Table of Contents Activity Name TEKS Description

The Real Number System:

Irrational Numbers

Support

Materials

Suggested pacing: 2 days

Austin ISD Middle School Math

Table of Contents

Activity Name TEKS Description

Introductory Activity Approximating Square Roots

8.2B

Students will use color tiles to model perfect

squares and then use the models to approximate

the value of irrational numbers, such as 10 .

Students also locate the approximation on a

number line.

Carnegie Lesson 1.3 Text

Assignments Skills Practice

8.2B

This lesson explores the decimal expansions of various numbers to develop an understanding for the set of irrational numbers.

http://curriculum.austinisd.org/schoolnetDocs/mathematics/generalResources/MiddleSchool/New_TEKS_Support_Documents/M_8_SupportDocument_8.2B.pdf


© C

arne

gie

Lear

ning

1

© 2

013

Car

negi

e Le

arni

ng

The Real Number System

1.1 So Many Numbers, So Little TimeNumber Sort ............................................................................... 3

1.2 Is It a Bird or a Plane?Rational Numbers .......................................................... 11

1.3 Sew What?Irrational Numbers ....................................................... 19

1.4 Worth 1000 WordsReal Numbers and Their Properties ...............................29

Pi is probably one of

the most famous numbers in all of history.

As a decimal, it goes on and on forever without repeating. Mathematicians have already

calculated trillions of the decimal digits of pi. It really is a fascinating

number. And it's delicious!

462351_CL_C3_CH01_pp1-42.indd   1 23/08/13   11:55 AM462352_C3_TIG_CH01_pp001-042.indd 1 24/08/13 3:19 PM

1A • Chapter 1 The Real Number System

© C

arne

gie

Lear

ning

Chapter 1 OverviewThis chapter extends the understanding of properties of numbers and number systems to include irrational and real numbers.

Lesson TEKS Pacing Highlights

Model

s

Work

ed E

xam

ple

s

Pee

r A

naly

sis

Talk

th

e Ta

lk

Tech

nolo

gy

1.1 Number Sort 8.2.A 1 This lesson provides exposure to all of the different sets of numbers. X

1.2 Rational Numbers 8.2.D 1

This lesson reviews the sets of natural numbers, whole numbers, integers, and rational numbers.

Questions develop student understanding of closure in each set.

X

1.3 Irrational Numbers

8.2.B8.2.D 1

This lesson explores the decimal expansions of various numbers to develop an understanding for the set of irrational numbers.

X X X

1.4

Real Numbers and Their Properties

8.2.A8.2.B8.2.D

1

This lesson defines the real number system, and a Venn diagram shows the relationship between the sets within the set of real numbers.

Questions ask students to identify properties.

X X X

462352_C3_TIG_CH01_pp001-042.indd 1 24/08/13 3:19 PM

© C

arne

gie

Lear

ning

Chapter 1 The Real Number System • 1B

Skills Practice Correlation for Chapter 1

LessonProblem

SetObjectives

1.1 Number Sort1 – 6 Provide a rationale for groups of numbers

7 – 12 List numbers that satisfy a given set

1.2 Rational Numbers

Vocabulary

1 – 6 Write fractions as decimals

7 – 12 Graph rational numbers on number lines and compare the numbers

13 – 18 Identify numbers as natural numbers, whole numbers, or integers, and identify the numbers as open or closed under the operation used

19 – 28 Add, subtract, multiply, and divide rational numbers

1.3 Irrational Numbers

Vocabulary

1 – 6 Convert fractions to decimals and identify decimals as terminating or repeating

7 – 16 Write repeating decimals as fractions

17 – 22 Calculate square roots of perfect squares

23 – 28 Use a calculator to calculate square roots

29 – 34 Estimate square roots

1.4

Real Numbers and Their Properties

Vocabulary

1 – 6 List the positive numbers, integers, irrational numbers, real numbers, and whole numbers given a larger set of numbers

7 – 16 Identify numbers as rational, irrational, integer, whole, or natural

17 – 28 Identify properties represented in problems

462352_C3_TIG_CH01_pp001-042.indd 2 24/08/13 3:19 PM

© C

arne

gie

Lear

ning

Key Terms irrational number

terminating decimal

repeating decimal

bar notation

Learning GoalsIn this lesson, you will:

Identify decimals as terminating or repeating.

Write repeating decimals as fractions.

Identify irrational numbers.

Essential Ideas

• An irrational number is a number that cannot be written in the form a __

b , where a and b are both integers

and b is not equal to 0.

• A repeating decimal is a decimal that has one or more digits repeat indefinitely.

• A terminating decimal is a decimal that has alast digit.

• All rational numbers can be written as terminating or repeating decimals.

• A decimal that is not terminating nor repeating is an irrational number.

• Square roots that are not perfect squares are irrational numbers.

Texas Essential Knowledge and Skills for MathematicsGrade8

(2) Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to:

(B) approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line

(D) order a set of real numbers arising from mathematical and real-world contexts

1.3 Irrational Numbers • 19A

Sew What?Irrational Numbers

462352_C3_TIG_CH01_pp001-042.indd 1 24/08/13 3:20 PM

19B • Chapter 1 The Real Number System

© C

arne

gie

Lear

ning

OverviewThe terms repeating decimal and terminating decimal are introduced. Students rewrite fractions as

repeating decimals. Bar notation is described and used to write repeating decimals. An example of

converting a repeating decimal into a fraction is provided and students will use the example to rewrite

several repeating decimals as fractions.

Students will conclude that square roots of numbers that are not perfect squares have no repeating

patterns of digits, and are therefore irrational numbers.

462352_C3_TIG_CH01_pp001-042.indd 2 24/08/13 3:20 PM

© C

arne

gie

Lear

ning

1.3 Irrational Numbers • 19C

Warm Up

Rewrite each fraction as a decimal.

1. 1 __ 2

1 __ 2

5 0.5

2. 1 __ 4

1 __ 4

5 0.25

3. 1 __ 3

1 __ 3

5 0.3333333...

4. 1 __ 9

1 __ 9

5 0.1111111...

5. How are the decimals of the first two fractions different from the decimals of the second two

fractions?

The decimals in the first two fractions have a last digit. The decimals in the second two

fractions do not have a last digit.

462352_C3_TIG_CH01_pp001-042.indd 3 24/08/13 3:20 PM

19D • Chapter 1 The Real Number System

© C

arne

gie

Lear

ning

462352_C3_TIG_CH01_pp001-042.indd 4 24/08/13 3:20 PM

© C

arne

gie

Lear

ning

1.3 Irrational Numbers • 19

© 2

013

Car

negi

e Le

arni

ng


Sew What?IrrationalNumbers

LearningGoalsInthislesson,youwill:




In 2006, a 60-year-old Japanese man named Akira Haraguchi publicly recited

the first 100,000 decimal places of p from memory.

The feat took him 16 hours to accomplish—from 9 a.m. on a Tuesday morning to

1:30 a.m. the next day.

Every one to two hours, Haraguchi took a break to use the restroom and have a

snack. And he was videotaped throughout the entire process—to make sure he

didn’t cheat!

KeyTerms irrational number

terminating decimal

repeating decimal

bar notation

462351_CL_C3_CH01_pp1-42.indd   19 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 19 24/08/13 3:20 PM

20 • Chapter 1 The Real Number System

© C

arne

gie

Lear

ning

• When converting a fraction to a decimal, where do you place the decimal in the quotient?

• How many 3’s are in the quotient?

• How is the decimal equivalent for 5 __ 6 different than the decimal equivalent for 1 __

3 ?

• Which fraction is larger: 3 __ 8 or 4 ___

10 ?

• How would you describe the pattern in this repeating decimal?

• How many decimal places do you need to determine if there is a pattern?

Problem 1Students rewrite fractions as repeating decimals. Terminating decimals are distinguished from repeating decimals and both are considered rational numbers. Bar notation is used to rewrite repeating decimals. An example of a repeating decimal rewritten as a fraction is given and students use the example to rewrite several repeating decimals as fractions.

20 • Chapter1 The Real Number System©

201

3 C

arne

gie

Lear

ning

Problem 1 RepeatingDecimals

You have worked with some numbers like p that are not rational numbers. For example, √

__ 2 and √

__ 5 are not the square roots of perfect squares and cannot be written in the

form a __ b

, where a and b are both integers.

Even though you often approximate square roots using a decimal, most square roots are

irrational numbers. Because all rational numbers can be written as a __ b

where a and b are

integers, they can be written as terminating decimals (e.g. 1 __ 4

5 0.25) or repeating decimals

(e.g., 1 __ 6

5 0.1666...). Therefore, all other decimals are irrationalnumbers because these

decimals cannot be written as fractions in the form a __ b

where a and b are integers and b is

not equal to 0.

1. Convert the fraction to a decimal by dividing the numerator by the denominator.

Continue to divide until you see a pattern.

0.3333 1 __

3 5 3)

_________ 1

2. Describe the pattern that you observed in Question 1.

1 __ 3

is equal to a decimal with an infinite number of 3s after the decimal point.

3. Order the fractions from least to greatest. Then, convert each fraction to a decimal by

dividing the numerator by the denominator. Continue to divide until you see a pattern.

3 ___ 22

, 2 __ 9

, 9 ___ 11

, 5 __ 6

0.83333 0.2222 a. 5 __

6 5 6)

__________ 5 b. 2 __

9 5 9)

________ 2

0.818181 0.1363636c. 9 ___

11 511)

________ 9 d. 3 ___

22 522)

___________ 3

462351_CL_C3_CH01_pp1-42.indd   20 23/08/13   11:55 AM

GroupingHave students complete Questions 1 through 4 with a partner. Then share the responses as a class.

Share Phase, Questions 1 through 4

• When converting a fraction to a decimal, how do you know how many zeros to add to the dividend?

Support students in connecting the term irrational with its common language use. Share with students the prefix ir as meaning “not” or “opposite.” Note that irrational numbers are those that are the opposite of being clear, having a pattern, or making sense. Connect this with examples of irrational numbers such as pi and non-repeating decimals.

Support students in

462352_C3_TIG_CH01_pp001-042.indd 20 24/08/13 3:20 PM

© C

arne

gie

Lear

ning


Grouping

• Ask a student to read the information following Question 4 aloud. Discuss the definitions and complete Question 5 as a class.

• Ask a student to read the information following Question 5 aloud. Discuss as a class.

Discuss Phase, Question 5

• How can you determine if a decimal has a last digit?

• How many dots are used to indicate a repeating decimal?

• Where is the bar placed to indicate a repeating decimal?

• What are some commonly used fractions that are repeating decimals that you are already familiar with?

• What are some commonly used fractions that are terminating decimals that you are already familiar with?

© 2

013

Car

negi

e Le

arni

ng


4. Explain why these decimal representations are called repeating decimals.

The decimals are called repeating decimals because one or more digits

repeat indefinitely.

A terminatingdecimal is a decimal that has a last digit. For instance, the decimal

0.125 is a terminating decimal because 125 _____ 1000

5 1 __ 8

. 1 divided by 8 is equal to 0.125.

A repeatingdecimalis a decimal with digits that repeat in sets of one or more. You can

use two different notations to represent repeating decimals. One notation shows one set

of digits that repeat with a bar over the repeating digits. This is called barnotation.

1 __ 3

5 0. __

3 7 ___ 22

5 0.3 ___

18

Another notation shows two sets of the digits that repeat with dots to indicate repetition.

You saw these dots as well when describing the number sets in the previous lesson.

1 __ 3

5 0.33… 7 ___ 22

5 0.31818…

5. Write each repeating decimal from Question 2 using both notations.

a. 5 __ 6

5 b. 2 __ 9

5

c. 9 ___ 11

5 d. 3 ___ 22

5

Some repeating decimals represent common fractions, such as 1 __ 3

, 2 __ 3

, and 1 __ 6

, and are used

often enough that you can recognize the fraction by its decimal representation. For most

repeating decimals, though, you cannot recognize the fraction that the decimal represents.

For example, can you tell which fraction is represented by the repeating decimal

0.44… or 0. ___

09 ?

0.8 __

3 0.833…

0. __

2 0.22…

0. ___

81 0.8181…

0.1 ___

36 0.13636…

462351_CL_C3_CH01_pp1-42.indd   21 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 21 24/08/13 3:20 PM


© C

arne

gie

Lear

ning

Grouping

• Ask students to read the worked example on the own. Then discuss the information as a class.

• Have students complete Questions 6 through 8 with a partner. Then share the responses as a class.

Discuss Phase, Worked Example

• Can this method be used to convert all repeating decimals to fractions? Explain.

• Can you rewrite the repeating decimal as a fraction without going through the procedure? Explain.


• How is the number that is written in the numerator of the fraction representing a repeating decimal determined?

• How is the number that is written in the denominator of the fraction representing a repeating decimal determined?


201

3 C

arne

gie

Lear

ning

You can use algebra to determine the fraction that is represented by the repeating

decimal 0.44… . First, write an equation by setting the decimal equal to a variable that

will represent the fraction.

w 5 0.44…

Next, write another equation by multiplying both sides of the equation by a power of 10.

The exponent on the power of 10 is equal to the number of decimal places until the

decimal begins to repeat. In this case, the decimal begins repeating after 1 decimal

place, so the exponent on the power of 10 is 1. Because 1 0 1 5 10, multiply both

sides by 10.

10w 5 4.4…

Then, subtract the first equation from the second equation.

10w 5 4.44…

2w 5 0.44…

9w 5 4

Finally, solve the equation by dividing both sides by 9.

6. What fraction is represented by the repeating decimal 0.44...? 4 __ 9

7. Complete the steps shown to determine the fraction that is represented by 0. ___

09 .

w 5 0.0909… 100w 5 9.0909… 100w 5 9.0909…

2w 5 0.0909…

99w 59

w 5 9 ___ 99

5 1 ___ 11

8. Repeat the procedure above to write the fraction that represents each repeating decimal.

a. 0.55… 5 5 __ 9 b. 0.0505… 5 5 ___ 99

c. 0. ___

12 5 12 ___ 99

5 4 ___ 33

d. 0. ___

36 5 36 ___ 99

5 4 ___ 11

462351_CL_C3_CH01_pp1-42.indd   22 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 22 24/08/13 3:20 PM

© C

arne

gie

Lear

ning


Problem 2Students calculate square roots of perfect squares. Then, students experience radicands that are not perfect squares. Estimation is used to determine square roots. They also estimate the location of square roots on a number line which provides them with a visual representation of the relationship of the square root and the closest perfect squares.



• What methods can be used to determine the square root of a number?

• What is the largest perfect square that you can think of?

© 2

013

Car

negi

e Le

arni

ng


Problem 2 Nobody’sPerfect...UnlessThey’reaPerfectSquare

Recall that a square root is one of two equal factors of a given number. Every positive

number has two square roots: a positive square root and a negative square root.

For instance, 5 is a square root of 25 because (5)(5) 5 25. Also, 25 is a square root of 25

because (25)(25) 5 25. The positive square root is called the principal square root. In this

course, you will only use the principal square root.

The symbol, , is called a radical and it is used to indicate square roots. The radicand

is the quantity under a radical sign.

radicand

This is read as “the square root of 25,” or as “radical 25.”

radical

√25

Remember that a perfect square is a number that is equal to the product of a distinct

factor multiplied by itself. In the example above, 25 is a perfect square because it is equal

to the product of 5 multiplied by itself.

1. Write the square root for each perfect square.

a. √__

1 5 1 b. √__

4 5 2 c. √__

9 5 3

d. √___

16 5 4 e. √___

25 5 5 f. √___

36 5 6

g. √___

49 5 7 h. √___

64 5 8 i. √___

81 5 9

j. √____

100 5 10 k. √____

121 5 11 l. √____

144 5 12

m. √____

169 5 13 n. √____

196 5 14 o. √____

225 5 15

2. What do you think is the value of √__

0 ? Explain your reasoning.

I think the value of √__

0 is 0. I know that 0 multiplied by itself is 0.

462351_CL_C3_CH01_pp1-42.indd   23 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 23 24/08/13 3:20 PM


© C

arne

gie

Lear

ning



• Do you notice any patterns in your answers to Question 4?

• In Question 4, were there any square roots that you could not evaluate?


201

3 C

arne

gie

Lear

ning

3. Notice that the square root of each expression in Question 1 resulted in a rational

number. Do you think that the square root of every number will result in a rational

number? Explain your reasoning.

Answers will vary.

No. I do not think that the square root of any number will result in a rational

number. For instance, √__

2 does not repeat and it goes on forever, so it is an

irrational number.

4. Use a calculator to evaluate each square root. Show each answer to the hundred-

thousandth.

√___

25 5 5 √_____

0.25 5 0.5 √____

250 5 15.81138 . . .

√__

5 5 2.23606 . . . √_____

225 5 cannot evaluate √___

2.5 5 1.58113 . . .

√_____

2500 5 50 √____

676 5 26 √_____

6760 5 82.21921 . . .

√_____

6.76 5 2.6 √_____

67.6 5 8.22192 . . . √______

26.76 5cannot evaluate

5. What do you notice about the square roots of rational numbers?

The square roots of rational numbers are sometimes rational numbers and

sometimes irrational numbers.

6. Is the square root of a whole number always a rational number?

The square root of a whole number is only a rational number when that whole

number is a perfect square.

7. Is the square root of a decimal always an irrational number?

The square root of a decimal is only an irrational number when that decimal is not

equal to a perfect square.

462351_CL_C3_CH01_pp1-42.indd   24 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 24 24/08/13 3:20 PM

© C

arne

gie

Lear

ning


© 2

013

Car

negi

e Le

arni

ng


8. Consider Penelope and Martin’s statements and reasoning, which are shown.

PenelopeI know that 144 is a perfect square, and so √

___ 144 is a rational

number. I can move the decimal point to the left and √____

14.4 and √

____ 1.44 will also be rational numbers.

Likewise, I can move the decimal point to the right so √____

1440 and

√______

14,400 will also be rational numbers.

MartinI know that 144 is a perfect square, and so √


number. I can move the decimal point two places to the right or left to get another perfect square rational number. For instance, √

____ 1.44 and √

______ 14,400 will also be rational numbers.

Moving the decimal two places at a time is like multiplying or dividing by 100. The square root of 100 is 10, which is also a rational number.

Who is correct? Explain your reasoning.

Martin is correct, and Penelope is incorrect. If a number is a perfect square, I can

move the decimal point two places to the right or left to get another perfect

square, which is a rational number.

462351_CL_C3_CH01_pp1-42.indd   25 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 25 24/08/13 3:20 PM


© C

arne

gie

Lear

ning

GroupingAsk a student to read the information about estimating the value of a square root that is not an integer aloud. Discuss the information and worked example as a class.

Discuss Phase, Worked Example

• How do you know what two numbers to choose when beginning to estimate the square root of a number?

• How do you determine what values to try for the tenth’s decimal place?

• Once you have the trial square values narrowed to the closest value less than the number and the closest value greater than the number, how do you determine which one to use as the best estimate for the square root of the number?


201

3 C

arne

gie

Lear

ning

The square root of most numbers is not an integer. You can estimate the square root of a

number that is not a perfect square. Begin by determining the two perfect squares closest

to the radicand so that one perfect square is less than the radicand, and one perfect

square is greater than the radicand. Then, use trial and error to determine the best

estimate for the square root of the number.

“It might be helpful to use the

grid you created in Question 1 to identify the perfect squares.”

So, √___

10

is between √__

0 and √

___ 10 . Why can't

I say it's between √

__ 0 and √

___ 10 ?

10

169

1 25

To estimate √___

10 to the nearest tenth, identify

the closest perfect square less than 10 and

the closest perfect square greater than 10.

The closest The closest

perfect square The square root perfect square

less than 10: you are estimating: greater than 10:

9 √___

10 16

You know:

√__

9 5 3 √___

16 5 4

This means the estimate of √___

10 is between 3 and 4.

Next, choose decimals between 3 and 4, and calculate the square

of each number to determine which one is the best estimate.

Consider: (3.1)(3.1) 5 9.61

(3.2)(3.2) 5 10.24

So, √___

10 3.2

The symbol means approximately equal to.

The location of √___

10 is closer to 3 than 4 when plotted on a

number line.

0 1 2 3 4 5 6 7 8 9 10

√10

462351_CL_C3_CH01_pp1-42.indd   26 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 26 24/08/13 3:20 PM

© C

arne

gie

Lear

ning


Share Phase, Questions 9 through 11Explain how plotting the values on the number line helped in determining what values to consider for the tenth’s decimal place when estimating the value of the square root.


© 2

013

Car

negi

e Le

arni

ng


9. Identify the two closest perfect squares, one greater than the radicand and one less

than the radicand.

a. √__

8

Eight is between the two perfect squares 4 and 9.

b. √___

45

Forty-five is between the two perfect squares 36 and 49.

c. √___

70

Seventy is between the two perfect squares 64 and 81.

d. √___

91

Ninety-one is between the two perfect squares 81 and 100.

10. Estimate the location of each square root in Question 9 on the number line.

Then, plot and label a point for your estimate.

0 1 2 3 4 5 6 7 8 9 10

a b c d

a. The square root of 8 will be located between 2 and 3 because 22 5 4 and 32 5 9.

The square root of 8 will be closer to 3.

b. The square root of 45 will be located between 6 and 7 because 62 5 36 and

72 5 49. The square root of 45 will be closer to 7.

c. The square root of 70 will be located between 8 and 9 because 82 5 64 and


d. The square root of 91 will be located between 9 and 10 because 92 5 81 and


462351_CL_C3_CH01_pp1-42.indd   27 23/08/13   11:55 AM

Guide students in explaining their process for estimating the location of each square root on the number line in Question 10. Prompt students to sequence their explanation by using statements such as “First, . . . ”, “Second, . . .”, “Next, . . .” and “Lastly, . . ..”

Guide students in

462352_C3_TIG_CH01_pp001-042.indd 27 24/08/13 3:20 PM


© C

arne

gie

Lear

ning


201

3 C

arne

gie

Lear

ning

11. Estimate each radical in Question 9 to the nearest tenth. Explain your reasoning.

a. √__

8 2.8. Because (2.8)(2.8) 5 7.84 and (2.9)(2.9) 5 8.41, 2.8 is closer to √__

8 .

b. √___

45 6.7. Because (6.7)(6.7) 5 44.89 and (6.8)(6.8) 5 46.24, 6.7 is closer to √___

45 .

c. √___


70 .

d. √___


91 .

Be prepared to share your solutions and methods.

462351_CL_C3_CH01_pp1-42.indd   28 23/08/13   11:55 AM

462352_C3_TIG_CH01_pp001-042.indd 28 24/08/13 3:20 PM

© C

arne

gie

Lear

ning

1.3 Irrational Numbers • 28A

Follow Up

AssignmentUse the Assignment for Lesson 1.3 in the Student Assignments book. See the Teacher’s Resources

and Assessments book for answers.

Skills PracticeRefer to the Skills Practice worksheet for Lesson 1.3 in the Student Assignments book for additional

resources. See the Teacher’s Resources and Assessments book for answers.

AssessmentSee the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 1.

Check for Students’ Understanding 1. Is √

___ 15 a repeating or terminating decimal? Explain your reasoning.

No. It is neither a repeating nor terminating decimal because it has no last digit and the digits

do not form a pattern.

2. Is √___

15 a rational or irrational number? Explain.

It is an irrational number because it is not a repeating or terminating decimal. Fifteen is not a

perfect square therefore it is an irrational number.

3. Approximate √___

15 to the nearest whole number.

4


15 to the nearest tenth.

3.9


15 to the nearest hundredth.

3.87


15 to the nearest thousandth.

3.873

462352_C3_TIG_CH01_pp001-042.indd 1 24/08/13 3:20 PM

28B • Chapter 1 The Real Number System

© C

arne

gie

Lear

ning

462352_C3_TIG_CH01_pp001-042.indd 2 24/08/13 3:20 PM

Chapter 1 Skills Practice • 309

©

Lear

ning

Car

negi

e


VocabularyMatch each term with the number that represents that term.

1. Irrational number a. 1 __ 2

5 0.5

c

2. Terminating decimal b. 0. __

3

a

3. Repeating decimal c. π

d

4. Bar notation d. 5 __ 9

5 0.555…

b

Problem SetConvert each fraction to a decimal. State whether the fraction is equivalent to a terminating or

repeating decimal.

1. 1 ___ 25

2. 7 __ 9

0.04 0.7 __

7 25 )

_____ 1.00 9 )

_____ 7.00

The fraction 1 ___ 25

is equivalent to The fraction 7 __ 9

is equivalent to

the terminating decimal 0.04. the repeating decimal 0.7 __ 7 .

3. 5 ___ 12

4. 5 __ 8

0.416 __

6 0.62512 )

_______ 5.0000 8 )

______ 5.000


is equivalent to The fraction 5 __ 8

is equivalent to

the repeating decimal 0.416 __

6 . the terminating decimal 0.625.

Lesson 1.3 Skills Practice

Name ________________________________________________________ Date _________________________

462354_C3_Skills_CH01_303-318.indd 309 28/03/13 12:07 PM

310 • Chapter 1 Skills Practice

©

Lear

ning

Car

negi

e

Lesson 1.3 Skills Practice page 2

5. 13 ___ 16

6. 8 ___ 11

0.8125 0. ___

72 16 )

________ 13.0000 11 )

_____ 8.00


is equivalent to The fraction 8 ___ 11

is equivalent to

the terminating decimal 0.8125. the repeating decimal 0. ___

72 .

Write each repeating decimal as a fraction.

7. 0.333 … 8. 0.888 …

10w 5 3.33... 10w 5 8.88...

2w 5 0.33... 2w 5 0.88...

9w 5 3 9w 5 8

w 5 3 __ 9 5 1 __

3 w 5 8 __

9

9. 0.0707 … 10. 0.5454 …

100w 5 7.07... 100w 5 54.54...

2w 5 0.07... 2w 5 0.54...

99w 5 7 99w 5 54

w 5 7 ___ 99

w 5 54 ___ 99

5 6 ___ 11



©

Lear

ning

Car

negi

e


Name ________________________________________________________ Date _________________________

11. 0.1515 … 12. 0.2727 …

100w 5 15.15... 100w 5 27.27...

2w 5 0.15... 2w 5 0.27...

99w 5 15 99w 5 27

w 5 15 ___ 99

5 5 ___ 33

w 5 27 ___ 99

5 3 ___ 11

13. 0.298298 … 14. 0.185185 …

1000w 5 298.298... 1000w 5 185.185...

2w 5 0.298... 2w 5 0.185...

999w 5 298 999w 5 185

w 5 298 ____ 999

w 5 185 ____ 999

5 5 ___ 27

15. 0.67896789… 16. 0.0243902439 …

10,000w 5 6789.6789... 100,000w 5 2439.02439...

2w 5 0.6789... 2w 5 0.02439...

9999w 5 6789 99,999w 5 2439

w 5 6789 _____ 9999

5 2263 _____ 3333

w 5 2439 _______ 99,999

5 1 ___ 41



©

Lear

ning

Car

negi

e


Calculate the square root for each perfect square.

Use a calculator to calculate each square root to the nearest thousandth.

17. √___

25

25 5 5 3 5

√___

25 5 √__

52

5 5

18. √__

9

9 5 3 3 3

√__

9 5 √__

32

5 3

23. √___

36 5 6 24. √___

3.6 5 1.897

25. √____

360 5 18.974 26. √_____

0.36 5 0.6

27. √_____

236 5 cannot evaluate 28. √_____

3600 5 60

19. √___

49

49 5 7 3 7

√___

49 5 √__

72

5 7

20. √____

225

225 5 15 3 15

√____

225 5 √____

152

5 15

21. √____

900

900 5 30 3 30

√____

900 5 √____

302

5 30

22. √____

625

625 5 25 3 25

√____

625 5 √____

252

5 25



©

Lear

ning

Car

negi

e


Name ________________________________________________________ Date _________________________

Estimate each square root to the nearest tenth.

29. √___

14

√__

9 ,√___

14 ,√___

16

√__

32 ,√___

14 ,√__

42

3,√___

14 ,4

(3.7)(3.7)513.69

(3.8)(3.8)514.44

√___

14 <3.7

30. √___

38

√___

36,√___

38,√___

49

√__

62,√___

38,√__

72

6,√___

38,7

(6.1)(6.1)537.21

(6.2)(6.2)538.44

√___

38<6.2

31. √__

7

√__

4,√__

7, √__

9

√__

22, √__

7,√__

32

2,√__

7,3

(2.6)(2.6)56.76

(2.7)(2.7)5 7.29

√__

7<2.6

32. √___

22

√___

16,√___

22,√___

25

√__

42,√___

22,√__

52

4,√___

22,5

(4.6)(4.6)521.16

(4.7)(4.7)522.09

√___

22<4.7

33. √___

93

√___

81, √___

93,√____

100

√__

92, √___

93, √____

102

9,√___

93,10

(9.6)(9.6)592.16

(9.7)(9.7)594.09

√___

93<9.6

34. √____

147

√____

144,√____

147,√____

169

√____

122, √____

147, √____

132

12, √____

147,13

(12.1)(12.1)5146.41

(12.2)(12.2),148.84

√____

147<12.1

462354_C3_Skills_CH01_303-318.indd 313 16/04/13 9:38 AM

©

Lear

ning

Car

negi

e

Chapter 1 Assignments • 5


1. Marcy Green is the manager for her high school softball team. She is in charge of equipment as

well as recording statistics for each player on the team. The table shows some batting statistics

for the four infielders on the team during the first 8 games of the season.

Player At Bats Hits

Brynn Thomas 36 16

Hailey Smith 32 12

Serena Rodrigez 33 11

Kata Lee 35 14

Lesson 1.3 Assignment

Name ________________________________________________________ Date _________________________

462353_C3_Assign_CH01_001-014.indd 5 28/03/13 10:52 AM

©

Lear

ning

Car

negi

e

6 • Chapter 1 Assignments

Lesson 1.3 Assignment page 2

In order to compare the batting averages of the players, Marcy must convert all of the ratios of hits to

at-bats to decimal form. Complete the table to determine the batting averages for each player. When

finding the decimal form, continue to divide until you see a pattern. Write your answer using both dots

and bar notation for repeating decimals.

PlayerFraction of Hits

to At-BatsReduced Fraction

Calculation Batting Average

Brynn Thomas 16 ___ 36

4 __ 9

0.449 )

_____ 4.00

236

40

0.44... or 0. __

4

Hailey Smith 12 ___ 32

3 __ 8

0.3758 )

______ 3.000

224

60 256

40240

0

0.375

Serena Rodriguez 11 ___ 33

1 __ 3

0.333 )

_____ 1.00

29

10

0.33... or 0. __

3

Kata Lee 14 ___ 35

2 __ 5

0.45 )

____ 2.0

220 0

0.4

a. Write the batting averages of the players in order from lowest to highest. Who has the best

batting average so far?

0. __

3 , 0.375, 0.4, 0. __

4

Brynn has the best batting average so far.


©

Lear

ning

Car

negi

e



Name ________________________________________________________ Date _________________________

b. Marcy keeps track of how many home runs each infielder hits on the high school softball team.

For each player, the fraction of home runs per at-bats is given in decimal form. Determine how

many home runs each player has had so far.

Brynn: 0.0 __

5

w 5 0.0555. . . 100w 5 5.55 . . . 100w 5 5.555 . . .

2w 5 0.0555 _____________ 99 w 5 5.5

99w 5 5 1 __ 2

w 5 5 1 __ 2

4 99

w 5 11 ___ 2

? 1 ___ 99

w 5 1 ___ 18

Brynn had 36 at-bats so she had 2 home runs because 2 ___ 36

5 1 ___ 18

.

Hailey: 0.15625

0.15625 5 x ___ 32

32(0.15625) 5 x

5 5 x

Hailey had 5 home runs.

1

9


©

Lear

ning

Car

negi

e



Serena: 0. ___

12

w 5 0.1212... 100w 5 12.1212... 100w 5 12.1212...

2w 5 0.1212... ______________ 99w 5 12

w 5 12 ___ 99

w 5 4 ___ 33

Serena had 4 home runs.

Kata: 0.2

0.2 5 x ___ 35

35(0.2 5 x)

7 5 x

Kata had 7 home runs.


©

Lear

ning

Car

negi

e



Name ________________________________________________________ Date _________________________

2. The distances from each base to the next is 60 feet on the field. The softball “diamond” is a square

shape. A diagram of the field is shown.

2nd

Base

1st

Base3rd

Base

Home

60 ft60 ft

60 ft60 ft

a. Is it possible to exactly measure the distance from home plate to second base? Use the

Pythagorean Theorem by filling in the blanks below. Explain your answer.

a2 1 b2 5 c2

602 1 602 5 c2

3600 1 3600 5 c2

7200 5 c2

√_______

7200 5 c

Yes, it is possible to find the exact distance; because the distance between home plate and

second base is the hypotenuse of a right triangle, and you know the lengths of the other two

sides, you can use the Pythagorean Theorem.


©

Lear

ning

Car

negi

e



b. Marcy uses an estimate of 84 feet for the distance between home plate and second base. One

of the coaches argues that she should be using 85 feet for the distance. Use a calculator to

square all the numbers shown. Who is correct? Explain your answer.

84.02 5 7056 84.12 5 7072.81 84.22 5 7089.64

84.32 5 7106.49 84.42 5 7123.36 84.52 5 7140.25

84.62 5 7157.16 84.72 5 7174.09 84.82 5 7191.04

84.92 5 7208.01 85.02 5 7225

The value of √_____

7200 appears to be somewhere between 84.8 and 84.9. So the coach is more correct in using 85 feet because it is closer to the true distance from home plate to second base.


© C

arne

gie

Lear

ning


Sew What?IrrationalNumbers

LearningGoalsInthislesson,youwill:




In 2006, a 60-year-old Japanese man named Akira Haraguchi publicly recited

the first 100,000 decimal places of p from memory.

The feat took him 16 hours to accomplish—from 9 a.m. on a Tuesday morning to

1:30 a.m. the next day.

Every one to two hours, Haraguchi took a break to use the restroom and have a

snack. And he was videotaped throughout the entire process—to make sure he

didn’t cheat!

KeyTerms irrational number

terminating decimal

repeating decimal

bar notation

462351_CL_C3_CH01_pp001-042.indd   19 28/08/13   2:12 PM

20 • Chapter1 The Real Number System

© C

arne

gie

Lear

ning

Problem 1 RepeatingDecimals

You have worked with some numbers like p that are not rational numbers. For example, √

__ 2 and √

__ 5 are not the square roots of perfect squares and cannot be written in the

form a __ b

, where a and b are both integers.

Even though you often approximate square roots using a decimal, most square roots are

irrational numbers. Because all rational numbers can be written as a __ b

where a and b are

integers, they can be written as terminating decimals (e.g. 1 __ 4

5 0.25) or repeating decimals

(e.g., 1 __ 6 5 0.1666...). Therefore, all other decimals are irrationalnumbers because these

decimals cannot be written as fractions in the form a __ b

where a and b are integers and b is

not equal to 0.

1. Convert the fraction to a decimal by dividing the numerator by the denominator.

Continue to divide until you see a pattern.

1 __ 3

5 3) _________

1

2. Describe the pattern that you observed in Question 1.

3. Order the fractions from least to greatest. Then, convert each fraction to a decimal by

dividing the numerator by the denominator. Continue to divide until you see a pattern.

a. 5 __ 6

5 6) __________

5 b. 2 __ 9

5 9) ________

2

c. 9 ___ 11

511) ________

9 d. 3 ___ 22

522) ___________

3

462351_CL_C3_CH01_pp001-042.indd   20 28/08/13   2:12 PM

© C

arne

gie

Lear

ning


4. Explain why these decimal representations are called repeating decimals.

A terminatingdecimal is a decimal that has a last digit. For instance, the decimal

0.125 is a terminating decimal because 125 _____ 1000

5 1 __ 8 . 1 divided by 8 is equal to 0.125.

A repeatingdecimalis a decimal with digits that repeat in sets of one or more. You can

use two different notations to represent repeating decimals. One notation shows one set

of digits that repeat with a bar over the repeating digits. This is called barnotation.

1 __ 3

5 0. __

3 7 ___ 22

5 0.3 ___

18

Another notation shows two sets of the digits that repeat with dots to indicate repetition.

You saw these dots as well when describing the number sets in the previous lesson.

1 __ 3

5 0.33… 7 ___ 22

5 0.31818…

5. Write each repeating decimal from Question 2 using both notations.

a. 5 __ 6 5 b. 2 __

9 5

c. 9 ___ 11

5 d. 3 ___ 22

5

Some repeating decimals represent common fractions, such as 1 __ 3 , 2 __

3 , and 1 __

6 , and are used

often enough that you can recognize the fraction by its decimal representation. For most

repeating decimals, though, you cannot recognize the fraction that the decimal represents.

For example, can you tell which fraction is represented by the repeating decimal

0.44… or 0. ___

09 ?

462351_CL_C3_CH01_pp001-042.indd   21 28/08/13   2:12 PM


© C

arne

gie

Lear

ning

You can use algebra to determine the fraction that is represented by the repeating

decimal 0.44… . First, write an equation by setting the decimal equal to a variable that

will represent the fraction.

w 5 0.44…

Next, write another equation by multiplying both sides of the equation by a power of 10.

The exponent on the power of 10 is equal to the number of decimal places until the

decimal begins to repeat. In this case, the decimal begins repeating after 1 decimal

place, so the exponent on the power of 10 is 1. Because 1 0 1 5 10, multiply both

sides by 10.

10w 5 4.4…

Then, subtract the first equation from the second equation.

10w 5 4.44…

2w 5 0.44…

9w 5 4

Finally, solve the equation by dividing both sides by 9.

6. What fraction is represented by the repeating decimal 0.44...?

7. Complete the steps shown to determine the fraction that is represented by 0. ___

09 .

8. Repeat the procedure above to write the fraction that represents each repeating decimal.

a. 0.55… 5 b. 0.0505… 5

c. 0. ___

12 5 d. 0. ___

36 5

462351_CL_C3_CH01_pp001-042.indd   22 28/08/13   2:12 PM

© C

arne

gie

Lear

ning


Problem 2 Nobody’sPerfect...UnlessThey’reaPerfectSquare

Recall that a square root is one of two equal factors of a given number. Every positive

number has two square roots: a positive square root and a negative square root.

For instance, 5 is a square root of 25 because (5)(5) 5 25. Also, 25 is a square root of 25

because (25)(25) 5 25. The positive square root is called the principal square root. In this

course, you will only use the principal square root.

The symbol, , is called a radical and it is used to indicate square roots. The radicand

is the quantity under a radical sign.

radicand

This is read as “the square root of 25,” or as “radical 25.”

radical

√25

Remember that a perfect square is a number that is equal to the product of a distinct

factor multiplied by itself. In the example above, 25 is a perfect square because it is equal

to the product of 5 multiplied by itself.

1. Write the square root for each perfect square.

a. √__

1 5 b. √__

4 5 c. √__

9 5

d. √___

16 5 e. √___

25 5 f. √___

36 5

g. √___

49 5 h. √___

64 5 i. √___

81 5

j. √____

100 5 k. √____

121 5 l. √____

144 5

m. √____

169 5 n. √____

196 5 o. √____

225 5

2. What do you think is the value of √__

0 ? Explain your reasoning.

462351_CL_C3_CH01_pp001-042.indd   23 28/08/13   2:12 PM


© C

arne

gie

Lear

ning

3. Notice that the square root of each expression in Question 1 resulted in a rational

number. Do you think that the square root of every number will result in a rational

number? Explain your reasoning.

4. Use a calculator to evaluate each square root. Show each answer to the hundred-

thousandth.

√___

25 5 √_____

0.25 5 √____

250 5

√__

5 5 √_____

225 5 √___

2.5 5

√_____

2500 5 √____

676 5 √_____

6760 5

√_____

6.76 5 √_____

67.6 5 √______

26.76 5

5. What do you notice about the square roots of rational numbers?

6. Is the square root of a whole number always a rational number?

7. Is the square root of a decimal always an irrational number?

462351_CL_C3_CH01_pp001-042.indd   24 28/08/13   2:12 PM

© C

arne

gie

Lear

ning


8. Consider Penelope and Martin’s statements and reasoning, which are shown.

PenelopeI know that 144 is a perfect square, and so √


number. I can move the decimal point to the left and √____

14.4 and √

____ 1.44 will also be rational numbers.

Likewise, I can move the decimal point to the right so √____

1440 and

√______

14,400 will also be rational numbers.

MartinI know that 144 is a perfect square, and so √


number. I can move the decimal point two places to the right or left to get another perfect square rational number. For instance, √

____ 1.44 and √

______ 14,400 will also be rational numbers.

Moving the decimal two places at a time is like multiplying or dividing by 100. The square root of 100 is 10, which is also a rational number.

Who is correct? Explain your reasoning.

462351_CL_C3_CH01_pp001-042.indd   25 28/08/13   2:12 PM


© C

arne

gie

Lear

ning

The square root of most numbers is not an integer. You can estimate the square root of a

number that is not a perfect square. Begin by determining the two perfect squares closest

to the radicand so that one perfect square is less than the radicand, and one perfect

square is greater than the radicand. Then, use trial and error to determine the best

estimate for the square root of the number.

“It might be helpful to use the

grid you created in Question 1 to identify the perfect squares.”

So, √___

10

is between √__

0 and √

___ 10 . Why can't

I say it's between √

__ 0 and √

___ 10 ?

10

169

1 25

To estimate √___

10 to the nearest tenth, identify

the closest perfect square less than 10 and

the closest perfect square greater than 10.

The closest The closest

perfect square The square root perfect square

less than 10: you are estimating: greater than 10:

9 √___

10 16

You know:

√__

9 5 3 √___

16 5 4

This means the estimate of √___

10 is between 3 and 4.

Next, choose decimals between 3 and 4, and calculate the square

of each number to determine which one is the best estimate.

Consider: (3.1)(3.1) 5 9.61

(3.2)(3.2) 5 10.24

So, √___

10 3.2

The symbol means approximately equal to.

The location of √___

10 is closer to 3 than 4 when plotted on a

number line.

0 1 2 3 4 5 6 7 8 9 10

√10

462351_CL_C3_CH01_pp001-042.indd   26 28/08/13   2:12 PM

© C

arne

gie

Lear

ning


9. Identify the two closest perfect squares, one greater than the radicand and one less

than the radicand.

a. √__

8

b. √___

45

c. √___

70

d. √___

91

10. Estimate the location of each square root in Question 9 on the number line.

Then, plot and label a point for your estimate.

0 1 2 3 4 5 6 7 8 9 10

462351_CL_C3_CH01_pp001-042.indd   27 28/08/13   2:12 PM


© C

arne

gie

Lear

ning

11. Estimate each radical in Question 9 to the nearest tenth. Explain your reasoning.

a. √__

8

b. √___

45

c. √___

70

d. √___

91

Be prepared to share your solutions and methods.

462351_CL_C3_CH01_pp001-042.indd   28 28/08/13   2:12 PM


©

Lear

ning

Car

negi

e


VocabularyMatch each term with the number that represents that term.

1. Irrational number a. 1 __ 2

5 0.5

2. Terminating decimal b. 0. __

3

3. Repeating decimal c. π

4. Bar notation d. 5 __ 9

5 0.555…

Problem SetConvert each fraction to a decimal. State whether the fraction is equivalent to a terminating or

repeating decimal.

1. 1 ___ 25

2. 7 __ 9

0.0425 )

_____ 1.00


is equivalent to

the terminating decimal 0.04.

3. 5 ___ 12

4. 5 __ 8

Lesson 1.3 Skills Practice

Name ________________________________________________________ Date _________________________



©

Lear

ning

Car

negi

e


5. 13 ___ 16

6. 8 ___ 11

Write each repeating decimal as a fraction.

7. 0.333 … 8. 0.888 …

10w 5 3.33...

2w 5 0.33...

9w 5 3

w 5 3 __ 9 5 1 __

3

9. 0.0707 … 10. 0.5454 …



©

Lear

ning

Car

negi

e


Name ________________________________________________________ Date _________________________

11. 0.1515 … 12. 0.2727 …

13. 0.298298 … 14. 0.185185 …

15. 0.67896789… 16. 0.0243902439 …



©

Lear

ning

Car

negi

e


Calculate the square root for each perfect square.

Use a calculator to calculate each square root to the nearest thousandth.

17. √___

25

25 5 5 3 5

√___

25 5 √__

52

5 5

18. √__

9

23. √___

36 5 6 24. √___

3.6 5

25. √____

360 5 26. √_____

0.36 5

27. √_____

236 5 28. √_____

3600 5

19. √___

49 20. √____

225

21. √____

900 22. √____

625



©

Lear

ning

Car

negi

e


Name ________________________________________________________ Date _________________________

Estimate each square root to the nearest tenth.

29. √___

14

√__

9 ,√___

14 ,√___

16

√__

32 ,√___

14 ,√__

42

3,√___

14 ,4

(3.7)(3.7)513.69

(3.8)(3.8)514.44

√___

14 <3.7

30. √___

38

31. √__

7 32. √___

22

33. √___

93 34. √____

147

462354_C3_Skills_CH01_303-318.indd 313 16/04/13 9:38 AM

©

Lear

ning

Car

negi

e



1. Marcy Green is the manager for her high school softball team. She is in charge of equipment as

well as recording statistics for each player on the team. The table shows some batting statistics

for the four infielders on the team during the first 8 games of the season.

Player At Bats Hits

Brynn Thomas 36 16

Hailey Smith 32 12

Serena Rodrigez 33 11

Kata Lee 35 14

Lesson 1.3 Assignment

Name ________________________________________________________ Date _________________________


©

Lear

ning

Car

negi

e



In order to compare the batting averages of the players, Marcy must convert all of the ratios of hits to

at-bats to decimal form. Complete the table to determine the batting averages for each player. When

finding the decimal form, continue to divide until you see a pattern. Write your answer using both dots

and bar notation for repeating decimals.

PlayerFraction of Hits

to At-BatsReduced Fraction

Calculation Batting Average

Brynn Thomas

Hailey Smith

Serena Rodriguez

Kata Lee

a. Write the batting averages of the players in order from lowest to highest. Who has the best

batting average so far?


©

Lear

ning

Car

negi

e



Name ________________________________________________________ Date _________________________

b. Marcy keeps track of how many home runs each infielder hits on the high school softball team.

For each player, the fraction of home runs per at-bats is given in decimal form. Determine how

many home runs each player has had so far.

Brynn: 0.0 __

5

Hailey: 0.15625


©

Lear

ning

Car

negi

e



Serena: 0. ___

12

Kata: 0.2


©

Lear

ning

Car

negi

e



Name ________________________________________________________ Date _________________________

2. The distances from each base to the next is 60 feet on the field. The softball “diamond” is a square

shape. A diagram of the field is shown.

2nd

Base

1st

Base3rd

Base

Home

60 ft60 ft

60 ft60 ft

a. Is it possible to exactly measure the distance from home plate to second base? Use the

Pythagorean Theorem by filling in the blanks below. Explain your answer.

a2 1 b2 5 c2

602 1 602 5 c2

3600 1 3600 5 c2

5 c2

√_______

5 c


©

Lear

ning

Car

negi

e



b. Marcy uses an estimate of 84 feet for the distance between home plate and second base. One

of the coaches argues that she should be using 85 feet for the distance. Use a calculator to

square all the numbers shown. Who is correct? Explain your answer.

84.02 5 84.12 5 84.22 5

84.32 5 84.42 5 84.52 5

84.62 5 84.72 5 84.82 5

84.92 5 85.02 5


Approximating Square Roots Introduction

Activity

1 Austin ISD Middle School Math

Standards

Content:

8.2B approximate the value of an irrational number,

including and square roots of numbers less than

225, and locate that rational number approximation

on a number line.

ELPS:

3(E) share information in cooperative learning

interactions

3(D) speak using grade-level content area

vocabulary in context to internalize new English

words and build academic language proficiency;

Objectives

Content Objective:

Students will use color tiles to model perfect

squares and then use the models to approximate

the value of irrational numbers, such as 10 .

Language Objective:

Students will explain using academic vocabulary

how to approximate the value of an irrational

number.

Vocabulary

Vocabulary: square, perfect square, side length, area, approximate, square root, number line Vocabulary Activity: Quick Definitions Provide a definition (orally and/or written on the board) of one of the word wall words. Students choose and write the word to match the definition. Repeat the process encouraging students to review all the words as they select the answer.

Guiding Questions

How does the side length of a square (the square root) compare to the area of the square?

Can you make a square with area 10?

What is the closest square we could make with an area of 10?

Where would the left over color tile (tile 10) fit on the square if it could be cut and placed around the square evenly?

How would that change the length of the side of the square?



Activity


Differentiation

Allow struggling students to use a multiplication chart or a calculator when needed.

Implementation

Materials:

Color tiles

Markers or map colors

Rulers

cm grid paper

Sentence strips

Perfect Squares and Square Roots (Region IV Accelerated Curriculum student page 125)

Estimating Square Roots (Region IV Accelerated Curriculum student page 126)

Preparation:

Prepare 100 color tiles (inch tiles) for each pair or group of 3 students.

Implementation:

Task 1

Distribute Perfect Squares and Square Roots to each pair of students.

Instruct students to use color tiles to complete the sheet.

o How does the side length of a square (the square root) compare to the area of the square?

Distribute cm grid paper to each student and lead them to draw the perfect squares to 15 x 15 =

225. Labeling the area and side lengths (square roots).

o Students will keep this as anchor of support in their notebooks.

o You can do the same on large grid chart paper for a classroom anchor as well. See examples

below.


Activity


Task 2

Bring the class back together after completing the anchor and work together to approximate the square root of 10.

Distribute another sheet of cm grid paper to each student and prompt students to tile this sheet “Approximating Square Roots”.

Using color tiles, build a square with an area of 10 color tiles. o Can it be done? This is why we have to approximate the length of the side of a square with

an area of 10? o What is the closest square we could make with an area of 10? o Where would the left over color tile (tile 10) fit on the square if it could be cut and placed

around the square evenly? o How would that change the length of the side of the square?

On centimeter grid paper and two colors of map colors, draw what you modeled with the tiles for a square with an area of 10 as shown in the diagram below.

How many more tiles are needed to make the next perfect square (16)? 7 Notice that the tenth tile is one of the seven tiles needed to make the next largest perfect square (16).

We can approximate that length of a side of a square with an area of 10 is about 3 and 1/7 because a square with an area of 10 is just a little bit bigger than a square with an area of 9. We can approximate the decimal as 3.1 or 3.2 centimeters.

Task 3

Instruct students to use the centimeter grid paper to draw square models to approximate the length

of the squares on the sheet Estimating Square Roots (Region IV Accelerated Curriculum student

page 126) and then measure with a ruler (centimeter side) to determine if their estimate was

accurate.


Activity


o How close was your estimate to the actual length?

Distribute sentence strips to each pair of students.

Instruct students to create a number line and approximate each square root from Estimating

Square Roots.

Students can keep this anchor in their notebooks as well. You may want to display one for a

classroom anchor.

Free Plain Graph Paper from http://incompetech.com/graphpaper/plain/