


Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number

Integrate Mathematical Practices MP6

This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots

Rational and Irrational Numbers 8


Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0

_ 59 must be entered as

0595959595959 not 059

YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10

EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number

bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1

Focus on Math Connections Make sure students understand the difference in finding radic

_ 121 and solving x 2 = 121 The

symbol radic_

indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite

YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512

Talk About ItCheck for Understanding

Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct

Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises

EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)

bull How do you know whether radic_

2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4

Connect Vocabulary EL

Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers

Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares

1 and 4 used to estimate radic_

7


ADDITIONAL EXAMPLE 3Solve each equation for x

A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

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9 Lesson 11



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9

The solution is 2 _ 5

C

D

729 = x 3

3 radic_ 729 = 3 radic

_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_

x 3 =thinsp 3 radic_ 8 ___ 125

x =thinsp 3 radic_ 8 ___ 125

x = 2 _ 5

Solve each equation for x

YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343

Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares

x 2 = 2 x = plusmn radic_

2


2

Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes

Now take the square root of each number

Simplify the square roots of perfect squares

radic_

2 is between and

A

B

C

8NS2 8EE2

Solve for x by taking the cube root of both sides


Apply the definition of cube root

Think What number cubed equals 729


Think What number cubed equals 8 ____ 125

radic_

2 is irrational

x = plusmn14 x = plusmn 3 __ 16

x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

Unit 110

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8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM


and Help

Personal

myhrwcom

Math On the Spot

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YOUR TURN

Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic

_ 5 indicates the positive

or principal square root

A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9

The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )

3

= 1 __ 27 The symbol 3 radic_ 1 indicates the

cube root

A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5


The solutions are 11 and minus11

The solutions are 4 __ 13 and minus 4 __ 13

EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14

Can you square an integer and get a negative number

What does this indicate about whether negative

numbers have square roots

Math TalkMathematical Practices

8EE2

Solve for x by taking the square root of both sides

Apply the definition of square root

Think What numbers squared equal 121



Think What numbers squared equal 16 ____ 169

3 __ 25 19 __ 33 1 2 _ 5

No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots

9Lesson 11

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Critical ThinkingIn the Explore Activity students estimated the location of radic

_ 2 on a number line Ask students

whether they think that it is possible to locate more precisely the point that represents radic

_ 2 In

other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic

_ 2

is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line

The Modeling note tells one way to do this

ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic

_ 2

Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic

_ 2 on a

number line

Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL

bull Reteach bull Challenge PRE-AP

DIFFERENTIATE INSTRUCTION



ElaborateTalk About ItSummarize the Lesson

Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational

You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero

GUIDED PRACTICEEngage with the Whiteboard

Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a

decimal

Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo

Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10

__ 11 Ask them to describe the patterns they find as a result of these investigations

11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =

Solve each equation for x (Example 3 and Explore Activity)

- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =

Write each fraction or mixed number as a decimal (Example 1)

1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6

Write each decimal as a fraction or mixed number in simplest form (Example 2)

13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216

Approximate each irrational number to one decimal place without a calculator

x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp

19 What is the difference between rational and irrational numbers

CHECK-INESSENTIAL QUESTION

x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325

Rational numbers can be written in the form a __ b where

a and b are integers and b ne 0 Irrational numbers cannot

be written in this form

Unit 112

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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15

Estimate that radic_

2 asymp 15

To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15

1 3 2 = 1 4 2 = 1 5 2 =

Is radic_

2 between 13 and 14 How do you know

Is radic_


2 is closer to than to so radic_

2 asymp

Locate and label this value on the number line

Reflect 11 How could you find an even better estimate of radic

_ 2

12 Find a better estimate of radic_

2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp

Draw a number line and locate and label your estimate

13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place

D

E

F

No 2 is not between 169 and 196

Yes 2 is between 196 and 225

196

19881

19881

225

20164

20164

14

141

20449

169 196 225

Test the squares of numbers between 14 and 15

x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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and Intervention

Personal

Online homework assignment available

myhrwcom

EvaluateGUIDED AND INDEPENDENT PRACTICE

Concepts amp Skills Practice

Example 1Expressing Rational Numbers as Decimals

Exercises 1ndash6 20ndash21 24ndash25

Example 2Expressing Decimals as Rational Numbers


Example 3Finding Square Roots and Cube Roots

Exercises 13ndash15 28 30ndash31 35

Explore ActivityEstimating Irrational Numbers

Exercises 13 16ndash18 29 32ndash34

Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets

Lesson Quiz available online

11 LESSON QUIZ

1 Write as a decimal 2 5 __ 8 1 7 __ 12

2 Write as a fraction 034 1 _

24

3 Solve x 2 = 9 __ 49 for x

4 Solve x 3 = 216 for x

5 Estimate the value of radic_

13 to one decimal place without using a calculator

myhrwcom


Exercise Depth of Knowledge (DOK) Mathematical Practices

20ndash27 2 SkillsConcepts MP4 Modeling

28 3 Strategic Thinking MP4 Modeling

29ndash32 2 SkillsConcepts MP6 Precision

33 3 Strategic Thinking MP7 Using Structure

34 2 SkillsConcepts MP3 Logic

35 2 SkillsConcepts MP4 Modeling

36 3 Strategic Thinking MP3 Logic


38 3 Strategic Thinking MP2 Reasoning

8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area

33 Analyze Relationships To find radic_

15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic

_ 15 must be between 3 and 4 He

thinks a good estimate for radic_

15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low

or correct Explain

34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate

35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer

36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning

37 Make a Conjecture Evaluate and compare the following expressions

radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49

Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots

38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct

FOCUS ON HIGHER ORDER THINKING

His estimate is low because 15 is very close to 16

so radic_

15 is very close to radic_

16 or 4 A better estimate

would be 38 or 39

Sample answer about 45 4 3 = 64 and 5 3 = 125

Because 95 is about halfway between 64 and 125 try 45

45 3 = 91125 which is a good estimate

3 feet

Yes the cube root of a negative number is negative

because a negative number cubed is always negative

and a nonnegative number cubed is always nonnegative

radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49

225 the solutions to x 2 = a are x = plusmn15 and

radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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8_MCABESE206984_U1M01L1indd 14 102913 1142 PM



Name Class Date

Independent Practice11

20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal

21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal

22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal

23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal

24 A heartbeat takes 08 second How many seconds is this written as a fraction

25 There are 262 miles in a marathon Write the number of miles using a fraction

26 The average score on a biology test was 72

_ 1 Write the average score using a

fraction

27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction

28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting

a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have

b Do all of the solutions that you found make sense in the context of the problem Explain

c What is the length of the wood trim needed to go around the painting

Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary

29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings

x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions

x = 20 makes sense but x = -20 doesnrsquot because a

painting cannot have a side length of -20 inches

4 times 20 = 80 inches

x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_

144 = plusmn12 x = plusmn radic_

29 asymp plusmn54

x = 3 radic_ 1331 = 11

4_5 second 26 1_5 mi

72 1 _ 9 101 ___ 200 cent

13Lesson 11

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odisc

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ages



myhrwcomActivity available onlineEXTEND THE MATH PRE-AP

Activity Write radic_

09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic

_ 09 As a hint point out that 09 is close to 10 and so they might

use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic

_ 09 is greater than 09 and less than 10 Try squaring 095 to get

09025 A good estimate for radic_

09 is 095



Integers

Rational Numbers IrrationalNumbers

Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2

Lesson Support Content Objective Students will learn to describe relationships between sets of numbers

Language Objective Students will explain how to describe relationships between sets of real numbers

LESSON 12 Sets of Real Numbers

Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic

_ 2 and -π

Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following

bull Identify all of the possible subsets of the real numbers for a given number

bull Decide whether a statement about a subset of the real numbers is true or false

bull Identify the set of numbers that best describes a real-world situation

Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson

Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers

Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work




MP7 Look for and make use of structure

15A


Math Talk

Language Support EL




Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as

The big oval represents __________The darklight blue color in the middle of the

big ovals represents __________These sets overlap because __________

In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset

Rules and Patterns

Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency

Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning


Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10

Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts

Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson

To help students answer the question posed in Math Talk provide a sentence frame for their answer

The numbers between 31 and 39 on a number line are __________ because __________

EL


Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process

Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process

Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes

Sets of Real Numbers 15B


12L E S S O N

Sets of Real Numbers


How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another

Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers

ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers

ExplainEXAMPLE 1

Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots

bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number

Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers

YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length

EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers

bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers

Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false


ADDITIONAL EXAMPLE 1Write all names that apply to each number

A -10integer rational real

B 12 _ 3

whole integer rational real

myhrwcom


ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice

No integers are whole numbers

False every whole number is also an integer

myhrwcom

Animated MathClassifying Numbers

Students build fluency in classifying numbers in this engaging fast-paced game

myhrwcom




Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices

MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

Personal

myhrwcom


Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false

Tell whether the given statement is true or false Explain your choice

All irrational numbers are real numbers

True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers

No rational numbers are whole numbers

False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers

EXAMPLE 2

A

B

Write all names that apply to each number

1 A baseball pitcher has pitched 12 2 _ 3 innings

2 The length of the side of a square that has an

area of 10 square yards

YOUR TURN


3 All rational numbers are integers

4 Some irrational numbers are integers

YOUR TURN

Give an example of a rational number that is a

whole number Show that the number is both whole

and rational


Give an example of a

8NS1

False Every integer is a rational number but not every

False Real numbers are either rational or irrational numbers

Integers are rational numbers so no integers are irrational numbers

rational real

irrational real

Sample answer 8 8 = 8_

1

and -thinsp 5 _ 2 are not integers

rational number is an integer Rational numbers such as 3 _ 5

Unit 116

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ewire



Math On the Spot

myhrwcom

Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom

Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine

You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers


radic_

5 irrational real

ndash1784rational real


EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N

12Sets of Real Numbers

ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers

Passerines such as the cardinal are also called ldquoperching birdsrdquo

What types of numbers are between 31 and 39 on a

number line


What types of numbers are

8NS1

8NS1

Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number

ndash1784 is a terminating decimal

5 is a whole number that is not a perfect square

radic_

81 _____ 9 = 9 __ 9 = 1 rational irrational real

15Lesson 12

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ons




Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi

a + bi

An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i

i = radic_

-1


This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas

Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations

Sets of Real Numbers 16


YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples

EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number

bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π

Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25

__ π 28 __ π

31 __ π and so on What set of numbers would

best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on

YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6


Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets


Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson

Integrating Language Arts EL

Encourage English learners to ask for clarification on any terms or phrases that they do not understand

Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false


ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice

A the amount of time that has passed since midnight

The set of real numbers time is continuous so the amount of time can be rational or irrational

B the number of tickets sold to a basketball game

The set of whole numbers the number of tickets sold may be 0 or a counting number

myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice

Write all names that apply to each number (Example 1)

1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6

Tell whether the given statement is true or false Explain your choice (Example 2)

9 All whole numbers are rational numbers

10 No irrational numbers are whole numbers

Identify the set of numbers that best describes each situation Explain your choice (Example 3)

11 the change in the value of an account when given to the nearest dollar

12 the markings on a standard ruler

13 What are some ways to describe the relationships between sets of numbers


rational real

rational real

True Whole numbers are rational numbers

Rational numbers the ruler is marked every 1 __ 16 th inch

Sample answer Describe one set as being a subset of

another or show their relationships in a Venn diagram

Integers the change can be a whole dollar amount

and can be positive negative or zero

True Whole numbers are a subset of the set of rational numbers

and can be written as a ratio of the whole number to 1

irrational real



rational real

integer rational real


Unit 118

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My Notes


and Help

Personal

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Math On the Spot

myhrwcom

Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour

Identify the set of numbers that best describes each situation Explain your choice

the number of people wearing glasses in a room

The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number

the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches

The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational

EXAMPLEXAMPLE 3

A

B

Identify the set of numbers that best describes the situation Explain your choice

5 the amount of water in a glass as it evaporates

6 the weight of a person in pounds

YOUR TURN

8NS1

Rational numbers a personrsquos weight can be a decimal

such as 835 pounds

Real numbers the amount can be any number greater

than 0

17Lesson 12

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Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below

Real Numbers

Rational numbers Irrational numbers

Integer numbers

Whole numbers

Ask students to write each number in the list in the correct section of the organizer

Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135







and Intervention

Personal


myhrwcom


12 LESSON QUIZ

1 Write all the names that apply to the number

2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers

3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points

-1 _

5

myhrwcom




Example 1Classifying Real Numbers


Example 2Understanding Sets and Subsets of Real Numbers

Exercises 9ndash10

Example 3Identifying Sets for Real-World Situations

Exercises 11ndash12 20ndash21 25



14ndash19 2 SkillsConcepts MP7 Using Structure


22ndash23 2 SkillsConcepts MP3 Logic

24 1 Recall of Information MP7 Using Structure

25 2 SkillsConcepts MP2 Reasoning

26ndash27 3 Strategic Thinking MP3 Logic

28 3 Strategic Thinking MP8 Patterns


8NS1

8NS1

Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo

Answers1 rational real

2 False radic_

2 is an example of an irrational number between 1 and 2

3 Integers each number is an integer but only three are whole numbers

19 Lesson 12


Work Area

π mi23 Critique Reasoning The circumference of a circular region is shown

What type of number best describes the diameter of the circle Explain

your answer

24 Critical Thinking A number is not an integer What type of number can it be

25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons

26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error

27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain

28 Draw Conclusions The decimal 0 _

3 represents 1 _ 3 What type of number best describes 0

_ 9 which is 3 middot 0

_ 3 Explain

29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this


It can be a rational number that is not an integer or an irrational number

rational number

The set of negative numbers also includes non-integer

rational numbers and irrational numbers

Sample answer If the calculator shows a decimal that

terminates in fewer digits than what the calculator screen

allows then you can tell that the number is rational If not

you cannot tell from the calculator display whether the

number terminates because you see a limited number

of digits It may be a repeating decimal (rational) or

non-terminating non-repeating decimal (irrational)

Whole 3 middot 0 _

3 represents 3 middot 1 _ 3 = 1 so 0 _

9 is exactly 1

Sample answer In decimal form irrational numbers never

terminate and never repeat Therefore no matter how

many decimal places you include the number will never

be precisely represented There are always more digits

Whole the diameter is π _ π = 1 mile

Unit 120

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Integers

Rational Numbers Irrational Numbers

Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date

Independent Practice


20 the height of an airplane as it descends to an airport runway

21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1

22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain

12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16

Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram

8NS1

Real numbers the height can be any number greater than zero

integer rational real whole integer rational real


irrational real

rational real

rational real

Integers the scores are counting numbers their

opposites and zero

Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction

19Lesson 12

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Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999

Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future



-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5

Lesson Support Content Objective Students will learn to order a set of real numbers

Language Objective Students will show and describe how to order a set of real numbers

LESSON 13 Ordering Real Numbers

Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic

_ 5 and -4 1 __ 2 Ask them to explain how

they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms

Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following

bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context

Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system

Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition

Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16

ϕ = 1 + radic_

5 _ 2




MP4 Model with mathematics

21A


Math Talk

Language Support EL




Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms


Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used


Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms

Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives

Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line

To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order

EL

Adjective Comparative Superlative

Far Farther Farthest

Large Larger Largest

Great Greater Greatest

Some Less Least

Some More Most


Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience

Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience

Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience

Ordering Real Numbers 21B


13L E S S O N

Ordering Real Numbers


ADDITIONAL EXAMPLE 1Compare Write lt gt or =

A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt


How do you order a set of real numbers Sample answer Find their approximate decimal values and order them

Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations

ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them

ExplainEXAMPLE 1

Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic

_ 5 and radic

_ 3

The difference between 5 and 3 is 2 the difference between radic_

5 and radic_

3 is approximately 1 So the difference between 5 and 3 is greater

Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal

YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal

EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic

_ 22 is less than or greater than 45 The square of 45 is

2025 which is less than 22 so the square root of 22 must be greater than 45

Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5

YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic

_ 1 radic

_ 4 and radic

_ 9


ADDITIONAL EXAMPLE 2Order 3π radic

_ 10 and 325 from greatest

to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1

Ordering Real Numbers You can compare and order real numbers and list them from least to greatest

Order radic_

22 π + 1 and 4 1 _ 2 from least to greatest

First approximate radic_

22

radic_

22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic

_ 22 so

you can compare it to 4 1 _ 2

Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic

_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304

Since 47 2 = 2209 an approximate value for radic_

22 is 47

An approximate value of π is 314 So an approximate value of π +1 is 414

Plot radic_

22 π + 1 and 4 1 _ 2 on a number line

Read the numbers from left to right to place them in order from least to greatest

From least to greatest the numbers are π + 1 4 1 _ 2 and radic_

22

EXAMPLE 2

STEP 1

STEP 2

Order the numbers from least to greatest Then graph them on the number line

YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75

If real numbers a b and c are in order from least to greatest what is the order

of their opposites from least to greatest

Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10

Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right

Unit 122

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My Notes


and Help

Personal

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Math On the Spot

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Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals

Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_


Next approximate radic_

5

radic_


Then use your approximations to simplify the expressions

radic_

3 + 5 is between 6 and 7

3 + radic_


So radic_

3 + 5 gt 3 + radic_

5

Reflect1 If 7 + radic

_ 5 is equal to radic

_ 5 plus a number what do you know about the

number Why

2 What are the closest two integers that radic_

300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2

Compare Write lt gt or =

YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N

13 Ordering Real Numbers

ESSENTIAL QUESTIONHow do you order a set of real numbers

8NS2


8NS2

Use perfect squares to estimate square roots

1 2 = 1 2 2 = 4 3 2 = 9

The number is 7 both expressions must equal 7 + radic_

5

17 and 18

gt lt

21Lesson 13

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Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic

_ 28 find a whole number whose perfect

square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56

_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy


This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols

Ordering Real Numbers 22



ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest

Joe radic_

18 mm Lisa 13 __ 3 mm

Pablo 46 mm Julien 4π __ 3 mm

Julien 4π __ 3 mm Lisa 13 __ 3 mm

Joe radic_

18 mm Pablo 46 mm

EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic

_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809

bull Explain how to determine which number is greater 5 _

5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater

Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance

YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3

_ 45 or 3450 3

_ 45 or 3455 and why Explain

that 3 _

45 can be written out as 34545hellipMake sure they understand that 3 _

45 is greater than 345 but less than 3455


Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional

number of decimal places


Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal

Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units

myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3

Compare Write lt gt or = (Example 1)

1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_

3 2π and 15 from least to greatest Then graph them on the number line (Example 2)

radic_

3 is between and so radic_

3 asymp

π asymp 314 so 2π asymp

From least to greatest the numbers are

10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)

11 Explain how to order a set of real numbers


Forest Perimeter (km)

Leon Mika Jason Ashley

radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628

Sample answer Convert each number to a decimal

equivalent using estimation to find equivalents for

irrational numbers Graph each number on a number line

Read the numbers from left to right for least to greatest

Read the numbers from right to left for greatest to least

lt gt

lt lt

ltgt

gt gt

24 Unit 1

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Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom

Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context

Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least

Distance Across Quarry Canyon (km)

Juana Lee Ann Ryne Jackson

radic_

28 23 __ 4 5 _

5 5 1 _ 2

Write each value as a decimal

radic_

28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic

_ 28 is 53

23 __ 4 = 575

5 _

5 is 5555hellip so 5 _

5 to the nearest hundredth is 556

5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _

5 and 5 1 _ 2 on a number line

From greatest to least the distances are

23 __ 4 km 5 _

5 km 5 1 _ 2 km radic_

28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2

7 Four people have found the distance in miles across a crater using different methods Their results are given below

Jonathan 10 __ 3 Elaine 3 _

45 Joseacute 3 1 _ 2 Lashonda radic_

10

Order the distances from greatest to least

YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed

Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points







and Intervention

Personal


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13 LESSON QUIZ

1 Compare Write lt gt or =

radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_

105 and 3π + 1 from greatest to least

3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest

KD 11 __ 2 cm Silvio 5 __ 3 π cm

Paula 5 _

4 cm Luis radic_

33 cm



Example 1Comparing Irrational Numbers

Exercises 1ndash8

Example 2Ordering Real Numbers

Exercises 9 12ndash15 18ndash21

Example 3Ordering Real Numbers in a Real-World Context

Exercises 10 16ndash17




12ndash15 1 Recall of Information MP5 Using Tools


17 2 SkillsConcepts MP6 Precision

18ndash21 2 SkillsConcepts MP2 Reasoning



8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105

3 Silvio 5 __ 3 π cm Paula 5 _

4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227

20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right

21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284

a Between which two square roots of integers could you find this number

b Between which two square roots of integers can you find π

22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979

a Label π and the two approximations on the number line

b Which of the two approximations is a better estimate for π Explain

c Find a whole number x so that the ratio x ___ 113 is a better estimate for π

than the two given approximations

23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain

24 Critique Reasoning Jill says that 12 _

6 is less than 1263 Explain her error


radic_

115 115 ___ 11 and 105624

between radic_

7 asymp 265 and radic_

8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316

22 __ 7 it is closer to π on the number line

She did not consider the repeating digit 1266

2 rational numbers can have the same location and

irrational numbers can have the same location but they

cannot share a location

355

Neither student is correct The answer

should be 115 ___ 11 105624 radic_

115

Unit 126

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Stoc

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ockP

hoto

com





Name Class Date


16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters

a Find the area of the square

b Find the area of the circle

c Compare your answers from parts a and b Which garden would give your sister the most space to plant

17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table

a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate

b Winniersquos father estimated the distance across his ranch to be radic_

56 km How does this distance compare to Winniersquos estimate

Give an example of each type of number

18 a real number between radic_

13 and radic_

14

19 an irrational number between 5 and 7

Order the numbers from least to greatest

12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4

Distance Across Fatherrsquos Ranch (km)

1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _

3 asymp 733 7 3 _ 5 = 760 so the average

π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2

4π m2 or approximately 126 m2

They are nearly identical radic_

56 is approximately 74833hellip

The circle would give her more space to plant because it has a

larger area

Sample answer 37

Sample answer radic_

31

25Lesson 13

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Activity available online myhrwcomEXTEND THE MATH PRE-AP

Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following

bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48

bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic

_ 11 and radic

_ 29

bull Between any two rational numbers there is at least one irrational number Sample answer radic


bull Between any two irrational numbers there is at least one irrational number Sample answer radic


_ 11 and radic

_ 29



ReadyMath Trainer

Online Practiceand Help

Personal

myhrwcom

Module Quiz

11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction

1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100

7 Asquarepatiohasanareaof200squarefeetHowlongiseachside

ofthepatiotothenearesttenth

12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number

8 121____radic

____121

9 π__2

10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice

13ensp OrderingenspRealenspNumbersCompare Write lt gt or =

11 radic__

8+3 8+radic__

3 12 radic__

5+11emsp emsp emsp 5+radic___

11


13 radic___

99π29__

8 14 radic___

1__251_40__

2

15 Howarerealnumbersusedtodescribereal-worldsituations

ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875

wholeintegerrationalreal

Trueintegerscanbewrittenasthequotientoftwointegers

SampleanswerRealnumberssuchastherational

π29__

8radic___

99

irrationalreal

lt gt

number1_4candescribeamountsusedincooking

radic___

1__250__

21_4

27Module1

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DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A

8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM


and Intervention

Personal

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1

2



Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment



bull Reteach worksheets


bull Success for English Learners EL



Extend the Math PRE-AP

Lesson Activities in TE

Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes

Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module


Lesson Exercises Common Core Standards

11 1ndash7 8NS1 8NS2 8EE2

12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1



Online Practice and HelpmyhrwcomAssessment Readiness

Module 1 MIXed ReVIeW

1 Look at each number Is the number between 2π and radic___

52

Select Yes or No for expressions AndashC

A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No

2 Consider the number -  11 __ 15

Choose True or False for each statement

A The number is rational True False

B The number can be written as True Falsea repeating decimal

C The number is less than ndash08 True False

3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem

4 A student says that radic___

83 is greater than 29 __ 3 Is the student correct Justify your

reasoning

1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used

to find the edge length in feet To solve the equation

write the mixed number as a fraction greater than 1

x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2

No Sample answer radic___

83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A

8_MCAAESE206984_U1M01RTindd 28 240413 946 AM


Online Assessment and

Interventionmyhrwcom

Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem

Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion

Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom

Assessment Readiness


Items Grade 8 Standards Mathematical Practices

1 8NS2 MP7

2 7NS2b 7NS2d 8NS1 MP7

3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3

Item integrates mixed review concepts from previous modules or a previous course

Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo

Real Numbers 28


YOUAre ReadyPersonal

Math Trainer


Complete these exercises to review skills you will need for this module

Find the Square of a NumberEXAMPLE Find the square of 2 _ 3

2 _ 3 times 2 _ 3 = 2 timesthinsp2 ____ 3 timesthinsp3

= 4 _ 9

Find the square of each number

1 7 2 21 3 -3 4 4 _ 5

5 27 6 thinsp- 1 _ 4 7 thinsp-57 8 1 2 _ 5

ExponentsEXAMPLE 5 3 = 5 times 5 times 5

thinsp = 25 times 5 thinsp = 125

Simplify each exponential expression

9 9 2 10 2 4 11 ( 1 _ 3 ) 2 12 (-7) 2

13 4 3 14 (-1) 5 15 45 2 16 10 5

Write a Mixed Number as an Improper FractionEXAMPLE 2 2 _ 5 = 2 + 2 _ 5

thinsp = 10 __ 5 + 2 _ 5

thinsp = 12 __ 5

Write each mixed number as an improper fraction

17 3 1 _ 3 18 1 5 _ 8 19 2 3 _ 7 20 5 5 _ 6

Write the mixed number as a sum of a whole number and a fractionWrite the whole number as an equivalent fraction with the same denominator as the fraction in the mixed numberAdd the numerators

Use the base 5 as a factor 3 timesMultiply from left to right

Multiply the number by itself

Simplify

49 441 9

729

81

64 -1

16

2025 100000

49

16 __ 25

1 _ 9

10 __ 3 13 __ 8 17 __ 7 35 __ 6

1 __ 16 1 24 __ 25 or 1963249

Unit 14

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8_MCAAESE206984_U1MO01indd 4 230513 448 PM


and Intervention

Personal

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1

2


Professional Development

PROFESSIONAL DEVELOPMENT VIDEO

Are You ReadyAssess ReadinessUse the assessment on this page to determine if students need intensive or strategic intervention for the modulersquos prerequisite skills

myhrwcom

myhrwcom

Interactive WhiteboardsEngage students with interactive whiteboard-ready lessons and activities

Personal Math Trainer Online Assessment and InterventionAssign automatically graded homework quizzes tests and intervention activitiesPrepare your students with updated practice tests aligned with Common Core

Online Teacher EditionAccess a full suite of teaching resources onlinemdashplan present and manage classes and assignments

ePlannerEasily plan your classes and access all your resources online

Interactive Answers and SolutionsCustomize answer keys to print or display in the classroom Choose to include answers only or full solutions to all lesson exercises


Access Are You Ready assessment online and receive instant scoring feedback and customized intervention or enrichment


Skills Intervention worksheets

bull Skill 11 Find the Square of a Number

bull Skill 12 Exponents

bull Skill 22 Write a Mixed Number as an Improper Fraction



Extend the Math PRE-AP Lesson Activities in TE

Real-World Video Viewing GuideAfter students have watched the video discuss the following bull What are some different ways that biologists classify animals bull What are some classifications of numbers mentioned in the video natural numbers integers rational numbers

Author Juli Dixon models successful teaching practices as she explores the concept of real numbers in an actual eighth-grade classroom

Real Numbers 4


Reading Start-Up





entero)








(decimal finito)






Integers

0 83 308

1 45 192

-21 -78 -93

square root

perfect square


whole numbers

negative numbers


5Module 1

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pany








EL

After




Before





5 Module 1


myhrwcom




8



_ 4 and radic

_ 9

radic_






8 is 285


0 _

3 = 1 _ 3 025 = 1 _ 4











EXAMPLE 8NS1

EXAMPLE 8NS2

8NS2

8NS1



Unit 16

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GETTING READY FOR


myhrwcom


Lesson 12

Lesson 13




_ 2 is irrational



Content Areas



Real Numbers 6















3 _ 4

= 075 1 2 _ 3

= 1 _

6

7 _ 10

= 07 45 = 4 1 _ 2

20 ft 2





_ 2 is irrational


7A


Math Talk

Language Support EL















EL







11L E S S O N






ExplainEXAMPLE 1









A 2 _ 5

04 B 5 _ 9

0 _

5

myhrwcom



A 0355 71 ___ 200

B 0 _

43 43 __ 99

myhrwcom









_ 2 is irrational


MP6 Precision



7 Lesson 11


My Notes



and Help

Personal

myhrwcom



0825


825 ____ 1000


825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _


_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _


37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37


99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B


YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1



100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37


0 _

45 0125 2 _

3

Unit 18

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My Notes

Math On the Spot

myhrwcom

= 033333333333331mdash3

ESSENTIAL QUESTION





1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B


025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N



8NS1


8NS1






7Lesson 11

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Com

pany











0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

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9 Lesson 11



and Help

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

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Math On the Spot

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


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myhrwcom



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MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

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EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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04 Ey

ewire



Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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Wiki

med

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mm

ons





a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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My Notes


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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

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MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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Eliss

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Alam

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Imag

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dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Reading Start-Up





entero)








(decimal finito)






Integers

0 83 308

1 45 192

-21 -78 -93

square root

perfect square


whole numbers

negative numbers


5Module 1

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EL

After




Before





5 Module 1


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8



_ 4 and radic

_ 9

radic_






8 is 285


0 _

3 = 1 _ 3 025 = 1 _ 4











EXAMPLE 8NS1

EXAMPLE 8NS2

8NS2

8NS1



Unit 16

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GETTING READY FOR


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Lesson 12

Lesson 13




_ 2 is irrational



Content Areas



Real Numbers 6















3 _ 4

= 075 1 2 _ 3

= 1 _

6

7 _ 10

= 07 45 = 4 1 _ 2

20 ft 2





_ 2 is irrational


7A


Math Talk

Language Support EL















EL







11L E S S O N






ExplainEXAMPLE 1









A 2 _ 5

04 B 5 _ 9

0 _

5

myhrwcom



A 0355 71 ___ 200

B 0 _

43 43 __ 99

myhrwcom









_ 2 is irrational


MP6 Precision



7 Lesson 11


My Notes



and Help

Personal

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0825


825 ____ 1000


825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _


_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _


37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37


99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B


YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1



100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37


0 _

45 0125 2 _

3

Unit 18

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My Notes

Math On the Spot

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= 033333333333331mdash3

ESSENTIAL QUESTION





1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B


025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N



8NS1


8NS1






7Lesson 11

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0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

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9 Lesson 11



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

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Math On the Spot

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




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8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


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MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

Personal

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EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

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radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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ons





a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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My Notes


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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

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Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

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MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


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8



_ 4 and radic

_ 9

radic_






8 is 285


0 _

3 = 1 _ 3 025 = 1 _ 4











EXAMPLE 8NS1

EXAMPLE 8NS2

8NS2

8NS1



Unit 16

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GETTING READY FOR


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Lesson 12

Lesson 13




_ 2 is irrational



Content Areas



Real Numbers 6















3 _ 4

= 075 1 2 _ 3

= 1 _

6

7 _ 10

= 07 45 = 4 1 _ 2

20 ft 2





_ 2 is irrational


7A


Math Talk

Language Support EL















EL







11L E S S O N






ExplainEXAMPLE 1









A 2 _ 5

04 B 5 _ 9

0 _

5

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A 0355 71 ___ 200

B 0 _

43 43 __ 99

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_ 2 is irrational


MP6 Precision



7 Lesson 11


My Notes



and Help

Personal

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0825


825 ____ 1000


825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _


_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _


37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37


99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B


YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1



100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37


0 _

45 0125 2 _

3

Unit 18

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My Notes

Math On the Spot

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= 033333333333331mdash3

ESSENTIAL QUESTION





1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B


025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N



8NS1


8NS1






7Lesson 11

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0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

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9 Lesson 11



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

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Math On the Spot

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




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8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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odisc

Get

ty Im

ages










09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

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radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

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MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

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5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28















3 _ 4

= 075 1 2 _ 3

= 1 _

6

7 _ 10

= 07 45 = 4 1 _ 2

20 ft 2





_ 2 is irrational


7A


Math Talk

Language Support EL















EL







11L E S S O N






ExplainEXAMPLE 1









A 2 _ 5

04 B 5 _ 9

0 _

5

myhrwcom



A 0355 71 ___ 200

B 0 _

43 43 __ 99

myhrwcom









_ 2 is irrational


MP6 Precision



7 Lesson 11


My Notes



and Help

Personal

myhrwcom



0825


825 ____ 1000


825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _


_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _


37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37


99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B


YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1



100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37


0 _

45 0125 2 _

3

Unit 18

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My Notes

Math On the Spot

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= 033333333333331mdash3

ESSENTIAL QUESTION





1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B


025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N



8NS1


8NS1






7Lesson 11

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0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

myhrwcom

9 Lesson 11



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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Phot

odisc

Get

ty Im

ages










09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

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EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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Wiki

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mm

ons





a + bi


i = radic_

-1










__ π 28 __ π

















myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Personal

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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Math Talk

Language Support EL















EL







11L E S S O N






ExplainEXAMPLE 1









A 2 _ 5

04 B 5 _ 9

0 _

5

myhrwcom



A 0355 71 ___ 200

B 0 _

43 43 __ 99

myhrwcom









_ 2 is irrational


MP6 Precision



7 Lesson 11


My Notes



and Help

Personal

myhrwcom



0825


825 ____ 1000


825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _


_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _


37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37


99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B


YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1



100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37


0 _

45 0125 2 _

3

Unit 18

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My Notes

Math On the Spot

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= 033333333333331mdash3

ESSENTIAL QUESTION





1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B


025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N



8NS1


8NS1






7Lesson 11

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pany











0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

myhrwcom

9 Lesson 11



and Help

Personal

myhrwcom

EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

Personal

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EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Imag

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dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


11L E S S O N






ExplainEXAMPLE 1









A 2 _ 5

04 B 5 _ 9

0 _

5

myhrwcom



A 0355 71 ___ 200

B 0 _

43 43 __ 99

myhrwcom









_ 2 is irrational


MP6 Precision



7 Lesson 11


My Notes



and Help

Personal

myhrwcom



0825


825 ____ 1000


825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _


_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _


37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37


99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B


YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1



100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37


0 _

45 0125 2 _

3

Unit 18

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My Notes

Math On the Spot

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= 033333333333331mdash3

ESSENTIAL QUESTION





1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B


025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N



8NS1


8NS1






7Lesson 11

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0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

myhrwcom

9 Lesson 11



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

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Math On the Spot

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

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Integer numbers

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12 LESSON QUIZ




-1 _

5

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Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

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MP4 Modeling


21 Lesson 13



and Help

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

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5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


My Notes



and Help

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0825


825 ____ 1000


825 divide 25 ________ 1000 divide 25 = 33 __ 40

0825 = 33 __ 40

0 _

37

Let x = 0 _

37 The number 0 _


_ 37 by 10 2 or 100

x = 0 _

37

(100)x = 100(0 _

37 )

100x = 37 _

37

Because x = 0 _


37 from the other

100x = 37 _

37

minusx minus0 _

37

99x = 37


99x ___ 99 = 37 __ 99

x = 37 __ 99

EXAMPLE 2

A

B


YOUR TURN

1 5 __ 11 2 1 _ 8 3 2 1 _ 3

8NS1



100 times 0 _

37 is 37 _

37

37 _

37 minus 0 _

37 is 37


0 _

45 0125 2 _

3

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My Notes

Math On the Spot

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= 033333333333331mdash3

ESSENTIAL QUESTION





1 _ 4

1 _ 4 = 025

1 _ 3

1 _ 3 = 0 _

3

EXAMPLEXAMPLE 1

A

B


025 4 ⟌ ⎯ 100 -8 20 -20

0

L E S S O N



8NS1


8NS1






7Lesson 11

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0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

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9 Lesson 11



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

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Math On the Spot

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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Phot

odisc

Get

ty Im

ages










09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


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myhrwcom



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MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

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EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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Wiki

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mm

ons





a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Personal

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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28




0595959595959 not 059






symbol radic_













7



A x 2 = 324 18 -18

B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12

C 343 = x 3 7

D x 3 = 125 ___ 512 5 __ 8

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9 Lesson 11



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

Unit 110

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




Unit 112

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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




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8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


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myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

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EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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Wiki

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ons





a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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My Notes


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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

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MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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Elena

Eliss

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Alam

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Har

cour

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Imag

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dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28



and Help

Personal

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EXPLORE ACTIVITY

lt 2 lt

radic_

lt radic

_ 2 lt

radic_

lt radic

_ 2 lt

The solution is 9


C

D

729 = x 3


_ x 3

3 radic_ 729 = x

9 = x

x 3 = 8 ___ 125

3 radic_



x = 2 _ 5


YOUR TURN

7 x 2 = 196 8 x 2 = 9 ___ 256

9 x 3 = 512 10 x 3 = 64 ___ 343



2


2




radic_

2 is between and

A

B

C

8NS2 8EE2







radic_

2 is irrational


x = 8 x = 4 _ 7

1 2

1 4

1 4

1 2

Unit 110

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YOUR TURN






3


cube root





EXAMPLEXAMPLE 3

A x 2 = 121

x 2 = 121

x = plusmn radic_

121

x = plusmn11

B x 2 = 16 ___ 169

x 2 = 16 ___ 169

x = plusmn radic_

16 ___ 169

x = plusmn 4 __ 13

4 012 5 0 _

57 6 14





8EE2







3 __ 25 19 __ 33 1 2 _ 5


9Lesson 11

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_ 2 In


_ 2




_ 2


_ 2 on a

number line











decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




Unit 112

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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

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radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

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Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

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MP4 Modeling


21 Lesson 13



and Help

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28







decimal




11 Lesson 11


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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Get

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


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myhrwcom



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MP7 Using Structure


15 Lesson 12



and Help

Personal

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and Help

Personal

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EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



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04 Ey

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

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MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Guided Practice

7 0675 8 56 9 044

10 0 _

4

10x =

x =

11 0 _

26

100x =

x =

12 0 _

325

1000x =

x =


- x

-

_______________

x =

- x

-

___________________

x =

- x

-

_______________________

x =


1 2 _ 5 2 8 _ 9 3 3 3 _ 4

4 7 __ 10 5 2 3 _ 8 6 5 _ 6


13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216


x = plusmn radic__

asymp plusmn x = 3

radic__

=

(Explore Activity)

16 radic_

5 asymp

17 radic_

3 asymp

18 radic_

10 asymp



x = plusmn radic__

__________ = plusmn _____

4 _

4

0 _

4

4 99

6216

269

41 25 5

17289

17

22 17 32

04

07

27__40

26 __ 99 325 ___ 999 4 _ 9

11__255 3_5

0 _

8

2375

375

08 _

3

26 _

26

0 _

26

325 _

325

0 _

325

999 325




Unit 112

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11 12 13 14 15

radic2 asymp 14

141 142 143 144 145

radic2 asymp 141

0 1 2 3 4

radic2 asymp 15


2 asymp 15


1 3 2 = 1 4 2 = 1 5 2 =

Is radic_


Is radic_



2 asymp



_ 2


2

1 41 2 = 1 42 2 = 1 43 2 =


2 asymp



D

E

F



196

19881

19881

225

20164

20164

14

141

20449

169 196 225


x = plusmn radic_

7 x asymp plusmn26

11Lesson 11

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28



and Intervention

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11 LESSON QUIZ



24

3 Solve x 2 = 9 __ 49 for x




myhrwcom












8NS1 8NS2 8EE2

8NS1 8NS2 8EE2

Answers1 2625 158

_ 3

2 17 __ 50 1 8 __ 33

3 x = plusmn 3 __ 7

4 x = 6

5 36

13 Lesson 11


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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a + bi


i = radic_

-1










__ π 28 __ π

















myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









and Intervention

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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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myhrwcom

Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Com

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

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Har

cour

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lishin

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pany

Imag

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copy3D

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Work Area






or correct Explain





radic_

4 __ 25 and radic

_ 4 ____

radic_

25 radic

_

16 __ 81 and radic_

16 ____

radic_

81 radic

_

36 __ 49 and radic_

36 ____

radic_

49





so radic_



would be 38 or 39




3 feet




radic_

4 __ 25 = 2 _ 5 = radic

_ 4 ____

radic_

25 radic

_

16 __ 81 = 4 _ 9 = radic_

16 ____

radic_

81 radic

_

36 __ 49 = 6 _ 7 = radic_

36 ____

radic_

49


radic_

a ___

radic_

b = radic

_ a __

b radic

_ a radic

_ b = radic

_ a b

15 - (-15) = 30

Unit 114

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Ilen

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Dona

ldA

lamy I

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es





Name Class Date










fraction







29 x 2 = 14 30 x 3 = 1331

31 x 2 = 144 32 x 2 = 29

8NS1 8NS2 8EE2

04375 in 01 _6

28 km 98 _6 innings





x = plusmn radic_

14 asymp plusmn37

x = plusmn radic_


29 asymp plusmn54

x = 3 radic_ 1331 = 11


72 1 _ 9 101 ___ 200 cent

13Lesson 11

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bull copy

Phot

odisc

Get

ty Im

ages










09 is 095



Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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ons





a + bi


i = radic_

-1










__ π 28 __ π

















myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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My Notes


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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









and Intervention

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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

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ifflin

Har

cour

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lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

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ton

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lin H

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Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Integers


Real Numbers

WholeNumbers

-3-4-5 -2-1 1 2 3 50 4

23

34-4 -π -1 25

radic2





_ 2 and -π












15A


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

copy H

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pany

bull Im

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04 Ey

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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ons





a + bi


i = radic_

-1










__ π 28 __ π

















myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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My Notes


and Help

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Math On the Spot

myhrwcom







EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









and Intervention

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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

copy H

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Com

pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

copy H

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Miff

lin H

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Com

pany








-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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myhrwcom

Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

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lin H

arco

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Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

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ublis

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Com

pany

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age C

redi

ts copy

Elena

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

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arco

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ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

myhrwcom

Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

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ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Math Talk

Language Support EL








Rules and Patterns









EL







12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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bull Im

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04 Ey

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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ons





a + bi


i = radic_

-1










__ π 28 __ π

















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17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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My Notes


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Math On the Spot

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EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Real Numbers


Integer numbers

Whole numbers









and Intervention

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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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and Intervention

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


12L E S S O N






ExplainEXAMPLE 1











B 12 _ 3


myhrwcom





myhrwcom



myhrwcom





MP7 Using Structure


15 Lesson 12



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

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Com

pany

bull Im

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ight

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04 Ey

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Math On the Spot

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Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

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ons





a + bi


i = radic_

-1










__ π 28 __ π

















myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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Com

pany



My Notes


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom







EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

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Com

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Real Numbers


Integer numbers

Whole numbers









and Intervention

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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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Com

pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

copy H

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Com

pany








-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

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hing

Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

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ublis

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Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

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Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

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Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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Com

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and Intervention

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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

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lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

myhrwcom

Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

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urt P

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hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28



and Help

Personal

myhrwcom


and Help

Personal

myhrwcom








EXAMPLE 2

A

B





YOUR TURN




YOUR TURN



and rational



8NS1




rational real

irrational real


1



Unit 116

copy H

ough

ton

Miff

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arco

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ublis

hing

Com

pany

bull Im

age C

redi

ts D

igita

l Im

age c

opyr

ight

copy20

04 Ey

ewire



Math On the Spot

myhrwcom

Vertebrates

Birds

Passerines

Animals

Integers


Real Numbers

WholeNumbers

1

45

3

0

274

67

radic4

-

-3

-2

-1

03

radic2

radic17

radic11-

π

Animated Math

myhrwcom




radic_

5 irrational real



EXAMPLEXAMPLE 1

A

B

C radic_ 81 ____ 9

L E S S O N





number line



8NS1

8NS1




radic_


15Lesson 12

copy H

ough

ton

Miff

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ublis

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Com

pany

bull Im

age C

redi

ts copy

Wiki

med

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mm

ons





a + bi


i = radic_

-1










__ π 28 __ π

















myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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ough

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Com

pany



My Notes


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom







EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

copy H

ough

ton

Miff

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arco

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ublis

hing

Com

pany




Real Numbers


Integer numbers

Whole numbers









and Intervention

Personal


myhrwcom


12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany








-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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pany



My Notes


and Help

Personal

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Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

copy H

ough

ton

Miff

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arco

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and Intervention

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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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arco

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pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

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pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28






__ π 28 __ π

















myhrwcom

17 Lesson 12


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

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ough

ton

Miff

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arco

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Com

pany



My Notes


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom







EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

copy H

ough

ton

Miff

lin H

arco

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Com

pany




Real Numbers


Integer numbers

Whole numbers









and Intervention

Personal


myhrwcom


12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany








-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

copy H

ough

ton

Miff

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arco

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ublis

hing

Com

pany



My Notes


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

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Com

pany











and Intervention

Personal


myhrwcom

myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

copy H

ough

ton

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


1IN

116 inch

Guided Practice


1 7 _ 8 2 radic_

36

3 radic_

24 4 075

5 0 6 - radic_ 100

7 5 _

45 8 - 18 __ 6









rational real

rational real









irrational real



rational real



Unit 118

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany



My Notes


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom







EXAMPLEXAMPLE 3

A

B




YOUR TURN

8NS1


such as 835 pounds


than 0

17Lesson 12

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany




Real Numbers


Integer numbers

Whole numbers









and Intervention

Personal


myhrwcom


12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany








-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany



My Notes


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

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Com

pany











and Intervention

Personal


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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

myhrwcom

Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

copy H

ough

ton

Miff

lin H

arco

urt P

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pany




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myhrwcom

1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28



and Intervention

Personal


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12 LESSON QUIZ




-1 _

5

myhrwcom







Exercises 9ndash10













8NS1

8NS1



2 False radic_



19 Lesson 12


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

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hing

Com

pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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ough

ton

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lin H

arco

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Com

pany








-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

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MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

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ough

ton

Miff

lin H

arco

urt P

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Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

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lin H

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pany

bull Im

age C

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Elena

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ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

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ough

ton M

ifflin

Har

cour

t Pub

lishin

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pany

Imag

e Cre

dits

copy3D

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kiSt

ockP

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

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Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Work Area



your answer








_ 3 Explain




rational number










Whole 3 middot 0 _


9 is exactly 1






Unit 120

copy H

ough

ton

Miff

lin H

arco

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ublis

hing

Com

pany



Integers


Real Numbers

Whole Numbers

257

radic16

166

radic9

128 radic50



Name Class Date






12

14 - radic_

9 15 257

16 radic_

50 17 8 1 _ 2

18 166 19 radic_

16


8NS1




irrational real

rational real

rational real


opposites and zero


19Lesson 12

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ough

ton

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lin H

arco

urt P

ublis

hing

Com

pany








-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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pany



My Notes


and Help

Personal

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Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

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Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









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23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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and Intervention

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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ton

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ublis

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Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


-3-4-5 -2-1 1 2 3 50 4

12-4 -radic5












ϕ = 1 + radic_

5 _ 2





21A


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


and Help

Personal

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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

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Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

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ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

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Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Math Talk

Language Support EL












EL





Some Less Least

Some More Most







13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

copy H

ough

ton

Miff

lin H

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pany



My Notes


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Math On the Spot

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Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

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hing

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pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

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urt P

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pany

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ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

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lishin

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Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

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_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

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1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


13L E S S O N




A radic_

8 - 2 4 - radic_

8 lt

B radic_

20 + 1 3 + radic_

2 gt





ExplainEXAMPLE 1


_ 5 and radic

_ 3


5 and radic_









_ 1 radic

_ 4 and radic

_ 9




to least

3π 325 radic_

10

myhrwcom

myhrwcom






MP4 Modeling


21 Lesson 13



and Help

Personal

myhrwcom


0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


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Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

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Com

pany

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age C

redi

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Elena

Eliss

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Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

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Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

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13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

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pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

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Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28



and Help

Personal

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0 05 1 15 2 25 3 35 4

radic5radic3

π2

8 85 9 95 10 105 11 115 12

radic75

4 42 44 46 48 5

radic224 12π + 1


Order radic_



22

radic_


_ 22 so



_ 22

45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304


22 is 47


Plot radic_




22

EXAMPLE 2

STEP 1

STEP 2


YOUR TURN

5 radic_

5 25 radic_

3

6 π 2 10 radic_

75



Explain


8NS2

radic_

3 radic_

5 25

radic_

75 π2 10


Unit 122

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My Notes


and Help

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myhrwcom

Math On the Spot

myhrwcom


Compare radic_

3 + 5 3 + radic_

5 Write lt gt or =


3

radic_



5

radic_



radic_


3 + radic_


So radic_

3 + 5 gt 3 + radic_

5




number Why


300 is between

EXAMPLEXAMPLE 1

STEP 1

STEP 2


YOUR TURN

3 radic_

2 + 4 2 + radic_

4 4 radic_

12 + 6 12 + radic_

6

L E S S O N



8NS2


8NS2


1 2 = 1 2 2 = 4 3 2 = 9


5

17 and 18

gt lt

21Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany














Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

copy H

ough

ton

Miff

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Com

pany











and Intervention

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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28




Joe radic_

18 mm Lisa 13 __ 3 mm



Joe radic_

18 mm Pablo 46 mm







_ 45 or 3450 3


that 3 _









myhrwcom

23 Lesson 13


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


and Help

Personal

myhrwcom

Math On the Spot

myhrwcom





radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

copy H

ough

ton

Miff

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pany











and Intervention

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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


0 05 1 15 2 25 3 35 4 45 5 55 6 65 7

2πradic3


1 radic_

3 + 2 radic_

3 + 3 2 radic_

8 + 17 radic_

11 + 15

3 radic_

6 + 5 6 + radic_

5 4 radic_

9 + 3 9 + radic_

3

5 radic_

17 - 3 -2 + radic_

5 6 12 - radic_

2 14 - radic_

8

7 radic_

7 + 2 radic_

10 - 1 8 radic_

17 + 3 3 + radic_

11

9 Order radic_


radic_


3 asymp








radic_

17 - 2 1 +thinsp π __ 2 12 ___ 5 25

Guided Practice

17

15

1 + π _ 2 km 25 km 12 __ 5 km radic_

17 - 2 km

2π radic

_ 3

18 175

628






lt gt

lt lt

ltgt

gt gt

24 Unit 1

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Miff

lin H

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urt P

ublis

hing

Com

pany

bull Im

age C

redi

ts copy

Elena

Eliss

eeva

Alam

y Im

ages



My Notes

5 52 54 56 58 6

radic28 5 12

23455


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Math On the Spot

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radic_

28 23 __ 4 5 _

5 5 1 _ 2


radic_


_ 28 is 53

23 __ 4 = 575

5 _



5 1 _ 2 = 55

Plot radic_

28 23 __ 4 5 _



23 __ 4 km 5 _


28 km

EXAMPLEXAMPLE 3

STEP 1

STEP 2




10


YOUR TURN

8NS2

3 1 _ 2 mi 3 _

45 mi 10 __ 3 mi radic_

10 mi

23Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

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pany











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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


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Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

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pany




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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28



and Intervention

Personal


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myhrwcom


13 LESSON QUIZ


radic_

95 - 5 radic_

62 - 2

2 Order 105 radic_




Paula 5 _

4 cm Luis radic_

33 cm




Exercises 1ndash8














8NS2

8NS2

Answers1 radic

_ 95 - 5 lt radic

_ 62 - 2

2 radic_

105 3π + 1 105


4 cm

KD 11

__ 2 cm Luis radic_

33 cm

25 Lesson 13


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

myhrwcom

Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany




and Intervention

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myhrwcom

1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


Work Area

3140 3141 3142 3143

314 π227














radic_

115 115 ___ 11 and 105624

between radic_


8 asymp 283

between radic_

9 = 3 and radic_

10 asymp 316






355


should be 115 ___ 11 105624 radic_

115

Unit 126

copy H

ough

ton M

ifflin

Har

cour

t Pub

lishin

g Com

pany

Imag

e Cre

dits

copy3D

Stoc

kiSt

ockP

hoto

com





Name Class Date












13 and radic_

14



12 radic_

7 2 radic_

8 ___ 2 13 radic_

10 π 35

14 radic_

220 -10 radic_

100 115 15 radic_

8 -375 3 9 _ 4


1 2 3 4

radic_

60 58 __ 8 7 _

3 7 3 _ 5

138NS2

radic_

8 ___ 2 2 radic_

7

-10 radic_

100 115 radic_

220

radic_

60 asymp 775 58 __ 8 = 725 7 _


π radic_

10 35

-375 9 _ 4 radic_

8 3

is 74825 km

1225 m2





larger area

Sample answer 37


31

25Lesson 13

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany







_ 11 and radic

_ 29





_ 11 and radic

_ 29



ReadyMath Trainer


Personal

myhrwcom

Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany




and Intervention

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1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


ReadyMath Trainer


Personal

myhrwcom

Module Quiz


1 7__20 2 1___

27 3 17_8


4 x2=81 5 x3=343 6 x2= 1___100




8 121____radic

____121

9 π__2



11 radic__

8+3 8+radic__

3 12 radic__


11


13 radic___

99π29__

8 14 radic___

1__251_40__

2


ESSENTIAL QUESTION

035

9-9

141ft

7 1__10- 1__10

14__11 1875




π29__

8radic___

99

irrationalreal

lt gt


radic___

1__250__

21_4

27Module1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany




and Intervention

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myhrwcom

1

2


















12 8ndash10 8NS1

13 11ndash14 8NS2

27 Unit 1 Module 1






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28






52


A 6 2 _ 3 Yes No

B 5π __ 2 Yes No

C 3 radic__

5 Yes No









reasoning






83 asymp 91 and 29 __ 3 = 9

__ 6

Because 91 lt 9 __

6 radic___

83 lt 29 __ 3

28 Unit 1

copy H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany












1 8NS2 MP7


3 8EE2 MP1 MP4

4 8NS1 8NS2 MP3



Real Numbers 28


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