Mathematical Modeling
Applications with GeoGebra

A logical problem-based introduction to the use of GeoGebra for mathematical modeling and problem solving within various areas of mathematics

A well-organized guide to mathematical modeling techniques for evaluating and solving problems in the diverse field of mathematics, Mathematical Modeling: Applications with GeoGebra presents a unique approach to software applications in GeoGebra and WolframAlpha. The software is well suited for modeling problems in numerous areas of mathematics including algebra, symbolic algebra, dynamic geometry, three-dimensional geometry, and statistics. Featuring detailed information on how GeoGebra can be used as a guide to mathematical modeling, the book provides comprehensive modeling examples that correspond to different levels of mathematical experience, from simple linear relations to differential equations.

Each chapter builds on the previous chapter with practical examples in order to illustrate the mathematical modeling skills necessary for problem solving. Addressing methods for evaluating models including relative error, correlation, square sum of errors, regression, and confidence interval, Mathematical Modeling: Applications with GeoGebra also includes:

• Over 400 diagrams and 300 GeoGebra examples with practical approaches to mathematical modeling that help the reader develop a full understanding of the content
• Numerous real-world exercises with solutions to help readers learn mathematical modeling techniques
• A companion website with GeoGebra constructions and screencasts

Mathematical Modeling: Applications with GeoGebrais ideal for upper-undergraduate and graduate-level courses in mathematical modeling, applied mathematics, modeling and simulation, operations research, and optimization. The book is also an excellent reference for undergraduate and high school instructors in mathematics.

Preface xi

Introduction xiii

About the Companion Website xxx

1 Some Introductory Problems 1

1.1 Ticket Prices, 3

1.2 How Long Will the Pasture Last in a Field?, 7

1.3 A Bit of Chemistry, 10

1.4 Sydney Harbor Bridge, 16

1.5 Perspective, 19

1.6 Lake Erie’s Area, 21

1.7 Zebra Crossing, 25

1.8 The Security Case, 31

1.9 Personal Measurements, 34

1.10 Height of the Body, 34

1.11 Lamp Pole, 35

1.12 The Skyscraper, 35

1.13 The Fence, 35

1.14 The Corridor, 35

1.15 Bird Feeders, 35

1.16 Golf, 36

2 Linear Models 37

2.1 Are Women Faster Than Men?, 38

2.2 Taxi Companies, 40

2.3 Crime Development, 47

2.4 The Metal Wire, 52

2.5 Options Trading, 57

2.6 Flying Foxes, 62

2.7 Knots on a Rope, 66

2.8 The Candle, 66

2.9 Hooke’s Law, 66

2.10 Ranking, 67

2.11 Dolbear’s Law, 67

2.12 Man at Office, 68

2.13 A Stack of Paper, 68

2.14 Milk Production in Cows, 69

3 Nonlinear Empirical Models I 70

3.1 Galaxy Rotation, 71

3.2 Olympic Pole Vaulting, 73

3.3 Kepler’s Third Law, 79

3.4 Density, 83

3.5 Yeast, 87

3.6 Cooling I, 88

3.7 Modeling the Population of Ireland, 93

3.8 The Rule of 72, 96

3.9 The Fish Farm I, 100

3.10 New Orleans Temperatures, 104

3.11 The Record Mile, 107

3.12 The Rocket, 107

3.13 Stopping Distances, 107

3.14 A Bottle with Holes, 108

3.15 The Pendulum, 108

3.16 Radio Range, 108

3.17 Running 400 Meters, 108

3.18 Blue Whale, 109

3.19 Used Cars, 109

3.20 Texts, 110

4 Nonlinear Empirical Models II 111

4.1 Cooling II, 112

4.2 Body Surface Area, 116

4.3 Warm]Blooded Animals, 120

4.4 Control of Insect Pests, 123

4.5 Selling Magazines for Christmas, 125

4.6 Tumor, 136

4.7 Free Fall, 141

4.8 Concentration, 145

4.9 Air Current, 150

4.10 Tides, 153

4.11 Fitness, 156

4.12 Life Expectancy versus Average Income, 157

4.13 Stockholm Center, 157

4.14 Workforce, 157

4.15 Population of Sweden, 158

4.16 Who Killed the Lion?, 158

4.17 AIDS in United States, 159

4.18 Thermal Comfort, 159

4.19 Watts and Lumen, 159

4.20 The Beaufort Scale, 160

4.21 The von Bertalanffy Growth Equation, 161

5 Modeling with Calculus 162

5.1 The Fish Farm II, 163

5.2 Titration, 169

5.3 The Bowl, 176

5.4 The Aircraft Wing, 180

5.5 The Gateway Arch in St. Louis, 182

5.6 Volume of a Pear, 187

5.7 Storm Flood, 190

5.8 Exercise, 193

5.9 Bicycle Reflectors, 202

5.10 Cardiac Output, 206

5.11 Medication, 210

5.12 New Song on Spotify, 215

5.13 Temperature Change, 221

5.14 Tar, 224

5.15 Bicycle Reflectors Revisited, 229

5.16 Gas Pressure, 229

5.17 Airborne Attacks, 229

5.18 Railroad Tracks, 230

5.19 Cobb–Douglas Production Functions, 230

5.20 Future Carbon Dioxide Emissions, 231

5.21 Overtaking, 232

5.22 Population Dynamics of India, 232

5.23 Drag Racing, 232

5.24 Super Eggs, 233

5.25 Measuring Sticks, 234

5.26 The Lecture Hall, 234

5.27 Progressive Braking Distances, 234

5.28 Cylinder in a Cone, 235

6 Using Differential Equations 236

6.1 Cooling III, 237

6.2 Moose Hunting, 241

6.3 The Water Container, 247

6.4 Skydiving, 250

6.5 Flu Epidemics, 256

6.6 USA’s Population, 263

6.7 Predators and Prey, 274

6.8 Smoke, 285

6.9 Alcohol Consumption, 289

6.10 Who Killed the Mathematics Teacher, 292

6.11 River Clams, 297

6.12 Contamination, 297

6.13 Damped Oscillation, 297

6.14 The Potassium–Argon Method, 298

6.15 Barium, Lanthanum, and Cerium, 298

6.16 Iodine, 298

6.17 Endemic Epidemics, 299

6.18 War, 299

6.19 Farmers, Bandits, and Rulers, 299

6.20 Epidemics Without Immunity, 300

6.21 Zombie Apocalypse I, 300

6.22 Zombie Apocalypse II, 300

7 Geometrical Models 301

7.1 The Looping Pen, 302

7.2 Comparing Areas, 304

7.3 Crossing Lines, 307

7.4 Points in a Triangle, 310

7.5 Trisected Area, 316

7.6 Spirograph, 320

7.7 Connected LP Players, 326

7.8 Folding Paper, 332

7.9 The Locomotive, 336

7.10 Maximum Volume, 340

7.11 Pascal’s Snail or Limaçon, 340

7.12 Equilateral Triangle Dissection, 341

7.13 Dividing the Sides of a Triangle, 341

7.14 The Pedal Triangle, 342

7.15 The Infinity Diagram, 343

7.16 Dissecting a Circular Segment, 344

7.17 Neuberg Cubic Art, 344

7.18 Phase Plots for Triangles, 345

7.19 The Joukowski Airfoil, 347

8 Discrete Models 348

8.1 The Cabinetmaker, 349

8.2 Weather, 358

8.3 Squirrels, 362

8.4 Chlorine, 365

8.5 The Deer Farm, 369

8.6 Analyzing a Number Sequence, 373

8.7 Inner Areas in a Square, 376

8.8 Inner Areas in a Triangle, 382

8.9 A Climate Model Based on Albedo, 387

8.10 Traffic Jam, 392

8.11 Wildfire, 399

8.12 A Modern Carpenter, 408

8.13 Conway’s Game of Life, 409

8.14 Matrix Taxis, 409

8.15 The Car Park, 409

8.16 Selecting a Collage, 410

8.17 Apportionment, 410

8.18 Steiner Trees for Regular Polygons, 410

8.19 Hugs and High Fives, 411

8.20 Pythagorean Triples, 411

8.21 Credits, 412

8.22 The Piano, 413

9 Modeling in the Classroom 415

9.1 The Teacher Creating Diagrams, 416

9.2 Student’s Lab Reports, 416

9.3 Making Screencast Instructions, 417

9.4 Demonstrations, 417

9.5 Students Investigating Constructions, 418

9.6 Working in Groups, 418

9.7 Students Constructing Models, 419

9.8 Broader Assignments, 420

9.9 The Same or Different Assignments, 421

9.10 Previous Assignments, 421

9.11 The Consultancy Bureau, 422

10 Assessing Modeling 425

10.1 To Evaluate Mathematical Modeling Assignments, 426

10.2 Concretizing Grading Criteria, 426

10.3 Evaluating Students’ Work, 431

11 Assessing Models 434

11.1 Relative Error, 435

11.2 Correlation, 435

11.3 Sum of Squared Errors, 436

11.4 Simple Linear Regression, 436

11.5 Multiple Regression Analysis, 438

11.6 Nonlinear Regression, 438

11.7 Confidence Intervals, 439

11.8 2D Confidence Interval Tools, 441

12 Interpreting Models 443

12.1 Mathematical Representations, 443

12.2 Graphical Representations, 444

12.3 A Sample Model Interpreted, 445

12.4 Creating the Model, 446

Appendix A: Introduction to GeoGebra 448

Appendix B: Function Library 485

Integer Properties 509

Index 523

Index of Problems by Name 535

Jonas Hall is Head of Mathematics at Rodengymnasiet in Norrtälje, Sweden, where he teaches mathematics and physics. His research interests include problem solving, the aesthetics of mathematics, and teaching with technology. He is a multiple finalist in Kappa, which is a competition for mathematics teachers in Sweden offered by the University of Stockholm.

Thomas Lingefjärd, PhD, is Associate Professor of Mathematics Education in the Department of Education at the University of Gothenburg. The author of more than 25 articles and 10 chapter contributions, Dr. Lingefjärd's research interests include mathematical modeling and advanced mathematical thinking.