michael serra - high school math | prek...

Fourth Edition

Michael Serra

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Project EditorAndres Marti

Project AdministratorBrady Golden

Consulting EditorsChristian Aviles-Scott, Elizabeth DeCarli, Kendra Lockman,

Ladie Malek

Mathematics ReviewersLarry Copes, Inner Grove Heights, Minnesota

Michael de Villiers, Ph.D., University of Durban, Westville,

Pinetown, South Africa

David Rasmussen, Neil’s Harbour, Nova Scotia

Abby Tanenbaum, Naperville, Illinois

Teacher ReviewersRich Crandall, Skyline High School, Oakland, California

Genie Dunn, Miami Killian Senior High School, Miami,

Florida

Judy Hicks, Ralston Valley High School, Arvada, Colorado

Susan May, University of Texas, Austin, Texas

Steve Phelps, Madeira High School, Cincinnati, Ohio

Adella Pietrzyk, Center Line High School, Center Line,

Michigan

Multicultural and Equity ReviewersDavid Keiser, Montclair State University, Upper Montclair,

New Jersey

Swapna Mukhopadhyay, Ph.D., San Diego State University,

San Diego, California

Accuracy CheckersDudley Brooks, Marcia Ellen Olmstead

Editorial Production ManagerChristine Osborne

Print Production SupervisorAnn Rothenbuhler

Production EditorAngela Chen

Production CoordinatorJennifer Young

CopyeditorJill Pellarin

Cover DesignersJill Kongabel, Marilyn Perry, Jensen Barnes

Text DesignerMarilyn Perry

IllustratorsJuan Alvarez, Andy Levine, Claudia Newell, Bill Pasini,

William Rieser, Sue Todd, Rose Zgodzinski

Compositor and Technical Art Interactive Composition Corporation

PrinterVon Hoffmann Corp.

Textbook Product ManagerJames Ryan

Executive EditorCasey FitzSimons

PublisherSteven Rasmussen

© 2008 by Michael Serra. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by anymeans, electronic, photocopying, recording, or otherwise,without the prior written permission of the publisher.

®The Geometer’s Sketchpad, Dynamic Geometry, and

Key Curriculum Press are registered trademarks of

Key Curriculum Press. ™The Discovering Mathematics logo

and Sketchpad are trademarks of Key Curriculum Press.

™Fathom Dynamic Data is a trademark of

KCP Technologies.

All other trademarks are held by their respective owners.

Key Curriculum Press1150 65th StreetEmeryville, CA [email protected]

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1 11 10 09 08 07

ISBN 978-1-55953-882-4

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Acknowledgments

First, to all the teachers who have used Discovering Geometry, a sincere thank you

for your wonderful support and encouragement. I wish to thank my always

delightful and ever-so-patient students for their insight, humor, and hard work.

I also wish to thank the many students across the country who have written to me

with their kind words, comments, and suggestions. And thanks to the marketing

and sales staff at Key Curriculum Press for their successful efforts in bringing the

first three editions into so many classrooms.

There are three people who have added their touch to earlier editions of Discovering

Geometry: Steve Rasmussen was editor on the first edition; Dan Bennett was editor

on the second edition; and Ladie Malek was editor on the third edition. Thank you,

Steve, Dan, and Ladie.

While working on this fourth edition of Discovering Geometry, I was fortunate to

have the assistance of Andres Marti as project editor. Thank you, Andres, for your

commitment to the pedagogy in Discovering Geometry, your cooperative

temperament in the editorial process, and your patience with me. To the editorial

and production staff and managers at Key, the field testers, the advisors, the

consultants, and the reviewers Rich Crandall, Genie Dunn, Judy Hicks, Susan May,

Steve Phelps, and Adella Pietrzyk, I am grateful for your quality work.

Michael Serra

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A Note from the Publisher

You probably know from your experience in school that you learn best when you

understand the concepts and are actively engaged in the learning process.

Studies confirm that understanding concepts and getting involved are important in

every subject at every grade level. This is the foundation that Discovering Geometry

is built on, and investigations are at its heart.

If you are a student, you’ll discover many important mathematical principles

by working together with other students in supportive, small groups doing

investigations. First you’ll explore concepts visually, looking for patterns, and then

you’ll learn to explain why these patterns exist. This approach is both challenging

and fun, and will make you believe in your ability to succeed at mathematics.

Rather than memorizing theorems and formulas, you’ll learn how to build what you

need from what you already know. You’ll find that Discovering Geometry is easy to

follow, and the design includes many interesting photographs and illustrations that

connect geometry with art, architecture, science, history, culture, and recreation.

If you are a parent, you’ll appreciate that Discovering Geometry continues to be the

leader in providing a successful, discovery-based approach. You’ll know that your

student is actively engaged in the process of learning and constructing his or her

own understanding of concepts, developing insight, confidence, and increasingly

sophisticated mathematical understanding along the way. The effectiveness of

Discovering Geometry’s approach has been substantiated in thousands of classrooms

by millions of students, and has changed the way geometry is taught today. If you

go to www.keymath.com/DG, you’ll find Dynamic Geometry Explorations that

explore important geometry concepts interactively, condensed lessons for when

your student is absent, downloadable worksheets to help your student practice his

or her skills, and resources designed especially for you as a parent.

If you are a teacher, you’ll see that this new edition balances the investigative

approach that is at the heart of the Discovering Mathematics series with an

emphasis on developing students’ ability to reason deductively. You’ll be supported

by a deliberate pedagogy based on educational research that carefully develops your

students’ ability to make sound logical arguments. Discovering Geometry introduces

students to reasoning strategies that help them explain their discoveries and enable

them to justify their conjectures through proof. There are also more opportunities

to review algebra, more ways to use technology, especially The Geometer’s

Sketchpad®, and there is an enhanced online textbook and other computer-based

resources. If you are familiar with earlier editions of Discovering Geometry, you’ll

still find the original and hallmark features, plus improvements based on feedback

from many of your colleagues in geometry classrooms.

If you are a student, we believe that as you work through this course you’ll gain

knowledge for a lifetime. If you are a parent, we believe you’ll enjoy watching your

student develop mathematical power. If you are a teacher, we believe you’ll find

that Discovering Geometry makes a significant positive impact in your classroom.

Whether you are learning, guiding, or teaching, we wish you success and urge you

to continue your involvement and interest in mathematics.

Steve Rasmussen, President

Key Curriculum Press

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A Note to Students from the Author xiv

0.1 Geometry in Nature and in Art 2

0.2 Line Designs 7

0.3 Circle Designs 10

0.4 Op Art 13

0.5 Knot Designs 16

Project: Symbolic Art 19

0.6 Islamic Tile Designs 20

Project: Photo or Video Safari 23

Chapter 0 Review 24

Assessing What You’ve Learned 26

1.1 Building Blocks of Geometry 28

Investigation: Mathematical Models 32

Project: Spiral Designs 35

Using Your Algebra Skills 1: Midpoint 36

1.2 Poolroom Math 38

Investigation: Virtual Pool 41

1.3 What’s a Widget? 47

Investigation: Defining Angles 49

1.4 Polygons 54

Investigation: Special Polygons 56

1.5 Triangles 59

Investigation: Triangles 60

1.6 Special Quadrilaterals 64

Investigation: Special Quadrilaterals 64

Project: Drawing the Impossible 68

1.7 Circles 69

Investigation: Defining Circle Terms 70

1.8 Space Geometry 75

Investigation: Space Geometry 77

1.9 A Picture Is Worth a Thousand Words 81

Exploration: Geometric Probability I 88

Activity: Chances Are 88

Chapter 1 Review 90


Contents

CHAPTER

1Introducing Geometry 27

CHAPTER

0Geometric Art 1

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2.1 Inductive Reasoning 96

Investigation: Shape Shifters 98

2.2 Finding the nth Term 102

Investigation: Finding the Rule 102

Project: Best-Fit Lines 107

2.3 Mathematical Modeling 108

Investigation: Party Handshakes 108

2.4 Deductive Reasoning 114

Investigation: Overlapping Segments 116

Exploration: The Seven Bridges of Königsberg 120

Activity: Traveling Networks 120

2.5 Angle Relationships 122

Investigation 1: The Linear Pair Conjecture 122

Investigation 2: Vertical Angles Conjecture 123

2.6 Special Angles on Parallel Lines 128

Investigation 1: Which Angles Are Congruent? 128

Investigation 2: Is the Converse True? 130

Project: Line Designs 134

Using Your Algebra Skills 2: Slope 135

Exploration: Patterns in Fractals 137

Activity: The Sierpinski Triangle 138

Chapter 2 Review 140


3.1 Duplicating Segments and Angles 144

Investigation 1: Duplicating a Segment 145

Investigation 2: Duplicating an Angle 146

3.2 Constructing Perpendicular Bisectors 149

Investigation 1: Finding the Right Bisector 149

Investigation 2: Constructing the Perpendicular Bisector 150

3.3 Constructing Perpendiculars to a Line 154

Investigation 1: Finding the Right Line 154

Investigation 2: Patty-Paper Perpendiculars 155

Project: Constructing a Tile Design 158

3.4 Constructing Angle Bisectors 159

Investigation 1: Angle Bisecting by Folding 159

Investigation 2: Angle Bisecting with Compass 160

3.5 Constructing Parallel Lines 163

Investigation: Constructing Parallel Lines by Folding 163

Using Your Algebra Skills 3: Slopes of Parallel and

Perpendicular Lines 167

3.6 Construction Problems 170

Exploration: Perspective Drawing 174

Activity: Boxes in Space 175

CHAPTER

3Using Tools of Geometry 143

CHAPTER

2Reasoning in Geometry 95

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CHAPTER

4Discovering and Proving Triangle Properties 199

3.7 Constructing Points of Concurrency 178

Investigation 1: Concurrence 178

Investigation 2: Circumcenter 179

Investigation 3: Incenter 180

3.8 The Centroid 185

Investigation 1: Are Medians Concurrent? 185

Investigation 2: Balancing Act 186

Exploration: The Euler Line 191

Activity: Three Out of Four 191

Project: Is There More to the Orthocenter? 192


Mixed Review 196


4.1 Triangle Sum Conjecture 200

Investigation: The Triangle Sum 200

4.2 Properties of Isosceles Triangles 206

Investigation 1: Base Angles in an Isosceles Triangle 207

Investigation 2: Is the Converse True? 208

Using Your Algebra Skills 4: Solving Equations 212

4.3 Triangle Inequalities 215

Investigation 1: What Is the Shortest Path from A to B? 216

Investigation 2: Where Are the Largest and Smallest Angles? 217

Investigation 3: Exterior Angles of a Triangle 217

Project: Random Triangles 220

4.4 Are There Congruence Shortcuts? 221

Investigation 1: Is SSS a Congruence Shortcut? 222

Investigation 2: Is SAS a Congruence Shortcut? 223

Investigation 3: Is SSA a Congruence Shortcut? 223

4.5 Are There Other Congruence Shortcuts? 227

Investigation 1: Is ASA a Congruence Shortcut? 227

Investigation 2: Is SAA a Congruence Shortcut? 228

Investigation 3: Is AAA a Congruence Shortcut? 228

4.6 Corresponding Parts of Congruent Triangles 232

Project: Polya’s Problem 236

4.7 Flowchart Thinking 237

4.8 Proving Special Triangle Conjectures 243

Investigation: The Symmetry Line in an Isosceles Triangle 244

Exploration: Napoleon’s Theorem 249

Activity: Napoleon Triangles 249

Project: Lines and Isosceles Triangles 250


Take Another Look 255


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CHAPTER

6Discovering and Proving Circle Properties 309

CHAPTER

5Discovering and Proving Polygon Properties 257

5.1 Polygon Sum Conjecture 258

Investigation: Is There a Polygon Sum Formula? 258

5.2 Exterior Angles of a Polygon 262

Investigation: Is There an Exterior Angle Sum? 262

Exploration: Star Polygons 266

Activity: Exploring Star Polygons 266

5.3 Kite and Trapezoid Properties 268

Investigation 1: What Are Some Properties of Kites? 268

Investigation 2: What Are Some Properties of Trapezoids? 270

5.4 Properties of Midsegments 275

Investigation 1: Triangle Midsegment Properties 275

Investigation 2: Trapezoid Midsegment Properties 276

Project: Building an Arch 280

5.5 Properties of Parallelograms 281

Investigation: Four Parallelogram Properties 281

Project: Drawing Regular Polygons 286

Using Your Algebra Skills 5: Writing Linear Equations 287

5.6 Properties of Special Parallelograms 291

Investigation 1: What Can You Draw with the

Double-Edged Straightedge? 291

Investigation 2: Do Rhombus Diagonals Have

Special Properties? 292

Investigation 3: Do Rectangle Diagonals Have

Special Properties? 293

5.7 Proving Quadrilateral Properties 298

Investigation: Finding the Square Route 300

Project: Japanese Puzzle Quilts 303




6.1 Tangent Properties 310

Investigation 1: Going Off on a Tangent 311

Investigation 2: Tangent Segments 312

6.2 Chord Properties 317

Investigation 1: Defining Angles in a Circle 317

Investigation 2: Chords and Their Central Angles 318

Investigation 3: Chords and the Center of the Circle 319

Investigation 4: Perpendicular Bisector of a Chord 319

6.3 Arcs and Angles 324

Investigation 1: Inscribed Angle Properties 324

Investigation 2: Inscribed Angles Intercepting the Same Arc 325

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Investigation 3: Angles Inscribed in a Semicircle 325

Investigation 4: Cyclic Quadrilaterals 326

Investigation 5: Arcs by Parallel Lines 326

6.4 Proving Circle Conjectures 330

6.5 The Circumference/Diameter Ratio 335

Investigation: A Taste of Pi 336

Project: Needle Toss 340

6.6 Around the World 341

Using Your Algebra Skills 6: Solving Systems of Linear Equations 345

6.7 Arc Length 349

Investigation: Finding the Arcs 350

Project: Racetrack Geometry 354

Exploration: Intersecting Lines Through a Circle 355

Activity 1: Exploring Secants and Chords 355

Activity 2: Exploring Tangents 357


Mixed Review 362



7.1 Transformations and Symmetry 368

Investigation: The Basic Property of a Reflection 370

7.2 Properties of Isometries 376

Investigation 1: Transformations on a Coordinate Plane 377

Investigation 2: Finding a Minimal Path 377

7.3 Compositions of Transformations 383

Investigation 1: Reflections across Two Parallel Lines 384

Investigation 2: Reflections across Two Intersecting Lines 385

7.4 Tessellations with Regular Polygons 389

Investigation: The Semiregular Tessellations 390

7.5 Tessellations with Nonregular Polygons 394

Investigation 1: Do All Triangles Tessellate? 394

Investigation 2: Do All Quadrilaterals Tessellate? 395

Project: Penrose Tilings 398

7.6 Tessellations Using Only Translations 399

Project: Kaleidoscopes 402

7.7 Tessellations That Use Rotations 403

Exploration: Tessellating with the Conway Criterion 408

Activity: Conway Hexagons 408

7.8 Tessellations That Use Glide Reflections 410

Using Your Algebra Skills 7: Finding Points of Concurrency 413




CHAPTER

7Transformations and Tessellations 367

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8.1 Areas of Rectangles and Parallelograms 422

Investigation: Area Formula for Parallelograms 424

Project: Random Rectangles 428

8.2 Areas of Triangles, Trapezoids, and Kites 429

Investigation 1: Area Formula for Triangles 429

Investigation 2: Area Formula for Trapezoids 429

Investigation 3: Area Formula for Kites 430

Project: Maximizing Area 433

8.3 Area Problems 434

Investigation: Solving Problems with Area Formulas 434

Using Your Algebra Skills 8: Products, Factors, and

Quadratic Equations 438

8.4 Areas of Regular Polygons 442

Investigation: Area Formula for Regular Polygons 442

Exploration: Pick’s Formula for Area 446

Activity: Dinosaur Footprints and Other Shapes 447

8.5 Areas of Circles 449

Investigation: Area Formula for Circles 449

8.6 Any Way You Slice It 453

Exploration: Geometric Probability II 458

Activity: Where the Chips Fall 458

Project: Different Dice 460

8.7 Surface Area 461

Investigation 1: Surface Area of a Regular Pyramid 464

Investigation 2: Surface Area of a Cone 465

Exploration: Alternative Area Formulas 469

Activity: Calculating Area in Ancient Egypt 469




9.1 The Theorem of Pythagoras 478

Investigation: The Three Sides of a Right Triangle 478

Project: Creating a Geometry Flip Book 483

9.2 The Converse of the Pythagorean Theorem 484

Investigation: Is the Converse True? 484

Using Your Algebra Skills 9: Radical Expressions 489

9.3 Two Special Right Triangles 491

Investigation 1: Isosceles Right Triangles 491

Investigation 2: 30°-60°-90° Triangles 492

Exploration: A Pythagorean Fractal 496

Activity: The Right Triangle Fractal 497

9.4 Story Problems 498

CHAPTER

9The Pythagorean Theorem 477

CHAPTER

8Area 421

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9.5 Distance in Coordinate Geometry 502

Investigation: The Distance Formula 502

Exploration: Ladder Climb 507

Activity: Climbing the Wall 507

9.6 Circles and the Pythagorean Theorem 508


Mixed Review 515



10.1 The Geometry of Solids 520

Exploration: Euler’s Formula for Polyhedrons 528

Activity: Toothpick Polyhedrons 528

10.2 Volume of Prisms and Cylinders 530

Investigation: The Volume Formula for Prisms and Cylinders 531

Project: The Soma Cube 537

10.3 Volume of Pyramids and Cones 538

Investigation: The Volume Formula for Pyramids and Cones 538

Project: The World’s Largest Pyramid 543

Exploration: The Five Platonic Solids 544

Activity: Modeling the Platonic Solids 544

10.4 Volume Problems 547

10.5 Displacement and Density 551

Project: Maximizing Volume 554

Exploration: Orthographic Drawing 555

Activity: Isometric and Orthographic Drawings 557

10.6 Volume of a Sphere 558

Investigation: The Formula for the Volume of a Sphere 558

10.7 Surface Area of a Sphere 562

Investigation: The Formula for the Surface Area of a Sphere 562

Using Your Algebra Skills 10: Solving for Any Variable 567

Exploration: Sherlock Holmes and Forms of Valid Reasoning 569

Activity: It’s Elementary! 570




CHAPTER

10Volume 519

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Using Your Algebra Skills 11: Proportion and Reasoning 578

11.1 Similar Polygons 581

Investigation 1: What Makes Polygons Similar? 582

Investigation 2: Dilations on the Coordinate Plane 585

Project: Making a Mural 588

11.2 Similar Triangles 589

Investigation 1: Is AA a Similarity Shortcut? 589

Investigation 2: Is SSS a Similarity Shortcut? 590

Investigation 3: Is SAS a Similarity Shortcut? 591

Exploration: Constructing a Dilation Design 595

Activity: Dilation Creations 595

11.3 Indirect Measurement with Similar Triangles 598

Investigation: Mirror, Mirror 598

11.4 Corresponding Parts of Similar Triangles 603

Investigation 1: Corresponding Parts 603

Investigation 2: Opposite Side Ratios 604

11.5 Proportions with Area 608

Investigation 1: Area Ratios 608

Investigation 2: Surface Area Ratios 609

Project: In Search of the Perfect Rectangle 613

11.6 Proportions with Volume 614

Investigation: Volume Ratios 615

Exploration: Why Elephants Have Big Ears 620

Activity: Convenient Sizes 620

11.7 Proportional Segments Between Parallel Lines 623

Investigation 1: Parallels and Proportionality 624

Investigation 2: Extended Parallel/Proportionality 626

Exploration: Two More Forms of Valid Reasoning 631

Activity: Symbolic Proofs 633




12.1 Trigonometric Ratios 640

Investigation: Trigonometric Tables 642

12.2 Problem Solving with Right Triangles 647

Project: Light for All Seasons 651

Exploration: Indirect Measurement 652

Activity: Using a Clinometer 652

12.3 The Law of Sines 654

Investigation 1: Area of a Triangle 654

Investigation 2: The Law of Sines 655

CHAPTER

12Trigonometry 639

CHAPTER

11Similarity 577

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12.4 The Law of Cosines 661

Project: Japanese Temple Tablets 666

12.5 Problem Solving with Trigonometry 667

Exploration: Trigonometric Ratios and the Unit Circle 671

Activity: The Unit Circle 671

Project: Trigonometric Functions 674

Using Your Algebra Skills 12: Transforming Functions 675

Exploration: Three Types of Proofs 679

Activity: Prove It! 681


Mixed Review 686



13.1 The Premises of Geometry 692

13.2 Planning a Geometry Proof 703

13.3 Triangle Proofs 710

13.4 Quadrilateral Proofs 716

Developing Proof: Proving Parallelogram Conjectures 716

Exploration: Proof as Challenge and Discovery 720

Activity: Exploring Properties of Special Constructions 720

13.5 Indirect Proof 722

Developing Proof: Proving the Tangent Conjecture 723

13.6 Circle Proofs 727

13.7 Similarity Proofs 730

Developing Proof: Proving the SSS Similarity Conjecture 732

Using Your Algebra Skills 13: Coordinate Proof 736

Project: Special Proofs of Special Conjectures 741

Exploration: Non-Euclidean Geometries 742

Activity: Spherical Geometry 743



Hints for Selected Exercises 749

Answers for Chapter Reviews 769

Glossary 787

Table of Symbols 812

Index 813

Photo Credits 832

CHAPTER

13Geometry as a Mathematical System 691

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A Note to Students from the Author

Michael Serra

What Makes Discovering Geometry Different?

Discovering Geometry is designed so that you can be actively engaged as you

learn geometry. In this book you “learn by doing.” You will learn to use the tools

of geometry and to perform geometry investigations with them. Many of the

investigations are carried out in small cooperative groups in which you jointly

plan and find solutions with other students. Your investigations will lead you to the

discovery of geometry properties. In addition, you will gradually learn about proof,

a form of reasoning that will help you explain why your discoveries are true,

through Developing Proof group activities and exercises.

Discovering Geometry is designed so that both you and your teacher can have fun

while you learn geometry. It has a lot of “extras.” Each lesson begins with a quote

that I hope you will find funny or thought provoking. I think you’ll enjoy the

extra challenges in the Improving Your…Skills puzzles at the end of most lessons.

To solve each puzzle, you’ll need clever visual thinking skills or sharp reasoning

skills or both. I hope you will find some of the illustrated word problems

humorous. In the explorations you will build geometric solids, find the height of

your school building, and discover why elephants have big ears. In the projects you

will draw the impossible, make kaleidoscopes, design a racetrack, and create a

mural. The online Dynamic Geometry Explorations help you visualize important

geometry concepts by putting them in motion, and there are graphing calculator

projects, Fathom Dynamic Data™ projects, The Geometer’s Sketchpad

explorations, and web links that will allow you to practice and improve

your skills.

Suggestions for Success

It is important to be organized. Keep a notebook with a section for definitions,

a section for your geometry investigations, a section for discoveries, and a section

for daily notes and exercises. Develop the habit of writing a summary page when

you have completed each chapter. Study your notebook regularly.

You will need four tools for the investigations: a compass, a protractor, a

straightedge, and a ruler. Some investigations use patty paper, small squares of

waxed paper usually used between burger patties, that can be used as a unique

geometry tool. Keep a calculator handy, too.

You will find hints for some exercises in the back of the book. Those exercises are

marked with an . Try to solve the problems on your own first. Refer to the hints

as a last resort if you can’t solve a problem. Solutions are provided for chapter

reviews so you can check your understanding and prepare for tests.

Discovering Geometry will ask you to work cooperatively with your classmates.

When you are working cooperatively, always be willing to listen to each other, to

actively participate, to ask each other questions, and to help each other when asked.

You can accomplish much more cooperatively than you can individually. And, best

of all, you’ll experience less frustration and have much more fun.

Michael Serra

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michael serra - high school math | prek...

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