101+ Great Ideas for Introducing Key Concepts in Mathematics

Alfred S. Posamentier is professor of mathematics education and dean of the School of Education at the City College of the City University of New York. He has authored and co-authored several resource books in mathematics education for Corwin Press. Herbert A. Hauptman is a world-renowned mathematician who pioneered and developed a mathematical method that has changed the whole field of chemistry. For this work he was recipient of the 1985 Nobel Price in chemistry. With this book Dr. Hauptman brings his highly sophisticated knowledge of mathematics and his many years of exploration in higher mathematics to the advantage of secondary school audience.

Praise for the First Edition:   "Written to appeal to all mathematics teachers. Teachers who are struggling with introducing these topical ideas will find the book is written in such a way as to facilitate their understanding of the topics. The language is easy to understand and the book is very user friendly. In addition, those teachers who have a sound grasp of these key concepts can find fresh ideas for teaching old concepts presented in a manner that is intellectual in design.”

• PrefaceAcknowledgmentsAbout the AuthorsIntroductory IdeaComing to Terms With Mathematical TermsAlgebra Ideas1. Introducing the Product of Two Negatives2. Multiplying Polynomials by Monomials (Introducing Algebra Tiles)3. Multiplying Binomials (Using Algebra Tiles)4. Factoring Trinomials (Using Algebra Tiles)5. Multiplying Binomials (Geometrically)6. Factoring Trinomials (Geometrically)7. Trinomial Factoring8. How Algebra Can Be Helpful9. Automatic Factoring of a Trinomial10. Reasoning Through Algebra11. Pattern Recognition Cautions12. Caution With Patterns13. Using a Parabola as a Calculator14. Introducing Literal Equations: Simple Algebra to Investigate an Arithmetic Phenomenon15. Introducing Nonpositive Integer Exponents16. Importance of Definitions in Mathematics (Algebra)17. Introduction to Functions18. When Algebra Explains Arithmetic19. Sum of an Arithmetic Progression20. Averaging Rates21. Using Triangular Numbers to Generate Interesting Relationships22. Introducing the Solution of Quadratic Equations Through Factoring23. Rationalizing the Denominator24. Paper Folding to Generate a Parabola25. Paper Folding to Generate an Ellipse26. Paper Folding to Generate a Hyperbola27. Using Concentric Circles to Generate a Parabola28. Using Concentric Circles to Generate an Ellipse29. Using Concentric Circles to Generate a Hyperbola30. Summing a Series of Powers31. Sum of Limits32. Linear Equations With Two Variables33. Introducing Compound Interest Using the "Rule of 72”34. Generating Pythagorean Triples35. Finding Sums of Finite Series Geometry IdeasGeometry Ideas1. Sum of the Measures of the Angles of a Triangle2. Introducing the Sum of the Measures of the Interior Angles of a Polygon3. Sum of the Measures of the Exterior Angles of a Polygon: I4. Sum of the Measures of the Exterior Angles of a Polygon: II5. Triangle Inequality6. Don’t Necessarily Trust Your Geometric Intuition7. Importance of Definitions in Mathematics (Geometry)8. Proving Quadrilaterals to Be Parallelograms9. Demonstrating the Need to Consider All Information Given10. Midlines of a Triangle11. Length of the Median of a Trapezoid12. Pythagorean Theorem13. Simple Proofs of the Pythagorean Theorem14. Angle Measurement With a Circle by Moving the Circle15. Angle Measurement With a Circle16. Introducing and Motivating the Measure of an Angle Formed by Two Chords17. Using the Property of the Opposite Angles of an Inscribed Quadrilateral18. Introducing the Concept of Slope19. Introducing Concurrency Through Paper Folding20. Introducing the Centroid of a Triangle21. Introducing the Centroid of a Triangle Via a Property22. Introducing Regular Polygons23. Introducing Pi24. The Lunes and the Triangle25. The Area of a Circle26. Comparing Areas of Similar Polygons27. Relating Circles28. Invariants in Geometry29. Dynamic Geometry to Find an Optimum Situation30. Construction-Restricted Circles31. Avoiding Mistakes in Geometric Proofs32. Systematic Order in Successive Geometric Moves: Patterns!33. Introducing the Construction of a Regular Pentagon34. Euclidean Constructions and the Parabola35. Euclidean Constructions and the Ellipse36. Euclidean Constructions and the Hyperbola37. Constructing Tangents to a Parabola From an External Point P38. Constructing Tangents to an Ellipse39. Constructing Tangents to a HyperbolaTrigonometry Ideas1. Derivation of the Law of Sines: I2. Derivation of the Law of Sines: II3. Derivation of the Law of Sines: III4. A Simple Derivation for the Sine of the Sum of Two Angles5. Introductory Excursion to Enable an Alternate Approach to Trigonometry Relationships6. Using Ptolemy’s Theorem to Develop Trigonometric Identities for Sums and Differences of Angles7. Introducing the Law of Cosines: I (Using Ptolemy’s Theorem)8. Introducing the Law of Cosines: II9. Introducing the Law of Cosines: III10. Alternate Approach to Introducing Trigonometric Identities11. Converting to Sines and Cosines12. Using the Double Angle Formula for the Sine Function13. Making the Angle Sum Function Meaningful14. Responding to the Angle-Trisection QuestionProbability and Statistics Ideas1. Introduction of a Sample Space2. Using Sample Spaces to Solve Tricky Probability Problems3. Introducing Probability Through Counting (or Probability as Relative Frequency)4. In Probability You Cannot Always Rely on Your Intuition5. When “Averages” Are Not Averages: Introducing Weighted Averages6. The Monty Hall Problem: “Let’s Make a Deal”7. Conditional Probability in Geometry8. Introducing the Pascal Triangle9. Comparing Means Algebraically10. Comparing Means Geometrically11. Gambling Can Be DeceptiveOther Topics Ideas1. Asking the Right Questions2. Making Arithmetic Means Meaningful3. Using Place Value to Strengthen Reasoning Ability4. Prime Numbers5. Introducing the Concept of Relativity6. Introduction to Number Theory7. Extracting a Square Root8. Introducing Indirect Proof9. Keeping Differentiation Meaningful10. Irrationality of the Square Root of an Integer That Is Not a Perfect Square11. Introduction to the Factorial Function x!12. Introduction to the Function x to the (n) Power13. Introduction to the Two Binomial Theorems14. Factorial Function Revisited15. Extension of the Factorial Function r! to the Case Where r Is Rational16. Prime Numbers Revisited17. Perfect Numbers