Using & 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 & 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.

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