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 & 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 

I. LOGIC AND PROBLEM SOLVING

1. Thinking Critically

Activity: Bursting Bubble 

1A. Living in the Media Age 

In Your World: Fact Checking on the Web

1B. Propositions and Truth Values

1C. Sets and Venn Diagrams

Brief Review: Sets of Numbers

1D. Analyzing Arguments

Mathematical Insight: Deductive Proof of the Pythagorean Theorem

1E. Critical Thinking in Everyday Life

In Your World: Beware of “Up to” Deals

2. Approaches to Problem Solving

Activity: Global Melting

2A. Understand, Solve, and Explain

Brief Review: Common Fractions 

Brief Review: Decimal Fractions 

Using Technology: Currency Exchange Rates

In Your World: Changing Money in Foreign Countries

2B. Extending Unit Analysis

In Your World: Gems and Gold Jewelry 

Brief Review: Powers of 10 

Using Technology: Metric Conversions 

In Your World: Save Money and Save the Earth

2C. Problem-Solving Hints

Mathematical Insight: Zeno’s Paradox

II. QUANTITATIVE INFORMATION IN EVERYDAY LIFE

3. Numbers in the Real World

Activity: Big Numbers

3A. Uses and Abuses of Percentages

Brief Review: Percentages 

Brief Review: What Is a Ratio?

3B. Putting Numbers in Perspective 

Brief Review: Working with Scientific Notation 

Using Technology: Scientific Notation

3C. Dealing with Uncertainty 

Brief Review: Rounding 

Using Technology: Rounding in Excel

3D. Index Numbers: The CPI and Beyond 

Using Technology: The Inflation Calculator 

In Your World: The Chained CPI and the Federal Budget

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

4. Managing Money

Activity: Student Loans

4A. Taking Control of Your Finances

4B. The Power of Compounding

Brief Review: Powers and Roots 

Using Technology: Powers 

Using Technology: The Compound Interest Formula 

Using Technology: The Compound Interest Formula for Interest Paid More Than Once a Year 

Using Technology: APY in Excel 

Using Technology: Powers of e 

Brief Review: Four Basic Rules of Algebra 

In Your World: Effects of Low Interest Rates 

4C. Savings Plans and Investments

Mathematical Insight: Derivation of the Savings Plan Formula 

Using Technology: The Savings Plan Formula 

Using Technology: Fractional Powers (Roots) 

In Your World: Building a Portfolio

4D. Loan Payments, Credit Cards, and Mortgages

Using Technology: The Loan Payment Formula (Installment Loans) 

Mathematical Insight: Derivation of the Loan Payment Formula 

Using Technology: Principal and Interest Portions of Loan Payments 

In Your World: Avoiding Credit Card Trouble 

In Your World: Choosing or Refinancing a Loan

4E. Income Taxes

4F. Understanding the Federal Budget

III. PROBABILITY AND STATISTICS

5. Statistical Reasoning

Activity: Cell Phones and Driving

5A. Fundamentals of Statistics

Using Technology: Random Numbers

5B. Should You Believe a Statistical Study?

In Your World: The Gun Debate: Defensive Gun Use

5C. Statistical Tables and Graphs

Using Technology: Frequency Tables in Excel 

Using Technology: Bar Graphs and Pie Charts in Excel 

Using Technology: Line Charts in Excel

5D. Graphics in the Media

Using Technology: Graphs with Multiple Data Sets

5E. Correlation and Causality

Using Technology: Scatterplots in Excel

6. Putting Statistics to Work

Activity: Are We Smarter Than Our Parents?

6A. Characterizing Data

Using Technology: Mean, Median, and Mode in Excel

6B. Measures of Variation

Using Technology: Standard Deviation in Excel

6C. The Normal Distribution

Using Technology: Standard Scores and Percentiles in Excel

6D. Statistical Inference

In Your World: Is Polling Reliable?

7. Probability: Living With The Odds

Activity: Lotteries

7A. Fundamentals of Probability

Brief Review: The Multiplication Principle

7B. Combining Probabilities

7C. The Law of Large Numbers

7D. Assessing Risk

In Your World: Terrorism, Risk, and Human Psychology

7E. Counting and Probability

Using Technology: Factorials 

Brief Review: Factorials 

Using Technology: Permutations 

Using Technology: Combinations

IV. MODELING

8. Exponential Astonishment

Activity: Towers of Hanoi

8A. Growth: Linear vs. Exponential

8B. Doubling Time and Half-Life

Using Technology: Logarithms 

Brief Review: Logarithms

8C. Real Population Growth

In Your World: Choosing Our Fate

8D. Logarithmic Scales: Earthquakes, Sounds, and Acids

In Your World: Ocean Acidification

9. Modeling Our World

Activity: Climate Modeling

9A. Functions: The Building Blocks of Mathematical Models

Brief Review: The Coordinate Plane

9B. Linear Modeling

Using Technology: Graphing Functions 

In Your World: Algebra’s Baghdad Connection

9C. Exponential Modeling

Brief Review: Algebra with Logarithms 

Mathematical Insight: Doubling Time and Half-Life Formulas 

In Your World: Changing Rates of Change 

10. Modeling With Geometry

Activity: Eyes in the Sky

10A. Fundamentals of Geometry

Mathematical Insight: Archimedes and Pi 

In Your World: Plato, Geometry, and Atlantis

10B. Problem Solving with Geometry

10C. Fractal Geometry

V. FURTHER APPLICATIONS

11. Mathematics and The Arts

Activity: Digital Music Files

11A. Mathematics and Music

In Your World: Music Just for You

11B. Perspective and Symmetry

11C. Proportion and the Golden Ratio

12. Mathematics and Politics

Activity: Partisan Redistricting

12A. Voting: Does the Majority Always Rule?

In Your World: Counting Votes–Not as Easy as It Sounds

12B. Theory of Voting

In Your World: The Electoral College and the Presidency

12C. Apportionment: The House of Representatives and Beyond

12D. Dividing the Political Pie

Credits 

Answers to Quick Quizzes and Odd-Numbered Exercises 

Index

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