Using and Understanding Mathematics

Jeffrey Bennett specializes in mathematics and science education. He has taught at every level from pre-school through graduate school, including more than 50 college courses in mathematics, physics, astronomy, and education. His work on Using and Understanding Mathematics began in 1987, when he helped create a new mathematics course for the University of Colorado's core curriculum. Variations on this course, with its quantitative reasoning approach, are now taught at hundreds of colleges nationwide. In addition to his work in mathematics, Dr. Bennett (whose PhD is in astrophysics) has written leading college-level textbooks in astronomy, statistics, and the new science of astrobiology, as well as books for the general public. He also proposed and developed both the Colorado Scale Model Solar System on the University of Colorado at Boulder campus and the Voyage Scale Model Solar System, a permanent, outdoor exhibit on the National Mall in Washington, DC. He has recently begun writing science books for children, including the award-winning Max Goes to the Moon and Max Goes to Mars. When not working, he enjoys swimming as well as hiking the trails of Boulder, Colorado with his family. William L. Briggs has been on the mathematics faculty at the University of Colorado at Denver for 22 years. He teaches numerous courses within the undergraduate and graduate curriculum, and has special interest in teaching calculus, differential equations, and mathematical modeling. He developed the quantitative reasoning course for liberal arts students at University of Colorado at Denver supported by his textbook Using and Understanding Mathematics. He has written two other tutorial monographs, The Multigrid Tutorial and The DFT: An Owner's Manual for the Discrete Fourier Transform, as well as Ants, Bikes, Clocks, a mathematical problem-solving text for undergraduates. He is a University of Colorado President's Teaching Scholar, an Outstanding Teacher awardee of the Rocky Mountain Section of the MAA, and the recipient of a Fulbright Fellowship to Ireland. Bill lives with his wife, Julie, and their Gordon setter, Seamus, in Boulder, Colorado. He loves to bake bread, run trails, and rock climb in the mountains near his home.

Preface

Acknowledgments

Prologue: Literacy for the Modern World



PART ONE: LOGIC AND PROBLEM SOLVING



1. Thinking Critically

1A Living in the Media Age

1B Propositions and Truth Values

1C Sets and Venn Diagrams

1D Analyzing Arguments

1E Critical Thinking in Everyday Life



2. Approaches to Problem Solving

2A Working with Units

2B Problem-Solving with Units

2C Problem-Solving Guidelines and Hints



PART TWO: QUANTITATIVE INFORMATION IN EVERYDAY LIFE



3. Numbers in the Real World

3A Uses and Abuses of Percentages

3B Putting Numbers in Perspective

3C Dealing with Uncertainty

3D Index Numbers: The CPI and Beyond

3E How Numbers Can Deceive: Polygraphs, Mammograms, and More



4. Managing Money

4A Taking Control of Your Finances

4B The Power of Compounding

4C Savings Plans and Investments

4D Loan Payments, Credit Cards, and Mortgages

4E Income Taxes

4F Understanding the Federal Budget



PART THREE: PROBABILITY AND STATISTICS



5. Statistical Reasoning

5A Fundamentals of Statistics

5B Should You Believe a Statistical Study?

5C Statistical Tables and Graphs

5D Graphics in the Media

5E Correlation and Causality



6. Putting Statistics to Work

6A Characterizing Data

6B Measures of Variation

6C The Normal Distribution

6D Statistical Inference



7. Probability: Living with the Odds

7A Fundamentals of Probability

7B Combining Probabilities

7C The Law of Large Numbers

7D Assessing Risk

7E Counting and Probability



PART FOUR: MODELING



8. Exponential Astonishment

8A Growth: Linear versus Exponential

8B Doubling Time and Half-Life

8C Real Population Growth

8D Logarithmic Scales: Earthquakes, Sounds, and Acids



9. Modeling Our World

9A Functions: The Building Blocks of Mathematical Models

9B Linear Modeling

9C Exponential Modeling



10. Modeling with Geometry

10A Fundamentals of Geometry

10B Problem Solving with Geometry

10C Fractal Geometry



PART FIVE: FURTHER APPLICATIONS



11. Mathematics and the Arts

11A Mathematics and Music

11B Perspective and Symmetry

11C Proportion and the Golden Ratio



12. Mathematics and Politics

12A Voting: Does the Majority Always Rule?

12B Theory of Voting

12C Apportionment: The House of Representatives and Beyond

12D Dividing the Political Pie



Credits

Answers

Index