phm07c3 te 0301.fm page 106 thursday, may 25, 2006 6:03...

106

1. Plan

3-1

106 Chapter 3 Real Numbers and the Coordinate Plane

3-1

Why Learn This?Not every situation can be modeled using

the four basic operations. For example,

you need square roots to relate the time

and distance a skydiver falls.

A number that is the square of a whole

number is a perfect square. The square root

of a number is another number that

when multiplied by itself is equal to the

given number.

In the diagram at the right, 16 square tiles form a square

with 4 tiles on each side. Since and

16 has two square roots, 4 and Since 16 is a

perfect square.

Finding Square Roots of Perfect Squares

Find the two square roots of 25.

and

The square roots of 25 are 5 and

1. Find the square roots of each number.

a. 36 b. 1 c.

The symbol means the square root of a number. In this book,

means the positive square root, unless stated otherwise. So

means the positive square root of 9, or 3, and means the opposite

of the positive square root of 9, or

4 4 16� � � � �4 4 16� ( ) ,

�4. 4 162 � ,

42 = 16

n n2

0

1

2

3

4

5

6

7

8

9

10

11

12

0

1

4

9

16

25

36

49

64

81

100

121

144

Perfect Squares

5 5 25� � � � �5 5 25� ( )

�5.

116

9

� 9

�3.

Exploring Square Roots and Irrational Numbers

1. Vocabulary Review In a power, the tells how many times a base is used as a factor.

Evaluate the expression x2 for each value of x.

2. 2 3.

4. 5. 10

Lesson 2–7

9

�2

�6

6, 6� 1, 1� 14

14, �

4

exponent

4

36 100

New Vocabulary perfect square, square root, irrational numbers,

real numbers

To find and estimate square roots and to classify numbers as rational

or irrational

What You’ll Learn

Objective

To find and estimate square roots and to classify numbers as rational or irrational

Examples

1

Finding Square Roots of Perfect Squares

2

Estimating a Square Root

3

Application: Skydiving

4

Classifying Real Numbers

Math Understandings:

p. 104C

Math Background

The equation has two possible solutions: and

However, there is only one solution to By convention, the radical sign, means the nonnegative square root only.

Nonnegative

describes all positive numbers and zero.

More Math Background

p. 104C

Lesson Planning and

Resources

See p. 104E for a list of the resources that support this lesson.

x2 9�x � 3

x � �3.x x� �9 3, .

,

Bell Ringer Practice

Check Skills You’ll Need

Use student page, transparency, or PowerPoint. For intervention, direct students to:

Powers and Exponents

Lesson 2-7Extra Skills and Word Problems

Practice, Ch. 2

Special Needs

Students draw a square and a square on grid paper. They count the square units.

9 and 36

Then they try to draw a square with 6 square units. Elicit the fact that some numbers cannot be drawn as perfect whole-number squares.

L13 3� 6 6�

learning style: visual

Below Level

Students are asked to notice what the sign should be in multiplications such as the ones below. Then, students identify which can be re-written as a number squared, and rewrite them.

positive, negative

learning style: visual

L2

( )( )� �2 2 (( 22))22� ( )( )�5 5

phm07c3_te_0301.fm 06 Thursday, May 25, 2006 6:03 PM

107

3-1 Exploring Square Roots and Irrational Numbers 107

For: Square Roots ActivityUse: Interactive

Textbook, 3-1

For help in using formulas, go to Lesson 2-6, Example 1.

To estimate the square root of a number that is not a perfect square,

use the square root of the nearest perfect square.

Estimating a Square Root

Estimate the value of to the nearest integer.

Since 28 is closer to 25 than it is to 36, is closer to 5 than to 6.

You can write

2. Estimate the value of to the nearest integer.

Finding a number’s square root is the inverse operation of finding the

number’s square. So

Application: Skydiving

The formula represents the approximate distance d in feet a

skydiver falls in t seconds before opening the parachute. The formula

assumes there is no air resistance. Find the time a skydiver takes to fall

816 feet before opening the parachute.

The skydiver takes about 7.1 seconds to fall 816 feet.

3. Find the time a skydiver takes to fall each distance. Round to the

nearest tenth of a second.

a. 480 ft b. 625 ft

Irrational numbers are numbers that cannot be written in the form

where a is any integer and b is any nonzero integer. Rational and

irrational numbers form the set of real numbers.

28

5 6

25� 28� 36�

28

28 5� .

38

3 32 � .

d t� 16 2

t2 51

d Use the formula for distance and time.

d Substitute 816 for d.

d Divide each side by 16 to isolate t.

d Simplify.

d Find the positive square root of each side.

d Use a calculator.

d Round to the nearest tenth.

d � 16t2

816 � 16t2

� t2

51 � t2

51

7.1 � t

81616

�

7 . 14 1428429

� �

ab,

6

5.5 s 6.3 s

Activity Lab

Use before the lesson.

Teaching Resources

Activity Lab 3-1:

Powerful Patterns

Guided Instruction

Example 1

To help students recognize perfect squares, have them make a table showing the squares of integers from 2 through 25.

Example 2

Remind students that the symbol means

approximately equal to.

Error Prevention!

Students may confuse squaring a number with multiplying a number by 2. To clarify this, write 3

2

and on the board. Elicit the fact that the first means which is not the same as Have students find the values for both expressions, and write them on the board.

Technology Tip

Note that, when presenting Example 3, on some calculators, taking the square root may be a 2nd function. This involves first pressing the key and then the key before entering the number. On other calculators, you may first enter the number and then press the key. Have students experiment with finding

to see what keystrokes their calculators require.

Additional Examples

Find the two square roots of 81

9 and

Estimate the value of to the nearest integer.

The math class drops a small ball from the top of a stairwell. They measure the distance to the basement as 48 feet. Use the formula

to find how long it takes the ball to fall.

�

3 2�3 3�

3 2� .

3 9 3 2 62 � �; ��

9

�9

� 70

� �70 8N

d t� 16 2

t N 1 7. s

Advanced Learners

Students find the side of a square with the given area:

81

9

121

11

400

20

L4

learning style: verbal

English Language Learners

Students draw a 2-column table on an index card and label the table

Real Numbers.

They label the columns

Rational Numbers

and

Irrational Numbers,

respectively.

Have them provide examples of rational numbers in one column and irrational numbers in the other.

learning style: verbal

2. Teach

phm07c3_te_0301.fm Page 107 Thursday, May 25, 2006 6:03 PM

108


A. rational number

B. irrational number

C. real number

D. perfect square

The diagram below shows the relationships among sets of numbers.

The decimal digits of irrational numbers do not terminate or repeat. The

decimal digits of do not terminate or repeat,

because is an irrational number. Irrational numbers can also include

decimals that have a pattern in their digits, like

For any integer n that is not a perfect square, is irrational.

Classifying Real Numbers

Is each number rational or irrational? Explain.

a. Irrational; the decimal does not terminate or repeat.

b. Rational; the decimal repeats.

c. Rational; the number can be written as the ratio

d. Irrational; 5 is not a perfect square.

4. Is rational or irrational? Explain.

Check Your UnderstandingCheck Your Understanding

Vocabulary Write all the possible names for each number. Choose from the terms at the right.

1. 2.

3. 4. 25

-3, 8, 0, -120 Integers

FractionsRationals

Irrationals

Reals

Terminating

and repeating

decimals

23

125

78

12 , , ,

�2, 0.1010010001 . . .

2 + p, , �11 15

1.25,-0.13, 0.2

-

p � 3 14159265359. . . .

p

0 02022022202222. . . .

n

0 818118111. . . .

�0 81.

129

119

.

5

0 6.

6 �0 6.

16

Find the positive and negative square roots of each number.

5. 4 6. 7. 100 8.14

1100

VVocabulary Tipocabulary TipThe word rational has the word ratio in it.

The word irrational means “not rational.”

For help with terminating and repeating decimals, go to Lesson 2–2, Example 3.

Rational; the decimal repeats.

irrational, real

rational, realrational, real, perfect square

rational, real

2, 2� 12

12, � 10, 10�

110

110, �

Guided Instruction

Connection to Physics

The formula in Example 3 is the same for objects of any size and weight. So, in the absence of air resistance, a feather and a hammer fall the same distance in a specified time. This was demonstrated by an astronaut on the moon.

Additional Examples

Identify each number as

rational

or

irrational.

Explain.

a.

Rational; the decimal repeats.

b.

Rational; the ratio is

c.

Irrational; 90 is not a perfect square.

d.

Irrational; the decimal does not terminate or repeat a group of digits.

Teaching Resources

• Daily Notetaking Guide 3-1• Adapted Notetaking 3-1

Closure

•

What is the square root of a given number?

A number that when multiplied by itself is equal to the given number.

•

Give several examples of irrational numbers.

Sample:

•

Give several examples of rational numbers.

Sample:

d t� 16 2

�9 3333.

4 79

439

.

90

6 36366366636666. . . .

L3L1

3 1 343344333444 12, . ,. . .

37

25 66666, , .0


109

Assignment Guide

Check Your Understanding

Go over Exercises 1–8 in class before assigning the Homework Exercises.

Homework Exercises

A

Practice by Example 9–31

B

Apply Your Skills 32–49

C

Challenge 50Test Prep and

Mixed Review 51–56

Homework Quick Check

To check students’ understanding of key skills and concepts, go over Exercises 23, 27, 34, 36, and 47.

3-1 Exploring Square Roots and Irrational Numbers 109

Homework ExercisesHomework ExercisesFor more exercises, see Extra Skills and Word Problems.

Find the square roots of each number.

9. 49 10. 900 11. 12. 13.

Estimate the value of each expression to the nearest integer.

14. 15. 16. 17.

18. 19. 20. 21.

Use to estimate the speed of sound s in meters per second for each Celsius temperature T. Round to the nearest integer.

22. 23. 24. 25.

Is each number rational or irrational? Explain.

26. 27. 28.

29. 30. 31.

32. Guided Problem Solving The area

of a square postage stamp is

What is the side length of the stamp?

• What is the formula for the area of

a square?

• How can you use the formula to

find the side length of a square?

33. Boxing The area of a square boxing ring is 484 ft2. What is the

perimeter of the boxing ring?

34. Geometry A tile is shown at the right.

The area of the larger square is 49 in.2.

Find the area of the smaller square.

35. Open-Ended Give an example of an

irrational number that is less than 2 and

greater than 1.5. Explain how you know

the number is irrational.

36. Writing in Math

Explain how you can approximate

37. The Closure Property states that a set of numbers is closed under a

given operation if the result of the operation is in the same set of

numbers. For example, the set of rational numbers is closed under

addition, because the sum of any two rational numbers is a rational

number. Is each set of numbers closed under addition? Explain.

a. even numbers b. irrational numbers c. prime numbers

136

1121

425

3 10 � 22 88

� 54 � 105 150 � 120

s T� 20 273 ��

0�C 20�C � �10 C 70�C

�0 6. 40 0 606606660. . . .

� 144 12 0 0203040506. . . .

81100

2in. .

2 in.

2 in.

2 in.2 in.

30.

For Exercises See Examples

9–13 1

14–21 2

22–25 3

26–31 4

nline

Visit: PHSchool.comWeb Code: ase-0301

lesson quiz, PHSchool.com, Web Code: asa-0301

7, 7�30, 30�

16

16, �

111

111, � 2

525, �

9

�1112

�53

�10

2

330 m/s 342 m/s324 m/s 370 m/s

26–31. See margin.

910 in.

88 ft

9 in.2

Answers may vary. Sample: 3 is not a perfect square.3;

See left.

37a–c. See left.

36. Find the closest perfect square to 30, which is 25. Then take the square root of 25, which is 5.

37a. Yes; the sum of even numbers is an even number.

b. Yes; the sum of two irrational numbers is an irrational number.

c. No; the sum of two prime numbers can be a composite number.

�7

3. Practice

Adapted Practice 3-1 L1

Find the two square roots of each number.

1. 81 2. 3. 4. 289

Find each square root. Round to the nearest tenth if necessary.

5. 6. 7. 8.

9. 10. 11. 12.

Identify each number as rational or irrational.

13. 14. 15.

16. 17. 18. -8

19. 20. 5.2 21. 0.1010010001 . . .

22. 23. 24. 2.7064

Use s � 20 to estimate the speed of sound s in meters persecond for each Celsius temperature T. Round to the nearest integer.

25. 37ºC 26. �1ºC 27. 15ºC 28. �18ºC

Find the value of each expression.

29. 30. ( )2 31. 32.

Estimate the value of each expression to the nearest integer.

33. 34. � 35.

36. � 37. � 38. !50!21!245

!3!4!5

"x2"(2.7)2!169"(49)2

!273 1 T

!3062!25

!3

0.71245

!196!11!16

!350!301!256!182

!160!144!8!130

1121

949

Practice 3-1 Exploring Square Roots and Irrational Numbers

9 17

11.4 2.8 12 12.6

13.5 16 17.3 18.7

rational irrational rational

rational rational rational

irrational rational irrational

rational irrational rational

352 330 339 319

49 169 2.7 x

111

37

2 �2 2

�16 �5 7

L3

3-1 • Guided Problem Solving

Student Page 110, Exercise 47:

Ferris Wheels The formula d � 1.23 represents the distance in

miles d you can see from h feet above ground. On the London Eye

Ferris wheel, you are 450 ft above ground. To the nearest tenth of a

mile, how far can you see?

Understand

1. What are you being asked to find?

Plan and Carry Out

2. What is the formula? 3. What is the height?

4. Substitute known values into the formula.

5. Simplify using a calculator. Round to the nearest tenth.

Check

6. Use estimation to check your answer.

Solve Another Problem

7. The formula d � 1.23 represents the distance in miles d youcan see from h feet above ground. At the top of the Ferris wheelat Cedar Point, you are 140 ft above ground. To the nearest tenthof a mile, how far can you see?

"h

"h

GPS

the distance in miles that one can see from

450 ft above ground on the London Eye

Ferris Wheel

d N 1.25 � 21 N (1 � 21) � ( � 21) N

21 � 5.25 N 26.25; It checks.

14

14.6 mi

d � 1.23"h

d � 1.23"450

26.1 mi

450 ft

L3

26. Rational; the decimal terminates.

27. Irrational; 40 is not a perfect square.

28. Irrational; the decimal does not terminate or repeat.

29. Rational; 144 is a perfect square.

30. Irrational; 12 is not a perfect square.

31. Irrational; the decimal does not terminate or repeat.


110


For Exercises See Lesson

54–56 2-8

Find the value of each expression.

38. 39. 40. 41.

A number that is used as a factor three times is the cube root of the product. Since 2 is the cube root of 8. Find each cube root n.

42. 43. 44. 45.

46. The area of a square is What is the length of its side?

47. Ferris Wheels The formula represents the distance in

miles d you can see from h feet above ground. On the London Eye

Ferris Wheel, you are 450 ft above ground. To the nearest tenth of a

mile, how far can you see?

48. Number Sense For what values of n is a rational number?

49. Error Analysis A student evaluated the expression and

got the answer 5. What error did the student make?

50. Challenge Explain how you know that the number

123,456,789,101,112 cannot be a perfect square. (Hint: What

is the units digit?)

Test Prep and Mixed ReviewTest Prep and Mixed Review Practice

51. The area of a square is 150 square centimeters. Which best

represents the side length of the square?

11.7 cm 12.2 cm 2.9 cm 13 cm

52. The diameter of a human hair is about Which of

the following represents this number in standard notation?

0.000017 0.00017 17,000 170,000

53. Which problem situation matches the equation

Jacob travels 5 more than twice as many miles to work as

Carrie travels. If Carrie travels 20 miles to work, how many

miles x does Jacob travel?

Dana’s arm is 5 inches longer than Collin’s arm. If Dana’s arm

is 20 inches long, what is twice the length x of Collin’s arm?

Joel made a $20 phone call to Spain. The call cost $2 per minute

plus a $5 connection fee. How many minutes x did the call last?

Alondra invited 20 people to a party. Two people arrived late,

and five people could not go. How many people x arrived on

time for the party?

Write each number in scientific notation.

54. 18,000 55. 6,038,000 56. 49,700

( )36 2 ( )10 2 ( . )3 2 2 ( )a 2

2 8,3 �

n3 27� n3 64� n3 125� n3 8� �

2536

in.2.

d h� 1 23.

n

4 9�

1 7 10 5. � � meters.

2 5 20x � � ?

Multiple Choice

36 10 3.2

�2543

56 in.

26.1 mi

when n is a perfect square, including 0

See margin.

No integer multiplied by itself ends in 2.

B

F

C

4 97 104. �6 038 106. �

1 8 104. �

… »a

• The square of 5 is 25. 12 � 15 · 5 � 52 � 25 22 � 4

• The square root of 25 is 5 32 � 9because 52 � 25. 42 � 16

52 � 25� 5

Example: You can use a calculator to find square roots.Find and to the nearest tenth.

36 � 6 21 � 4.5825757 � 4.6

You can estimate square roots like and .

49 � 7 � 7

Perfect 52 Estimate � 7 Estimate � 8squares

64 � 8 � 8

Find each square root. Estimate to the nearest integer if necessary.Use ≈ to show that a value is estimated.

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. If a whole number is not a perfect square, its square root is an irrational number. List the numbers from exercises 1–16 that are irrational.

Á75Á37Á64Á29

Á144Á68Á121Á5

Á18Á100Á40Á98

Á36Á26Á85Á16

Á64Á64

Á61Á52

Á49Á49

Á61Á52

Á21Á36

Á25

Reteaching 3-1 Exploring Square Roots and Irrational Numbers

4

N 10

N 2

N 5 8 N 6 N 9

11 N 8 12

N 6 10 N 4

N 9 N 5 6

!75!85, !26, !98, !40, !18, !5, !68, !29, !37,

5

5 perfect squares�L2

Use a calculator or a table of square roots to find the square root ofeach integer below. Round each answer to the nearest thousandth.The first ten are done for you.

1. Use the square roots in the table to find each product. Round theproduct to the nearest thousandth.

a. b. c.

d. e. f.

2. Look at your answers in Exercise 1. Compare them to the square roots of other numbers in the table. Describe the pattern you see.

3. Choose two pairs of two numbers from the table. Multiply to seeif your conjecture is true for these numbers.

!2 3 !13!3 3 !5!3 3 !4

!2 3 !5!2 3 !4!2 3 !3

Enrichment 3-1 Exploring Square Roots and Irrational Numbers

Patterns in Numbers

N !N N !N N

2 1.414 12 3.464 22

3 1.732 13 3.606 23

4 2.000 14 3.742 24

7 2.646 17 4.123 27

8 2.828 18 4.243 28

5 2.236 15 3.873 25

6 2.449 16 4.000 26

9 3.000 19 4.359 29

10 3.162 20 4.472 30

!N

4.690

4.796

4.899

5.196

5.292

5.000

5.099

5.385

5.47711 3.317 21 4.583 31 5.568

2.449

3.464

Sample answer: The product of the square roots of two integers is

equal to the square root of the product of the two integers.

2.828

3.873

3.162

5.099

Sample answer: � � 1.414 � 3.317 � 4.960 �

!5 3 !5 5 2.236 3 2.236 5 5.000 5 !25

"22"11"2

L4

Lesson Quiz

1.

Find the two square roots of 400.

20 and

2.

Estimate to the nearest integer.

6

3.

Using find how long it takes a skydiver to fall 676 ft from an airplane.

6.5 s

4.

Is rational or irrational? Explain.

Rational; it can be written as

�20

34

d t� 16 2,

645

85.

Alternative Assessment

Each student in a pair writes an irrational number. Then each partner decides which two whole numbers the other partner’s value falls between.

4. Assess & Reteach

Test Prep

Resources

For additional practice with a variety of test item formats:• Test-Taking Strategies, p. 151• Test Prep, p. 155• Test-Taking Strategies with Transparencies

49. The student took the square root of 4 and added it to the square root of 9. You must add

first and then take the square root.4 9�


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