Mathematical Modeling

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.

• Preface xiIntroduction xiiiAbout the Companion Website xxx1 Some Introductory Problems 11.1 Ticket Prices, 31.2 How Long Will the Pasture Last in a Field?, 71.3 A Bit of Chemistry, 101.4 Sydney Harbor Bridge, 161.5 Perspective, 191.6 Lake Erie’s Area, 211.7 Zebra Crossing, 251.8 The Security Case, 311.9 Personal Measurements, 341.10 Height of the Body, 341.11 Lamp Pole, 351.12 The Skyscraper, 351.13 The Fence, 351.14 The Corridor, 351.15 Bird Feeders, 351.16 Golf, 362 Linear Models 372.1 Are Women Faster Than Men?, 382.2 Taxi Companies, 402.3 Crime Development, 472.4 The Metal Wire, 522.5 Options Trading, 572.6 Flying Foxes, 622.7 Knots on a Rope, 662.8 The Candle, 662.9 Hooke’s Law, 662.10 Ranking, 672.11 Dolbear’s Law, 672.12 Man at Office, 682.13 A Stack of Paper, 682.14 Milk Production in Cows, 693 Nonlinear Empirical Models I 703.1 Galaxy Rotation, 713.2 Olympic Pole Vaulting, 733.3 Kepler’s Third Law, 793.4 Density, 833.5 Yeast, 873.6 Cooling I, 883.7 Modeling the Population of Ireland, 933.8 The Rule of 72, 963.9 The Fish Farm I, 1003.10 New Orleans Temperatures, 1043.11 The Record Mile, 1073.12 The Rocket, 1073.13 Stopping Distances, 1073.14 A Bottle with Holes, 1083.15 The Pendulum, 1083.16 Radio Range, 1083.17 Running 400 Meters, 1083.18 Blue Whale, 1093.19 Used Cars, 1093.20 Texts, 1104 Nonlinear Empirical Models II 1114.1 Cooling II, 1124.2 Body Surface Area, 1164.3 Warm]Blooded Animals, 1204.4 Control of Insect Pests, 1234.5 Selling Magazines for Christmas, 1254.6 Tumor, 1364.7 Free Fall, 1414.8 Concentration, 1454.9 Air Current, 1504.10 Tides, 1534.11 Fitness, 1564.12 Life Expectancy versus Average Income, 1574.13 Stockholm Center, 1574.14 Workforce, 1574.15 Population of Sweden, 1584.16 Who Killed the Lion?, 1584.17 AIDS in United States, 1594.18 Thermal Comfort, 1594.19 Watts and Lumen, 1594.20 The Beaufort Scale, 1604.21 The von Bertalanffy Growth Equation, 1615 Modeling with Calculus 1625.1 The Fish Farm II, 1635.2 Titration, 1695.3 The Bowl, 1765.4 The Aircraft Wing, 1805.5 The Gateway Arch in St. Louis, 1825.6 Volume of a Pear, 1875.7 Storm Flood, 1905.8 Exercise, 1935.9 Bicycle Reflectors, 2025.10 Cardiac Output, 2065.11 Medication, 2105.12 New Song on Spotify, 2155.13 Temperature Change, 2215.14 Tar, 2245.15 Bicycle Reflectors Revisited, 2295.16 Gas Pressure, 2295.17 Airborne Attacks, 2295.18 Railroad Tracks, 2305.19 Cobb–Douglas Production Functions, 2305.20 Future Carbon Dioxide Emissions, 2315.21 Overtaking, 2325.22 Population Dynamics of India, 2325.23 Drag Racing, 2325.24 Super Eggs, 2335.25 Measuring Sticks, 2345.26 The Lecture Hall, 2345.27 Progressive Braking Distances, 2345.28 Cylinder in a Cone, 2356 Using Differential Equations 2366.1 Cooling III, 2376.2 Moose Hunting, 2416.3 The Water Container, 2476.4 Skydiving, 2506.5 Flu Epidemics, 2566.6 USA’s Population, 2636.7 Predators and Prey, 2746.8 Smoke, 2856.9 Alcohol Consumption, 2896.10 Who Killed the Mathematics Teacher, 2926.11 River Clams, 2976.12 Contamination, 2976.13 Damped Oscillation, 2976.14 The Potassium–Argon Method, 2986.15 Barium, Lanthanum, and Cerium, 2986.16 Iodine, 2986.17 Endemic Epidemics, 2996.18 War, 2996.19 Farmers, Bandits, and Rulers, 2996.20 Epidemics Without Immunity, 3006.21 Zombie Apocalypse I, 3006.22 Zombie Apocalypse II, 3007 Geometrical Models 3017.1 The Looping Pen, 3027.2 Comparing Areas, 3047.3 Crossing Lines, 3077.4 Points in a Triangle, 3107.5 Trisected Area, 3167.6 Spirograph, 3207.7 Connected LP Players, 3267.8 Folding Paper, 3327.9 The Locomotive, 3367.10 Maximum Volume, 3407.11 Pascal’s Snail or Limaçon, 3407.12 Equilateral Triangle Dissection, 3417.13 Dividing the Sides of a Triangle, 3417.14 The Pedal Triangle, 3427.15 The Infinity Diagram, 3437.16 Dissecting a Circular Segment, 3447.17 Neuberg Cubic Art, 3447.18 Phase Plots for Triangles, 3457.19 The Joukowski Airfoil, 3478 Discrete Models 3488.1 The Cabinetmaker, 3498.2 Weather, 3588.3 Squirrels, 3628.4 Chlorine, 3658.5 The Deer Farm, 3698.6 Analyzing a Number Sequence, 3738.7 Inner Areas in a Square, 3768.8 Inner Areas in a Triangle, 3828.9 A Climate Model Based on Albedo, 3878.10 Traffic Jam, 3928.11 Wildfire, 3998.12 A Modern Carpenter, 4088.13 Conway’s Game of Life, 4098.14 Matrix Taxis, 4098.15 The Car Park, 4098.16 Selecting a Collage, 4108.17 Apportionment, 4108.18 Steiner Trees for Regular Polygons, 4108.19 Hugs and High Fives, 4118.20 Pythagorean Triples, 4118.21 Credits, 4128.22 The Piano, 4139 Modeling in the Classroom 4159.1 The Teacher Creating Diagrams, 4169.2 Student’s Lab Reports, 4169.3 Making Screencast Instructions, 4179.4 Demonstrations, 4179.5 Students Investigating Constructions, 4189.6 Working in Groups, 4189.7 Students Constructing Models, 4199.8 Broader Assignments, 4209.9 The Same or Different Assignments, 4219.10 Previous Assignments, 4219.11 The Consultancy Bureau, 42210 Assessing Modeling 42510.1 To Evaluate Mathematical Modeling Assignments, 42610.2 Concretizing Grading Criteria, 42610.3 Evaluating Students’ Work, 43111 Assessing Models 43411.1 Relative Error, 43511.2 Correlation, 43511.3 Sum of Squared Errors, 43611.4 Simple Linear Regression, 43611.5 Multiple Regression Analysis, 43811.6 Nonlinear Regression, 43811.7 Confidence Intervals, 43911.8 2D Confidence Interval Tools, 44112 Interpreting Models 44312.1 Mathematical Representations, 44312.2 Graphical Representations, 44412.3 A Sample Model Interpreted, 44512.4 Creating the Model, 446Appendix A: Introduction to GeoGebra 448Appendix B: Function Library 485Integer Properties 509Index 523Index of Problems by Name 535