Mathematical Ideas

1. The Art of Problem Solving.


Solving Problems by Inductive Reasoning.



An Application of Inductive Reasoning: Number Patterns.



Strategies for Problem Solving.



Calculating, Estimating, and Reading Graphs.



Extension: Writing to Learn about Mathematics.



Collaborative Investigation: Discovering Mathematics in Pascal's Triangle.



2. The Basic Concepts of Set Theory.


Symbols and Terminology.



Venn Diagrams and Subsets.



Set Operations and Cartesian Products.



Cardinal Numbers and Surveys.



Infinite Sets and Their Cardinalities.



Collaborative Investigation: A Survey of Your Class.



3. Introduction to Logic.


Statements and Quantifiers.



Truth Tables and Equivalent Statements.



The Conditional and Circuits.



More on the Conditional.



Analyzing Arguments with Euler Diagrams.



Extension: Logic Puzzles.



Analyzing Arguments with Truth Tables.



Collaborative Investigation: Logic Puzzles Revisited.



4. Numeration and Mathematical Systems.


Historical Numeration Systems.



Arithmetic in the Hindu-Arabic System.



Converting Between Number Bases.



Other Finite Mathematical Systems.



Groups.



Collaborative Investigation: A Perpetual Calendar Algorithm.



5. Number Theory.


Prime and Composite Numbers.



Selected Topics from Number Theory.



Greatest Common Factor and Least Common Multiple.



Modular Systems.



The Fibonacci Sequence and the Golden Ratio.



Extension: Magic Squares.



Collaborative Investigation: Investigating an Interesting Property of Number Squares.



6. The Real Number System.


Real Numbers, Order, and Absolute Value.



Operations, Properties, and Applications of Real Numbers.



Rational Numbers and Decimal Representation.



Irrational Numbers and Decimal Representation.



Applications of Decimals and Percents.



Extension: Complex Numbers.



Collaborative Investigation: Budgeting to Buy a Car.



7. The Basic Concepts of Algebra.


Linear Equations.



Applications of Linear Equations.



Ratio, Proportion, and Variation.



Linear Inequalities.



Properties of Exponents and Scientific Notation.



Polynomials and Factoring.



Quadratic Equations and Applications.



Collaborative Investigation: Calculating the Magic Number in Sports.



8. Graphs, Functions, and Systems of Equations and Inequalities.


The Rectangular Coordinate System and Circles.



Lines and Their Slopes.



Equations of Lines.



An Introduction to Functions: Linear Functions and Applications.



Quadratic Functions and Their Tables.



Exponential and Logarithmic Functions and Applications.



Systems of Equations and Applications.



Extension: Using Matrix Row Operations to Solve Systems.



Linear Inequalities and Systems of Inequalities.



Collaborative Investigation: Olympic Track and Field Results.



9. Geometry.


Points, Lines, Planes, and Angles.



Curves, Polygons, and Circles.



Perimeter, Area, and Circumference.



The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem.



Extension: Right Angle Trigonometry.



Space Figures, Volume, and Surface Area.



Non-Euclidean Geometry, Topology, and Networks.



Chaos and Fractal Geometry.



Collaborative Investigation: Generalizing the Angle Sum Concept.



10. Counting Methods.


Counting by Systematic Listing.



Using the Fundamental Counting Principle.



Using Permutations and Combinations.



Using Pascal's Triangle and the Binomial Theorem.



Counting Problems Involving "Not" and "Or".



Collaborative Investigation: Approximating Factorials Using Stirling's Formula.



11. Probability.


Basic Concepts.



Events Involving "Not" and "Or".



Events Involving "And".



Binomial Probability.



Expected Value.



Estimating Probabilities by Simulation.



Collaborative Investigation: Finding Empirical Values of pi.



12. Statistics.


Frequency Distributions and Graphs.



Measures of Central Tendency.



Measures of Dispersion.



Measures of Position.



The Normal Distribution.



Extension: How to Lie with Statistics.



Regression and Correlation.



Collaborative Investigation: Combining Sets of Data.



13. Consumer Mathematics.


Interest and Inflation.



Extension: Annuities.



Consumer Credit.



Truth in Lending.



Buying a Home.



Investing in the Stock Market.



Collaborative Investigation: To Buy or to Rent?



Appendix: The Metric System.