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Enrichment Masters Course 2 Applications and Connections Applications and Connections

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Page 1: Enrichment Masters - Wikispacesand... · 4-9 A Cross-Number Puzzle.....32 4-10 Intersection and Union of Sets.....33 5-1 ... When you have solved the puzzle below, the letters in

Enrichment Masters

Course 2

Applications and ConnectionsApplications and Connections

Page 2: Enrichment Masters - Wikispacesand... · 4-9 A Cross-Number Puzzle.....32 4-10 Intersection and Union of Sets.....33 5-1 ... When you have solved the puzzle below, the letters in

Copyright © by the McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the materialcontained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solelyin conjunction with Glencoe’s Mathematics: Applications and Connections. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill936 Eastwind DriveWesterville, Ohio 43081-3329

ISBN: 0-02-833075-7 Enrichment Masters, Course 2

3 4 5 6 7 8 9 10 11 066 05 04 03 02 01 00 99

Glencoe/McGraw-Hill

Page 3: Enrichment Masters - Wikispacesand... · 4-9 A Cross-Number Puzzle.....32 4-10 Intersection and Union of Sets.....33 5-1 ... When you have solved the puzzle below, the letters in

Lesson Title Page1-1 Bargain Hunt....................................11-2 Nested Expressions.........................21-3 Albert Einstein’s Famous Theory .....31-4 The Four-Digits Problem..................41-5 Equations as Models .......................51-6 Tangrams .........................................61-7 Two Area Puzzles.............................72-1 The Binary Number System ............82-2 Record-Breaking Rides ...................92-3 How Much Change?......................102-4 Decimation Problems ....................112-5 Squaring Larger Numbers .............122-6 Estimating Quotients .....................132-7 Making a Line Design ....................142-8 Using a Measurement

Conversion Chart ..................... 152-9 The Speed of Light ....................... 163-1 Breaking the Code........................ 173-2 Number Patterns .......................... 183-3 Queues ......................................... 193-4 Quartiles ....................................... 203-5 Making Pictographs ..................... 213-6 The Trimmed Mean....................... 223-7 Misleading Dissections................. 234-1 Perfect Numbers .......................... 244-2 Figurate Numbers......................... 254-3 Nested Magic Squares ................. 264-4 Sundaram’s Sieve......................... 274-5 A Two-Clock Code ....................... 284-6 Double Bar Graphs....................... 294-7 Dividing a Line Segment into

Equal Parts ............................... 304-8 Coin-Tossing Experiments............ 314-9 A Cross-Number Puzzle ............... 32

4-10 Intersection and Union of Sets..... 335-1 Integer Dominoes ......................... 345-2 Quantitative Comparisons............ 355-3 Latitude and Longitude ................ 365-4 Dart Board Puzzles....................... 375-5 Distance on the Number Line....... 385-6 Integer Maze................................. 395-7 Division by Zero?.......................... 405-8 Transformations ............................ 416-1 Equation Hexa-Maze .................... 426-2 Describing Variation...................... 436-3 Combining Like Terms.................. 446-4 Expressions for Figurate

Numbers................................... 456-5 Compound Inequalities ................ 466-6 Functioning with Machines........... 476-7 Equations with Two Variables....... 487-1 Fractional Areas............................ 497-2 Fractions Maze ............................. 507-3 Arithmetic Sequences of

Fractions....................................51

Lesson Title Page7-4 Changing Measures of Length ..... 527-5 Changing Measurement with

Factors of 1 .............................. 537-6 Networks ...................................... 547-7 A Circle Puzzle.............................. 557-8 Moebius Bands ............................ 567-9 Continued Fractions ..................... 578-1 Squares and Rectangles .............. 588-2 An Educated Consumer ............... 598-3 What Am I?................................... 608-4 Scale Drawings............................. 618-5 Shaded Regions ........................... 628-6 Mental Math Magic....................... 638-7 The Colormatch Square ............... 648-8 Model Behavior ............................ 658-9 Working Backward ....................... 669-1 Compass Directions ..................... 679-2 Star Polygons ............................... 689-3 Similar Figures and Area .............. 699-4 Dissecting Squares....................... 709-5 Tessellated Patterns for Solid

Shapes ..................................... 719-6 Knight Moves................................ 729-7 The Twelve-Dot Puzzle ................. 73

10-1 The Geometric Mean.................... 7410-2 World Series Records................... 7510-3 Pythagoras in the Air .................... 7610-4 Finding the Area from the

Vertex Coordinates................... 7710-5 Heron’s Formula ........................... 7810-6 Extending the Pythagorean

Theorem ................................... 7910-7 Area Formula for Regular

Polygons................................... 8011-1 Made in the Shade ....................... 8111-2 Just the Facts............................... 8211-3 Relative Frequency and Circle

Graphs...................................... 8311-4 Table of Random Digits ................ 8411-5 A Taxing Exercise ......................... 8511-6 Missing Fact Math ........................ 8611-7 Taking an Interest ......................... 8712-1 Counting Cubes............................ 8812-2 Volumes of Pyramids.................... 8912-3 Volumes of Non-Right Solids ....... 9012-4 Pattern Puzzles............................. 9112-5 Cross Sections ............................. 9213-1 Rolling a Dodecahedron............... 9313-2 Probabilities and Regions............. 9413-3 Curious Cubes.............................. 9513-4 Independent Events...................... 9613-5 Permutation Puzzles..................... 9713-6 From Impossible to Certain

Events....................................... 98

Contents

iii

Page 4: Enrichment Masters - Wikispacesand... · 4-9 A Cross-Number Puzzle.....32 4-10 Intersection and Union of Sets.....33 5-1 ... When you have solved the puzzle below, the letters in

Mathematics: Applications© Glencoe/McGraw-Hill 1 and Connections, Course 2

Enrichment

Name Date

Bargain Hunt

Use the For Sale signs on this page to solve each problem. If information you need is not given, write “cannot be solved.”

1. Kiko works Saturday mornings at the videotape store.She bought ten videotapes on sale and used a $10employee discount coupon to help pay for the tapes. How much did she spend in all?

2. Toni bought six handbags at the store that is going outof business. How much did she spend for each handbag?

3. Sid earned $40 working after school. How much money will he have left if he buys a sweatshirt and four jigsaw puzzles?

4. Suzette bought six jigsaw puzzles and a model airplane kit. How much change did she receive from a $20 bill?

5. Last week Norrine bought a model airplane kit for $18.67. How much would she have saved if she had waited until this week to buy the kit?

6. How much would you save if you bought three sweatshirts and two jigsaw puzzles?

Handbags—3 for $15

Going Out of Business

Model Airplane Kits $3.99Reg. $4.99–$24.99

This Week Only!

Videotapes!!!

5 for $45.95!

Sweatshirts$9.99 each

Regularly $11.99Save!

Regularly $3.49 each

JIGSAW PUZZLES

2 for $5

11--11

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Mathematics: Applications© Glencoe/McGraw-Hill T1 and Connections, Course 2

Enrichment

Name Date

Bargain Hunt

Use the For Sale signs on this page to solve each problem. If information you need is not given, write “cannot be solved.”

1. Kiko works Saturday mornings at the videotape store.She bought ten videotapes on sale and used a $10employee discount coupon to help pay for the tapes. How much did she spend in all?

$81.90

2. Toni bought six handbags at the store that is going outof business. How much did she spend for each handbag?

$5.00

3. Sid earned $40 working after school. How much money will he have left if he buys a sweatshirt and four jigsaw puzzles?

$20.01

4. Suzette bought six jigsaw puzzles and a model airplane kit. How much change did she receive from a $20 bill?

$1.01

5. Last week Norrine bought a model airplane kit for $18.67. How much would she have saved if she had waited until this week to buy the kit?

$14.68

6. How much would you save if you bought three sweatshirts and two jigsaw puzzles?

$7.98 Handbags—3 for $15

Going Out of Business

Model Airplane Kits $3.99Reg. $4.99–$24.99

This Week Only!

Videotapes!!!

5 for $45.95!

Sweatshirts$9.99 each

Regularly $11.99Save!

Regularly $3.49 each

JIGSAW PUZZLES

2 for $5

11--11

Page 6: Enrichment Masters - Wikispacesand... · 4-9 A Cross-Number Puzzle.....32 4-10 Intersection and Union of Sets.....33 5-1 ... When you have solved the puzzle below, the letters in

Mathematics: Applications© Glencoe/McGraw-Hill 2 and Connections, Course 2

Enrichment

Name Date

Nested Expressions

Sometimes more than one set of parentheses are used to group thequantities in an expression. These expressions are said to have “nested”parentheses. The expression below has “nested” parentheses.

(4 � (3 � (2 � 3)) � 8) � 9

Expressions with several sets of grouping symbols are clearer if bracessuch as {} or brackets such as [ ] are used. Here is the same examplewritten with brackets and braces.

{4 + [3 � (2 + 3)] + 8} � 9

To evaluate expressions of this type, work from the inside out.

{4 � [3 � (2 � 3)] � 8} � 9 � {4 � [3 � 5] � 8} � 9� [4 � 15 � 8] � 9� 27 � 9� 3

Evaluate each expression.

1. 3 � [(24 � 8) � 7] � 20 2. [(16 � 7 � 5) � 2] � 7

3. [2 � (23 � 6) � 14] � 6 4. 50 � [3 � (15 � 5)] � 25

5. 12 � {28 � [2 � (11 � 7)] � 3} 6. {75 � 3 � [(17 � 9) � 2]} � 2

7. 20 � {3 � [6 � (56 � 8)]} 8. {4 � [5 � (12 � 5)] � 15} � 10

9. {15 � [(38 � 26) � 4]} � 15 10. {[34 � (6 � 5)] � 8} � 40

11--22

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Mathematics: Applications© Glencoe/McGraw-Hill T2 and Connections, Course 2

Enrichment

Name Date

Nested Expressions

Sometimes more than one set of parentheses are used to group thequantities in an expression. These expressions are said to have “nested”parentheses. The expression below has “nested” parentheses.

(4 � (3 � (2 � 3)) � 8) � 9

Expressions with several sets of grouping symbols are clearer if bracessuch as {} or brackets such as [ ] are used. Here is the same examplewritten with brackets and braces.

{4 + [3 � (2 + 3)] + 8} � 9

To evaluate expressions of this type, work from the inside out.

{4 � [3 � (2 � 3)] � 8} � 9 � {4 � [3 � 5] � 8} � 9� [4 � 15 � 8] � 9� 27 � 9� 3

Evaluate each expression.

1. 3 � [(24 � 8) � 7] � 20 4 2. [(16 � 7 � 5) � 2] � 7 0

3. [2 � (23 � 6) � 14] � 6 8 4. 50 � [3 � (15 � 5)] � 25 45

5. 12 � {28 � [2 � (11 � 7)] � 3} 35 6. {75 � 3 � [(17 � 9) � 2]} � 2 174

7. 20 � {3 � [6 � (56 � 8)]} 59 8. {4 � [5 � (12 � 5)] � 15} � 10 540

9. {15 � [(38 � 26) � 4]} � 15 30 10. {[34 � (6 � 5)] � 8} � 40 48

11--22

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Mathematics: Applications© Glencoe/McGraw-Hill T3 and Connections, Course 2

Enrichment

Name Date

Albert Einstein’s Famous Theory

When you have solved the puzzle below, the letters in the heavy blackboxes will spell the name of an important scientific theory proposed byAlbert Einstein. His theory relates mass and energy.

Use these clues to complete the puzzle below.

1.

2.

3.

5.

9.

8.

10.

6.

7.

4.

11--33

1. A language of symbols.

3. To find a specific numerical value for analgebraic expression.

5. The order of ________ helps you to knowwhich operation to do first when evaluatingan expression.

7. The worth of something.

9. The process of finding the product of twonumbers is called ________.

2. To evaluate an expression, you ________the variable with a number.

4. An expression must contain at ________one operation as well as variables ornumbers.

6. A symbol that stands for an unknownquantity.

8. You call 2a � 3b an algebraic ________.

10. In an algebraic expression, you should________ or divide before you add orsubtract.

Page 9: Enrichment Masters - Wikispacesand... · 4-9 A Cross-Number Puzzle.....32 4-10 Intersection and Union of Sets.....33 5-1 ... When you have solved the puzzle below, the letters in

Mathematics: Applications© Glencoe/McGraw-Hill T3 and Connections, Course 2

Enrichment

Name Date

Albert Einstein’s Famous Theory

When you have solved the puzzle below, the letters in the heavy blackboxes will spell the name of an important scientific theory proposed byAlbert Einstein. His theory relates mass and energy.

Use these clues to complete the puzzle below.

A1. L G E B R AR2. E P L A C E

E3. V A L U A T E

E RO5. P A T I O N S

L II PL TM9. U C A T I OR EX PE8. S S I

T IU LM10. P L Y

O NN

V6. A R I A B L EV7. A L U E

L4. E A S T

11--33

1. A language of symbols.

3. To find a specific numerical value for analgebraic expression.

5. The order of ________ helps you to knowwhich operation to do first when evaluatingan expression.

7. The worth of something.

9. The process of finding the product of twonumbers is called ________.

2. To evaluate an expression, you ________the variable with a number.

4. An expression must contain at ________one operation as well as variables ornumbers.

6. A symbol that stands for an unknownquantity.

8. You call 2a � 3b an algebraic ________.

10. In an algebraic expression, you should________ or divide before you add orsubtract.

Page 10: Enrichment Masters - Wikispacesand... · 4-9 A Cross-Number Puzzle.....32 4-10 Intersection and Union of Sets.....33 5-1 ... When you have solved the puzzle below, the letters in

Mathematics: Applications© Glencoe/McGraw-Hill 4 and Connections, Course 2

Enrichment

Name Date

The Four-Digits Problem

Use the digits 1, 2, 3, and 4 to write expressions for the numbers1 through 50. Each digit is used exactly once in each expression. (Theremight be more than one expression for a given number.)

You can use addition, subtraction, multiplication (not division), exponents,and parentheses in any way you wish. Also, you can use two digits to makeone number, as in 34. A few expressions are given to get you started.

1 � (3 � 1) � (4�2) 18 � 35 � 2(4�1) � 3

2 � 19 � 3(2 � 4) � 1 36 �

3 � 20 � 37 �

4 � 21 � 38 �

5 � 22 � 39 �

6 � 23 � 31 � (4 � 2) 40 �

7 � 24 � 41 �

8 � 25 � 42 �

9 � 26 � 43 � 42 � 13

10 � 27 � 44 �

11 � 28 � 45 �

12 � 29 � 2(4 � 1) � 3 46 �

13 � 30 � 47 �

14 � 31 � 48 �

15 �2(3 � 4) � 1 32 � 49 �

16 � 33 � 50 �

17 � 34 �

11--44

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Mathematics: Applications© Glencoe/McGraw-Hill T4 and Connections, Course 2

Enrichment

Name Date

The Four-Digits Problem

Use the digits 1, 2, 3, and 4 to write expressions for the numbers1 through 50. Each digit is used exactly once in each expression. (Theremight be more than one expression for a given number.)

You can use addition, subtraction, multiplication (not division), exponents,and parentheses in any way you wish. Also, you can use two digits to makeone number, as in 34. A few expressions are given to get you started.

1 � (3 � 1) � (4�2) 18 � (2 � 3) � (4 � 1) 35 � 2(4�1) � 3

2 � (4 � 3) � (2 � 1) 19 � 3(2 � 4) � 1 36 � 34 � (2 � 1)

3 � (4 � 3) � (2 � 1) 20 � 21 � (4 � 3) 37 � 31 � 4 � 2

4 � (4 � 2) � (3 � 1) 21 � (4 � 3) � (2 � 1) 38 � 42 � (1 � 3)

5 � (4 � 2) � (3 � 1) 22 � 21 � (4 � 3) 39 � 42 � (1 � 3)

6 � 4 � 3 � 1 � 2 23 � 31 � (4 � 2) 40 � 41 � (3 � 2)

7 � 3(4 � 1) � 2 24 � (4 � 2) � (3 � 1) 41 � 43 � (2 � 1)

8 � 4 � 3 � 2 � 1 25 � (2 � 3) � (4 � 1) 42 � 43 � (2 � 1)

9 � 4 � 2 � (3 � 1) 26 � 24 � (3 � 1) 43 � 42 � 13

10 � 4 � 3 � 2 � 1 27 � 32 � (4 � 1) 44 � 43 � (2 � 1)

11 � (4 � 3) � (2 � 1) 28 � 21 � 4 � 3 45 � 43 � (2 � 1)

12 � (4 � 3) � (2 � 1) 29 � 2(4 � 1) � 3 46 � 43 � (2 � 1)

13 � (4 � 3) � (2 � 1) 30 � (2 � 3) � (4 � 1) 47 � 31 � 42

14 � (4 � 3) � (2 � 1) 31 � 34 � (2 � 1) 48 � 42 � (3 � 1)

15 �2(3 � 4) � 1 32 � 42 � (3 � 1) 49 � 41 � 23

16 � (4 � 2) � (3 � 1) 33 � 21 � (4 � 3) 50 � 41 � 32

17 � 3(4 � 2) � 1 34 � 2 � (14 � 3)

11--44

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Enrichment

Name Date

Equations as Models

When you write an equation that represents the information in a problem,the equation serves as a model for the problem. One equation can be amodel for several different problems.

Each of Exercises 1-8 can be modeled by one of theseequations.

n � 2 � 10 n � 2 � 10 2n � 10 �n2� � 10

Choose the correct equation. Then solve the problem.

11--55

1. Chum earned $10 for working two hours.How much did he earn per hour?

3. Kathy and her brother won a contest andshared the prize equally. Each received $10.What was the amount of the prize?

5. In the figure below, the length of A�C� is 10 cm. The length of B�C� is 2 cm. What is the length of A�B�?

7. The width of the rectangle below is2 inches less than the length. What is thelength?

n � 2 � 10: 12 inches9. CHALLENGE On a separate sheet of

paper, write a problem that can be modeledby the equation 3a � 5 � 29.

2. Ana needs $2 more to buy a $10 scarf. Howmuch money does she already have?

4. Jameel loaned two tapes to a friend. He hasten tapes left. How many tapes did Jameeloriginally have?

6. Ray AC bisects ∠BAD. The measure of∠BAC is 10°. What is the measure of∠BAD?

�n2� � 10; 20°8. In the triangle below, the length of P�Q� is

twice the length of Q�R�. What is the lengthof Q�R�?

P R

10 cm

Q

60˚

30˚

A D

C

B

10 in.

A CB

Mathematics: Applications© Glencoe/McGraw-Hill 5 and Connections, Course 2

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Enrichment

Name Date

Equations as Models

When you write an equation that represents the information in a problem,the equation serves as a model for the problem. One equation can be amodel for several different problems.

Each of Exercises 1-8 can be modeled by one of theseequations.

n � 2 � 10 n � 2 � 10 2n � 10 �n2� � 10

Choose the correct equation. Then solve the problem.

11--55

1. Chum earned $10 for working two hours.How much did he earn per hour?

2n � 10; $53. Kathy and her brother won a contest and

shared the prize equally. Each received $10.What was the amount of the prize?

�n2� � 10; $205. In the figure below, the length of A�C� is

10 cm. The length of B�C� is 2 cm. What is the length of A�B�?

n � 2 � 10; 8 cm

7. The width of the rectangle below is2 inches less than the length. What is thelength?

n � 2 � 10: 12 inches9. CHALLENGE On a separate sheet of

paper, write a problem that can be modeledby the equation 3a � 5 � 29.

Answers will vary.

2. Ana needs $2 more to buy a $10 scarf. Howmuch money does she already have?

n � 2 � 10; $84. Jameel loaned two tapes to a friend. He has

ten tapes left. How many tapes did Jameeloriginally have?

n � 2 � 10; 12 tapes6. Ray AC bisects ∠BAD. The measure of

∠BAC is 10°. What is the measure of∠BAD?

�n2� � 10; 20°8. In the triangle below, the length of P�Q� is

twice the length of Q�R�. What is the lengthof Q�R�?

2n � 10; 5 cm

P R

10 cm

Q

60˚

30˚

A D

C

B

10 in.

A CB

Mathematics: Applications© Glencoe/McGraw-Hill T5 and Connections, Course 2

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Mathematics: Applications© Glencoe/McGraw-Hill 6 and Connections, Course 2

Enrichment

Name Date

Tangrams

Here is a chance for you to practice some visual estimation. The puzzle at the right is called a tangram. Each of the seven pieces is a tan. The tangram pieces can be arranged as shown to form a square.

Trace the square and cut it into sevenpieces as shown.

1. Separate the tans and put them back togetherto form the square. Try this first withoutlooking at the solution.

2. Describe the seven tangram pieces in words.

Right triangles: two large,one middle-sized, twosmall; one square, oneparallelogram.

The tans can be used to make many other shapes. Use all seven pieces to make each shape shown. Record your solutions. Other arrangements are possible.3. 4.

5. 6.

2

5 6

7

4

13

11--66

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Mathematics: Applications© Glencoe/McGraw-Hill T6 and Connections, Course 2

Enrichment

Name Date

Tangrams

Here is a chance for you to practice some visual estimation. The puzzle at the right is called a tangram. Each of the seven pieces is a tan. The tangram pieces can be arranged as shown to form a square.

Trace the square and cut it into sevenpieces as shown.

1. Separate the tans and put them back togetherto form the square. Try this first withoutlooking at the solution.

2. Describe the seven tangram pieces in words.

Right triangles: two large,one middle-sized, twosmall; one square, oneparallelogram.

The tans can be used to make many other shapes. Use all seven pieces to make each shape shown. Record your solutions. Other arrangements are possible.3. 4.

5. 6.

2

5 6

7

4

13

11--66

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Mathematics: Applications© Glencoe/McGraw-Hill 7 and Connections, Course 2

Enrichment

Name Date

Two Area Puzzles

Cut out the five puzzle pieces at the bottom of this page. Thenuse them to solve these two puzzles.

2 in.

1 in.

1 in.

2 in.

2 in.

1 in2

11--77

1. Use all five puzzle pieces to make a square with an area of 9 square inches. Record yoursolution below.

2. Use the four largest pieces to make asquare with an area of 8 square inches.Record your solution below.

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Mathematics: Applications© Glencoe/McGraw-Hill T7 and Connections, Course 2

Enrichment

Name Date

Two Area Puzzles

Cut out the five puzzle pieces at the bottom of this page. Thenuse them to solve these two puzzles.

2 in.

1 in.

1 in.

2 in.

2 in.

1 in2

11--77

1. Use all five puzzle pieces to make a square with an area of 9 square inches. Record yoursolution below.

2. Use the four largest pieces to make asquare with an area of 8 square inches.Record your solution below.

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Mathematics: Applications© Glencoe/McGraw-Hill 8 and Connections, Course 2

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Name Date

The Binary Number System

The numbers you use everyday are written in the decimal system. Thevalue of each place is based on a power of 10. So, this system is alsocalled the base ten system. Ten different digits are used in the decimalsystem, 0 through 9. The chart below shows the number 45,603 in thedecimal system.

Numbers can also be written in the binary system. In this sytem, the valueof each place is based on a power of 2. The base of the system is 2, so it isalso called the base-two system. Just two digits are used in the binarysystem, 0 and 1. The chart below shows the number 10101 in the binarysystem.

The total value of the binary number 10101 is 16 � 4 � 1, or 21. This canbe written 10101two � 21ten.

Find the base ten number that equals each of these binarynumbers.

1. 11two 2. 101two 3. 1000two

4. 111two 5. 1010two 6. 1111two

7. 10000two 8. 10010two 9. 11011two

Find the binary number that equals each of these base tennumbers.

10. 4ten 11. 9ten 12. 12ten

13. 17ten 14. 24ten 15. 20ten

16. 19ten 17. 30ten 18. 31ten

22--11

Digit 4 5 6 0 3Place Value 104 � 10,000 103 � 1,000 102 � 100 101 � 10 100 � 1

Digit Value 40,000 5,000 600 0 3

Digit 1 0 1 0 1Place Value 24 � 16 23 � 8 22 � 4 21 � 2 20 � 1

Digit Value 16 0 4 0 1

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The Binary Number System

The numbers you use everyday are written in the decimal system. Thevalue of each place is based on a power of 10. So, this system is alsocalled the base ten system. Ten different digits are used in the decimalsystem, 0 through 9. The chart below shows the number 45,603 in thedecimal system.

Numbers can also be written in the binary system. In this sytem, the valueof each place is based on a power of 2. The base of the system is 2, so it isalso called the base-two system. Just two digits are used in the binarysystem, 0 and 1. The chart below shows the number 10101 in the binarysystem.

The total value of the binary number 10101 is 16 � 4 � 1, or 21. This canbe written 10101two � 21ten.

Find the base ten number that equals each of these binarynumbers.

1. 11two 3ten 2. 101two 5ten 3. 1000two 8ten

4. 111two 7ten 5. 1010two 10ten 6. 1111two 15ten

7. 10000two 16ten 8. 10010two 18ten 9. 11011two 27ten

Find the binary number that equals each of these base tennumbers.

10. 4ten 100two 11. 9ten 1001two 12. 12ten 1100two

13. 17ten 10001two 14. 24ten 11000two 15. 20ten 10100two

16. 19ten 10011two 17. 30ten 11110two 18. 31ten 11111two

22--11

Digit 4 5 6 0 3Place Value 104 � 10,000 103 � 1,000 102 � 100 101 � 10 100 � 1

Digit Value 40,000 5,000 600 0 3

Digit 1 0 1 0 1Place Value 24 � 16 23 � 8 22 � 4 21 � 2 20 � 1

Digit Value 16 0 4 0 1

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Record-Breaking Rides

Round each decimal to the nearest tenth. Then use the letterto show its location on the number lines below.

Located in Cedar Point Park, Sandusky, Ohio, this roller coaster once hadthe highest vertical drop.

This Japanese roller coaster is one of the tallest in the world. It is 207 feettall.

This large, looping roller coaster is found at Six Flags Magic Mountain inValencia, California.

With a run of 1.4 miles, this roller coaster at Kings Island near Cincinnati,Ohio, was once the longest in the world.

7.0 7.2 7.4 7.6 7.8 8.06.86.66.46.26.0

5.0 5.2 5.4 5.6 5.8 6.04.84.64.44.24.0

3.0 3.2 3.4 3.6 3.8 4.02.82.62.42.22.0

1.0 1.2 1.4 1.6 1.8 2.00.80.60.40.20

22--22

Letter Decimal

A 0.41

A 3.316

A 7.28

B 6.71

E 4.377

E 6.93

E 5.43

E 6.646

Letter Decimal

G 0.738

H 6.252

H 4.236

I 4.99

L 3.752

L 1.82

M 1.365

M 2.103

Letter Decimal

M 0.067

N 2.909

N 0.98

O 2.475

O 2.68

P 5.06

R 5.707

S 3.08

Letter Decimal

S 7.72

T 4.11

T 7.78

T 3.86

T 6.181

U 1.23

U 3.639

V 4.667

X 1.608

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Name Date

Record-Breaking Rides

Round each decimal to the nearest tenth. Then use the letterto show its location on the number lines below.

Located in Cedar Point Park, Sandusky, Ohio, this roller coaster once hadthe highest vertical drop.

This Japanese roller coaster is one of the tallest in the world. It is 207 feettall.

This large, looping roller coaster is found at Six Flags Magic Mountain inValencia, California.

With a run of 1.4 miles, this roller coaster at Kings Island near Cincinnati,Ohio, was once the longest in the world.

S TEH7.0

T B E A7.2 7.4 7.6 7.8 8.06.86.66.46.26.0

E REH5.0

T V I P5.2 5.4 5.6 5.8 6.04.84.64.44.24.0

M O O N S A U L T3.0 3.2 3.4 3.6 3.8 4.02.82.62.42.22.0

M A G N U M X L1.0 1.2 1.4 1.6 1.8 2.00.80.60.40.20

22--22

Letter Decimal

A 0.41

A 3.316

A 7.28

B 6.71

E 4.377

E 6.93

E 5.43

E 6.646

Letter Decimal

G 0.738

H 6.252

H 4.236

I 4.99

L 3.752

L 1.82

M 1.365

M 2.103

Letter Decimal

M 0.067

N 2.909

N 0.98

O 2.475

O 2.68

P 5.06

R 5.707

S 3.08

Letter Decimal

S 7.72

T 4.11

T 7.78

T 3.86

T 6.181

U 1.23

U 3.639

V 4.667

X 1.608

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How Much Change?

When you pay for an item in a store, you need to recognize quickly if youhave received the right amount of change. Here is how estimation can helpyou.

You gave the clerk: $40.00Amount of your purchase: $23.15 about $23Change received: $6.85 about $7

The sum is only $30. You gave the clerk $40, so the amount of changeis not reasonable.

Circle the amount of change that is most reasonable.

Make a reasonable estimate of the amount of change.

5. The amount of your purchase is $33.25, and you give the clerk two $20bills.

6. You order a hamburger that costs $2.19, French fries that cost $1.15,and milk that costs $0.79. You give the cashier $10.

7. You are purchasing a cassette that costs $5.98 and headphones that cost$7.98. You give the clerk a $10 bill and a $5 bill.

8. You are purchasing two sweaters that cost $17.99 each and a skirt thatcosts $21.99. You give the salesperson three $20 bills.

22--33

1. The amount of your purchase is $6.74, andyou give the clerk $20.

$3.26 $4.26 $13.26 $14.26

3. The amount of your purchase is $1.89, andyou give the clerk $5.

$4.11 $3.89 $3.11 $2.11

2. The cost of dinner is $21.15, and you givethe server $25.

$3.15 $3.85 $4.85 $8.85

4. You are purchasing a box of cereal thatcosts $3.19 and a roast that costs $7.98,and you give the cashier $20.

$17.81 $12.02 $9.83 $8.83

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How Much Change?

When you pay for an item in a store, you need to recognize quickly if youhave received the right amount of change. Here is how estimation can helpyou.

You gave the clerk: $40.00Amount of your purchase: $23.15 about $23Change received: $6.85 about $7

The sum is only $30. You gave the clerk $40, so the amount of changeis not reasonable.

Circle the amount of change that is most reasonable.

Make a reasonable estimate of the amount of change.

5. The amount of your purchase is $33.25, and you give the clerk two $20bills.

about $7

6. You order a hamburger that costs $2.19, French fries that cost $1.15,and milk that costs $0.79. You give the cashier $10.

about $6

7. You are purchasing a cassette that costs $5.98 and headphones that cost$7.98. You give the clerk a $10 bill and a $5 bill.

about $1

8. You are purchasing two sweaters that cost $17.99 each and a skirt thatcosts $21.99. You give the salesperson three $20 bills.

about $2

22--33

1. The amount of your purchase is $6.74, andyou give the clerk $20.

$3.26 $4.26 $13.26 $14.26

3. The amount of your purchase is $1.89, andyou give the clerk $5.

$4.11 $3.89 $3.11 $2.11

2. The cost of dinner is $21.15, and you givethe server $25.

$3.15 $3.85 $4.85 $8.85

4. You are purchasing a box of cereal thatcosts $3.19 and a roast that costs $7.98,and you give the cashier $20.

$17.81 $12.02 $9.83 $8.83

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Decimation Problems

You might want to use small objects or counters to help yousolve the problems on this page.

1. Half of a group of 30 people at a party will each receive a party gift.Annette arranges the men and women as shown. Starting with person 1,the people count off from 1 to 10 and then start over with 1. Eachperson who says “10” drops out of the ring. Who will get the partygifts?

2. A group of 15 people arranged in a circle count off as in Exercise 1. List the order in which the people drop out of the circle.

10, 5, 1, 13, 11, 9, 12, 15, 4, 14, 8,3, 6, 2, 7

3. What is the number of the last person left?

4. Hal has number 14. In order for Hal to be the lastperson left, what number person should start thecounting?

5. Why do you suppose that problems of this type arecalled “decimation” problems?

6. Thirty people sit in a circle like the one above and count off startingwith the first person. Every tenth person drops out. What is the numberof the last person left?

7. There are 40 people in the chess club. They agree that the two newofficers will be chosen by trimation; that is, they will count off so thatevery third person drops out. What are the numbers of the newpresident and vice-president of the club?

1312

11

10

9

8

7

14

15

4

56

1

3

2

6 7

3

30

8 9 10

202223242627

11

14

15menwomen

54

21

1213

1617

18192125

2829

22--44

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Decimation Problems

You might want to use small objects or counters to help yousolve the problems on this page.

1. Half of a group of 30 people at a party will each receive a party gift.Annette arranges the men and women as shown. Starting with person 1,the people count off from 1 to 10 and then start over with 1. Eachperson who says “10” drops out of the ring. Who will get the partygifts? the fifteen women

2. A group of 15 people arranged in a circle count off as in Exercise 1. List the order in which the people drop out of the circle.

10, 5, 1, 13, 11, 9, 12, 15, 4, 14, 8,3, 6, 2, 7

3. What is the number of the last person left? 74. Hal has number 14. In order for Hal to be the last

person left, what number person should start thecounting? 8

5. Why do you suppose that problems of this type arecalled “decimation” problems?

Every tenth person is removed.6. Thirty people sit in a circle like the one above and count off starting

with the first person. Every tenth person drops out. What is the numberof the last person left? 28

7. There are 40 people in the chess club. They agree that the two newofficers will be chosen by trimation; that is, they will count off so thatevery third person drops out. What are the numbers of the newpresident and vice-president of the club? 13 and 28

1312

11

10

9

8

7

14

15

4

56

1

3

2

6 7

3

30

8 9 10

202223242627

11

14

15menwomen

54

21

1213

1617

18192125

2829

22--44

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Squaring Larger Numbers

Many people have found it worthwhile to memorizethe squares of the counting numbers through 20. Ifyou are really ambitious, you might memorize themup to 30.

To square a two-digit number ending in 5, use thistrick. Multiply the tens digit times itself plus 1. Thenwrite 25 to the right of that product.

Number Think: Product

452 4 � (4 � 1) � 4 � 5 � 20 2,025852 8 � (8 � 1) � 8 � 9 � 72 7,225

The examples below show how to square a numberending in 0.

Number Think: Product

802 82 � 102 6,4002402 242 � 102 57,600

Write the square of each number.

1. 14 2. 21 3. 17 4. 26

5. 24 6. 28 7. 12 8. 22

Square each number mentally.

9. 95 10. 25 11. 15 12. 65

13. 75 14. 35 15. 55 16. 105

17. 40 18. 110 19. 250 20. 70

21. 160 22. 750 23. 180 24. 350

22--55

Squares of Numbers

112 = 121 212 = 441

122 = 144 222 = 484

132 = 169 232 = 529

142 = 196 242 = 576

152 = 225 252 = 625

162 = 256 262 = 676

172 = 289 272 = 729

182 = 324 282 = 784

192 = 361 292 = 841

202 = 400 302 = 900

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Name Date

Squaring Larger Numbers

Many people have found it worthwhile to memorizethe squares of the counting numbers through 20. Ifyou are really ambitious, you might memorize themup to 30.

To square a two-digit number ending in 5, use thistrick. Multiply the tens digit times itself plus 1. Thenwrite 25 to the right of that product.

Number Think: Product

452 4 � (4 � 1) � 4 � 5 � 20 2,025852 8 � (8 � 1) � 8 � 9 � 72 7,225

The examples below show how to square a numberending in 0.

Number Think: Product

802 82 � 102 6,4002402 242 � 102 57,600

Write the square of each number.

1. 14 196 2. 21 441 3. 17 289 4. 26 676

5. 24 576 6. 28 784 7. 12 144 8. 22 484

Square each number mentally.

9. 95 9,025 10. 25 625 11. 15 225 12. 65 4,225

13. 75 5,625 14. 35 1,225 15. 55 3,025 16. 105

11,025

17. 40 1,600 18. 110 12,100 19. 250 62,500 20. 70 4,900

21. 160 25,600 22. 750 562,500 23. 180 32,400 24. 350

122,500

22--55

Squares of Numbers

112 = 121 212 = 441

122 = 144 222 = 484

132 = 169 232 = 529

142 = 196 242 = 576

152 = 225 252 = 625

162 = 256 262 = 676

172 = 289 272 = 729

182 = 324 282 = 784

192 = 361 292 = 841

202 = 400 302 = 900

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Estimating Quotients

Estimating a quotient is easy when the divisor is near 1, 10, 100, or 1,000.For example, here is how you can estimate 4.08 � 9.375.

4.08 � 9.3759.375 is close to 10.4.08 � 10 � 0.408 Mentally move the decimal point0.408 is about 0.4 one place to the left.So 4.08 � 9.375 is about 0.4.

Estimate each quotient.

1. 0.684 � 10.0035 2. 0.2807 � 0.94

3. 22.3 � 98.05 4. 6.7 � 0.9331

5. 879.4 � 1,005 6. 134.038 � 9.9781

7. 51.5 � 0.95 8. 4,295 � 992.8

Another useful skill in estimating quotients is simply recognizing whetherthe quotient will be greater than 1 or less than 1.

Estimate: 0.5 � 0.42 Estimate: 4.08 � 4.35

0.5 � 0.42, so the quotient 4.08 � 4.35, so the quotientwill be greater than 1. will be less than 1.

Tell whether each quotient is greater than 1 or less than 1.

9. 0.608 � 0.695 10. 0.3009 � 0.24

11. 10.482 � 9.4 12. 26 � 25.9007

13. 0.0084 � 0.009 14. 0.005 � 0.0046

15. 0.8075 � 0.81 16. 9.0049 � 10.8

22--66

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Estimating Quotients

Estimating a quotient is easy when the divisor is near 1, 10, 100, or 1,000.For example, here is how you can estimate 4.08 � 9.375.

4.08 � 9.3759.375 is close to 10.4.08 � 10 � 0.408 Mentally move the decimal point0.408 is about 0.4 one place to the left.So 4.08 � 9.375 is about 0.4.

Estimate each quotient. Estimates may vary.1. 0.684 � 10.0035 about 0.07 2. 0.2807 � 0.94 about 0.3

3. 22.3 � 98.05 about 0.2 4. 6.7 � 0.9331 about 7

5. 879.4 � 1,005 about 0.9 6. 134.038 � 9.9781 about 13

7. 51.5 � 0.95 about 52 8. 4,295 � 992.8 about 4.3

Another useful skill in estimating quotients is simply recognizing whetherthe quotient will be greater than 1 or less than 1.

Estimate: 0.5 � 0.42 Estimate: 4.08 � 4.35

0.5 � 0.42, so the quotient 4.08 � 4.35, so the quotientwill be greater than 1. will be less than 1.

Tell whether each quotient is greater than 1 or less than 1.

9. 0.608 � 0.695 less than 1 10. 0.3009 � 0.24 greater than 1

11. 10.482 � 9.4 greater than 1 12. 26 � 25.9007 greater than 1

13. 0.0084 � 0.009 less than 1 14. 0.005 � 0.0046 greater than 1

15. 0.8075 � 0.81 less than 1 16. 9.0049 � 10.8 less than 1

22--66

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Making a Line Design

Connect each pair of equivalent numbers with a straight line segment.Although you will draw only straight lines, the finished design will appear curved!

0.05 0.125

0.5 0.2

0.3 0.1

0.6 0.63

0.03 0.384615

0.16 0.142857

0.5 0.25

0.875 0.05

0.375 0.0625

0.75 0.083

18

120

15

12

19

13

711

23

513

130

17

16

14

59

118

78

116

38

112

34

22--77

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Making a Line Design

Connect each pair of equivalent numbers with a straight line segment.Although you will draw only straight lines, the finished design will appear curved!

0.05 0.125

0.5 0.2

0.3 0.1

0.6 0.63

0.03 0.384615

0.16 0.142857

0.5 0.25

0.875 0.05

0.375 0.0625

0.75 0.083

18

120

15

12

19

13

711

23

513

130

17

16

14

59

118

78

116

38

112

34

22--77

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Using a Measurement Conversion Chart

You may sometimes want to convert customary measurements to metricmeasurements. For example, suppose you are reading about horses and wantto know how long 5 furlongs are.

Start by finding a conversion table such as the one shown here.(Dictionaries often include such tables.)

To change from a large unit to a small unit, multiply. To change from asmall unit to a large one, divide.

Example Change 5 furlongs to meters.

5 � 201.168 = 1,005.84So, 5 furlongs is about 1,000 meters, or 1 kilometer.

Change each measurement to a metric measurement. Roundeach answer to the nearest tenth.

1. 10 yards 2. 100 leagues 3. 10 inches 4. 100 rods

5. 1,000 mils 6. 10 feet 7. 50 miles 8. 50 furlongs

25.4 mm 3.0 m 80.5 km 10,058.4 m,or 10.1 km

9. 50 inches 10. 200 feet 11. 200 miles 12. 200 yards

22--88

1 mil = 0.001 inch = 0.0254 millimeter

1 inch = 1,000 mil = 2.54 centimeters

12 inches = 1 foot = 0.3048 meter

3 feet = 1 yard = 0.9144 meter

5�12� yards, or 16�12� feet = 1 rod = 5.029 meters

40 rods = 1 furlong = 201.168 meters

8 furlongs

5,280 feet = 1 (statute) mile = 1.6093 kilometers

1,760 yards

3 miles = 1 (land) league = 4.828 kilometers

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Using a Measurement Conversion Chart

You may sometimes want to convert customary measurements to metricmeasurements. For example, suppose you are reading about horses and wantto know how long 5 furlongs are.

Start by finding a conversion table such as the one shown here.(Dictionaries often include such tables.)

To change from a large unit to a small unit, multiply. To change from asmall unit to a large one, divide.

Example Change 5 furlongs to meters.

5 � 201.168 = 1,005.84So, 5 furlongs is about 1,000 meters, or 1 kilometer.

Change each measurement to a metric measurement. Roundeach answer to the nearest tenth.

1. 10 yards 2. 100 leagues 3. 10 inches 4. 100 rods

9.1 m 482.8 km 25.4 cm 502.9 m

5. 1,000 mils 6. 10 feet 7. 50 miles 8. 50 furlongs

25.4 mm 3.0 m 80.5 km 10,058.4 m,or 10.1 km

9. 50 inches 10. 200 feet 11. 200 miles 12. 200 yards

127 cm 61.0 m 321.9 km 182.9 m

22--88

1 mil = 0.001 inch = 0.0254 millimeter

1 inch = 1,000 mil = 2.54 centimeters

12 inches = 1 foot = 0.3048 meter

3 feet = 1 yard = 0.9144 meter

5�12� yards, or 16�12� feet = 1 rod = 5.029 meters

40 rods = 1 furlong = 201.168 meters

8 furlongs

5,280 feet = 1 (statute) mile = 1.6093 kilometers

1,760 yards

3 miles = 1 (land) league = 4.828 kilometers

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The Speed of Light

Light travels at approximately 186,000 miles per second. You can use the formula below to find how long it takes light to travel from one place to another.

� time

For example, the sun is about 9.3 � 107 miles from Earth. If a giganticexplosion were to occur on the sun, how long would it take to see it from Earth?

= 500 Write 9.3 � 107 as 93,000,000.

It would take about 500 seconds to see the explosion.

Now you need to change seconds to minutes, since minutes is a moresensible unit for time in this case. To change seconds to minutes, divide.

8�56000� 60�5�0�0�

48020

It would take about 8 minutes to see the explosion from Earth.

Compute each amount of time it takes for light to travel to Earth from each place. Then change seconds to a sensible unit.

1.

2.

3.

4.

5.

6.

93,000,000��186,000

distance��speed of light

22--99

Closest Distance Time TimeLocation to Earth (in seconds) (in a sensible unit)

moon 2.2 � 105 mi

Mars 3.46 � 107 mi

Venus 2.57 � 107 mi

Jupiter 3.67 � 108 mi

Pluto 2.67 � 109 mi

nearest star 2.48 � 1013 mi

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The Speed of Light

Light travels at approximately 186,000 miles per second. You can use the formula below to find how long it takes light to travel from one place to another.

� time

For example, the sun is about 9.3 � 107 miles from Earth. If a giganticexplosion were to occur on the sun, how long would it take to see it from Earth?

= 500 Write 9.3 � 107 as 93,000,000.

It would take about 500 seconds to see the explosion.

Now you need to change seconds to minutes, since minutes is a moresensible unit for time in this case. To change seconds to minutes, divide.

8�56000� 60�5�0�0�

48020

It would take about 8 minutes to see the explosion from Earth.

Compute each amount of time it takes for light to travel to Earth from each place. Then change seconds to a sensible unit.

1.

2.

3.

4.

5.

6.

93,000,000��186,000

distance��speed of light

22--99

Closest Distance Time TimeLocation to Earth (in seconds) (in a sensible unit)

moon 2.2 � 105 mi 1 s 1 sMars 3.46 � 107 mi 186 s 3 minVenus 2.57 � 107 mi 138 s 2 minJupiter 3.67 � 108 mi 1,973 s 33 minPluto 2.67 � 109 mi 14,355 s 4 hnearest star 2.48 � 1013 mi 133,333,333 s 4 y

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Breaking the Code

Many secret messages are written in code. One way toconstruct a code is to use a substitution alphabet. For example,the letter A might be coded into Y, the letter B into R, and soon until every letter is coded.

To break a code of this type, it is helpful to know that theletters of the alphabet occur with different frequencies. Forexample, the letter E occurs an average of 13 times out ofevery 100 letters. In any message, however, the frequencieswill vary.

Use the clues below to break this coded message.

“FOZ BUJRSJBKD CJMMJSGDFE,” UZQKUAZC VOZUDNSAONDQZV, “DKE JR FOZ MKSF NM FOZUZ HZJRT FNN QGSOZYJCZRSZ. POKF PKV YJFKD PKV NYZUDKJC KRC OJCCZR HEPOKF PKV JUUZDZYKRF.”1. On another sheet of paper, make a frequency distribution chart for the

letters in the message. Which seven letters in the message appear mostfrequently?

2. The most frequently used letter in the alphabet is E, so write an Eunderneath each place this letter occurs in the message.

3. The word the is very common and appears twice in the message. Usethis fact to determine which letters stand for T and H.

4. The word was occurs three times in the last sentence. What lettersrepresent W, A, and S in the message?

5. The message is a quote from a famous detective whose last namebegins with H. Complete the detective’s name and you will have threemore letters.

6. What is the message?

33--11

Letters Frequency(per Hundred)

E 13T 9A, O 8N 7I, R 6.5S, H 6

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Breaking the Code

Many secret messages are written in code. One way toconstruct a code is to use a substitution alphabet. For example,the letter A might be coded into Y, the letter B into R, and soon until every letter is coded.

To break a code of this type, it is helpful to know that theletters of the alphabet occur with different frequencies. Forexample, the letter E occurs an average of 13 times out ofevery 100 letters. In any message, however, the frequencieswill vary.

Use the clues below to break this coded message.

“FOZ BUJRSJBKD CJMMJSGDFE,” UZQKUAZC VOZUDNSAONDQZV, “DKE JR FOZ MKSF NM FOZUZ HZJRT FNN QGSOZYJCZRSZ. POKF PKV YJFKD PKV NYZUDKJC KRC OJCCZR HEPOKF PKV JUUZDZYKRF.”1. On another sheet of paper, make a frequency distribution chart for the

letters in the message. Which seven letters in the message appear mostfrequently?

Z (16); K (13); J (11); F (10); O (9); D, U (8)2. The most frequently used letter in the alphabet is E, so write an E

underneath each place this letter occurs in the message. Z � E3. The word the is very common and appears twice in the message. Use

this fact to determine which letters stand for T and H.

F � T, O � H4. The word was occurs three times in the last sentence. What letters

represent W, A, and S in the message?

P � W, K � A, V � S5. The message is a quote from a famous detective whose last name

begins with H. Complete the detective’s name and you will have threemore letters. N � O, D � L, Q � M

6. What is the message?

“The principal difficulty,” remarked SherlockHolmes, “lay in the fact of there being toomuch evidence. What was vital was overlaidand hidden by what was irrelevant.”

33--11

Letters Frequency(per Hundred)

E 13T 9A, O 8N 7I, R 6.5S, H 6

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Number Patterns

The dot diagram below illustrates a number pattern.

You can discover what number in the pattern comes next by drawing thenext figure in the dot pattern. You can also use thinking with numbers. Tryto see how two consecutive numbers in the pattern are related.

It looks like the next number in the pattern is obtained by adding 6 to 15.The next number in the pattern is 21. You can check this by drawing thenext figure in the dot pattern.

Write the next two numbers in the number pattern for eachdot diagram.

1. 25, 36

2. 13, 17

3. A staircase is being built from cubes. How many cubes will it take to make a staircase 25 cubes high?

1 3 6 10 15

�2 �3 �4 �5

33--22

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Number Patterns

The dot diagram below illustrates a number pattern.

You can discover what number in the pattern comes next by drawing thenext figure in the dot pattern. You can also use thinking with numbers. Tryto see how two consecutive numbers in the pattern are related.

It looks like the next number in the pattern is obtained by adding 6 to 15.The next number in the pattern is 21. You can check this by drawing thenext figure in the dot pattern.

Write the next two numbers in the number pattern for eachdot diagram.

1. 25, 36

2. 13, 17

3. A staircase is being built from cubes. How many cubes will it take to make a staircase 25 cubes high?

325

1 3 6 10 15

�2 �3 �4 �5

33--22

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Queues

The word queue is commonly used in Great Britain to mean a line of peoplewaiting for service. Here in the United States we might “line up” to buytickets; in Great Britain, people “queue up.” There is an entire branch ofmathematics devoted to the study of queues.

This chart shows the arrival times for five different customers. Exercises 1and 2 refer to this chart. You are to assume that it takes 2 minutes for eachcustomer to be served.

1. A line plot for the information in the chart has been started below. Itshows what happens to customers A and B. Finish the plot.

2. Refer to your line plot in Exercise 1. What is the greatest number ofpeople waiting in line at any one time?

3. A store owner believes that customers will leave without making apurchase if there are more than two people ahead of them in line. Is thestore losing customers? To find out, make a line plot for the chart below.Assume that no customer leaves and that it takes 3 minutes for each to beserved.

3:05 3:06 3:07 3:08 3:09 3:103:043:033:023:013:00

BBA

B

A

33--33

Arrival Time 3:00 3:01 3:02 3:03 3:04 3:05 3:06

Customer A B C, D E

Arrival Time 1:00 1:01 1:02 1:03 1:04

Customer A B, C D

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Queues

The word queue is commonly used in Great Britain to mean a line of peoplewaiting for service. Here in the United States we might “line up” to buytickets; in Great Britain, people “queue up.” There is an entire branch ofmathematics devoted to the study of queues.

This chart shows the arrival times for five different customers. Exercises 1and 2 refer to this chart. You are to assume that it takes 2 minutes for eachcustomer to be served.

1. A line plot for the information in the chart has been started below. Itshows what happens to customers A and B. Finish the plot.

2. Refer to your line plot in Exercise 1. What is the greatest number ofpeople waiting in line at any one time? 3

3. A store owner believes that customers will leave without making apurchase if there are more than two people ahead of them in line. Is thestore losing customers? To find out, make a line plot for the chart below.Assume that no customer leaves and that it takes 3 minutes for each to beserved. no

B

C

D

B

C

D

B

C

A

B

C

A

B

C

A C C

D D

C

D

D

1:05 1:06 1:07 1:08 1:09

D

1:10

D

1:111:041:031:021:011:00

C

DC

D

C D D

D

E

E E

E E

3:05 3:06 3:07 3:08 3:09 3:103:043:033:023:013:00

BBA

B

A

33--33

Arrival Time 3:00 3:01 3:02 3:03 3:04 3:05 3:06

Customer A B C, D E

Arrival Time 1:00 1:01 1:02 1:03 1:04

Customer A B, C D

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Quartiles

The median is a number that describes the “center” of a set of data. Hereare two sets with the same median, 50, indicated by .

But, sometimes a single number may not be enough. The numbers shownin the triangles can also be used to describe the data. They are calledquartiles. The lower quartile is the median of the lower half of the data. Itis indicated by . The upper quartile is the median of the upper half. It isindicated by .

Circle the median in each set of data. Draw triangles aroundthe quartiles.

1.

2.

3.

4.

Use the following set of test scores to solve the problems. 71 57 29 37 53 41 25 37 53 27 62 55 75 48 66 53 66 48 75 66

5. Which scores are “in the lower quartile”?

6. How high would you have to score to be “in the upper quartile”?

5 2 9 7 9 3 7 8 7 2 5 6 9 5 1

1,150 1,600 1,450 1,750 1,500 1,300 1,200

1.7 0.4 1.4 2.3 0.3 2.7 2.0 0.9 2.7 2.6 1.2

29 52 44 37 27 46 43 60 31 54 36

0 10 20 30 40 50 60 70 80 90 100

25 30 35 40 45 50 55 60 65 70 75

33--44

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Quartiles

The median is a number that describes the “center” of a set of data. Hereare two sets with the same median, 50, indicated by .

But, sometimes a single number may not be enough. The numbers shownin the triangles can also be used to describe the data. They are calledquartiles. The lower quartile is the median of the lower half of the data. Itis indicated by . The upper quartile is the median of the upper half. It isindicated by .

Circle the median in each set of data. Draw triangles aroundthe quartiles.

1.

2.

3.

4.

Use the following set of test scores to solve the problems. 71 57 29 37 53 41 25 37 53 27 62 55 75 48 66 53 66 48 75 66

5. Which scores are “in the lower quartile”? 25, 27, 29, and 376. How high would you have to score to be “in the upper quartile”?

66 or higher

5 2 9 7 9 3 7 8 7 2 5 6 9 5 1

1,150 1,600 1,450 1,750 1,500 1,300 1,200

1.7 0.4 1.4 2.3 0.3 2.7 2.0 0.9 2.7 2.6 1.2

29 52 44 37 27 46 43 60 31 54 36

0 10 20 30 40 50 60 70 80 90 100

25 30 35 40 45 50 55 60 65 70 75

33--44

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Making Pictographs

A pictograph uses pictures or symbols to stand for numbers of objects.When making a pictograph, it is good to choose a symbol that can beeasily divided into fractional parts. A key should be provided to tell whateach symbol stands for.

1. Use the data given in parentheses to complete the pictograph.

2. On a separate sheet of paper, make a pictograph for the data in this chart. Create your own symbol to use in the graph.

Attendance at Professional Sports

Farming(4,300,000)

Civilian Labor Force, by OccupationKey: Each Dot Figure � 1 Million People

Laborers(5,457,000)

Transportation(5,079,000)

Machine Operators(7,776,000)

Technicians(3,350,000)

33--55

Baseball 55,173,000

Basketball 15,465,000

Football 17,399,000

Hockey 12,579,000

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Making Pictographs

A pictograph uses pictures or symbols to stand for numbers of objects.When making a pictograph, it is good to choose a symbol that can beeasily divided into fractional parts. A key should be provided to tell whateach symbol stands for.

1. Use the data given in parentheses to complete the pictograph.

2. On a separate sheet of paper, make a pictograph for the data in this chart. Create your own symbol to use in the graph.

Attendance at Professional Sports

Farming(4,300,000)

Civilian Labor Force, by OccupationKey: Each Dot Figure � 1 Million People

Laborers(5,457,000)

Transportation(5,079,000)

Machine Operators(7,776,000)

Technicians(3,350,000)

33--55

Baseball 55,173,000

Basketball 15,465,000

Football 17,399,000

Hockey 12,579,000

Graphs willvary.

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The Trimmed Mean

Sometimes a mean can be distorted by outliers. To avoid this, exclude anyoutliers and compute a new mean. This new measure is called the trimmedmean.

Construct a box-and-whisker plot for each frequency table.Mark the mean M and the trimmed mean TM.

1. Areas of 48 States, Rounded to Nearest 25 Thousand Square Miles

2. Projected Populations in 2010, Rounded to Nearest 2 Million

Use reference materials to predict the outliers.3. Problem 1 4. Problem 2

0 2 6 10 14 18 22 26 30 344 8 12 16 20 24 28 32 36

150 175 200 225 250 275 3000 25 50 75 100 125

0 1 2 3 4 5 6 7 8

*MTM

*9 10 11 12 13 14 15 16 17 18 19 20

33--66

Data Points 0 2 3 4 5 6 9 10 11 12 17 19

Frequency 6 3 6 1 5 5 1 1 1 2 1 1

Area (Thousands) 0 25 50 75 100 125 150 275

Number of States 8 4 17 10 4 2 2 1

Population (Millions) 0 2 4 6 8 10 12 16 18 22 36

Number of States 6 13 10 7 3 3 2 1 1 1 1

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The Trimmed Mean

Sometimes a mean can be distorted by outliers. To avoid this, exclude anyoutliers and compute a new mean. This new measure is called the trimmedmean.

Construct a box-and-whisker plot for each frequency table.Mark the mean M and the trimmed mean TM.

1. Areas of 48 States, Rounded to Nearest 25 Thousand Square Miles

2. Projected Populations in 2010, Rounded to Nearest 2 Million

Use reference materials to predict the outliers.3. Problem 1 4. Problem 2

Montana, California, Texas Florida, New York, Texas,California

**MTM

* *0 2 6 10 14 18 22 26 30 344 8 12 16 20 24 28 32 36

*MTM

*150 175 200 225 250 275 3000 25 50 75 100 125

0 1 2 3 4 5 6 7 8

*MTM

*9 10 11 12 13 14 15 16 17 18 19 20

33--66

Data Points 0 2 3 4 5 6 9 10 11 12 17 19

Frequency 6 3 6 1 5 5 1 1 1 2 1 1

Area (Thousands) 0 25 50 75 100 125 150 275

Number of States 8 4 17 10 4 2 2 1

Population (Millions) 0 2 4 6 8 10 12 16 18 22 36

Number of States 6 13 10 7 3 3 2 1 1 1 1

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Misleading Dissections

Dissection puzzles involve cutting one figure into parts so that the partscan be rearranged to form a second figure.

Some dissection puzzles involve vanishing squares. Here are two puzzlesof this type. Your job is to explain what happens to the “disappearing”square. In Exercises 3 and 4, use the figure below.

A B

D

C

33--77

1. Copy the Greek cross and cut it into5 pieces as shown. Then rearrange thepieces to form a square.

2. Rearrange the 4 pieces shown to form asquare. Does this square have the same areaas the one in Exercise 1?

A

B

DC

A

B

D

E

C

3. The rectangle below has an area of65 square units, but the area of the square isonly 64 square units.

4. The square has area 64 square units, but thearea of the new figure is only 63 squareunits.

D

A

C

B

DC A

B

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Misleading Dissections

Dissection puzzles involve cutting one figure into parts so that the partscan be rearranged to form a second figure.

Some dissection puzzles involve vanishing squares. Here are two puzzlesof this type. Your job is to explain what happens to the “disappearing”square. In Exercises 3 and 4, use the figure below.

A B

D

C

33--77

1. Copy the Greek cross and cut it into5 pieces as shown. Then rearrange thepieces to form a square.

2. Rearrange the 4 pieces shown to form asquare. Does this square have the same areaas the one in Exercise 1? yes

AC

D B

A

B

DC

AC

E

D

A

B

B

D

E

C

3. The rectangle below has an area of65 square units, but the area of the square isonly 64 square units.

The pieces actuallyoverlap along thediagonal lines.

4. The square has area 64 square units, but thearea of the new figure is only 63 squareunits.

D

A

C

B

DC A

B

There is aslight overlapalong the diagonal.

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Perfect Numbers

A positive integer is perfect if it equals the sum of its factors that are lessthan the integer itself.

If the sum of the factors (excluding the integer itself) is greater than theinteger, the integer is called abundant.

If the sum of the factors (excluding the integer itself) is less than theinteger, the integer is called deficient.

The factors of 28 (excluding 28 itself) are 1, 2, 4, 7, and 14.Since 1 � 2 � 4 � 7 � 14 � 28, 28 is a perfect number.

Complete the chart to classify each number as perfect,abundant, or deficient.

1.

2.

3.

4.

5.

Show that each number is perfect.

6. 496

1 � 2 � 4 � 8 � 16 � 31 � 62 � 124 � 248 � 496

7. 8,128

1 � 2 � 4 � 8 � 16 � 32 � 64 � 127 � 254 � 508 �1,016 � 2,032 � 4,064 � 8,128

8. CHALLENGE 33,550,336

1 � 2 � 4 � 8 � 16 � 32 � 64 � 128 � 256 � 512 � 1,024 �2,048 � 4,096 � 8,191 � 16,382 � 32,764 � 65,528 �131,056 � 262,112 � 524,224 � 1,048,448 � 2,096,896 �4,193,792 � 8,387,584 � 16,775,168 � 33,550,336

44--11

Divisors (Excluding theNumber Number Itself ) Sum Classification

14 1, 2, 7 10 deficient6 1, 2, 3 6 perfect12 1, 2, 3, 4, 6 16 abundant20 1, 2, 4, 5, 10 22 abundant10 1, 2, 5 8 deficient

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Perfect Numbers

A positive integer is perfect if it equals the sum of its factors that are lessthan the integer itself.

If the sum of the factors (excluding the integer itself) is greater than theinteger, the integer is called abundant.

If the sum of the factors (excluding the integer itself) is less than theinteger, the integer is called deficient.

The factors of 28 (excluding 28 itself) are 1, 2, 4, 7, and 14.Since 1 � 2 � 4 � 7 � 14 � 28, 28 is a perfect number.

Complete the chart to classify each number as perfect,abundant, or deficient.

1.

2.

3.

4.

5.

Show that each number is perfect.

6. 496

1 � 2 � 4 � 8 � 16 � 31 � 62 � 124 � 248 � 496

7. 8,128

1 � 2 � 4 � 8 � 16 � 32 � 64 � 127 � 254 � 508 �1,016 � 2,032 � 4,064 � 8,128

8. CHALLENGE 33,550,336

1 � 2 � 4 � 8 � 16 � 32 � 64 � 128 � 256 � 512 � 1,024 �2,048 � 4,096 � 8,191 � 16,382 � 32,764 � 65,528 �131,056 � 262,112 � 524,224 � 1,048,448 � 2,096,896 �4,193,792 � 8,387,584 � 16,775,168 � 33,550,336

44--11

Divisors (Excluding theNumber Number Itself ) Sum Classification

14 1, 2, 7 10 deficient6 1, 2, 3 6 perfect12 1, 2, 3, 4, 6 16 abundant20 1, 2, 4, 5, 10 22 abundant10 1, 2, 5 8 deficient

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Figurate Numbers

Figurate numbers are numbers associated with a pattern of geometricfigures. Triangular and square numbers are examples of figurate numbers.

The diagram below shows the first four hexagonal numbers. Thesefigurate numbers are based on hexagons (six-sided figures). Notice howeach number is built onto the one before it.

Solve.

1. Draw the 5th 2. Pentagonal numbers are 3. Draw the 4th squarehexagonal number. are based on a five- number.

sided figure. Draw the3rd pentagonal number.

Complete the chart below and look for patterns in thenumbers to help you.

4.

5.

6.

7.

1 6 15 28

44--22

Name 1st 2nd 3rd 4th 5th 6th

Triangular 1 3 6 10 15 21Square 1 4 9 16 25 36Pentagonal 1 5 12 22 35 51Hexagonal 1 6 15 28 45 66

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Figurate Numbers

Figurate numbers are numbers associated with a pattern of geometricfigures. Triangular and square numbers are examples of figurate numbers.

The diagram below shows the first four hexagonal numbers. Thesefigurate numbers are based on hexagons (six-sided figures). Notice howeach number is built onto the one before it.

Solve.

1. Draw the 5th 2. Pentagonal numbers are 3. Draw the 4th squarehexagonal number. are based on a five- number.

sided figure. Draw the3rd pentagonal number.

Complete the chart below and look for patterns in thenumbers to help you.

4.

5.

6.

7.

1 6 15 28

44--22

Name 1st 2nd 3rd 4th 5th 6th

Triangular 1 3 6 10 15 21Square 1 4 9 16 25 36Pentagonal 1 5 12 22 35 51Hexagonal 1 6 15 28 45 66

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Nested Magic Squares

A magic square is a square arrangement of numbers in which the sum ofthe numbers in every row, column, and diagonal is the same number. Thenumbers 1 through 49 can be arranged to make three nested magicsquares. First, the large 7-by-7 outer square is magic. Remove its borderand you get a 5-by-5 magic square. Finally, remove the border again to geta 3-by-3 magic square.

In the figure below insert the rest of the numbers 1 through 49 to makethree nested magic squares.

46 1 2 3 42 41 40

24 29 22

44--33

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Nested Magic Squares

A magic square is a square arrangement of numbers in which the sum ofthe numbers in every row, column, and diagonal is the same number. Thenumbers 1 through 49 can be arranged to make three nested magicsquares. First, the large 7-by-7 outer square is magic. Remove its borderand you get a 5-by-5 magic square. Finally, remove the border again to geta 3-by-3 magic square.

In the figure below insert the rest of the numbers 1 through 49 to makethree nested magic squares.

46 1 2 3 42 41 40

45

44

7

12

11

10

35

34

20

19

49

13

37

48

14

36

47

32

18

8

31

16

30

15

9

5

6

17 33 43

38

39

4

24 29 22

28 21 26

23 25 27

44--33

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Sundaram’s Sieve

This arrangement of numbers is called Sundaram’s Sieve. Like the Sieveof Eratosthenes, Sundaram’s arrangement can be used to find primenumbers.

Here’s how to use Sundaram’s Sieve to find prime numbers. If a number, n,is not in the Sieve, then 2n � 1 is a prime number. If a number, n, is in theSieve, then 2n � 1 is not a prime number.

32 is in the sieve. 2 � 32 � 1 � 65 65 is not prime.

35 is not in the sieve. 2 � 35 � 1 � 71 71 is prime.

1. Does the sieve give all primes up to 99? all the composites?

2. Sundaram’s Sieve is constructed from arithmetic sequences. Describethe pattern used to make the first row.

3. How is the first column constructed?

4. How are the second through fifth rows constructed?

5. How would you add a sixth row to the sieve?

6. Use Sundaram’s Sieve to find 5 four-digit prime numbers. You willneed to add more numbers to the sieve to do this.

44--44

4 7 10 13 16 19 22 25 28 31

7 12 17 22 27 32 37 42 47 52

10 17 24 31 38 45 52 59 66 73

13 22 31 40 49 58 67 76 85 94

16 27 38 49 60 71 82 93 104 115

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Mathematics: Applications© Glencoe/McGraw-Hill T27 and Connections, Course 2

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Name Date

Sundaram’s Sieve

This arrangement of numbers is called Sundaram’s Sieve. Like the Sieveof Eratosthenes, Sundaram’s arrangement can be used to find primenumbers.

Here’s how to use Sundaram’s Sieve to find prime numbers. If a number, n,is not in the Sieve, then 2n � 1 is a prime number. If a number, n, is in theSieve, then 2n � 1 is not a prime number.

32 is in the sieve. 2 � 32 � 1 � 65 65 is not prime.

35 is not in the sieve. 2 � 35 � 1 � 71 71 is prime.

1. Does the sieve give all primes up to 99? all the composites?

All primes except 2; only 25 of the composites.

2. Sundaram’s Sieve is constructed from arithmetic sequences. Describethe pattern used to make the first row.

Start with 4 and add 3 each time.3. How is the first column constructed?

It is the same as the first row.4. How are the second through fifth rows constructed?

Arithmetic sequences using 5, 7, 9, and 11.5. How would you add a sixth row to the sieve?

Start with 19 and add 13 each time.6. Use Sundaram’s Sieve to find 5 four-digit prime numbers. You will

need to add more numbers to the sieve to do this.

Answers will vary.

44--44

4 7 10 13 16 19 22 25 28 31

7 12 17 22 27 32 37 42 47 52

10 17 24 31 38 45 52 59 66 73

13 22 31 40 49 58 67 76 85 94

16 27 38 49 60 71 82 93 104 115

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Mathematics: Applications© Glencoe/McGraw-Hill 28 and Connections, Course 2

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A Two-Clock Code

Two clock faces can be used to create coded secret messages.

To encode a message, write each letter of the message as a fraction. Use the hour next to the letter as the denominator and the number in the center ofthat clock as the numerator.

For example, the letter G will be encoded as the fraction �1

7�. The letter

R becomes �25

�.

Notice the Y and Z are both writtenwith the same fraction. The same is true for P and Q.

1. Decode this message. The result will be a “secret” from a well-knownpoem written by Henry Wadsworth Longfellow.

2. Use the two-clock code to create a secret message of your own.

2–3

1–9

1–6

1––12

1––12

1––12

1––12

2––12

1–1

2–3

1–9

1–7

1–8

2–7

2–3

2–2

1–5

1–9

1–6

1–2

2––12

1––12

1–1

2–2

1–4

1–1

2–2

1–4

2–7

2––10

1–9

1–6

1–2

2––12

2–6

1–5

1–1

1–6

2–7

1–8

1–5

2–2

2–3

2–5

2–7

1–8

1–3

1–8

2–8

2–5

1–3

1–8

2–7

2–3

2––10

1–5

2–5

1–1

2–6

1–1

2–6

1–9

1–7

2–2

1–1

1––12

2–2

2–7

1–5

2–5

2–2

1–1

2–3

1–6

2–7

1–9

2–2

2–7

1–8

1–5

1–2

1–6

2–5

1–1

2–5

1–3

1–8

1–5

2–5

2–3

2–1

2–7

1–8

1–5

2–7

2–3

2––10

2–2

2–7

2–3

2–2

1–9

1–7

1–8

2–7

1–8

1–1

2–2

1–7

1–1

1–6

2–7

1–8

1–5

1–2

2–5

1–9

2–7

1–9

2–6

1–8

2–1

1–1

2–5

1–3

1–8

1–2

2––12

1––12

1–1

2–2

1–4

2–3

2–5

2–6

1–5

1–1

1 2

12 1

2

3

4

56

7

8

9

10

1112 1

2

3

4

56

7

8

9

10

11A

B

C

D

EF

G

H

I

J

K LM

N

O

P/Q

RS

T

U

V

W

XY/Z

44--55

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Mathematics: Applications© Glencoe/McGraw-Hill T28 and Connections, Course 2

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A Two-Clock Code

Two clock faces can be used to create coded secret messages.

To encode a message, write each letter of the message as a fraction. Use the hour next to the letter as the denominator and the number in the center ofthat clock as the numerator.

For example, the letter G will be encoded as the fraction �1

7�. The letter

R becomes �25

�.

Notice the Y and Z are both writtenwith the same fraction. The same is true for P and Q.

1. Decode this message. The result will be a “secret” from a well-knownpoem written by Henry Wadsworth Longfellow.

2. Use the two-clock code to create a secret message of your own.

Answers will vary.

If the British march by land or sea

from the town tonight, hang a

lantern aloft in the belfry arch

of the North Church tower as a signal

light, one if by land, and two, if by sea.

2–3

1–9

1–6

1––12

1––12

1––12

1––12

2––12

1–1

2–3

1–9

1–7

1–8

2–7

2–3

2–2

1–5

1–9

1–6

1–2

2––12

1––12

1–1

2–2

1–4

1–1

2–2

1–4

2–7

2––10

1–9

1–6

1–2

2––12

2–6

1–5

1–1

1–6

2–7

1–8

1–5

2–2

2–3

2–5

2–7

1–8

1–3

1–8

2–8

2–5

1–3

1–8

2–7

2–3

2––10

1–5

2–5

1–1

2–6

1–1

2–6

1–9

1–7

2–2

1–1

1––12

2–2

2–7

1–5

2–5

2–2

1–1

2–3

1–6

2–7

1–9

2–2

2–7

1–8

1–5

1–2

1–6

2–5

1–1

2–5

1–3

1–8

1–5

2–5

2–3

2–1

2–7

1–8

1–5

2–7

2–3

2––10

2–2

2–7

2–3

2–2

1–9

1–7

1–8

2–7

1–8

1–1

2–2

1–7

1–1

1–6

2–7

1–8

1–5

1–2

2–5

1–9

2–7

1–9

2–6

1–8

2–1

1–1

2–5

1–3

1–8

1–2

2––12

1––12

1–1

2–2

1–4

2–3

2–5

2–6

1–5

1–1

1 2

12 1

2

3

4

56

7

8

9

10

1112 1

2

3

4

56

7

8

9

10

11A

B

C

D

EF

G

H

I

J

K LM

N

O

P/Q

RS

T

U

V

W

XY/Z

44--55

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Double Bar Graphs

Complicated sets of data, such as in the chart below, are often illustratedby using double bar graphs. In graphs of this type, different colors orpatterns are used to distinguish the bars. You will need to provide a keythat tells what each color or pattern stands for.

1. Complete this double bar graph. It uses the percents given for allstudents and compares the teachers’ reports with the students’ reports.

2. On a separate sheet of paper, make a double bar graph to compare themale and female students’ reports.

0

5

10

15

20

25

30

35

40

45

None 15 Minutes 30 Minutes 45 Minutes An Hour or More

Time Spent On Math Homework Each Day (Eighth Graders)

Key:

Percents

Teachers'Reports

Students'Reports

44--66

15 30 45 An HourNone Minutes Minutes Minutes or More

All Students 1% 42% 43% 10% 4%

Male 1% 43% 42% 9% 5%

Female 1% 41% 43% 11% 4%

All Students 9% 31% 32% 16% 12%

Male 11% 34% 29% 15% 11%

Female 7% 28% 35% 17% 13%

Math HomeworkTime per Day

(Eighth Graders)

Teac

hers

’R

epo

rts

Stu

den

ts’

Rep

ort

s

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Double Bar Graphs

Complicated sets of data, such as in the chart below, are often illustratedby using double bar graphs. In graphs of this type, different colors orpatterns are used to distinguish the bars. You will need to provide a keythat tells what each color or pattern stands for.

1. Complete this double bar graph. It uses the percents given for allstudents and compares the teachers’ reports with the students’ reports.

2. On a separate sheet of paper, make a double bar graph to compare themale and female students’ reports.

Check students’ graphs.

0

5

10

15

20

25

30

35

40

45

None 15 Minutes 30 Minutes 45 Minutes An Hour or More

Time Spent On Math Homework Each Day (Eighth Graders)

Key:

Percents

Teachers'Reports

Students'Reports

44--66

15 30 45 An HourNone Minutes Minutes Minutes or More

All Students 1% 42% 43% 10% 4%

Male 1% 43% 42% 9% 5%

Female 1% 41% 43% 11% 4%

All Students 9% 31% 32% 16% 12%

Male 11% 34% 29% 15% 11%

Female 7% 28% 35% 17% 13%

Math HomeworkTime per Day

(Eighth Graders)

Teac

hers

’R

epo

rts

Stu

den

ts’

Rep

ort

s

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Dividing a Line Segment into Equal Parts

Notice in the drawing at the right how the parallel lines divide segment AB and segment AC into equal parts. This property ofparallel lines can be used to divide any segment into any numberof equal parts. Follow these steps to divide the segment at theright below in five segments each the same length.

Step 1 Lay a sheet of notebook paper over the linesegment. Line up one edge with the left endpoint of the segment.

Step 2 Hold the notebook paper at the left endpoint and then turn it until the fifth line of the paper touches the right endpoint of the segment.

Step 3 On the notebook paper, place a dot at each point where the line segment touches a line on the notebook paper. There should be six dots.

Step 4 Fold the notebook paper just above the six dots. Then use it to mark the divisions on the line segment.

Divide each line segment into the given number of equal parts.

1. three equal parts

2. seven equal parts

3. five equal parts

4. eleven equal parts

5. thirteen equal parts

6. seventeen equal parts

A B

C

44--77

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Dividing a Line Segment into Equal Parts

Notice in the drawing at the right how the parallel lines divide segment AB and segment AC into equal parts. This property ofparallel lines can be used to divide any segment into any numberof equal parts. Follow these steps to divide the segment at theright below in five segments each the same length.

Step 1 Lay a sheet of notebook paper over the linesegment. Line up one edge with the left endpoint of the segment.

Step 2 Hold the notebook paper at the left endpoint and then turn it until the fifth line of the paper touches the right endpoint of the segment.

Step 3 On the notebook paper, place a dot at each point where the line segment touches a line on the notebook paper. There should be six dots.

Step 4 Fold the notebook paper just above the six dots. Then use it to mark the divisions on the line segment.

Divide each line segment into the given number of equal parts.

1. three equal parts

2. seven equal parts

3. five equal parts

4. eleven equal parts

5. thirteen equal parts

6. seventeen equal parts

A B

C

44--77

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Coin-Tossing ExperimentsIf a coin is tossed 3 times, there are 8 possible outcomes. They are listed inthe table below.

Once all the outcomes are known, the probability of any event can be

found. For example, the probability of getting 2 heads is �38

�. Notice thatthis is the same as getting 1 tail.

1. A coin is tossed 4 times. Complete this chart to show the possibleoutcomes.

2. What is the probability of getting all tails? �116�

3. Now complete this table. Make charts like the one in Exercise 1 to helpfind the answers. Look for patterns in the numbers.

4. What happens to the number of outcomes? the probability of all tails?

44--88

Number of Heads 0 1 2 3

Outcomes TTT HTT HHT HHH

THT THH

TTH HTH

Number of Coin Tosses 2 3 4 5 6 7 8

Total Outcomes 4 8 16 32 64 128 256

Probability of Getting �14� �18� �116� �3

12� �6

14� �1

128� �2

156�All Tails

Number of Heads 0 1 2 3 4

Outcomes TTTT HTTT HHTT THHH HHHHTHTT HTHT HTHHTTHT THHT HHTHTTTH TTHH HHHT

THTHHTTH

Mathematics: Applications© Glencoe/McGraw-Hill 31 and Connections, Course 2

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Name Date

Coin-Tossing ExperimentsIf a coin is tossed 3 times, there are 8 possible outcomes. They are listed inthe table below.

Once all the outcomes are known, the probability of any event can be

found. For example, the probability of getting 2 heads is �38

�. Notice thatthis is the same as getting 1 tail.

1. A coin is tossed 4 times. Complete this chart to show the possibleoutcomes.

2. What is the probability of getting all tails? �116�

3. Now complete this table. Make charts like the one in Exercise 1 to helpfind the answers. Look for patterns in the numbers.

4. What happens to the number of outcomes? the probability of all tails?

It is doubling; it is halving.

44--88

Number of Heads 0 1 2 3

Outcomes TTT HTT HHT HHH

THT THH

TTH HTH

Number of Coin Tosses 2 3 4 5 6 7 8

Total Outcomes 4 8 16 32 64 128 256

Probability of Getting �14� �18� �116� �3

12� �6

14� �1

128� �2

156�All Tails

Number of Heads 0 1 2 3 4

Outcomes TTTT HTTT HHTT THHH HHHHTHTT HTHT HTHHTTHT THHT HHTHTTTH TTHH HHHT

THTHHTTH

Mathematics: Applications© Glencoe/McGraw-Hill T31 and Connections, Course 2

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A Cross-Number Puzzle

Use the clues at the bottom of the page to complete the puzzle. You are towrite one digit in each box.

A2 9 3

B

E5

8

C D

F G H I

J K

L M

N O P

Q R

44--99

Across

C largest number less than 200 that isdivisible by 29

E square of first prime greater than 20

F sum of first seven Fibonacci numbers

H next term in sequence 61, 122, 244, 488

J greatest common factor of 141 and 329

K the eighth power of 2

L least common multiple of 2, 7, and 13

M numerator of fraction equal to 0.8125

N least common multiple of 86 and 5

O smallest prime greater than 60

P largest two-digit prime

Q next term in sequence 4, 15, 26, 37

R largest two-digit composite less than 40

Down

B smallest number divisible by 3 and 5

D the sixth power of 4

G least common multiple of 2 and 179

H the number of two-digit positive integers

I smallest number over 600 divisible by 89

L smallest three-digit number divisible by13.

M the smallest two-digit prime number

N largest prime factor of 82

O perfect square between 60 and 70

P largest two-digit number divisible by 3

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A Cross-Number Puzzle

Use the clues at the bottom of the page to complete the puzzle. You are towrite one digit in each box.

A2 9 3

B

E5

8

1

1

1

1

1

5 2 9

9 9

9

9

1 7

7

7

4

0

8

8

3

3

3

3

3

6

6

6

4

4

4 4

7 5 50

0

2

2

C D

F G H I

J K

L M

N O P

Q R

44--99

Across

C largest number less than 200 that isdivisible by 29

E square of first prime greater than 20

F sum of first seven Fibonacci numbers

H next term in sequence 61, 122, 244, 488

J greatest common factor of 141 and 329

K the eighth power of 2

L least common multiple of 2, 7, and 13

M numerator of fraction equal to 0.8125

N least common multiple of 86 and 5

O smallest prime greater than 60

P largest two-digit prime

Q next term in sequence 4, 15, 26, 37

R largest two-digit composite less than 40

Down

B smallest number divisible by 3 and 5

D the sixth power of 4

G least common multiple of 2 and 179

H the number of two-digit positive integers

I smallest number over 600 divisible by 89

L smallest three-digit number divisible by13.

M the smallest two-digit prime number

N largest prime factor of 82

O perfect square between 60 and 70

P largest two-digit number divisible by 3

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Intersection and Union of Sets

The shaded areas in the Venn diagrams show the union and intersection ofsets A and B.

For example, if A � {1, 2, 3, 4} and B � {3, 4, 5, 6}, then their union and intersection are written as:

Union: A � B � {1, 2, 3, 4, 5, 6} Intersection: A � B � {3, 4}

Draw a Venn diagram for sets A and B. Then write thenumbers included in A � B and A � B. In Exercises 2 and 4,record the numbers as decimals.

1. A � {integers between 0 and 7}B � {factors of 12}

A � B � {1, 2, 3, 4, 5, 6, 12}A � B � {1, 2, 3, 4, 6}

2. A � {one-place decimals between 0 and 0.5}B � {fractions with 1, 2, 3, or 4 as numerator

and 5 as a denominator}

A � B � {0.1, 0.2, 0.3, 0.4, 0.6,0.8}A � B � {0.2, 0.4}

3. A � {perfect squares between 0 and 30}B � {odd whole numbers less than 10}

A � B � {1, 3, 4, 5, 7, 9, 16, 25}A � B � {1, 9}

4. A � ��12

�, �13

�, �14

�, �15

��B � {0.1�, 0.2�, 0.3�, 0.4�, 0.5�, 0.6�, 0.7�, 0.8�, 0.9�}

A � B � {0.1�, 0.2, 0.2�, 0.25,0.3�, 0.4�, 0.5, 0.5�, 0.6�, 0.7�, 0.8�,0.9�}A � B � {0.3�}

A B

Intersection A B

A B

Union A B

44--1100

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Intersection and Union of Sets

The shaded areas in the Venn diagrams show the union and intersection ofsets A and B.

For example, if A � {1, 2, 3, 4} and B � {3, 4, 5, 6}, then their union and intersection are written as:

Union: A � B � {1, 2, 3, 4, 5, 6} Intersection: A � B � {3, 4}

Draw a Venn diagram for sets A and B. Then write thenumbers included in A � B and A � B. In Exercises 2 and 4,record the numbers as decimals.

1. A � {integers between 0 and 7}B � {factors of 12}

A � B � {1, 2, 3, 4, 5, 6, 12}A � B � {1, 2, 3, 4, 6}

2. A � {one-place decimals between 0 and 0.5}B � {fractions with 1, 2, 3, or 4 as numerator

and 5 as a denominator}

A � B � {0.1, 0.2, 0.3, 0.4, 0.6,0.8}A � B � {0.2, 0.4}

3. A � {perfect squares between 0 and 30}B � {odd whole numbers less than 10}

A � B � {1, 3, 4, 5, 7, 9, 16, 25}A � B � {1, 9}

4. A � ��12

�, �13

�, �14

�, �15

��B � {0.1�, 0.2�, 0.3�, 0.4�, 0.5�, 0.6�, 0.7�, 0.8�, 0.9�}

A � B � {0.1�, 0.2, 0.2�, 0.25,0.3�, 0.4�, 0.5, 0.5�, 0.6�, 0.7�, 0.8�,0.9�}A � B � {0.3�}

0.5

0.25

0.2

0.3

0.1 0.2 0.4

0.5 0.6

0.7 0.8 0.9

– – –

– –

– –

164

25

19

35

7

0.1

0.3

0.2

0.4

0.6

0.8

51 2 3

6 4

12

A B

Intersection A B

A B

Union A B

44--1100

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Integer DominoesIn integer dominoes, black dots on a white background represent positive integers and white dots on a black background represent negative integers.

Cut out the 15 dominoes at the bottom right of this page. Then use them to solve the two puzzles. To solve each puzzle, the ends of adjacent dominoes must match.

Solve each puzzle.1. The spokes of the wheel begin 2.

with the five numbers shown.

2

1

0

�1�2

�2 �2

55--11

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Mathematics: Applications© Glencoe/McGraw-Hill T34 and Connections, Course 2

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Integer DominoesIn integer dominoes, black dots on a white background represent positive integers and white dots on a black background represent negative integers.

Cut out the 15 dominoes at the bottom right of this page. Then use them to solve the two puzzles. To solve each puzzle, the ends of adjacent dominoes must match.

Solve each puzzle.1. The spokes of the wheel begin 2.

with the five numbers shown.

2

1

0

�1�2

�2 �2

55--11

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Mathematics: Applications© Glencoe/McGraw-Hill 35 and Connections, Course 2

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Quantitative Comparisons

An unusual type of problem is found on some standardized multiple-choice tests. This problem type is called the quantitative comparison.

In each quantitative comparison question, you are given two quantities,one in Column A and one in Column B. You are to compare the twoquantities and shade one of four circles on an answer sheet.

Shade circle A if the quantity in Column A is greater;

Shade circle B if the quantity in Column B is greater;

Shade circle C if the two quantities are equal;

Shade circle D if the relationship cannot be determinedfrom the information given.

Shade the correct circle to the left of each problem number.

55--22

Column A Column B

0.006 � 2 0.002 � 6

�x if x is less than 0 x if x is less than 0

the greatest possible the greatest possibleproduct of two odd positive product of two even positivenumbers less than 20 numbers less than 20

ten billion dollars 1,000 million dollars

20 inches the perimeter of a squarewith an area of 25 squareinches

0.000000001 �x is x if greater than 0

half of one third one fifth

A B C D 1.

A B C D 2.

A B C D 3.

A B C D 4.

A B C D 5.

A B C D 6.

A B C D 7.

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Mathematics: Applications© Glencoe/McGraw-Hill T35 and Connections, Course 2

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Name Date

Quantitative Comparisons

An unusual type of problem is found on some standardized multiple-choice tests. This problem type is called the quantitative comparison.

In each quantitative comparison question, you are given two quantities,one in Column A and one in Column B. You are to compare the twoquantities and shade one of four circles on an answer sheet.

Shade circle A if the quantity in Column A is greater;

Shade circle B if the quantity in Column B is greater;

Shade circle C if the two quantities are equal;

Shade circle D if the relationship cannot be determinedfrom the information given.

Shade the correct circle to the left of each problem number.

55--22

Column A Column B

0.006 � 2 0.002 � 6

�x if x is less than 0 x if x is less than 0

the greatest possible the greatest possibleproduct of two odd positive product of two even positivenumbers less than 20 numbers less than 20

ten billion dollars 1,000 million dollars

20 inches the perimeter of a squarewith an area of 25 squareinches

0.000000001 �x is x if greater than 0

half of one third one fifth

A B C D 1.

A B C D 2.

A B C D 3.

A B C D 4.

A B C D 5.

A B C D 6.

A B C D 7.

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Latitude and Longitude

This world map shows some of the latitude and longitude lines. Latitude ismeasured in degrees north and south of the equator. Longitude ismeasured in degrees east and west of the prime meridian, a line passingthrough Greenwich, England. (Greenwich is a suburb of London.)

The latitude is usually given first. For example, the location of 30°S, 60°Wis lower South America.

(Sample answers are given. Students mayprovide more specific places.)Name a place near each location. Use an atlas or otherreference source to check your answers.

1. 30°N, 30°W 2. 30°S, 30°E 3. 60°N, 120°W

Atlantic Ocean South Africa Canada4. 15°N, 150°W 5. 30°S, 140°E 6. 25°N, 100°W

Hawaii Australia Mexico7. 40°N, 120°W 8. 45°N, 90°W 9. 40°N, 5°W

California Wisconsin Spain10. 60°N, 45°W 11. 35°N, 140°E 12. 0°, 60°E

Greenland Japan Indian Ocean

Equator

Prim

e M

erid

ian

60˚N

30˚N

30˚S, 60˚W

30˚N

30˚S 30˚S

60˚S 60˚S

60˚N

30˚E

30˚ W

60˚E

90˚E

120̊

E

150̊

E

180̊

E

60˚ W

90˚ W

120˚ W

150W̊

55--33

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Latitude and Longitude

This world map shows some of the latitude and longitude lines. Latitude ismeasured in degrees north and south of the equator. Longitude ismeasured in degrees east and west of the prime meridian, a line passingthrough Greenwich, England. (Greenwich is a suburb of London.)

The latitude is usually given first. For example, the location of 30°S, 60°Wis lower South America.

(Sample answers are given. Students mayprovide more specific places.)Name a place near each location. Use an atlas or otherreference source to check your answers.

1. 30°N, 30°W 2. 30°S, 30°E 3. 60°N, 120°W

Atlantic Ocean South Africa Canada4. 15°N, 150°W 5. 30°S, 140°E 6. 25°N, 100°W

Hawaii Australia Mexico7. 40°N, 120°W 8. 45°N, 90°W 9. 40°N, 5°W

California Wisconsin Spain10. 60°N, 45°W 11. 35°N, 140°E 12. 0°, 60°E

Greenland Japan Indian Ocean

Equator

Prim

e M

erid

ian

60˚N

30˚N

30˚S, 60˚W

30˚N

30˚S 30˚S

60˚S 60˚S

60˚N

30˚E

30˚ W

60˚E

90˚E

120̊

E

150̊

E

180̊

E

60˚ W

90˚ W

120˚ W

150W̊

55--33

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Mathematics: Applications© Glencoe/McGraw-Hill 37 and Connections, Course 2

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Dart Board Puzzles

Three darts are thrown. Each dart must land on a different spacein order to count. Find the highest and the lowest possible scores.

1. 2. 3.

highest score: highest score: highest score:

lowest score: lowest score: lowest score:

In these problems, five darts are thrown. Each dart must land ona different space in order to count. Solve each puzzle.

4. Find three ways to make the score �5. 5. Find three ways to make the score 0.

−5

−5

−5

−5

−5

−5

−5

−50

−5

−10

−10

−10−10

−15

−15

−10

−20

−20

−20

−25

0

0

10

10

10

10

105

55

55

5

5

50

35

25

25

20

20 2015

1515

15

15

15

20

50

−10

−10

−1

−3

−3

−3

−2

−2

−8

−7

−7

−9−8

−9

−5

−5

−6

−6

−4

−4

−1

−1

0

0

0

0

5

5

8

8

2

22

9

10

10

3

3

3

7

7

6

6

4

4

1

1

1

9

100

−75−50

−50

25

25

−150

−150

75

75

−10

−10−75

10

10

200

200

0

−1

−4−1

−4

−3

−3

−8

−8

−7

−7−2

−2

−6−6−5

−5

−8

−1 −4

−6−7

−2

26

3

5

4

3

170

−6

−5

55--44

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Mathematics: Applications© Glencoe/McGraw-Hill T37 and Connections, Course 2

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Dart Board Puzzles

Three darts are thrown. Each dart must land on a different spacein order to count. Find the highest and the lowest possible scores.

1. 2. 3.

highest score: 18 highest score: �2 highest score: 500lowest score: �21 lowest score: �23 lowest score: �375

In these problems, five darts are thrown. Each dart must land ona different space in order to count. Solve each puzzle.

4. Find three ways to make the score �5. 5. Find three ways to make the score 0.

Answers will vary.

−5

−5

−5

−5

−5

−5

−5

−50

−5

−10

−10

−10−10

−15

−15

−10

−20

−20

−20

−25

0

0

10

10

10

10

105

55

55

5

5

50

35

25

25

20

20 2015

1515

15

15

15

20

50

−10

−10

−1

−3

−3

−3

−2

−2

−8

−7

−7

−9−8

−9

−5

−5

−6

−6

−4

−4

−1

−1

0

0

0

0

5

5

8

8

2

22

9

10

10

3

3

3

7

7

6

6

4

4

1

1

1

9

100

−75−50

−50

25

25

−150

−150

75

75

−10

−10−75

10

10

200

200

0

−1

−4−1

−4

−3

−3

−8

−8

−7

−7−2

−2

−6−6−5

−5

−8

−1 −4

−6−7

−2

26

3

5

4

3

170

−6

−5

55--44

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Mathematics: Applications© Glencoe/McGraw-Hill 38 and Connections, Course 2

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55--55Distance on the Number LineTo find the distance between two points on a number line, subtract theircoordinates. Then, take the absolute value of the difference.

�4 � 3 � �7�7= 7

You can also find the distance by finding the absolute value of thedifference of the coordinates.

�4 � 3 � 7

Graph each pair of points. Then write an expression usingabsolute value to find the distance between the points.

1. A at �5 and B at 2

2. C at �7 and D at �1

3. E at �5 and F at 5

4. W at 0 and X at 6

5. Y at �4 and Z at 0

�8 �7 �6 �5 �4 �3 �2 �1 876543210

�8 �7 �6 �5 �4 �3 �2 �1 876543210

�8 �7 �6 �5 �4 �3 �2 �1 876543210

�8 �7 �6 �5 �4 �3 �2 �1 876543210

�8 �7 �6 �5 �4 �3 �2 �1 876543210

�8 �7 �6 �5 �4 �3 �2 �1 876543210

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55--55Distance on the Number LineTo find the distance between two points on a number line, subtract theircoordinates. Then, take the absolute value of the difference.

�4 � 3 � �7�7= 7

You can also find the distance by finding the absolute value of thedifference of the coordinates.

�4 � 3 � 7

Graph each pair of points. Then write an expression usingabsolute value to find the distance between the points.

1. A at �5 and B at 2 �5 � 2 � 7

2. C at �7 and D at �1 �7 � (�1) � 6

3. E at �5 and F at 5 �5 � 5 � 10

4. W at 0 and X at 6 0 � 6 � 6

5. Y at �4 and Z at 0 �4 � 0 � 4Y Z

�8 �7 �6 �5 �4 �3 �2 �1 876543210

W X�8 �7 �6 �5 �4 �3 �2 �1 876543210

E F�8 �7 �6 �5 �4 �3 �2 �1 876543210

C D�8 �7 �6 �5 �4 �3 �2 �1 876543210

A B�8 �7 �6 �5 �4 �3 �2 �1 876543210

�8 �7 �6 �5 �4 �3 �2 �1 876543210

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Mathematics: Applications© Glencoe/McGraw-Hill 39 and Connections, Course 2

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Integer Maze

Find your way through the maze by moving to the expression with the nexthighest value.

Start−50

−55

20 − 20

−32 + 28

−12 + 2

−3 + (−4)

4 + (−12)

−13 + 12−10 + 16

−35 + 5

6 − 8

5(−9)

9(−6)

−8(5)

−3(5)

−5(−6)

−1(3)

−4(2)

−3(−5)

3(−3)(−5)

−2(−5)(−1)

6(−10)9(−1)

3 + (−3)

30 − (−10)

−4(5)(−1)

3(5)(−5)

5(−4 + 9)

(−4)2

−52

−[12 − (−8)]

2[−5 • (−5)]

−2(−12 + 7)

−2(−1)

−5

0 − 6

15 − 50

55--66

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Integer Maze

Find your way through the maze by moving to the expression with the nexthighest value.

Start−50

−55

20 − 20

−32 + 28

−12 + 2

−3 + (−4)

4 + (−12)

−13 + 12−10 + 16

−35 + 5

6 − 8

5(−9)

9(−6)

−8(5)

−3(5)

−5(−6)

−1(3)

−4(2)

−3(−5)

3(−3)(−5)

−2(−5)(−1)

6(−10)9(−1)

3 + (−3)

30 − (−10)

−4(5)(−1)

3(5)(−5)

5(−4 + 9)

(−4)2

−52

−[12 − (−8)]

2[−5 • (−5)]

−2(−12 + 7)

−2(−1)

−5

0 − 6

15 − 50

55--66

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Division by Zero?

Some interesting things happen when you try to divide by zero. Forexample, look at these two equations.

�50

� � x �00

� � y

If you can write the equations above, you can also write the two equationsbelow.

0 � x � 5 0 � y � 0

However, there is no number that will make the left equation true. Thisequation has no solution. For the right equation, every number will make ittrue. The solutions for this equation are “all numbers.”

Because division by zero leads to impossible situations, it is not a “legal”step in solving a problem. People say that division by zero is undefined, ornot possible, or simply not allowed.

Describe the solution set for each equation.

1. 4x � 0 2. x � 0 � 0 3. x � 0 � x

4. �0x

� � 0 5. �0x

� � x 6. �0x

� � 5

What values for x must be excluded to prevent division by 0?

7. 8. 9.

10. �20x� 11. 12.

Explain what is wrong with this “proof.”

13. Step 1 0 � 1 � 0 and 0 � (�1) � 0

Step 2 Therefore, �00

� � 1 and �00

� � �1. Step 2 involvesStep 3 Therefore, 1 � �1. division by zero.

1�3x � 6

1�2x � 2

1�x � 1

1�x � 1

1�x2

55--77

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Mathematics: Applications© Glencoe/McGraw-Hill T40 and Connections, Course 2

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Division by Zero?

Some interesting things happen when you try to divide by zero. Forexample, look at these two equations.

�50

� � x �00

� � y

If you can write the equations above, you can also write the two equationsbelow.

0 � x � 5 0 � y � 0

However, there is no number that will make the left equation true. Thisequation has no solution. For the right equation, every number will make ittrue. The solutions for this equation are “all numbers.”

Because division by zero leads to impossible situations, it is not a “legal”step in solving a problem. People say that division by zero is undefined, ornot possible, or simply not allowed.

Describe the solution set for each equation.

1. 4x � 0 0 2. x � 0 � 0 all numbers 3. x � 0 � x 0

4. �0x

� � 0 5. �0x

� � x 6. �0x

� � 5

all numbers but 0 no solution no solution

What values for x must be excluded to prevent division by 0?

7. 0 8. 1 9. �1

10. �20x� 0 11. 1 12. �2

Explain what is wrong with this “proof.”

13. Step 1 0 � 1 � 0 and 0 � (�1) � 0

Step 2 Therefore, �00

� � 1 and �00

� � �1. Step 2 involvesStep 3 Therefore, 1 � �1. division by zero.

1�3x � 6

1�2x � 2

1�x � 1

1�x � 1

1�x2

55--77

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TransformationsThe figure at the right shows a partially filled TIC-TAC-TOE board. If the board is rotated 90° in the clockwise direction,the result is the second figure at the right.

The six diagrams below show what happens when the TIC-TAC-TOE board is transformed in other ways.

A larger version of a TIC-TAC-TOE board is shown below.Which kind of transformation was used to obtain each boardfrom the board given?

1. 2. 3.

180° rotation vertical flip diagonal flip I4. 5. 6.

horizontal flip diagonal flip II 270° rotation

X

X X

X

X

X

X

XO

O

O

O O

O

O

O

X

X X

X

X

X

X

XO

O

O

O O

O

O

O

X

X X

XX X

X

XO

OO

O O

O

O

O

X

X X

X

X

X

X

XO

O

O

O O

O

O

O

X

X X

X

X

X X

XO

O

O

O O

O O

OX

X X

X

X

X X

XO

O

O

OO

O O

O

X

X X

X

X

X X

XO

O

O

OO

O O

O

O

X X

vertical flip

OX

X

diagonal flip II

OX

X

270˚ rotation

O

X X

horizontal flip

O

X

X

diagonal flip I

O

XX

180˚ rotation

O

X X

O X

X

55--88

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Mathematics: Applications© Glencoe/McGraw-Hill T41 and Connections, Course 2

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Name Date

TransformationsThe figure at the right shows a partially filled TIC-TAC-TOE board. If the board is rotated 90° in the clockwise direction,the result is the second figure at the right.

The six diagrams below show what happens when the TIC-TAC-TOE board is transformed in other ways.

A larger version of a TIC-TAC-TOE board is shown below.Which kind of transformation was used to obtain each boardfrom the board given?

1. 2. 3.

180° rotation vertical flip diagonal flip I4. 5. 6.

horizontal flip diagonal flip II 270° rotation

X

X X

X

X

X

X

XO

O

O

O O

O

O

O

X

X X

X

X

X

X

XO

O

O

O O

O

O

O

X

X X

XX X

X

XO

OO

O O

O

O

O

X

X X

X

X

X

X

XO

O

O

O O

O

O

O

X

X X

X

X

X X

XO

O

O

O O

O O

OX

X X

X

X

X X

XO

O

O

OO

O O

O

X

X X

X

X

X X

XO

O

O

OO

O O

O

O

X X

vertical flip

OX

X

diagonal flip II

OX

X

270˚ rotation

O

X X

horizontal flip

O

X

X

diagonal flip I

O

XX

180˚ rotation

O

X X

O X

X

55--88

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Equation Hexa-maze

This figure is called a hexa-maze because each cell has the shape of ahexagon, or six-sided figure.

To solve the maze, start with the number in the center. This number is thesolution to the equation in one of the adjacent cells. Move to that cell. Thenumber in the new cell will then be the solution to the equation in the nextcell. At each move, you may only move to an adjacent cell. Each cell isused only once.

n � 3.7 � 7

7 � n � 33

16 � n � 23

n � 4.5 � 6

3 � 4.5 � n

41 � 30 � n

n � 16 � 33

12 � n � 18

9 � n � 4.5

n � 4.5 � 10

19 � n � 17.9

3 � n � 1.5

22 � n � 18

75 � n � 55

n � 36 � 40

28 � n � 12

8 � n � 3

150 � n � 175

75 � 80 � n

40 � 40 � n

n � 11� 16

5.2 � n � 3.7

29.2 � 36.2 � n

75 � n � 25

11

3.3

17

4.5

1.5

1.1

40

7

30

1.5

1.5

4

Start Here

40

25

5.5

0

5

4

100

20

End

1.5

7

11

5

66--11

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Equation Hexa-maze

This figure is called a hexa-maze because each cell has the shape of ahexagon, or six-sided figure.

To solve the maze, start with the number in the center. This number is thesolution to the equation in one of the adjacent cells. Move to that cell. Thenumber in the new cell will then be the solution to the equation in the nextcell. At each move, you may only move to an adjacent cell. Each cell isused only once.

n � 3.7 � 7

7 � n � 33

16 � n � 23

n � 4.5 � 6

3 � 4.5 � n

41 � 30 � n

n � 16 � 33

12 � n � 18

9 � n � 4.5

n � 4.5 � 10

19 � n � 17.9

3 � n � 1.5

22 � n � 18

75 � n � 55

n � 36 � 40

28 � n � 12

8 � n � 3

150 � n � 175

75 � 80 � n

40 � 40 � n

n � 11� 16

5.2 � n � 3.7

29.2 � 36.2 � n

75 � n � 25

11

3.3

17

4.5

1.5

1.1

40

7

30

1.5

1.5

4

Start Here

40

25

5.5

0

5

4

100

20

End

1.5

7

11

5

66--11

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Describing Variation

Equations of the form y � ax and y � x � a can be used to show how onequantity varies with another. Here are two examples.

Driving at a speed of 50 miles per hour, the distance you travel (d ) varies d � 50tdirectly with the time you are on the road (t). The longer you drive, the farther you get.

It is also the case that the time (t) varies directly with the distance (d ). The t � �5d0�

farther you drive, the more time it takes.

Complete the equation for each situation. Then describe therelationship in words.

1. If you go on a diet and lose 2 pounds a month, after a certain number p �of months (m), you will have lost p pounds.

The longer you diet, the more weight you will lose.

2. You and your family are deciding between two different places for t �your summer vacation. You plan to travel by car and estimate you willaverage 55 miles per hour. The distance traveled (d ) will result in a travel time of t hours.

The farther you drive, the more time it will take.

3. You find that you are spending more than you had planned on renting m �video movies. It costs $2.00 to rent each movie. You can use the total amount spent (a) to find the number of movies you have rented (m).

The greater the amount spent, the more movies rented.

4. You spend $30 a month to take the bus to school. After a certain d �number of months (m), you will have spent a total of d dollars on transportation to school.

The longer you ride the bus, the more you will spend.

5. You are saving money for some new athletic equipment and have s �12 weeks before the season starts. The amount you need to save each week (s) will depend on the cost (c) of the equipment you want to buy.

The more expensive the equipment, the more money must be saved each week.

66--22

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Mathematics: Applications© Glencoe/McGraw-Hill T43 and Connections, Course 2

Enrichment

Name Date

Describing Variation

Equations of the form y � ax and y � x � a can be used to show how onequantity varies with another. Here are two examples.

Driving at a speed of 50 miles per hour, the distance you travel (d ) varies d � 50tdirectly with the time you are on the road (t). The longer you drive, the farther you get.

It is also the case that the time (t) varies directly with the distance (d ). The t � �5d0�

farther you drive, the more time it takes.

Complete the equation for each situation. Then describe therelationship in words.

1. If you go on a diet and lose 2 pounds a month, after a certain number p � 2mof months (m), you will have lost p pounds.

The longer you diet, the more weight you will lose.

2. You and your family are deciding between two different places for t � �5d5�

your summer vacation. You plan to travel by car and estimate you willaverage 55 miles per hour. The distance traveled (d ) will result in a travel time of t hours.

The farther you drive, the more time it will take.

3. You find that you are spending more than you had planned on renting m � �a2�video movies. It costs $2.00 to rent each movie. You can use the total amount spent (a) to find the number of movies you have rented (m).

The greater the amount spent, the more movies rented.

4. You spend $30 a month to take the bus to school. After a certain d � 30mnumber of months (m), you will have spent a total of d dollars on transportation to school.

The longer you ride the bus, the more you will spend.

5. You are saving money for some new athletic equipment and have s � �1c2�

12 weeks before the season starts. The amount you need to save each week (s) will depend on the cost (c) of the equipment you want to buy.

The more expensive the equipment, the more money must be saved each week.

66--22

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Combining Like Terms

Some equations contain two or more expressions that are called like terms.For example, in the equation 3a � 2a � 4 � 14, the expressions 2a and 3aare like terms. When you see like terms, you can combine them into oneexpression.

2a � 3a � 5a

When you solve an equation containing like terms, combine them firstbefore continuing to solve the equation. To solve 2a � 3a � 4 � 14,proceed as follows.

2a � 3a � 4 � 14

Combine like terms.

5a � 4 � 14

5a � 4 � 4 � 14 � 4

5a � 10

�55a� � �1

50�

a � 2

Solve each equation. Then locate the solution on the numberline below. Place the letter corresponding to the answer on theline at the right of the exercise.

1. 3x � 4x � 3 � �39

2. �3x � 2 � 5x � 12

3. �5 � 4x � 7x � 1

4. ��12

�x � 6x � 2 � 20

5. �2.4x � 1.2 � 1.2x � 4.8

6. �13

�(6 � x) � �1

7. 1 � ��14

�x � 5 � �34

�x

8. 7x � (�2x) � x � 42

9. �25

�(5x � 5x) � �20

W I H F L J O P N V G C D T M K A B R

�9 �8 �7 �6 �5 �4 �3 �2 �1 9876543210

66--33

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Mathematics: Applications© Glencoe/McGraw-Hill T44 and Connections, Course 2

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Name Date

Combining Like Terms

Some equations contain two or more expressions that are called like terms.For example, in the equation 3a � 2a � 4 � 14, the expressions 2a and 3aare like terms. When you see like terms, you can combine them into oneexpression.

2a � 3a � 5a

When you solve an equation containing like terms, combine them firstbefore continuing to solve the equation. To solve 2a � 3a � 4 � 14,proceed as follows.

2a � 3a � 4 � 14

Combine like terms.

5a � 4 � 14

5a � 4 � 4 � 14 � 4

5a � 10

�55a� � �1

50�

a � 2

Solve each equation. Then locate the solution on the numberline below. Place the letter corresponding to the answer on theline at the right of the exercise.

1. 3x � 4x � 3 � �39

2. �3x � 2 � 5x � 12

3. �5 � 4x � 7x � 1

4. ��12

�x � 6x � 2 � 20

5. �2.4x � 1.2 � 1.2x � 4.8

6. �13

�(6 � x) � �1

7. 1 � ��14

�x � 5 � �34

�x

8. 7x � (�2x) � x � 42

9. �25

�(5x � 5x) � �20

W I H F L J O P N V G C D T M K A B R

�9 �8 �7 �6 �5 �4 �3 �2 �1 9876543210

66--33

FACTORIAL

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Expressions for Figurate Numbers

Figurate numbers are numbers that can be shown with dots arranged inspecific geometric patterns. Below are the first five square numbers.

The expression n2 will give you the number of dots in the nth squarenumber. The variable n takes on the values 1, 2, 3, 4, and so on. So, to findthe 10th square number, you would use 10 for n.

1. Match each set of dot patterns with its name and expression. Writeexercise numbers in the boxes to show the matchings.

Dot Patterns for Name ofSecond and Third Numbers Figurate Number

Expression

a. pentagonal n(2n � 1)

b. hexagonal

c. triangular

Use the algebraic expressions on this page to compute eachnumber. Then make a drawing of the number on a separatesheet of paper.

2. 6th square 3. 4th triangular 4. 4th pentagonal

5. 4th hexagonal 6. 5th triangular 7. 5th pentagonal

28 15 35

n(3n � 1)��2

n(n � 1)�2

66--44

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Expressions for Figurate Numbers

Figurate numbers are numbers that can be shown with dots arranged inspecific geometric patterns. Below are the first five square numbers.

The expression n2 will give you the number of dots in the nth squarenumber. The variable n takes on the values 1, 2, 3, 4, and so on. So, to findthe 10th square number, you would use 10 for n.

1. Match each set of dot patterns with its name and expression. Writeexercise numbers in the boxes to show the matchings.

Dot Patterns for Name ofSecond and Third Numbers Figurate Number

Expression

a. pentagonal n(2n � 1)

b. hexagonal

c. triangular

Use the algebraic expressions on this page to compute eachnumber. Then make a drawing of the number on a separatesheet of paper.

2. 6th square 3. 4th triangular 4. 4th pentagonal

36 10 22

5. 4th hexagonal 6. 5th triangular 7. 5th pentagonal

28 15 35

bn(3n � 1)��2a

an(n � 1)�2c

cb

66--44

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Compound Inequalities

Statements that consist of two or more inequalities are called compoundinequalities. When you graph a compound inequality, you need to payspecial attention to the words that connect the inequalities.

The graph includes all numbers that are The graph includes all numbers that areeither less than �3 or greater than 2. both greater than �3 and less than 2.

Graph each compound inequality.

1. h � �5 and h � 4

2. q � �7 or q � 6

3. x 0 and x 8

4. k 4 or k �2

5. r �3 or r � 0

6. a � 8 and a �4

7. CHALLENGE Describe the graph of each inequality.a. m � �4 and m � 4 b. m � �4 or m � 4

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

t � �3 or t � 2

�4 �3 �2 �1 43210 �4 �3 �2 �1 43210

t � �3 and t � 2

66--55

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Compound Inequalities

Statements that consist of two or more inequalities are called compoundinequalities. When you graph a compound inequality, you need to payspecial attention to the words that connect the inequalities.

The graph includes all numbers that are The graph includes all numbers that areeither less than �3 or greater than 2. both greater than �3 and less than 2.

Graph each compound inequality.

1. h � �5 and h � 4

2. q � �7 or q � 6

3. x 0 and x 8

4. k 4 or k �2

5. r �3 or r � 0

6. a � 8 and a �4

7. CHALLENGE Describe the graph of each inequality.a. m � �4 and m � 4 b. m � �4 or m � 4

number line with number line withno points graphed all points graphed

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

�10 �9 �8 �7 �6 �5 �4 �3 �2 �1 109876543210

t � �3 or t � 2

�4 �3 �2 �1 43210 �4 �3 �2 �1 43210

t � �3 and t � 2

66--55

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Functioning with Machines

Two students were playing a game. The game worked this way. The firststudent picked three integers, say 2, 6, and �4 to put into a functionmachine. The second student said that the output was 7, 11, and 1,respectively.

The first student was challenged to state a rule by which each output wasobtained. “That’s easy,” replied the first student. “Just add 5.”

2 � 5 � 7 6 � 5 � 11 �4 � 5 � 1

This function rule can be written as x � 5.

Write a function rule for each machine.

1.

2.

3.

4.

5. Write a function rule for this assignment.2 → 1 5 → 0 7 → 0 8 → 1 4 → 1 3 → 0

2 4 6 16 �4 �14

2 5 6 13 �4 �7

2 �3 6 �11 �4 9

2 3 6 11 �4 �9

2 7 6 11 �4 1

66--66

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Mathematics: Applications© Glencoe/McGraw-Hill T47 and Connections, Course 2

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Name Date

Functioning with Machines

Two students were playing a game. The game worked this way. The firststudent picked three integers, say 2, 6, and �4 to put into a functionmachine. The second student said that the output was 7, 11, and 1,respectively.

The first student was challenged to state a rule by which each output wasobtained. “That’s easy,” replied the first student. “Just add 5.”

2 � 5 � 7 6 � 5 � 11 �4 � 5 � 1

This function rule can be written as x � 5.

Write a function rule for each machine.

1.

2x � 1

2.

�2x � 1

3.

2x � 1

4.

3x � 2

5. Write a function rule for this assignment.2 → 1 5 → 0 7 → 0 8 → 1 4 → 1 3 → 0

1 if the input is even; 0 if the input is odd

2 4 6 16 �4 �14

2 5 6 13 �4 �7

2 �3 6 �11 �4 9

2 3 6 11 �4 �9

2 7 6 11 �4 1

66--66

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Equations With Two Variables

To solve equations containing two variables, find ordered pair solutions y � 2xfor the equation by selecting values for x and completing a table. Although any value can be selected for x, values usually selected include �2, �1, 0, 1, and 2.

For example, to solve the equation y � 2x given below in Exercise 1, first select values for x, then complete a table.

Ordered pair solutions for the equation y � 2x include (�2, �4), (�1, �2), (0, 0), (1, 2), and (2, 4).

Match each equation with the point whose coordinates are a solution of theequation. Then, at the bottom of the page, write the letter of the point onthe line directly above the number of the equation each time it appears.(The first one has been done as an example.) If you have matched theequations and solutions correctly, the letters below will reveal a message.

1. y � 2x A(�3, 8) N(�1, 0)

2. y � x � 3 B(0, 2) O(3, 0)

3. y � �x � 1 C(�2, 1) P(1, 5)

4. y � 3x � 2 D(0, �5) Q(0, 6)

5. y � �2x � 4 E(�1, �5) R(1, 6)

6. y � x � (�2) F(1, 3) S(2, 1)

7. y � �4x � 1 G(0, �4) T(�2, 3)

8. y � �12

�x H(�1, 3) U(1, 2)

9. y � x � 3 I(2, 0) V(�3, 5)

10. y � 7x � 7 J(0, 4) W(0, �7)

11. y � �2x � 6 K(�3, 1) X(�3, �3)

12. y � �x � 5 L(�4, 2) Y(1, 8)

13. y � �5x � 8 M(�2, 2) Z(0, �8)

14. y � �x

x y�2 �4�1 �2

0 01 22 4

66--77

M14

A12

T3

H7

E4

M14

A12

T3

I6

C9

S8

I6

S8

T3

H7

E4

L11

A12

N10

G5 1

A12

G5

E4

O2

F13

S8

C9

I6

E4

N10

C9

E4

U

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Mathematics: Applications© Glencoe/McGraw-Hill T48 and Connections, Course 2

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Name Date

Equations With Two Variables

To solve equations containing two variables, find ordered pair solutions y � 2xfor the equation by selecting values for x and completing a table. Although any value can be selected for x, values usually selected include �2, �1, 0, 1, and 2.

For example, to solve the equation y � 2x given below in Exercise 1, first select values for x, then complete a table.

Ordered pair solutions for the equation y � 2x include (�2, �4), (�1, �2), (0, 0), (1, 2), and (2, 4).

Match each equation with the point whose coordinates are a solution of theequation. Then, at the bottom of the page, write the letter of the point onthe line directly above the number of the equation each time it appears.(The first one has been done as an example.) If you have matched theequations and solutions correctly, the letters below will reveal a message.

1. y � 2x A(�3, 8) N(�1, 0)

2. y � x � 3 B(0, 2) O(3, 0)

3. y � �x � 1 C(�2, 1) P(1, 5)

4. y � 3x � 2 D(0, �5) Q(0, 6)

5. y � �2x � 4 E(�1, �5) R(1, 6)

6. y � x � (�2) F(1, 3) S(2, 1)

7. y � �4x � 1 G(0, �4) T(�2, 3)

8. y � �12

�x H(�1, 3) U(1, 2)

9. y � x � 3 I(2, 0) V(�3, 5)

10. y � 7x � 7 J(0, 4) W(0, �7)

11. y � �2x � 6 K(�3, 1) X(�3, �3)

12. y � �x � 5 L(�4, 2) Y(1, 8)

13. y � �5x � 8 M(�2, 2) Z(0, �8)

14. y � �x

x y�2 �4�1 �2

0 01 22 4

66--77

M14

A12

T3

H7

E4

M14

A12

T3

I6

C9

S8

I6

S8

T3

H7

E4

L11

A12

N10

G5

U1

A12

G5

E4

O2

F13

S8

C9

I6

E4

N10

C9

E4

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Fractional Areas

The figure at the right shows one square inch. Each small square equals �116� of a square inch.

Write a fraction or mixed number for the shaded area of eachdrawing.

1. 2. 3.

4. 5. 6.

7. 8.

9. 10.

77--11

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Fractional Areas

The figure at the right shows one square inch. Each small square equals �116� of a square inch.

Write a fraction or mixed number for the shaded area of eachdrawing.

1. 2. 3.

4. 5. 6.

�176� in2

7. 8.

1�392� in2

9. 10.

1�12� in2

77--11

�136� in2 �14� in2

�1116� in2 �34� in2

�1332� in2

�34� in2

�12� in2

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Fractions Maze

To solve this maze, start at the upper left corner. Then, draw a line to thenext circle with the smallest sum or difference. The answers written in orderwill form a pattern.

Describe the pattern in the fractions along the line you drew from start tofinish.

13

19

78

38

14

130

130

118

12

12

413

310

67

12

57

16

120

522

112

23

18

16

�� �

��

Start Here Finish

37�

� �

��

�120

722

14

59

513

77--22

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Fractions Maze

To solve this maze, start at the upper left corner. Then, draw a line to thenext circle with the smallest sum or difference. The answers written in orderwill form a pattern.

Describe the pattern in the fractions along the line you drew from start tofinish.

Numerator is always 1; denominatorsdecrease by one from 15 to 2.

13

19

78

38

14

130

130

118

12

12

413

310

67

12

57

16

120

522

112

23

18

16

�� �

��

Start Here Finish

37�

� �

��

�120

722

14

59

513

77--22

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Arithmetic Sequences of Fractions

Each term in an arithmetic sequence is created by adding or subtractingthe same number to the term before. The number added or subtracted iscalled the common difference.

The sequence below is an increasing arithmetic sequence with a commondifference of �1

4�.

�18

�, �38

�, �58

�, �78

�, 1�18

Below is a decreasing arithmetic sequence with a common difference of 1�1

5�.

7�35

�, 6�52,� 5�1

5�, 4, 2�4

5�

Write the common difference for each arithmetic sequence.

1. �12

�, �58

�, �34

�, �78

�, 1, 1�18

� 2. 1�13

�, 3�56

�, 6�13

�, 8�56

3. 4�12

�, 4�25

�, 4�130�, 4�1

5� � 4. 11, 9�2

3�, 8�1

3�, 7, 5�2

3�

Write the next term in each arithmetic sequence.

5. �23

�, �34

�, �56

�, �1112�, 1 6. �1

230�, �1

210�, �

290�, �

270�

7. 5�15

�, 5�170�, 6�1

5�, 6�

170� 8. 4�1

112�, 3�3

4�, 2�

172�, 1�

152�

Write the first five terms in each sequence.

9. This increasing sequence starts with �16

� and has a common differenceof 1�1

5�.

10. This decreasing sequence starts with 6�13

� and has a common differenceof �3

4�.

77--33

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Arithmetic Sequences of Fractions

Each term in an arithmetic sequence is created by adding or subtractingthe same number to the term before. The number added or subtracted iscalled the common difference.

The sequence below is an increasing arithmetic sequence with a commondifference of �1

4�.

�18

�, �38

�, �58

�, �78

�, 1�18

Below is a decreasing arithmetic sequence with a common difference of 1�1

5�.

7�35

�, 6�52,� 5�1

5�, 4, 2�4

5�

Write the common difference for each arithmetic sequence.

1. �12

�, �58

�, �34

�, �78

�, 1, 1�18

� �18� 2. 1�13

�, 3�56

�, 6�13

�, 8�56

� 2�12�

3. 4�12

�, 4�25

�, 4�130�, 4�1

5� �1

10� 4. 11, 9�2

3�, 8�1

3�, 7, 5�2

3� 1�13�

Write the next term in each arithmetic sequence.

5. �23

�, �34

�, �56

�, �1112�, 1 1�1

12� 6. �1

230�, �1

210�, �

290�, �

270� �14�

7. 5�15

�, 5�170�, 6�1

5�, 6�

170� 7�15� 8. 4�1

112�, 3�3

4�, 2�

172�, 1�

152� �1

32�, or �14�

Write the first five terms in each sequence.

9. This increasing sequence starts with �16

� and has a common differenceof 1�1

5�.

�16�, 1�1310�, 2�13

70�, 3�23

30�, 4�23

90�

10. This decreasing sequence starts with 6�13

� and has a common differenceof �3

4�.

6�13�, 5�172�, 4�56�, 4�1

12�, 3�13�

77--33

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Changing Measures of Length

Fractions and mixed numbers are frequently used with customary measures.

The problems on this page will give you a chance to practice using multiplication of fractions as you change measures of lengths to different equivalent forms.

Use a fraction or a mixed number to complete each statement.Refer to the table above as needed.

1. 12 ft 6 in. � ft 2. 1 rod � ft

3. �58

� yd � in. 4. 10 ft � yd

5. 7 yd 2 ft � yd 6. 1,540 yd � mi

7. 1,000 rd � mi 8. 27 in. � yd

Use a whole number to complete each statement. Refer to thetable above as needed.

9. 10�12

� ft � 10 ft in. 10. 12�12

� yd � in.

11. 1 mi � ft 12. 1 mi � yd

13. �110� mi � yd 14. �3

4� ft � in.

15. 10 rd � ft 16. �38

� mi � ft

12 inches (in.) � 1 foot (ft)

3 feet � 1 yard (yd)

5�12� yards � 1 rod (rd)

320 rods � 1 mile (mi)

77--44

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Changing Measures of Length

Fractions and mixed numbers are frequently used with customary measures.

The problems on this page will give you a chance to practice using multiplication of fractions as you change measures of lengths to different equivalent forms.

Use a fraction or a mixed number to complete each statement.Refer to the table above as needed.

1. 12 ft 6 in. � ft 2. 1 rod � ft

3. �58

� yd � in. 4. 10 ft � yd

5. 7 yd 2 ft � yd 6. 1,540 yd � mi

7. 1,000 rd � mi 8. 27 in. � yd

Use a whole number to complete each statement. Refer to thetable above as needed.

9. 10�12

� ft � 10 ft in. 10. 12�12

� yd � in.

11. 1 mi � ft 12. 1 mi � yd

13. �110� mi � yd 14. �3

4� ft � in.

15. 10 rd � ft 16. �38

� mi � ft1,980165

9176

1,7605,280

4506

�34�3�18�

�78�7�23�

3�13�22�12�

16�12�12�12�

12 inches (in.) � 1 foot (ft)

3 feet � 1 yard (yd)

5�12� yards � 1 rod (rd)

320 rods � 1 mile (mi)

77--44

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Changing Measurements with Factors of 1

Multiplying an expression by the number 1 does not change its value. Thisproperty of multiplication can be used to change measurements.

Let’s say you wanted to change 4.5 hours to seconds. Start by multiplying 4.5 by the number 1 written in the form �60

1mhionuurtes�. This first step changes

4.5 hours to minutes.

4.5 hours � �601mhionuurtes�

Now, multiply by the number 1 again. This time use the fact that

1 � �610

mse

icnountdes�.

4.5 hours � �601mhionuurtes� � �6

10

mse

icnountdes� � 16,200 seconds

Complete by writing the last factor and the answer. You mayneed to use a table of measurements to find the factors.

1. Change 5 pints to fluid ounces.

5 pints � �

2. Change 0.8 miles to inches.

0.8 mile � �

3. Change 4 square yards to square inches.

4 yd2 � �

4. Change 12 bushels to pints.

12 bushels � � �

5. Change one-half of an acre to square inches.

�12

� acre � � �9 ft2

�1 yd2

4,840 yd2

��1 acre

8 quarts�1 peck

4 pecks�1 bushel

9 ft2�1 yd2

5,280 feet��1 mile

8 fluid ounces��1 cup2 cups�1 pint

77--55

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Changing Measurements with Factors of 1

Multiplying an expression by the number 1 does not change its value. Thisproperty of multiplication can be used to change measurements.

Let’s say you wanted to change 4.5 hours to seconds. Start by multiplying 4.5 by the number 1 written in the form �60

1mhionuurtes�. This first step changes

4.5 hours to minutes.

4.5 hours � �601mhionuurtes�

Now, multiply by the number 1 again. This time use the fact that

1 � �610

mse

icnountdes�.

4.5 hours � �601mhionuurtes� � �6

10

mse

icnountdes� � 16,200 seconds

Complete by writing the last factor and the answer. You mayneed to use a table of measurements to find the factors.

1. Change 5 pints to fluid ounces.

5 pints � � � 80 fluid ounces

2. Change 0.8 miles to inches.

0.8 mile � � � 50,688 inches

3. Change 4 square yards to square inches.

4 yd2 � � � 5,184 in2

4. Change 12 bushels to pints.

12 bushels � � � � 768 pints

5. Change one-half of an acre to square inches.

�12

� acre � � � � 3,136,320 in2144 in2�1 ft2

9 ft2

�1 yd2

4,840 yd2

��1 acre

2 pints�1 quart8 quarts�1 peck

4 pecks�1 bushel

144 in2�1 ft2

9 ft2�1 yd2

12 inches��1 foot5,280 feet��

1 mile

8 fluid ounces��1 cup2 cups�1 pint

77--55

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Networks

A network is a collection of points, segments, and arcs. An example of anetwork is shown at the left below. Often you can begin at one point of thenetwork and trace it without lifting your pencil and without going over anysegment or arc twice.

One such tracing is shown at the right above.

Identify which networks can be traced without lifting your pencil and without tracing any segment or arc more than once.

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

Start

77--66

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Networks

A network is a collection of points, segments, and arcs. An example of anetwork is shown at the left below. Often you can begin at one point of thenetwork and trace it without lifting your pencil and without going over anysegment or arc twice.

One such tracing is shown at the right above.

Identify which networks can be traced without lifting your pencil and without tracing any segment or arc more than once.

1. 2. 3. 4.

yes yes no no

5. 6. 7. 8.

yes yes yes no

9. 10. 11. 12.

yes no yes yes

Start

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A Circle Puzzle

The circle at the right has been divided into ten pieces. Notice that the vertical diameter is marked off into four congruent segments.

Trace the circle and cut it into ten parts to make a set ofpuzzle pieces.

1. Separate the pieces and put them back together to form the circle. Trythis first without looking at the solution.

The puzzle pieces can be used to make many shapes. Use all ten pieces to make each shape shown. Record your solutions.

2. 3. 4.

5. 6.

7. 8.

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A Circle Puzzle

The circle at the right has been divided into ten pieces. Notice that the vertical diameter is marked off into four congruent segments.

Trace the circle and cut it into ten parts to make a set ofpuzzle pieces.

1. Separate the pieces and put them back together to form the circle. Trythis first without looking at the solution.

The puzzle pieces can be used to make many shapes. Use all ten pieces to make each shape shown. Record your solutions.

2. 3. 4.

5. 6.

7. 8.

77--77

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Trail Blazers

Each puzzle on this page is called a trail blazer. To solve it, you must find a trail that begins at any one of the small squares and ends at thegoal square, following these rules.

1. The sum of all the fractions on the trail must equal the number inthe goal square.

2. The trail can only go horizontally or vertically.

3. The trail cannot retrace or cross itself.

When you are solving a trail blazer, try to eliminate possibilities. Forinstance, in the puzzle at the right, you know that you cannot include

�34

�: �34

� � �14

� � 1 and �34

� � �12

� � 1�14

�, while the goal for the entire

trail is only 1.

1. 2. 3.

4. 5.

4

34

14

15

35

15

12

45

15

14

25

14

12

720

110

1320

120

1920

320

1120

710

1720

110

120

310

920

6

58

38

18

14

34

18

12

14

12

12

14

78

12

14

78

78

18

14

18

34

38

38

12

34

18

512

23

78

38

34

12

14

13

3 12

58

710

910

310

110

2

15

45

35

12

12

25

512

112

712

112

1

23

56

12

16

13

34

316

116

1 goalsquare

12

14

14

18

18

58

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Trail Blazers

Each puzzle on this page is called a trail blazer. To solve it, you must find a trail that begins at any one of the small squares and ends at thegoal square, following these rules.

1. The sum of all the fractions on the trail must equal the number inthe goal square.

2. The trail can only go horizontally or vertically.

3. The trail cannot retrace or cross itself.

When you are solving a trail blazer, try to eliminate possibilities. Forinstance, in the puzzle at the right, you know that you cannot include

�34

�: �34

� � �14

� � 1 and �34

� � �12

� � 1�14

�, while the goal for the entire

trail is only 1. Answers may vary.1. 2. 3.

4. 5.

4

34

14

15

35

15

12

45

15

14

25

14

12

720

110

1320

120

1920

320

1120

710

1720

110

120

310

920

6

58

38

18

14

34

18

12

14

12

12

14

78

12

14

78

78

18

14

18

34

38

38

12

34

18

512

23

78

38

34

12

14

13

3 12

58

710

910

310

110

2

15

45

35

12

12

25

512

112

712

112

1

23

56

12

16

13

34

316

116

1 goalsquare

12

14

14

18

18

58

77--88

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Continued Fractions

The expression at the right is an example of a continued fraction. Although 1 + 1

continued fractions may look complicated, they are just a combination of 1 � 1

addition and division. Here is one way to simplify a continued fraction. 1 + �19

1 � 1 � 1 � [1 � (1 � [1 � (1 � �19

�)])]

1 + 1

1 � �19

� 1 � [1 � (1 � [1 � �190�])]

� 1 � [1 � (1 � �190�)]

� 1 � [1 � �1190�]

� 1 � �1109�

� �2199�

Write each continued fraction as an improper fraction.

1. 1 � 1 2. 2 � 1 3. 1 � 2

3 � �13

� 2 � �12

� 3 � �23

4. 1 � 3 5. 5 � 1 6. 2 � 2

3 � �14

� 1 � �15

� 2 � �25

7. 1 � 1 8. 1 � 1 9. 1 � 1

1 � 1 1 � 1 1 � 1

1 � �12

� 1 � �13

� 1 � �15

10. 2 � 1 11. 3 � 1 12. 6 � 1

2 � 1 3 � 2 1 � 1

2 � �12

� 1 � �13

� 3 � �13

77--99

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Continued Fractions

The expression at the right is an example of a continued fraction. Although 1 + 1

continued fractions may look complicated, they are just a combination of 1 � 1

addition and division. Here is one way to simplify a continued fraction. 1 + �19

1 � 1 � 1 � [1 � (1 � [1 � (1 � �19

�)])]

1 + 1

1 � �19

� 1 � [1 � (1 � [1 � �190�])]

� 1 � [1 � (1 � �190�)]

� 1 � [1 � �1190�]

� 1 � �1109�

� �2199�

Write each continued fraction as an improper fraction.

1. 1 � 1 2. 2 � 1 3. 1 � 2

3 � �13

� �1130� 2 � �1

2� �15

2� 3 � �23

� �1171�

4. 1 � 3 5. 5 � 1 6. 2 � 2

3 � �14

� �2153� 1 � �1

5� �36

5� 2 � �25

� �167�

7. 1 � 1 8. 1 � 1 9. 1 � 1

1 � 1 1 � 1 1 � 1

1 � �12

� �85� 1 � �13

� �171� 1 � �1

5� �11

71�

10. 2 � 1 11. 3 � 1 12. 6 � 1

2 � 1 3 � 2 1 � 1

2 � �12

� �2192� 1 � �1

3� �29

9� 3 � �13

� �8183�

77--99

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Squares and Rectangles

The perimeter and area of a square are found by using the formulas P � 4sand A � s2. The perimeter and area of a rectangle are found by usingformulas P � 2l � 2w and A � lw.

Use these formulas to help answer the following questions.

1. A piece of rope 72 inches long must be cut into two pieces. Each pieceof rope will be used to form a square. Where should the rope be cut ifthe perimeter of one square must be �

13� of the perimeter of the other

square?

2. A piece of rope 60 inches long must be cut into two pieces. Each pieceof rope will be used to form a rectangle. Where should the rope be cutif the perimeter of one rectangle must be twice as great as the perimeterof the other rectangle?

3. A piece of rope 100 inches long must be cut into two pieces. Eachpiece of rope will be used to form a square. Where should the rope becut if the sides of one square must be 4 inches longer than the sides ofthe other square?

4. A piece of rope 80 centimeters long must be cut into two pieces. Eachpiece of rope will be used to form a rectangle. Where would the rope be

cut if the area of one rectangle must be �14� of the area of the other

rectangle?

5. A piece of rope 144 inches long must be cut into two pieces. Eachpiece of rope will be used to form a square. Where should the rope becut if the area of one square must be 9 times greater than the area of theother square?

88--11

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Squares and Rectangles

The perimeter and area of a square are found by using the formulas P � 4sand A � s2. The perimeter and area of a rectangle are found by usingformulas P � 2l � 2w and A � lw.

Use these formulas to help answer the following questions.

1. A piece of rope 72 inches long must be cut into two pieces. Each pieceof rope will be used to form a square. Where should the rope be cut ifthe perimeter of one square must be �

13� of the perimeter of the other

square?

18 in. from the end

2. A piece of rope 60 inches long must be cut into two pieces. Each pieceof rope will be used to form a rectangle. Where should the rope be cutif the perimeter of one rectangle must be twice as great as the perimeterof the other rectangle?

20 in. from the end

3. A piece of rope 100 inches long must be cut into two pieces. Eachpiece of rope will be used to form a square. Where should the rope becut if the sides of one square must be 4 inches longer than the sides ofthe other square?

42 in. from the end

4. A piece of rope 80 centimeters long must be cut into two pieces. Eachpiece of rope will be used to form a rectangle. Where would the rope be

cut if the area of one rectangle must be �14� of the area of the other

rectangle?

Sample: 30 cm from the end

5. A piece of rope 144 inches long must be cut into two pieces. Eachpiece of rope will be used to form a square. Where should the rope becut if the area of one square must be 9 times greater than the area of theother square?

36 in. from the end

88--11

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An Educated Consumer

Choosing a checking account is something that most people do at somepoint in their lives. Because checking accounts vary from institution toinstitution, and from one type of account to another, you will need toconsider the options associated with each account before choosing one ofthem.

Suppose a bank offers two kinds of checking accounts.

Account A: a $0.20 charge for writing each check and no servicecharge

Account B: a $0.10 charge for writing each check and a monthlyservice charge of $1.50

1. Which account would cost less if a person were to write 10 checks in amonth?

2. Which account would cost less if a person were to write 20 checks in amonth?

3. Using the guess-and-check strategy, find the number of checks thatwould have to be written for the cost of Account A to equal the cost ofAccount B. What is that cost?

4. Which account would cost less if a person were to write 250 checks in ayear? By how much?

5. Diana Durbin wrote 300 checks in one year. Her total charge for the useof the account that year was $72.00. The bank charges $0.15 for writingone check and charges a fixed amount each month for the use of theaccount. What is that monthly service charge?

88--22

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An Educated Consumer

Choosing a checking account is something that most people do at somepoint in their lives. Because checking accounts vary from institution toinstitution, and from one type of account to another, you will need toconsider the options associated with each account before choosing one ofthem.

Suppose a bank offers two kinds of checking accounts.

Account A: a $0.20 charge for writing each check and no servicecharge

Account B: a $0.10 charge for writing each check and a monthlyservice charge of $1.50

1. Which account would cost less if a person were to write 10 checks in amonth?

Account A

2. Which account would cost less if a person were to write 20 checks in amonth?

Account B

3. Using the guess-and-check strategy, find the number of checks thatwould have to be written for the cost of Account A to equal the cost ofAccount B. What is that cost?

It would take 15 checks; $3.00.

4. Which account would cost less if a person were to write 250 checks in ayear? By how much?

Account B; $7.00

5. Diana Durbin wrote 300 checks in one year. Her total charge for the useof the account that year was $72.00. The bank charges $0.15 for writingone check and charges a fixed amount each month for the use of theaccount. What is that monthly service charge?

$2.25

88--22

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What Am I?

Solve each proportion. Then, starting at the box marked with the heavyoutline, draw an arrow to the adjacent box containing the variable with theleast value. (You may move horizontally or vertically. You may use eachbox at most once.)

Now fill in the table below with the letters in the order in which you foundthem. Now you can say what I am.

88--33

� �o2�

3.5��14

�2z� � �4

1� �0

0..34� � �1

o8� �

5.382� � �

48n.5� �

5p5� �

�25

��1

�0i.5� � �

32844

��27

� � �1p4��a

5� � �

195� �

3z2� � �7

8� �4

138.2� � �u

5�

� �1t8�

�29

��13

��36..48� � �2

r.5��4

o� � 20�

30 �

�23

��60

�13

��z �

t��14

72��12

�35..55� � �7

r��

2o4� � �1

306.5�� �

126�

p��12

� � �5l�

2�12

�9

Stop here.

� 12.5��12

e��15

�0o.2� � �0

3.55�� �

2r0�

1�25

��1.4�

15e0

� � �212.5.5� �3

8� � �6

d� � �

936�

�14

��i

�5e

� �

2�12

��21

�3n7� � 54�

55�12

� �1366.5� � �1

a1�

�0h.7� � �

120.18

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Mathematics: Applications© Glencoe/McGraw-Hill T60 and Connections, Course 2

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What Am I?

Solve each proportion. Then, starting at the box marked with the heavyoutline, draw an arrow to the adjacent box containing the variable with theleast value. (You may move horizontally or vertically. You may use eachbox at most once.)

Now fill in the table below with the letters in the order in which you foundthem. Now you can say what I am.

88--33

A P R O P O R T I O N P U Z Z L E

� �o2�

3.5��14

�2z� � �4

1� �0

0..34� � �1

o8� �

5.382� � �

48n.5� �

5p5� �

�25

��1

�0i.5� � �

32844

��27

� � �1p4��a

5� � �

195� �

3z2� � �7

8� �4

138.2� � �u

5�

� �1t8�

�29

��13

��36..48� � �2

r.5��4

o� � 20�

30 �

�23

��60

�13

��z � t�

�14

72��12

�35..55� � �7

r��

2o4� � �1

306.5�� �

126�

p��12

� � �5l�

2�12

�9

Stop here.

� 12.5��12

e��15

�0o.2� � �0

3.55�� �

2r0�

1�25

��1.4�

15e0

� � �212.5.5� �3

8� � �6

d� � �

936�

�14

��i

�5e

� �

2�12

��21

�3n7� � 54�

55�12

� �1366.5� � �1

a1�

�0h.7� � �

120.18

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Mathematics: Applications© Glencoe/McGraw-Hill 61 and Connections, Course 2

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Scale Drawings

Use the scale drawings of two different apartments to answerthe questions.

1. Which apartment has the greater area?

2. What is the difference in square feet between Apartment A andApartment B?

3. How much more closet space is offered by Apartment B thanApartment A?

4. How much more bathroom space is offered by Apartment B thanApartment A?

5. A one-year lease for Apartment A costs $450 per month. A one-yearlease for Apartment B costs $525 per month. Which apartment offersthe greatest value in terms of the cost per square foot?

Scale: 1 inch � 16 feet

Apartment BApartment A

KitchenKitchen

Living Room

LivingRoom

BedroomBedroom Bedroom Bath

Bathroom

Closet

Closet

Closet

Closet

Closet

Scale: 1 inch � 12 feet

88--44

Scale: 1 inch � 16 feet

Apartment B

Kitchen

Living Room

BedroomBath

Closet

Closet

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Mathematics: Applications© Glencoe/McGraw-Hill T61 and Connections, Course 2

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Scale Drawings

Use the scale drawings of two different apartments to answerthe questions.

1. Which apartment has the greater area? Apartment B

2. What is the difference in square feet between Apartment A andApartment B? 48 ft2

3. How much more closet space is offered by Apartment B thanApartment A? 56 ft2

4. How much more bathroom space is offered by Apartment B thanApartment A? 6 ft2

5. A one-year lease for Apartment A costs $450 per month. A one-yearlease for Apartment B costs $525 per month. Which apartment offersthe greatest value in terms of the cost per square foot?

Apartment A

Scale: 1 inch � 16 feet

Apartment BApartment A

KitchenKitchen

Living Room

LivingRoom

BedroomBedroom Bedroom Bath

Bathroom

Closet

Closet

Closet

Closet

Closet

Scale: 1 inch � 12 feet

88--44

Scale: 1 inch � 16 feet

Apartment B

Kitchen

Living Room

BedroomBath

Closet

Closet

�12

�""

�12

�""

2�12

�""

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Mathematics: Applications© Glencoe/McGraw-Hill 62 and Connections, Course 2

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Shaded Regions

The fractions or percents listed below each represent one of the shadedregions.

Match each fraction or percent with the shaded region itrepresents.

1. �12

� a. b. c.

2. �2654�

3. �1116�

4. 25% d. e. f.

5. �34

6. 62�12

�%

7. �2694� g. h. i.

8. 37.5%

9. �176�

88--55

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Shaded Regions

The fractions or percents listed below each represent one of the shadedregions.

Match each fraction or percent with the shaded region itrepresents.

1. �12

� d a. b. c.

2. �2654� i

3. �1116� h

4. 25% b d. e. f.

5. �34

� a

6. 62�12

�% g

7. �2694� f g. h. i.

8. 37.5% c

9. �176� e

88--55

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Mental Math Magic

You can do this trick with a calendar and mental math.

Magician Hand your spectator a calendar.Ask the spectator to mark off a four-by-four square, such asthe one at the left below.Mentally, find the sum of the two numbers in oppositecorners of the square. Double the sum.Write it on a piece of paper and hand it to a third person.

Spectator Circle any number. Cross out the row and columncontaining it. Choose a number not circled or crossed out.Circle it and cross out the row and column containing it.Repeat this again. Circle the one remaining number. Theresult is a diagram like the one at the right below. Add thefour circled numbers.

Magician Show everyone the number written on the folded paper. Thenumber matches the sum that the spectator got.

1. Make a clean copy of the square at the left above.a. Add the numbers found in two opposite corners of the square.

Double the sum and write it down.b. Follow the directions for the spectator, but with different numbers

than those above. What is the sum?c. Do your answers to parts a and b match?

2. Repeat Exercise 1, but, this time, choose different numbers in part b.Did you get the same result?

3. On a calendar, mark off a four-by-four square of numbers that isdifferent from the one above. Repeat Exercise 1 three times. What is theresult?

The answers to parts a and b alwaysmatch.

4. Try this magic trick with some of your friends.

2524 2726

1817 2019

1110 1312

43 65

2524 2726

1817 2019

1110 1312

43 65

88--66

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Mathematics: Applications© Glencoe/McGraw-Hill T63 and Connections, Course 2

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Mental Math Magic

You can do this trick with a calendar and mental math.

Magician Hand your spectator a calendar.Ask the spectator to mark off a four-by-four square, such asthe one at the left below.Mentally, find the sum of the two numbers in oppositecorners of the square. Double the sum.Write it on a piece of paper and hand it to a third person.

Spectator Circle any number. Cross out the row and columncontaining it. Choose a number not circled or crossed out.Circle it and cross out the row and column containing it.Repeat this again. Circle the one remaining number. Theresult is a diagram like the one at the right below. Add thefour circled numbers.

Magician Show everyone the number written on the folded paper. Thenumber matches the sum that the spectator got.

1. Make a clean copy of the square at the left above.a. Add the numbers found in two opposite corners of the square.

Double the sum and write it down. 60b. Follow the directions for the spectator, but with different numbers

than those above. What is the sum? 60c. Do your answers to parts a and b match? yes

2. Repeat Exercise 1, but, this time, choose different numbers in part b.Did you get the same result? yes

3. On a calendar, mark off a four-by-four square of numbers that isdifferent from the one above. Repeat Exercise 1 three times. What is theresult?

The answers to parts a and b alwaysmatch.

4. Try this magic trick with some of your friends.

2524 2726

1817 2019

1110 1312

43 65

2524 2726

1817 2019

1110 1312

43 65

88--66

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Mathematics: Applications© Glencoe/McGraw-Hill 64 and Connections, Course 2

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The Colormatch Square

To work this puzzle, cut out the 16 tiles at the bottom of this page. Thegoal of the puzzle is to create a square so that the sides of any pair ofadjacent tiles match. You are not allowed to rotate any of the tiles.

1. Complete the solution to the colormatch square puzzle below.

2. Find at least one other solution in which the A tile is in the upper leftcorner.

There are 10 other solutions with 0 in theupper left corner and 50 unique solutionsin all.

A D

PN

B G

L O

H

M

C

I KJ

FE

A

D

Whitesquaresmatch.

P

N

B

L

O

M

I

K

J

E

88--77

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Mathematics: Applications© Glencoe/McGraw-Hill T64 and Connections, Course 2

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Name Date

The Colormatch Square

To work this puzzle, cut out the 16 tiles at the bottom of this page. Thegoal of the puzzle is to create a square so that the sides of any pair ofadjacent tiles match. You are not allowed to rotate any of the tiles.

1. Complete the solution to the colormatch square puzzle below.

2. Find at least one other solution in which the A tile is in the upper leftcorner.

There are 10 other solutions with A in theupper left corner and 50 unique solutionsin all.

A D

PN

B G

L O

H

M

C

I KJ

FE

A

D

Whitesquaresmatch.

P

N

B

G

L

O

H

M

C

I

K

J

F

E

88--77

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Model Behavior

When a block is painted and then separated into small cubes, some of thefaces of the cubes will have paint on them and some will not.

For each set of blocks determine the percent of cubes that arepainted on the given number of faces.

1. 0 faces

2. 1 face

3. 2 faces

4. 3 faces

5. 4 faces

6. 5 faces

7. 6 faces

8. 0 faces

9. 1 face

10. 2 faces

11. 3 faces

12. 4 faces

13. 5 faces

14. 6 faces

15. 0 faces

16. 1 face

17. 2 faces

18. 3 faces

19. 4 faces

20. 5 faces

21. 6 faces

88--88

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Model Behavior

When a block is painted and then separated into small cubes, some of thefaces of the cubes will have paint on them and some will not.

For each set of blocks determine the percent of cubes that arepainted on the given number of faces.

1. 0 faces 02. 1 face 03. 2 faces 644. 3 faces 325. 4 faces 46. 5 faces 07. 6 faces 0

8. 0 faces 09. 1 face 0

10. 2 faces 011. 3 faces 012. 4 faces 9013. 5 faces 1014. 6 faces 0

15. 0 faces 016. 1 face 017. 2 faces 018. 3 faces 019. 4 faces 10020. 5 faces 021. 6 faces 0

88--88

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Working Backward

Working backward can be a helpful problem-solving tool, especially inproblems where the answer is given and information you would expect tohave is omitted.

A large corporation reports that % of its employees exerciseon a regular basis. If 2,120 employees exercise regularly, how manyemployees does the corporation have? Answer 2,650 employees

Use the percent proportion to solve for the missing percent.

�22,,162500

� � �1x00�

2,120 � 100 � 2,650x

x � 80

80% of the employees exercise on a regular basis.

Write the missing information for each exercise.

1. A progressive community states that 96% of its households recycle materials at least once monthly. If households recycle at least once monthly, how many households are in the community? Answer: 15,480 households

2. The purchase price of a cassette tape deck is $139.00. The sales tax rate is %. Find the cost of the cassette tape deck. Answer: $148.73

3. In a seventh grade class, 60% of the students participate in extra-curricular activities. The class has students. How many students participate in extra-curricular activities? Answer: 15 students

4. Claims by a manufacturer state that 3 out of 4 people prefer their product when compared to a similar product of another manufacturer. If people were surveyed, how many did not prefer the product? Answer: 35

5. Seventy percent of the students entering a certain high school complete their studies and graduate. If students did not complete their studies and graduate, how many students earned a diploma? Answer: 455

6. A middle school survey discovered that 15% of the student body watched two hours or less of television each week and 45% watched ten or more hours each week. If students were surveyed, how many students watched between 2 and 10 hours of television each week? Answer: 48.

88--99

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Mathematics: Applications© Glencoe/McGraw-Hill T66 and Connections, Course 2

Enrichment

Name Date

Working Backward

Working backward can be a helpful problem-solving tool, especially inproblems where the answer is given and information you would expect tohave is omitted.

A large corporation reports that % of its employees exerciseon a regular basis. If 2,120 employees exercise regularly, how manyemployees does the corporation have? Answer 2,650 employees

Use the percent proportion to solve for the missing percent.

�22,,162500

� � �1x00�

2,120 � 100 � 2,650x

x � 80

80% of the employees exercise on a regular basis.

Write the missing information for each exercise.

1. A progressive community states that 96% of its households recycle 14,860.8materials at least once monthly. If households recycle at least once monthly, how many households are in the community? Answer: 15,480 households

2. The purchase price of a cassette tape deck is $139.00. The sales tax rate 7is %. Find the cost of the cassette tape deck. Answer: $148.73

3. In a seventh grade class, 60% of the students participate in 25extra-curricular activities. The class has students. How many students participate in extra-curricular activities? Answer: 15 students

4. Claims by a manufacturer state that 3 out of 4 people prefer their 140product when compared to a similar product of another manufacturer. If people were surveyed, how many did not prefer the product? Answer: 35

5. Seventy percent of the students entering a certain high school complete 195their studies and graduate. If students did not complete their studies and graduate, how many students earned a diploma? Answer: 455

6. A middle school survey discovered that 15% of the student body 120watched two hours or less of television each week and 45% watched ten or more hours each week. If students were surveyed, how many students watched between 2 and 10 hours of television each week? Answer: 48.

88--99

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Mathematics: Applications© Glencoe/McGraw-Hill 67 and Connections, Course 2

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Compass Directions

There are 360° in a complete rotation. The directions north, east, south, and west areshown on the compass at the right.

To find the direction a boat or airplane isheading, measure clockwise from northaround the compass. The example shows aheading of 150°.

Use a protractor. Write the compass heading in degrees foreach diagram.

1. 2. 3.

4. 5. 6.

The drawing at the right is called a compass rose. Use the compass rose to translate each direction into degrees.7. East 8. North

9. Northeast (NE) 10. Southwest (SW)

11. South 12. Southeast (SE)

13. Northwest (NW) 14. North northeast (NNE)

S

N

SW

SWW

SSW

SE

SEE

SSE

NW

NWW

NNW

NE

NEE

NNE

W E

N

S

W E

N

S

W E

N

S

W E

N

S

W E

N

S

W E

N

S

W E

330˚300˚

210˚

240˚

30˚150˚

150˚ � 90˚ � 60˚

N N

S

W E

S

W E

60˚

150˚

120˚

99--11

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Mathematics: Applications© Glencoe/McGraw-Hill T67 and Connections, Course 2

Enrichment

Name Date

Compass Directions

There are 360° in a complete rotation. The directions north, east, south, and west areshown on the compass at the right.

To find the direction a boat or airplane isheading, measure clockwise from northaround the compass. The example shows aheading of 150°.

Use a protractor. Write the compass heading in degrees foreach diagram.

1. 2. 3.

4. 5. 6.

The drawing at the right is called a compass rose. Use the compass rose to translate each direction into degrees.7. East 8. North

90° 0°, or 360°9. Northeast (NE) 10. Southwest (SW)

45° 225°11. South 12. Southeast (SE)

180° 135°13. Northwest (NW) 14. North northeast (NNE)

315° 22.5° S

N

SW

SWW

SSW

SE

SEE

SSE

NW

NWW

NNW

NE

NEE

NNE

W E

N

S

W E

N

S

W E

N

S

W E

N

S

W E

N

S

W E

N

S

W E

330˚300˚

210˚

240˚

30˚150˚

150˚ � 90˚ � 60˚

N N

S

W E

S

W E

60˚

150˚

120˚

99--11

70°

20°

230°

280°

300°

140°

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Star Polygons

Any polygon can be turned into a star polygon by extending its sides. Astar polygon is also called a stellated polygon.

Octagon Extend the sides to Extend the sides againmake the first star. to make a second star.

Make a star by extending the sides of each polygon.

1. 2.

Trace each polygon on a separate sheet of paper. Then, makethree different stars by extending the sides three times.

3. 4.

Show all the different stars that can be made from each polygon.

5. 6.

99--22

hexagon

decagon

dodecagon

pentagon

nonagon

heptagon

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Star Polygons

Any polygon can be turned into a star polygon by extending its sides. Astar polygon is also called a stellated polygon.

Octagon Extend the sides to Extend the sides againmake the first star. to make a second star.

Make a star by extending the sides of each polygon.

1. 2.

Trace each polygon on a separate sheet of paper. Then, makethree different stars by extending the sides three times.

3. 4.

Show all the different stars that can be made from each polygon.

5. 6.

C C

99--22

hexagon

decagon

dodecagon

pentagon

nonagon

heptagon

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Similar Figures and Areas

The areas of two similar figures are related in a special way. Suppose that rectangle A is 2 units by 3 units and rectangle B is 4 units by 6 units.

The area of rectangle A is 2 � 3 � 6 units2.

The area of rectangle B is 4 � 6 � 24 units2.

The lengths of the sides of rectangle B are twice those of rectangle A and the area of rectangle B is four times that of rectangle A.

Sketch figure B similar to figure A and satisfying the givencondition.1. Rectangle B has sixteen times the area of rectangle A.

2. Square B has an area that is 4 times that of square A.

3. Circle B has an area four times that of circle A.

Circle A

1.5

Square A

2

2

Rectangle A

31

Rectangle BRectangle A

36

42

99--33

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Similar Figures and Areas

The areas of two similar figures are related in a special way. Suppose that rectangle A is 2 units by 3 units and rectangle B is 4 units by 6 units.

The area of rectangle A is 2 � 3 � 6 units2.

The area of rectangle B is 4 � 6 � 24 units2.

The lengths of the sides of rectangle B are twice those of rectangle A and the area of rectangle B is four times that of rectangle A.

Sketch figure B similar to figure A and satisfying the givencondition.1. Rectangle B has sixteen times the area of rectangle A.

2. Square B has an area that is 4 times that of square A.

3. Circle B has an area four times that of circle A.

Circle BCircle A

1.5

Square BSquare A

2

2

Rectangle BRectangle A

31

Rectangle BRectangle A

36

42

99--33

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Dissecting Squares

In a dissection puzzle, the pieces of one shape are rearranged to make adifferent shape. Draw a square and then make a set of pieces to solve eachdissection puzzle. Record your answers.

1. Rearrange the pieces to make a figure 2. Rearrange the pieces to make a figureshaped like the one at the right below. shaped like the one at the right below.

3. Rearrange the pieces to make an octagon 4. Rearrange the pieces to make a figurewith sides of equal length. shaped like a plus sign.

5. Rearrange the pieces to make two 6. Rearrange the pieces to make threenew squares. squares of equal size.

99--44

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Dissecting Squares

In a dissection puzzle, the pieces of one shape are rearranged to make adifferent shape. Draw a square and then make a set of pieces to solve eachdissection puzzle. Record your answers.

1. Rearrange the pieces to make a figure 2. Rearrange the pieces to make a figureshaped like the one at the right below. shaped like the one at the right below.

3. Rearrange the pieces to make an octagon 4. Rearrange the pieces to make a figurewith sides of equal length. shaped like a plus sign.

5. Rearrange the pieces to make two 6. Rearrange the pieces to make threenew squares. squares of equal size.

99--44

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Tessellated Patterns for Solid Shapes

Tessellations made from equilateral triangles can be used to build three-dimensional shapes. In Exercise 1, you should get a shape like the one shown at the right. It is called a pyramid.

Copy each pattern. Crease the pattern along the lines. Thenfollow directions for folding the pattern. Use tape to securethe folded parts. When you have finished each model, describeit in words.1. Fold 5 over 1.

Repeat, in this order:fold 6 over 7,fold 2 over 6.

pyramid with fourtriangular faces

2. Cut between 4 and 5. Then fold 5 over 3.Repeat in this order:

fold 6 over 5,fold 7 over 12, andfold 2 over 9.

double pyramid with sixtriangular faces

3. Cut between 1 and 2 and between 14 and 15. Then fold 15 over 14.Repeat, in this order:

fold 1 over 2,fold 4 over 3,fold 11 over 1,fold 16 over 5, andfold 12 over 13.

pentagonal shape withten triangular faces

1

5 710

14

69 11

12 13 15

16

2

8

3 4

3

8 107 9

12

2

4

11

5 6

1

1

3 52 4

7

6

99--55

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Tessellated Patterns for Solid Shapes

Tessellations made from equilateral triangles can be used to build three-dimensional shapes. In Exercise 1, you should get a shape like the one shown at the right. It is called a pyramid.

Copy each pattern. Crease the pattern along the lines. Thenfollow directions for folding the pattern. Use tape to securethe folded parts. When you have finished each model, describeit in words.1. Fold 5 over 1.

Repeat, in this order:fold 6 over 7,fold 2 over 6.

pyramid with fourtriangular faces

2. Cut between 4 and 5. Then fold 5 over 3.Repeat in this order:

fold 6 over 5,fold 7 over 12, andfold 2 over 9.

double pyramid with sixtriangular faces

3. Cut between 1 and 2 and between 14 and 15. Then fold 15 over 14.Repeat, in this order:

fold 1 over 2,fold 4 over 3,fold 11 over 1,fold 16 over 5, andfold 12 over 13.

pentagonal shape withten triangular faces

1

5 710

14

69 11

12 13 15

16

2

8

3 4

3

8 107 9

12

2

4

11

5 6

1

1

3 52 4

7

6

99--55

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Knight Moves

In the game of chess, a knight can move several different ways. It can move twospaces vertically or horizontally, thenone space at a 90° angle. It can alsomove one space vertically orhorizontally, then two spaces at a 90°angle. Several examples of a knight’smoves are indicated on the grid at theright.

1. Use the diagram at the right. Place a knight or other piece in the square marked 1. Move the knight so that it lands on each of the remaining white squares only once. Mark each square in which the knight lands with 2, then 3, and so on.

Sample answer given. Otheranswers are possible.

2. Use the diagram below. Place a knight or other piece in the squaremarked 1. Move the knight so that it lands on each of the remainingsquares only once. Mark each square in which the knight lands with 2,then 3, and so on.

Sample answer given. Otheranswers are possible.

1

1

99--66

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Knight Moves

In the game of chess, a knight can move several different ways. It can move twospaces vertically or horizontally, thenone space at a 90° angle. It can alsomove one space vertically orhorizontally, then two spaces at a 90°angle. Several examples of a knight’smoves are indicated on the grid at theright.

1. Use the diagram at the right. Place a knight or other piece in the square marked 1. Move the knight so that it lands on each of the remaining white squares only once. Mark each square in which the knight lands with 2, then 3, and so on.

Sample answer given. Otheranswers are possible.

2. Use the diagram below. Place a knight or other piece in the squaremarked 1. Move the knight so that it lands on each of the remainingsquares only once. Mark each square in which the knight lands with 2,then 3, and so on.

Sample answer given. Otheranswers are possible.

21 6

27 17

18 7 22

2

15

12

3

4

23

16

8

3019

28 13 24 11926

25 10 5 142029

1

6 3

4 8

7 2 5

1

99--66

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The Twelve Dot Puzzle

In this puzzle, a broken line made up of 5 segments must pass through each of 12 dots. The line cannot go through a dot more than once,although it may intersect itself. The line must start at one dot and end at a different dot.

One solution to this puzzle is shown at the right. Two solutions to thepuzzle are not “different” if one is just a reflection or rotation of the other.

Find 18 other solutions.1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

99--77

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The Twelve Dot Puzzle

In this puzzle, a broken line made up of 5 segments must pass through each of 12 dots. The line cannot go through a dot more than once,although it may intersect itself. The line must start at one dot and end at a different dot.

One solution to this puzzle is shown at the right. Two solutions to thepuzzle are not “different” if one is just a reflection or rotation of the other.

Find 18 other solutions.1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

99--77

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The Geometric Mean

The square root of the product of two numbers is called their geometricmean. The geometric mean of 12 and 48 is �1�2� �� 4�8� � �5�7�6� or 24.

Find the geometric mean for each pair of numbers.

1. 2 and 8 2. 4 and 9 3. 9 and 16

4. 16 and 4 5. 16 and 36 6. 12 and 3

7. 18 and 8 8. 2 and 18 9. 27 and 12

Recall the definition of a geometric sequence. Each term is found bymultiplying the previous term by the same number. A missing term in ageometric sequence equals the geometric mean of the two terms on either side.

Find the missing term in each geometric sequence.

10. 4, 12, , 108, 324 11. 10, , 62.5, 156.25, 390.625

12. 1, 0.4, , 0.064, 0.0256 13. 700, 70, 7, 0.7, , 0.007

14. 6, , 24 15. 18, , 32??

??

??

1100--11

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The Geometric Mean

The square root of the product of two numbers is called their geometricmean. The geometric mean of 12 and 48 is �1�2� �� 4�8� � �5�7�6� or 24.

Find the geometric mean for each pair of numbers.

1. 2 and 8 2. 4 and 9 3. 9 and 16

4 6 12

4. 16 and 4 5. 16 and 36 6. 12 and 3

8 24 6

7. 18 and 8 8. 2 and 18 9. 27 and 12

12 6 18

Recall the definition of a geometric sequence. Each term is found bymultiplying the previous term by the same number. A missing term in ageometric sequence equals the geometric mean of the two terms on either side.

Find the missing term in each geometric sequence.

10. 4, 12, , 108, 324 11. 10, , 62.5, 156.25, 390.625

36 25

12. 1, 0.4, , 0.064, 0.0256 13. 700, 70, 7, 0.7, , 0.007

0.16 0.07

14. 6, , 24 15. 18, , 32

12 24

??

??

??

1100--11

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World Series Records

Each problem gives the name of a famous baseball player. Tofind who set each record, graph the points on the number line.

1. pitched 23 strikeouts in one World Series

U at �3�, X at 3.3, K at 0.75, O at �32

�, F at �6�, A at 2�78

2. 71 base hits in his appearances in World Series

B at �5�, R at �1�2�, A at 3.75, G at �1163�, E at �5

2�, Y at 0.375, R at �1

43�,

I at 1.6, and O at 0.7�

3. 10 runs in a single World Series

N at �6�0�, K at �3�0�, A at 4.3, S at 6.2, C at �496�, O at �4�5�, and J at �1�7�

4. Batting average of 0.625 in a single World Series

E at �3�2�, U at 6�56

�, A at �134�, T at �5�5�, B at 5.3, R at �4�0�, H at 7.75,

B at �251�

5. 42 World Series runs in his career

E at �1�4�0�, Y at 9.6, I at 8.6, E at �9�0�, A at �221�, M at �7�0�, C at 8�7

8�,

M at �1�0�0�, N at 10.7, K at 9�111�, T at �1�2�0�, L at 11.4

8 1211109

4 8765

4 8765

0 4321

0 4321

1100--22

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World Series Records

Each problem gives the name of a famous baseball player. Tofind who set each record, graph the points on the number line.

1. pitched 23 strikeouts in one World Series

U at �3�, X at 3.3, K at 0.75, O at �32

�, F at �6�, A at 2�78

2. 71 base hits in his appearances in World Series

B at �5�, R at �1�2�, A at 3.75, G at �1163�, E at �5

2�, Y at 0.375, R at �1

43�,

I at 1.6, and O at 0.7�

3. 10 runs in a single World Series

N at �6�0�, K at �3�0�, A at 4.3, S at 6.2, C at �496�, O at �4�5�, and J at �1�7�

4. Batting average of 0.625 in a single World Series

E at �3�2�, U at 6�56

�, A at �134�, T at �5�5�, B at 5.3, R at �4�0�, H at 7.75,

B at �251�

5. 42 World Series runs in his career

E at �1�4�0�, Y at 9.6, I at 8.6, E at �9�0�, A at �221�, M at �7�0�, C at 8�7

8�,

M at �1�0�0�, N at 10.7, K at 9�111�, T at �1�2�0�, L at 11.4

M KCI YE M A TN EL8 1211109

4 8765

B A B E R U HT

4 8765

J A C K S O N

Y O G I B E R R A0 4321

K O U F A X0 4321

1100--22

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Pythagoras in the Air

In the diagram at the right, an airplane heads north at 180 mi/h. But, the wind is blowing towards the east at 30 mi/h. So, theairplane is really traveling east of north. The middle arrow in thediagram shows the actual direction of the airplane.

The actual speed of the plane can be found using the PythagoreanTheorem.

�3�0�2��� 1�8�0�2� � �9�0�0� �� 3�2�,4�0�0�

� �3�3�,3�0�0�

� 182.5

The plane’s actual speed is about 182.5 mi/h.

Find the actual speed of each airplane. Round answers to thenearest tenth. (You might wish to draw a diagram to help yousolve the problem.)

1. An airplane travels at 240 mi/h east. 2. An airplane travels at 620 mi/h west.A wind is blowing at 20 mi/h toward A wind is blowing at 35 mi/h towardthe south. the south.

3. An airplane travels at 450 mi/h south. 4. An airplane travels at 1,200 mi/h east.A wind is blowing at 40 mi/h toward A wind is blowing at 30 mi/h towardthe east. the north.

E

N(not drawn to scale)

1100--33

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Pythagoras in the Air

In the diagram at the right, an airplane heads north at 180 mi/h. But, the wind is blowing towards the east at 30 mi/h. So, theairplane is really traveling east of north. The middle arrow in thediagram shows the actual direction of the airplane.

The actual speed of the plane can be found using the PythagoreanTheorem.

�3�0�2��� 1�8�0�2� � �9�0�0� �� 3�2�,4�0�0�

� �3�3�,3�0�0�

� 182.5

The plane’s actual speed is about 182.5 mi/h.

Find the actual speed of each airplane. Round answers to thenearest tenth. (You might wish to draw a diagram to help yousolve the problem.)

1. An airplane travels at 240 mi/h east. 2. An airplane travels at 620 mi/h west.A wind is blowing at 20 mi/h toward A wind is blowing at 35 mi/h towardthe south. the south.

240.8 mi/h 621 mi/h

3. An airplane travels at 450 mi/h south. 4. An airplane travels at 1,200 mi/h east.A wind is blowing at 40 mi/h toward A wind is blowing at 30 mi/h towardthe east. the north.

451.8 mi/h 1,200.4 mi/h

E

N(not drawn to scale)

1100--33

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Finding the Area From the Vertex Coordinates

Here is a method for finding the area of a polygon whose vertices have integer coordinates.

Step 1List the vertices in counterclockwise order. Repeat the firstvertex at the bottom of the list.

Step 2Find D, the sum of the downward products. (See the solidarrows).D � (5 � 5) � (2 � 1) � (2 � 3) �(6 � 7) or 75

Step 3Find U, the sum of the upward products. (See the dashed arrows.)U � (5 � 3) � (6 � 1) � (2 � 5) � (2 � 7) or 45

Step 4The formula A � (D � U) � 2 gives the area.A � (75 � 45) � 2 or 15 units2

Find each area using the steps shown above.

1. 2.

3. 4. y

x642

2

4

6

O 8 10

y

x642

2

4

6

O 8 10

y

x642

2

4

6

8

O 8 10

y

x642

2

4

6

8

O 8 10

y

x642

2

4

6

8

O

(5, 7)

(5, 7)

(2, 5)

(2, 1)

(6, 3)

(5, 7)

(6, 3)

(2, 1)

(2, 5)

1100--44

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Finding the Area From the Vertex Coordinates

Here is a method for finding the area of a polygon whose vertices have integer coordinates.

Step 1List the vertices in counterclockwise order. Repeat the firstvertex at the bottom of the list.

Step 2Find D, the sum of the downward products. (See the solidarrows).D � (5 � 5) � (2 � 1) � (2 � 3) �(6 � 7) or 75

Step 3Find U, the sum of the upward products. (See the dashed arrows.)U � (5 � 3) � (6 � 1) � (2 � 5) � (2 � 7) or 45

Step 4The formula A � (D � U) � 2 gives the area.A � (75 � 45) � 2 or 15 units2

Find each area using the steps shown above.

1. 36 units2 2.

3. 4.

14 units2 36.5 units2

y

x642

2

4

6

O 8 10

y

x642

2

4

6

O 8 10

y

x642

2

4

6

8

O 8 10

y

x642

2

4

6

8

O 8 10

y

x642

2

4

6

8

O

(5, 7)

(5, 7)

(2, 5)

(2, 1)

(6, 3)

(5, 7)

(6, 3)

(2, 1)

(2, 5)

1100--44

20 units2

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Heron’s Formula

A formula named after Heron of Alexandria, Egypt, can be used to findthe area of a triangle given the lengths of its sides.

Heron’s formula states that the area A of a triangle whose sides measurea, b, and c is given by

A � �s(�s��� a�)�(s� �� b�)�(s� �� c�)�,

where s is the semiperimeter:

s � .

Estimate the area of each triangle by finding the mean of theinner and outer measures. Then use Heron’s Formula tocompute a more exact area. Give each answer to the nearesttenth of a square unit.

1. 2. 3.

Estimated area: Estimated area: Estimated area:

Computed area: Computed area: Computed area:

4. 5. 6.

Estimated area: Estimated area: Estimated area:

Computed area: Computed area: Computed area:

9

57

8

83

7

7

7

6

8

109

9

106

6

6

a � b � c��2

1100--55

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Heron’s Formula

A formula named after Heron of Alexandria, Egypt, can be used to findthe area of a triangle given the lengths of its sides.

Heron’s formula states that the area A of a triangle whose sides measurea, b, and c is given by

A � �s(�s��� a�)�(s� �� b�)�(s� �� c�)�,

where s is the semiperimeter:

s � .

Estimate the area of each triangle by finding the mean of theinner and outer measures. Then use Heron’s Formula tocompute a more exact area. Give each answer to the nearesttenth of a square unit.

1. 2. 3.

Estimated area: 15 Estimated area: 38 Estimated area: 25Computed area: 15.6 Computed area: 37.4 Computed area: 24.0

4. 5. 6.

Estimated area: 20.5 Estimated area: 12.5 Estimated area: 18Computed area: 21.2 Computed area: 11.8 Computed area: 17.4

9

57

8

83

7

7

7

6

8

109

9

106

6

6

a � b � c��2

1100--55

Estimateswill vary.

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Extending the Pythagorean Theorem

The Pythagorean Theorem says that the sum of the areas of the two smaller squares is equal to the area of the largest square. Show that thePythagorean Theorem can be extended to include other shapes on thesides of a triangle. To do so, find the areas of the two smaller shapes.Then, check that their sum equals the area of the largest shape.

1. area of smallest shape: 2. area of smallest shape:

area of middle shape: area of middle shape:

area of largest shape: area of largest shape:

3. area of smallest shape: 4. area of smallest shape:

area of middle shape: area of middle shape:

area of largest shape: area of largest shape:

3 in.

3 in.

3 in.

5 in.

5 in.

5 in.

4 in.

4 in.4 in.

3 in.

3 in.

5 in.5 in.

4 in.

4 in.

1.5in.

3 in.

5 in.

4 in.

2.5 in.

2 in.

5 in.

4 in.

3 in.

55

44 3

3

1100--66

(Hint: for an

equilateral triangle,

A � �s42� �3�.)

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Extending the Pythagorean Theorem

The Pythagorean Theorem says that the sum of the areas of the two smaller squares is equal to the area of the largest square. Show that thePythagorean Theorem can be extended to include other shapes on thesides of a triangle. To do so, find the areas of the two smaller shapes.Then, check that their sum equals the area of the largest shape.

1. area of smallest shape: 3.5 in2 2. area of smallest shape: 2.25 in2

area of middle shape: 6.3 in2 area of middle shape: 4 in2

area of largest shape: 9.8 in2 area of largest shape: 6.25 in2

3. area of smallest shape: 4.5 in2 4. area of smallest shape: 3.9 in2

area of middle shape: 8 in2 area of middle shape: 6.9 in2

area of largest shape: 12.5 in2 area of largest shape: 10.8 in2

3 in.

3 in.

3 in.

5 in.

5 in.

5 in.

4 in.

4 in.4 in.

3 in.

3 in.

5 in.5 in.

4 in.

4 in.

1.5in.

3 in.

5 in.

4 in.

2.5 in.

2 in.

5 in.

4 in.

3 in.

55

44 3

3

1100--66

(Hint: for an

equilateral triangle,

A � �s42� �3�.)

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Area Formulas for Regular Polygons

Recall that the sides of a regular polygon are all the same length. Here aresome area formulas for four of the regular polygons. The variable s stands for the length of one side.

triangle pentagon hexagon octagon

A � �3� A � �2�5� �� 1�0� ��5�� A � �3� A � 2s2 (�2� � 1)

Find the area of each polygon with the side of given length. Use acalculator and round each answer to the nearest tenth.

1.

2.

3.

4.

Now use your chart to find the area of each shaded region below.Each small segment is 1 cm long.

5. 6. 7.

3.8 cm2 1.3 cm2 1.8 cm2

8. 9. 10.

5.7 cm2 5.4 cm2 6.2 cm2

3s2�2

s2�4

s2�4

1100--77

Length of Triangle Pentagon Hexagon Octagona Side

1 cm 0.4 cm2 1.7 cm2 2.6 cm2 4.8 cm2

2 cm 1.7 cm2 6.9 cm2 10.4 cm2 19.3 cm2

3 cm 3.9 cm2 15.5 cm2 23.4 cm2 43.5 cm2

4 cm 6.9 cm2 27.5 cm2 41.6 cm2 77.3 cm2

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Area Formulas for Regular Polygons

Recall that the sides of a regular polygon are all the same length. Here aresome area formulas for four of the regular polygons. The variable s stands for the length of one side.

triangle pentagon hexagon octagon

A � �3� A � �2�5� �� 1�0� ��5�� A � �3� A � 2s2 (�2� � 1)

Find the area of each polygon with the side of given length. Use acalculator and round each answer to the nearest tenth.

1.

2.

3.

4.

Now use your chart to find the area of each shaded region below.Each small segment is 1 cm long.

5. 6. 7.

3.8 cm2 1.3 cm2 1.7 cm2

8. 9. 10.

5.7 cm2 5.4 cm2 6.2 cm2

3s2�2

s2�4

s2�4

1100--77

Length of Triangle Pentagon Hexagon Octagona Side

1 cm 0.4 cm2 1.7 cm2 2.6 cm2 4.8 cm2

2 cm 1.7 cm2 6.9 cm2 10.4 cm2 19.3 cm2

3 cm 3.9 cm2 15.5 cm2 23.4 cm2 43.5 cm2

4 cm 6.9 cm2 27.5 cm2 41.6 cm2 77.3 cm2

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Made in the Shade

To shade 25% of the figure at the right below, ask yourself how many ofthe eight squares need to be shaded. Then use the percent proportion tofind the answer.

�8x� � �

12050

100x � 8 � 25

�110000x� � �2

10000

x � 2

If you shade two squares, you have shaded 25% of the figure.

Shade the indicated percent of each diagram.

1. Shade 40%. 2. Shade 37.5%. 3. Shade 16�23

�%.

Shade the indicated percent of each diagram. You will need todivide the squares in each diagram into smaller squares.

4. Shade 30%. 5. Shade 62.5%.

6. Shade 27.5%. 7. Shade 28.125%.

1111--11

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Made in the Shade

To shade 25% of the figure at the right below, ask yourself how many ofthe eight squares need to be shaded. Then use the percent proportion tofind the answer.

�8x� � �

12050

100x � 8 � 25

�110000x� � �2

10000

x � 2

If you shade two squares, you have shaded 25% of the figure.

Shade the indicated percent of each diagram.

1. Shade 40%. 2. Shade 37.5%. 3. Shade 16�23

�%.

Shade the indicated percent of each diagram. You will need todivide the squares in each diagram into smaller squares.

4. Shade 30%. 5. Shade 62.5%.

6. Shade 27.5%. 7. Shade 28.125%.

1111--11

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Just The Facts

Use the percent proportion and your calculator to helpdiscover some interesting facts about the United States.Round your answers to the nearest tenth of a percent.

1. The United States produced 67,832 million eggs in 1990. The state ofIndiana produced 5,445 million eggs—more than any other state. Whatpercent of the eggs produced in the United States were producedoutside Indiana?

2. The population of the United States in 1990 was 248,709,873 people.Of these people, 121,239,418 were male. What percent of thepopulation was female?

3. In the 1988 presidential election, 48,881,221 people voted for GeorgeBush, and 41,805,422 people voted for Michael Dukakis. Of the peoplethat voted for these two candidates, what percent voted for GeorgeBush?

4. During the period 1980–1990, Moreno Valley, California, was thefastest-growing city in the United States. Its population grew from28,309 to 118,779 people. By what percent did the population increaseduring this period?

5. The public debt of the United States in 1980 was 907.7 billion dollars.In 1990, it was 3,233.3 billion dollars. By what percent did the publicdebt increase from 1980 to 1990?

6. In 1986, the average annual pay in the United States was $19,966. In1990, it rose to $22,563. By what percent did the average annual payincrease from 1986 to 1990?

7. During the period 1980–1990, Naples, Florida, was the fastest-growingmetropolitan area in the United States. The 1980 population of Napleswas 85,980, and its 1990 population was 152,099. By what percent didthe population of Naples increase from 1980 to 1990?

1111--22

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Just The Facts

Use the percent proportion and your calculator to helpdiscover some interesting facts about the United States.Round your answers to the nearest tenth of a percent.

1. The United States produced 67,832 million eggs in 1990. The state ofIndiana produced 5,445 million eggs—more than any other state. Whatpercent of the eggs produced in the United States were producedoutside Indiana? 92.0%

2. The population of the United States in 1990 was 248,709,873 people.Of these people, 121,239,418 were male. What percent of thepopulation was female? 51.3%

3. In the 1988 presidential election, 48,881,221 people voted for GeorgeBush, and 41,805,422 people voted for Michael Dukakis. Of the peoplethat voted for these two candidates, what percent voted for GeorgeBush? 53.9%

4. During the period 1980–1990, Moreno Valley, California, was thefastest-growing city in the United States. Its population grew from28,309 to 118,779 people. By what percent did the population increaseduring this period? 319.6%

5. The public debt of the United States in 1980 was 907.7 billion dollars.In 1990, it was 3,233.3 billion dollars. By what percent did the publicdebt increase from 1980 to 1990? 256.2%

6. In 1986, the average annual pay in the United States was $19,966. In1990, it rose to $22,563. By what percent did the average annual payincrease from 1986 to 1990? 13.0%

7. During the period 1980–1990, Naples, Florida, was the fastest-growingmetropolitan area in the United States. The 1980 population of Napleswas 85,980, and its 1990 population was 152,099. By what percent didthe population of Naples increase from 1980 to 1990? 76.9%

1111--22

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Relative Frequency and Circle Graphs

The relative frequency tells how the frequency of one item compares to the total of all the frequencies. Relative frequencies are written as fractions, decimals, or percents.

For example, in Exercise 1 below, the total of all thefrequencies is 50. So, the relative frequency of thegrade A is 8 � 50, or 0.16.

The circle at the right is divided into 20 equal parts.You can trace this circle and then use relativefrequencies to make circle graphs.

Complete each chart to show the relativefrequencies. Then sketch a circle graph forthe data. Use decimals rounded to the nearest hundredth.

1. History Grades for 50 Students

2. Steve's Budget

0 0.050.1

0.15

0.2

0.3

0.35

0.40.450.50.55

0.25

0.6

0.65

0.7

0.75

0.8

0.85

0.90.95

1111--33

Grade Frequency RelativeFrequency

A 8 0.16

B 16

C 18

D 6

F 2 0.04

Item Amount RelativeSpent Spending

Telephone $26

Movies $46

Books $24

Car $38

Other $66 0.33

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Relative Frequency and Circle Graphs

The relative frequency tells how the frequency of one item compares to the total of all the frequencies. Relative frequencies are written as fractions, decimals, or percents.

For example, in Exercise 1 below, the total of all thefrequencies is 50. So, the relative frequency of thegrade A is 8 � 50, or 0.16.

The circle at the right is divided into 20 equal parts.You can trace this circle and then use relativefrequencies to make circle graphs.

Complete each chart to show the relativefrequencies. Then sketch a circle graph forthe data. Use decimals rounded to the nearest hundredth.

1. History Grades for 50 Students

2. Steve's Budget Steve's Budget

Other

CarBooks

Movies

Tele-phone

F

A

BC

D

History Grades for 50 Students

0 0.050.1

0.15

0.2

0.3

0.35

0.40.450.50.55

0.25

0.6

0.65

0.7

0.75

0.8

0.85

0.90.95

1111--33

Grade Frequency RelativeFrequency

A 8 0.16

B 16 0.32C 18 0.36D 6 0.12F 2 0.04

Item Amount RelativeSpent Spending

Telephone $26 0.13Movies $46 0.23Books $24 0.12Car $38 0.19Other $66 0.33

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Table of Random Digits

A table of random digits can be used to simulate probability experiments.This table of random digits contains 50 digits.

For example, how often might someone expect a coin to land heads up twoconsecutive times or more in 50 tosses? Our table can be used to make thisprediction. Since the table contains the digits 0 through 9, let’s say a tossof heads represents the digits 0, 2, 4, 6, and 8, or �1

2� of the possible digits

that appear in the table.

Using the table to imitate the 50 tosses, we must look for the digits 0, 2, 4,6, and 8 that occur two or more times consecutively. These have beencircled in the table at the right above, and we would expect to toss two ormore consecutive heads 4 times in 50 trials.

Use the table of random digits to answer the followingquestions.

1. How many times might a coin toss of 3 or more consecutive tails occurin 50 trials? (Hint: Let 1, 3, 5, 7, and 9 represent a toss of tails.)

2. How many times might a coin toss of 4 or more consecutive heads occurin 50 trials? (Hint: Let 0, 2, 4, 6, and 8 represent a toss of heads.)

3. Letting the digits 1, 3, 5, 7, and 9 represent a coin toss of tails, what isthe maximum number of consecutive tails that could be expected in 50tosses?

4. Letting the digits 0, 2, 4, 6, and 8 represent a coin toss of heads, what isthe maximum number of consecutive heads that could be expected in50 tosses?

8 1 5 8 6 7 9 9 8 0

9 9 3 7 3 3 1 8 7 4

7 3 0 9 9 2 4 6 2 4

4 0 5 2 9 9 6 3 8 2

8 4 2 1 6 3 7 0 3 1

8 1 5 8 6 7 9 9 8 0

9 9 3 7 3 3 1 8 7 4

7 3 0 9 9 2 4 6 2 4

4 0 5 2 9 9 6 3 8 2

8 4 2 1 6 3 7 0 3 1

1111--44

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Table of Random Digits

A table of random digits can be used to simulate probability experiments.This table of random digits contains 50 digits.

For example, how often might someone expect a coin to land heads up twoconsecutive times or more in 50 tosses? Our table can be used to make thisprediction. Since the table contains the digits 0 through 9, let’s say a tossof heads represents the digits 0, 2, 4, 6, and 8, or �1

2� of the possible digits

that appear in the table.

Using the table to imitate the 50 tosses, we must look for the digits 0, 2, 4,6, and 8 that occur two or more times consecutively. These have beencircled in the table at the right above, and we would expect to toss two ormore consecutive heads 4 times in 50 trials.

Use the table of random digits to answer the followingquestions.

1. How many times might a coin toss of 3 or more consecutive tails occurin 50 trials? (Hint: Let 1, 3, 5, 7, and 9 represent a toss of tails.) 2

2. How many times might a coin toss of 4 or more consecutive heads occurin 50 trials? (Hint: Let 0, 2, 4, 6, and 8 represent a toss of heads.) 2

3. Letting the digits 1, 3, 5, 7, and 9 represent a coin toss of tails, what isthe maximum number of consecutive tails that could be expected in 50tosses? 7

4. Letting the digits 0, 2, 4, 6, and 8 represent a coin toss of heads, what isthe maximum number of consecutive heads that could be expected in50 tosses? 7

8 1 5 8 6 7 9 9 8 0

9 9 3 7 3 3 1 8 7 4

7 3 0 9 9 2 4 6 2 4

4 0 5 2 9 9 6 3 8 2

8 4 2 1 6 3 7 0 3 1

8 1 5 8 6 7 9 9 8 0

9 9 3 7 3 3 1 8 7 4

7 3 0 9 9 2 4 6 2 4

4 0 5 2 9 9 6 3 8 2

8 4 2 1 6 3 7 0 3 1

1111--44

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A Taxing Exercise

People who earn income are required by law to pay taxes. The amount oftax a person owes is computed by first subtracting the amount of allexemptions and deductions from the amount of income, then using a taxtable like this.

Schedule X—Use if your filing status is Single

Compute each person’s income. Subtract $5,550 for eachperson’s exemption and deduction. Then use the tax rateschedule to compute the amount of federal tax owed.

1. A cashier works 40 hours each week, earns $7.50 per hour, and works50 weeks each year.

2. A newspaper carrier works each day, delivers 154 papers daily, andearns $0.12 delivering each paper.

3. A babysitter earns $3.50 per hour per child. During a year, thebabysitter works with two children every Saturday for 8 hours and withthree children every other Sunday for 6 hours.

4. While home from college for the summer, a painter earns $17.00 perhour, working 45 hours each week for 15 weeks.

5. Working before and after school in the school bookstore, an employeeworks 2.5 hours each day for 170 days and earns $4.60 per hour.

6. After graduating from college, a computer programmer accepts aposition earning $2,450 monthly.

1111--55

If the amount on Enter onForm 1040, line Form 1040, of the37, is: But not line 38 amountOver— over— over—

$0 $20,350 ------------15% $0

20,350 49,300 $3,052.50 + 28% 20,350

49,300 -------- 11,158.50 + 31% 49,300

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A Taxing Exercise

People who earn income are required by law to pay taxes. The amount oftax a person owes is computed by first subtracting the amount of allexemptions and deductions from the amount of income, then using a taxtable like this.

Schedule X—Use if your filing status is Single

Compute each person’s income. Subtract $5,550 for eachperson’s exemption and deduction. Then use the tax rateschedule to compute the amount of federal tax owed.

1. A cashier works 40 hours each week, earns $7.50 per hour, and works50 weeks each year. $1,417.50

2. A newspaper carrier works each day, delivers 154 papers daily, andearns $0.12 delivering each paper. $179.28

3. A babysitter earns $3.50 per hour per child. During a year, thebabysitter works with two children every Saturday for 8 hours and withthree children every other Sunday for 6 hours. $0.00

4. While home from college for the summer, a painter earns $17.00 perhour, working 45 hours each week for 15 weeks. $888.75

5. Working before and after school in the school bookstore, an employeeworks 2.5 hours each day for 170 days and earns $4.60 per hour.

$0.00

6. After graduating from college, a computer programmer accepts aposition earning $2,450 monthly. $4,032.50

1111--55

If the amount on Enter onForm 1040, line Form 1040, of the37, is: But not line 38 amountOver— over— over—

$0 $20,350 ------------15% $0

20,350 49,300 $3,052.50 + 28% 20,350

49,300 -------- 11,158.50 + 31% 49,300

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Missing Fact Match

The problems on this page are missing a key fact and cannot be solved.

Find the missing fact in Column 2 that completes eachproblem in Column 1. After each missing fact has beenmatched to its problem in Column 1, find each answer.

Problem Missing Fact Answer

1. The school band held a fund raiser by ________ The team 1.selling band buttons. Each button sold had 12 players.for $1.50, which included a 20% profit.How much profit did the fund raiser earn?

2. The athletic department received a bill of ________ Regular 2.$153.36, including tax, for extra uniforms. price is $22.50.Find the cost of the uniforms before tax.

3. If everyone is present, there are 25 students ________ Paper 3.in a mathematics class. How many students products arewere in class on Monday? 25% off.

4. The volleyball team stopped after the game ________ They 4.to eat. The bill was $57.60, not including reached 150%a 15% tip. If the bill was split equally among of their goal.the players, what was each player’s share?

5. The school bookstore does not tax supplies ________ Sales 5.and is having a spring sale. Find the cost of tax on clothingtwo spiral notebooks that regularly sell is 8%.for $1.40 each.

6. A department store advertises 40% off jeans ________ Total 6.in a back-to-school sale. If sales tax is 5%, sales werewhat is the cost of two pairs of jeans? 1,200 buttons.

7. Last season, the goal of the basketball team ________ Monday’s 7.was to win 12 games. How many games did attendance wasthey win last season? 92%.

1111--66

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Missing Fact Match

The problems on this page are missing a key fact and cannot be solved.

Find the missing fact in Column 2 that completes eachproblem in Column 1. After each missing fact has beenmatched to its problem in Column 1, find each answer.

Problem Missing Fact Answer

1. The school band held a fund raiser by ________ The team 1. $360selling band buttons. Each button sold had 12 players.for $1.50, which included a 20% profit.How much profit did the fund raiser earn?

2. The athletic department received a bill of ________ Regular 2. $142$153.36, including tax, for extra uniforms. price is $22.50.Find the cost of the uniforms before tax.

3. If everyone is present, there are 25 students ________ Paper 3. 23in a mathematics class. How many students products arewere in class on Monday? 25% off.

4. The volleyball team stopped after the game ________ They 4. $5.52to eat. The bill was $57.60, not including reached 150%a 15% tip. If the bill was split equally among of their goal.the players, what was each player’s share?

5. The school bookstore does not tax supplies ________ Sales 5. $2.10and is having a spring sale. Find the cost of tax on clothingtwo spiral notebooks that regularly sell is 8%.for $1.40 each.

6. A department store advertises 40% off jeans ________ Total 6. $28.35in a back-to-school sale. If sales tax is 5%, sales werewhat is the cost of two pairs of jeans? 1,200 buttons.

7. Last season, the goal of the basketball team ________ Monday’s 7. 18was to win 12 games. How many games did attendance wasthey win last season? 92%.

1111--66

4

6

5

7

2

1

3

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Taking an Interest

When interest is paid on both the amount of the deposit and any interestalready earned, interest is said to be compounded. You can use theformula below to find out how much money is in an account for whichinterest is compounded.

A � P(1 � r)n

In the formula, P represents the principal, or amount deposited, rrepresents the rate applied each time interest is paid, n represents the number of times interest is given, and A represents the amount in the account.

Example A customer deposited $1,500 in an account that earns8% per year. If interest is compounded and earnedsemiannually, how much is in the account after 1 year?

Use the formula A � P(1 � r)n.Since interest is earned semiannually, r � 8 � 2 or 4% andn � 2.

A � 1,500(1 � 0.04)2 Use a calculator.� 1,622.40

After 1 year, there is $1,622.40 in the account.

Use the compound interest formula and a calculator to findthe value of each of these investments. Round each answer tothe nearest cent.

1. $2,500 invested for 1 year at 6% interestcompounded semiannually

2. $3,600 invested for 2 years at 7% interestcompounded semiannually

3. $1,000 invested for 5 years at 8% interestcompounded annually

4. $2,000 invested for 6 years at 12% interest compounded quarterly

5. $4,800 invested for 10 years at 9% interestcompounded annually

6. $10,000 invested for 15 years at 7.5% interestcompounded semiannually

1111--77

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Taking an Interest

When interest is paid on both the amount of the deposit and any interestalready earned, interest is said to be compounded. You can use theformula below to find out how much money is in an account for whichinterest is compounded.

A � P(1 � r)n

In the formula, P represents the principal, or amount deposited, rrepresents the rate applied each time interest is paid, n represents the number of times interest is given, and A represents the amount in the account.

Example A customer deposited $1,500 in an account that earns8% per year. If interest is compounded and earnedsemiannually, how much is in the account after 1 year?

Use the formula A � P(1 � r)n.Since interest is earned semiannually, r � 8 � 2 or 4% andn � 2.

A � 1,500(1 � 0.04)2 Use a calculator.� 1,622.40

After 1 year, there is $1,622.40 in the account.

Use the compound interest formula and a calculator to findthe value of each of these investments. Round each answer tothe nearest cent.

1. $2,500 invested for 1 year at 6% interest $2,652.25compounded semiannually

2. $3,600 invested for 2 years at 7% interest $4,131.08compounded semiannually

3. $1,000 invested for 5 years at 8% interest $1,469.33compounded annually

4. $2,000 invested for 6 years at 12% interest $4,065.59compounded quarterly

5. $4,800 invested for 10 years at 9% interest $11,363.35compounded annually

6. $10,000 invested for 15 years at 7.5% interest $30,174.71compounded semiannually

1111--77

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Counting Cubes

The figures on this page have been built by gluing cubes together. Use yourvisual imagination to count the total number of cubes as well as the numberof cubes with glue on 1, 2, 3, 4, or 5, or 6 faces.

Complete this chart for the figures below.

1. 2. 3.

4. 5. 6.

C C

1122--11

Number of Faces with Glue on ThemTotal NumberFigure of Cubes 1 face 2 faces 3 faces 4 faces 5 faces 6 faces

1 13 0 3 7 2 1 02 17 0 1 8 6 2 03 14 5 0 8 0 1 04 36 0 0 16 20 0 05 22 4 6 6 4 2 06 49 1 5 12 21 9 1

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Counting Cubes

The figures on this page have been built by gluing cubes together. Use yourvisual imagination to count the total number of cubes as well as the numberof cubes with glue on 1, 2, 3, 4, or 5, or 6 faces.

Complete this chart for the figures below.

1. 2. 3.

4. 5. 6.

C C

1122--11

Number of Faces with Glue on ThemTotal NumberFigure of Cubes 1 face 2 faces 3 faces 4 faces 5 faces 6 faces

1 13 0 3 7 2 1 02 17 0 1 8 6 2 03 14 5 0 8 0 1 04 36 0 0 16 20 0 05 22 4 6 6 4 2 06 35 1 2 9 16 6 1

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Volumes of Pyramids

A pyramid and a prism with the same base and height are shown below.

The exercises on this page will help you discover how their volumes are related.

Make copies of the two patterns below to make the openpyramid and the open prism shown above. (Each equilateraltriangle should measure 8 centimeters on a side.)

1. Describe the bases of the two solids. equilateral triangles

2. How do the heights of the solids compare?

3. Fill the open pyramid with sand or sugar. Pour the contents into the open prism. How many times must you do this to fill the open prism?

4. Describe how you would find the volume of the pyramid shown at the right.

Divide the volume of a prism with the same base and height by 3.

5. Generalize: State a formula for the volume of a pyramid.

OpenBottom

OpenTop

1122--22

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Volumes of Pyramids

A pyramid and a prism with the same base and height are shown below.

The exercises on this page will help you discover how their volumes are related.

Make copies of the two patterns below to make the openpyramid and the open prism shown above. (Each equilateraltriangle should measure 8 centimeters on a side.)

1. Describe the bases of the two solids. equilateral triangles

2. How do the heights of the solids compare?

They are the same.3. Fill the open pyramid with sand or sugar. Pour the contents

into the open prism. How many times must you do this to fill the open prism? three times

4. Describe how you would find the volume of the pyramid shown at the right.

Divide the volume of a prism with the same base and height by 3.

5. Generalize: State a formula for the volume of a pyramid.

The volume is �13

� � area of the base � height.

OpenBottom

OpenTop

1122--22

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Volumes of Non-Right Solids

Imagine a stack of ten pennies. By pushing against the stack, you can change its shape as shown at the right. But, the volume of the stack does not change.

The diagrams below show prisms and cylinders that have thesame volume but do not have the same shape.

Find the volume of each solid figure.

1. 2. 3.

4. 5. 6.

15 cm

4 cm

4 cm10 cm

2 cm

10 yd

5 yd

3 yd

7 yd

6 yd

12 in.

10 in.

5 in.

10 in.

11 in.

2 in.3 in.

2 m

3 m2.5 m

height height height

radiusradius

height

lengthlength

Right Prism Non-right Prism Right Cylinder Non-right Cylinder

width width

1122--33

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Volumes of Non-Right Solids

Imagine a stack of ten pennies. By pushing against the stack, you can change its shape as shown at the right. But, the volume of the stack does not change.

The diagrams below show prisms and cylinders that have thesame volume but do not have the same shape.

Find the volume of each solid figure.

1. 2. 3.

7.9 m3 60 in3 785 in3

4. 5. 6.

180 yd3 125.6 cm3 240 cm3

15 cm

4 cm

4 cm10 cm

2 cm

10 yd

5 yd

3 yd

7 yd

6 yd

12 in.

10 in.

5 in.

10 in.

11 in.

2 in.3 in.

2 m

3 m2.5 m

height height height

radiusradius

height

lengthlength

Right Prism Non-right Prism Right Cylinder Non-right Cylinder

width width

1122--33

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Pattern Puzzles

1. Make three copies of this pattern. Fold each pattern tomake a pyramid. Then, put thethree pyramids together to makea cube. Draw a sketch of thecompleted cube.

Sketches will vary depending onthe view chosen.

2. Make four copies of this pattern. Fold each pattern to make a solidfigure. Then, put the four solidstogether to make a pyramid.Make a sketch of the finishedpyramid.

Sketches will vary depending onthe view chosen.

3. Find the surface area of the cube in Exercise 1.

1122--44

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Mathematics: Applications© Glencoe/McGraw-Hill T91 and Connections, Course 2

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Pattern Puzzles

1. Make three copies of this pattern. Fold each pattern tomake a pyramid. Then, put thethree pyramids together to makea cube. Draw a sketch of thecompleted cube.

Sketches will vary depending onthe view chosen.

2. Make four copies of this pattern. Fold each pattern to make a solidfigure. Then, put the four solidstogether to make a pyramid.Make a sketch of the finishedpyramid.

Sketches will vary depending onthe view chosen.

3. Find the surface area of the cube in Exercise 1. 54 cm2

1122--44

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Cross Sections

In each diagram on this page, a plane cuts through a solid figure. Theintersection of the plane with the solid figure is called a cross section.

Sketch the cross section formed in each diagram.

1. 2.

3. 4.

5. 6.

7. 8. (pyramid with atriangular base)

(pyramid with atriangular base)

(pyramid with asquare base)

1122--55

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Cross Sections

In each diagram on this page, a plane cuts through a solid figure. Theintersection of the plane with the solid figure is called a cross section.

Sketch the cross section formed in each diagram.

1. 2.

3. 4.

5. 6.

7. 8. (pyramid with atriangular base)

(pyramid with atriangular base)

(pyramid with asquare base)

1122--55

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Rolling a Dodecahedron

A dodecahedron is a solid. It has twelve faces, and each face is a is a pentagon.

At the right, you see a dodecahedron whose faces are marked with the integers from 1 through 12. You can roll this dodecahedron justas you roll a number cube. With the dodecahedron, however, thereare twelve equally likely outcomes.

Refer to the dodecahedron shown at the right. Find theprobability of each event.

1. P(5) 2. P(odd)

3. P(prime) 4. P(divisible by 5)

5. P(less than 4) 6. P(fraction)

You can make your own dodecahedron by cutting outthe pattern at the right. Fold along each of the solidlines. Then use tape to join the faces together so thatyour dodecahedron looks like the one shown above.

7. Roll your dodecahedron 100 times. Record yourresults on a separate sheet of paper, using a tablelike this.

8. Use your results from Exercise 7. Find theexperimental probability for each of the eventsdescribed in Exercises 1-6.

Outcome

1

2

Tally Frequency

1 2

3

45

67

11

12

8

9

10

2

11 10

1

36

1133--11

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Rolling a Dodecahedron

A dodecahedron is a solid. It has twelve faces, and each face is a is a pentagon.

At the right, you see a dodecahedron whose faces are marked with the integers from 1 through 12. You can roll this dodecahedron justas you roll a number cube. With the dodecahedron, however, thereare twelve equally likely outcomes.

Refer to the dodecahedron shown at the right. Find theprobability of each event.

1. P(5) �112� 2. P(odd) �

12

3. P(prime) �152� 4. P(divisible by 5) �

16

5. P(less than 4) �14

� 6. P(fraction) 0

You can make your own dodecahedron by cutting outthe pattern at the right. Fold along each of the solidlines. Then use tape to join the faces together so thatyour dodecahedron looks like the one shown above.

7. Roll your dodecahedron 100 times. Record yourresults on a separate sheet of paper, using a tablelike this.

Answers will vary.8. Use your results from Exercise 7. Find the

experimental probability for each of the eventsdescribed in Exercises 1-6.

Answers will vary.

Outcome

1

2

Tally Frequency

1 2

3

45

6

7

11

128

9

10

2

11 10

1

36

1133--11

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Probabilities and Regions

The spinner at the right can be used to indicate that the probability of

landing in either of two regions is �12

�.

P(A) � �12

� P(B) � �12

Read the description of each spinner. Using a protractor andruler, divide each spinner into regions that show the indicatedprobability.

5. The spinner at the right is an equilateral triangle, divided into regions by line segments that divide the sides in half. Is the spinner divided intoregions of equal probability?

A

B

1133--22

1. Two regions A and B: the probability oflanding in region A is �

34

�. What is theprobability of landing in region B?

3. Three regions A, B, and C: the probability of landing in region A is �

38

� and theprobability of landing in region B is �

18

�.What is the probability of landing in region C?

2. Three regions A, B, and C: the probability of landing in region A is �

12

� and theprobability of landing in region B is �

14

�.What is the probability of landing in region C?

4. Four regions A, B, C, and D: the probability of landing in region A is �

116�, the probability

of landing in region B is �18

�, and theprobability of landing in region C is �

14

�.What is the probability of landing in region D?

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Probabilities and Regions

The spinner at the right can be used to indicate that the probability of

landing in either of two regions is �12

�.

P(A) � �12

� P(B) � �12

Read the description of each spinner. Using a protractor andruler, divide each spinner into regions that show the indicatedprobability.

5. The spinner at the right is an equilateral triangle, divided into regions by line segments that divide the sides in half. Is the spinner divided intoregions of equal probability?

yes

A

B

1133--22

1. Two regions A and B: the probability oflanding in region A is �

34

�. What is theprobability of landing in region B?

P(B) � �14

3. Three regions A, B, and C: the probability of landing in region A is �

38

� and theprobability of landing in region B is �

18

�.What is the probability of landing in region C?

P(C) � �12

2. Three regions A, B, and C: the probability of landing in region A is �

12

� and theprobability of landing in region B is �

14

�.What is the probability of landing in region C?

P(C) � �14

4. Four regions A, B, C, and D: the probability of landing in region A is �

116�, the probability

of landing in region B is �18

�, and theprobability of landing in region C is �

14

�.What is the probability of landing in region D?

P(D) � �196�

B

CD

A

C

A

B

B

C

A

B

A

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Curious Cubes

If a six-faced die is rolled any number of times, the theoretical probabilityof the die landing on any given face is �

16

�.

Each die below has six faces and has been rolled 100 times.The outcomes have been tallied and recorded in a frequencytable. Based on the data in each frequency table, what canyou say are probably on the unseen faces of each cube?

1.

The faces are numbered 2, 3, and 6.

2.

There are two yellowfaces and one red face.

3.

The faces are blank.

4.

The faces are numbered1, 4, and 5.

5.

One face is numbered 2 and two faces are blank.

14

5

14

5

BlueRed

Red

YellowBlue

Red

14

5

1133--33

Outcome Tally1 152 143 184 165 19

6 18

Outcome Tally1 344 325 34

Outcome Tallyred 30blue 16blank 54

Outcome Tallyblue 17red 30yellow 53

Outcome Tally1 145 134 182 16

blank 39

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Curious Cubes

If a six-faced die is rolled any number of times, the theoretical probabilityof the die landing on any given face is �

16

�.

Each die below has six faces and has been rolled 100 times.The outcomes have been tallied and recorded in a frequencytable. Based on the data in each frequency table, what canyou say are probably on the unseen faces of each cube?

1.

The faces are numbered 2, 3, and 6.

2.

There are two yellowfaces and one red face.

3.

The faces are blank.

4.

The faces are numbered1, 4, and 5.

5.

One face is numbered 2 and two faces are blank.

14

5

14

5

BlueRed

Red

YellowBlue

Red

14

5

1133--33

Outcome Tally1 152 143 184 165 19

6 18

Outcome Tally1 344 325 34

Outcome Tallyred 30blue 16blank 54

Outcome Tallyblue 17red 30yellow 53

Outcome Tally1 145 134 182 16

blank 39

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Independent Events

The game of roulette is played by dropping a ballinto a spinning, bowl-shaped wheel. When the wheelstops spinning, the ball will come to rest in any of 38locations.

On a roulette wheel, the eighteen even numbers from 2 through 36 are colored red and the eighteen oddnumbers from 1 through 35 are colored black. Thenumbers 0 and 00 are colored green.

To find the probability of two independent events, the results of two spins, find the probability of eachevent first.

P(red) � �13

88� or �

199�

P(black) � �13

88� or �

199�

Then multiply.

P(red, then black) � �199� � �

199� or �

38611

Find each probability.

1. black, then black

2. prime number, then a composite number

3. a number containing at least one 0, then a number containing at leastone 2

4. red, then black

5. the numbers representing your age, month of birth, and then day ofbirth

0 3 9 15 20 3226

410

1621

3327

5

1117223428006122335

2918

713

124

3036

8

214 19 31 25

1133--44

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Independent Events

The game of roulette is played by dropping a ballinto a spinning, bowl-shaped wheel. When the wheelstops spinning, the ball will come to rest in any of 38locations.

On a roulette wheel, the eighteen even numbers from 2 through 36 are colored red and the eighteen oddnumbers from 1 through 35 are colored black. Thenumbers 0 and 00 are colored green.

To find the probability of two independent events, the results of two spins, find the probability of eachevent first.

P(red) � �13

88� or �

199�

P(black) � �13

88� or �

199�

Then multiply.

P(red, then black) � �199� � �

199� or �

38611

Find each probability.

1. black, then black �38611

2. prime number, then a composite number �36661

3. a number containing at least one 0, then a number containing at leastone 2

�1,

64544�

4. red, then black �38611

5. the numbers representing your age, month of birth, and then day ofbirth

�54,

1872�

0 3 9 15 20 3226

410

1621

3327

5

1117223428006122335

2918

713

124

3036

8

214 19 31 25

1133--44

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Permutation Puzzles

When you change the order of a set of objects in a permutation byswitching the places of two items next to one another, you transpose twoitems in the permutation. In the permutation at the left below, switch B andC to get the permutation at the right below.

For each arrangement at the left show how to switch twoletters at a time to get the arrangement at the right. Show yourswitches in drawings.

1.

2.

3.

EG

AD

FCB I

HEH

DG

FBA C

I

AD

BE

CF

BA

FD

CE

A CBDA

CBD

A B C D A C B D

1133--55

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Permutation Puzzles

When you change the order of a set of objects in a permutation byswitching the places of two items next to one another, you transpose twoitems in the permutation. In the permutation at the left below, switch B andC to get the permutation at the right below.

For each arrangement at the left show how to switch twoletters at a time to get the arrangement at the right. Show yourswitches in drawings.

1.

2.

3.

EH

GD

FBA C

IEI

GD

FBA C

HEG

ID

FBA C

HEG

BD

FIA C

HEG

AD

FIB C

HEG

AD

FCB I

H

EG

AD

FCB I

HEH

DG

FBA C

I

AD

BE

CF

BA

FD

CE

BF

AD

CE

BD

AF

CE

BA

DF

CE

BA

FD

CE

A CBDA

CBD

ACB

D A CBD

A B C D A C B D

1133--55

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From Impossible to Certain Events

A probability is often expressed as a fraction. As you know, an event that isimpossible is given a probability of 0 and an event that is certain is given aprobability of 1. Events that are neither impossible nor certain are given aprobability somewhere between 0 and 1. The probability line below showsrelative probabilities.

Determine the probability of an event by considering its place on the diagram above.

1. Medical research will find a cure for all diseases.

2. There will be a personal computer in each home by the year 2000.

3. One day, people will live in space or under the sea.

4. Wildlife will disappear as Earth’s human population increases.

5. There will be a fifty-first state in the United States.

6. The sun will rise tomorrow morning.

7. Most electricity will be generated by nuclear power by the year 2000.

8. The fuel efficiency of automobiles will increase as the supply of gasoline decreases.

9. Astronauts will land on Mars.

10. The percent of high school students who graduate and enter college will increase.

11. Global warming problems will be solved.

12. All people in the United States will exercise regularly within the near future.

equally likelyprettylikely

1

certainnot solikely

14

12

34

0

impossible

1133--66

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From Impossible to Certain Events

A probability is often expressed as a fraction. As you know, an event that isimpossible is given a probability of 0 and an event that is certain is given aprobability of 1. Events that are neither impossible nor certain are given aprobability somewhere between 0 and 1. The probability line below showsrelative probabilities.

Determine the probability of an event by considering its place on the diagram above. Accept logical responses.

1. Medical research will find a cure for all diseases.

2. There will be a personal computer in each home by the year 2000.

3. One day, people will live in space or under the sea.

4. Wildlife will disappear as Earth’s human population increases.

5. There will be a fifty-first state in the United States.

6. The sun will rise tomorrow morning.

7. Most electricity will be generated by nuclear power by the year 2000.

8. The fuel efficiency of automobiles will increase as the supply of gasoline decreases.

9. Astronauts will land on Mars.

10. The percent of high school students who graduate and enter college will increase.

11. Global warming problems will be solved.

12. All people in the United States will exercise regularly within the near future.

equally likelyprettylikely

1

certainnot solikely

14

12

34

0

impossible

1133--66