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101+  Great Ideas for Introducing Key Concepts in Mathematics
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101+ Great Ideas for Introducing Key Concepts in Mathematics
A Resource for Secondary School Teachers

Second Edition


May 2006 | 296 pages | Corwin

Multiply math mastery and interest with these inspired teaching tactics!

Invigorate instruction and engage students with this treasure trove of "Great Ideas" compiled by two of the greatest minds in mathematics. From commonly taught topics in algebra, geometry, trigonometry, and statistics, to more advanced explorations into indirect proofs, binomial theorem, irrationality, relativity, and more, this guide outlines concepts and techniques that will inspire veteran and new educators alike.

This updated second edition offers more proven practices for bringing math concepts to life in the classroom, including:

  • 114 innovative strategies organized by subject area
  • User-friendly content identifying "objective," "materials," and "procedure" for each technique
  • A range of teaching models, including hands-on and computer-based methods
  • Specific and straightforward examples with step-by-step lessons

Written by two distinguished leaders in the field-mathematician, author, professor, university dean, and popular commentator Alfred S. Posamentier, along with mathematical pioneer and Nobel Prize recipient Herbert A. Hauptman-this guide brings a refreshing perspective to secondary math instruction to spark renewed interest and success among students and teachers.


 
Preface
 
Acknowledgments
 
About the Authors
 
Introductory Idea
 
Coming to Terms With Mathematical Terms
 
Algebra Ideas
 
1. Introducing the Product of Two Negatives
 
2. 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 Factoring
 
8. How Algebra Can Be Helpful
 
9. Automatic Factoring of a Trinomial
 
10. Reasoning Through Algebra
 
11. Pattern Recognition Cautions
 
12. Caution With Patterns
 
13. Using a Parabola as a Calculator
 
14. Introducing Literal Equations: Simple Algebra to Investigate an Arithmetic Phenomenon
 
15. Introducing Nonpositive Integer Exponents
 
16. Importance of Definitions in Mathematics (Algebra)
 
17. Introduction to Functions
 
18. When Algebra Explains Arithmetic
 
19. Sum of an Arithmetic Progression
 
20. Averaging Rates
 
21. Using Triangular Numbers to Generate Interesting Relationships
 
22. Introducing the Solution of Quadratic Equations Through Factoring
 
23. Rationalizing the Denominator
 
24. Paper Folding to Generate a Parabola
 
25. Paper Folding to Generate an Ellipse
 
26. Paper Folding to Generate a Hyperbola
 
27. Using Concentric Circles to Generate a Parabola
 
28. Using Concentric Circles to Generate an Ellipse
 
29. Using Concentric Circles to Generate a Hyperbola
 
30. Summing a Series of Powers
 
31. Sum of Limits
 
32. Linear Equations With Two Variables
 
33. Introducing Compound Interest Using the "Rule of 72”
 
34. Generating Pythagorean Triples
 
35. Finding Sums of Finite Series Geometry Ideas
 
Geometry Ideas
 
1. Sum of the Measures of the Angles of a Triangle
 
2. Introducing the Sum of the Measures of the Interior Angles of a Polygon
 
3. Sum of the Measures of the Exterior Angles of a Polygon: I
 
4. Sum of the Measures of the Exterior Angles of a Polygon: II
 
5. Triangle Inequality
 
6. Don’t Necessarily Trust Your Geometric Intuition
 
7. Importance of Definitions in Mathematics (Geometry)
 
8. Proving Quadrilaterals to Be Parallelograms
 
9. Demonstrating the Need to Consider All Information Given
 
10. Midlines of a Triangle
 
11. Length of the Median of a Trapezoid
 
12. Pythagorean Theorem
 
13. Simple Proofs of the Pythagorean Theorem
 
14. Angle Measurement With a Circle by Moving the Circle
 
15. Angle Measurement With a Circle
 
16. Introducing and Motivating the Measure of an Angle Formed by Two Chords
 
17. Using the Property of the Opposite Angles of an Inscribed Quadrilateral
 
18. Introducing the Concept of Slope
 
19. Introducing Concurrency Through Paper Folding
 
20. Introducing the Centroid of a Triangle
 
21. Introducing the Centroid of a Triangle Via a Property
 
22. Introducing Regular Polygons
 
23. Introducing Pi
 
24. The Lunes and the Triangle
 
25. The Area of a Circle
 
26. Comparing Areas of Similar Polygons
 
27. Relating Circles
 
28. Invariants in Geometry
 
29. Dynamic Geometry to Find an Optimum Situation
 
30. Construction-Restricted Circles
 
31. Avoiding Mistakes in Geometric Proofs
 
32. Systematic Order in Successive Geometric Moves: Patterns!
 
33. Introducing the Construction of a Regular Pentagon
 
34. Euclidean Constructions and the Parabola
 
35. Euclidean Constructions and the Ellipse
 
36. Euclidean Constructions and the Hyperbola
 
37. Constructing Tangents to a Parabola From an External Point P
 
38. Constructing Tangents to an Ellipse
 
39. Constructing Tangents to a Hyperbola
 
Trigonometry Ideas
 
1. Derivation of the Law of Sines: I
 
2. Derivation of the Law of Sines: II
 
3. Derivation of the Law of Sines: III
 
4. A Simple Derivation for the Sine of the Sum of Two Angles
 
5. Introductory Excursion to Enable an Alternate Approach to Trigonometry Relationships
 
6. Using Ptolemy’s Theorem to Develop Trigonometric Identities for Sums and Differences of Angles
 
7. Introducing the Law of Cosines: I (Using Ptolemy’s Theorem)
 
8. Introducing the Law of Cosines: II
 
9. Introducing the Law of Cosines: III
 
10. Alternate Approach to Introducing Trigonometric Identities
 
11. Converting to Sines and Cosines
 
12. Using the Double Angle Formula for the Sine Function
 
13. Making the Angle Sum Function Meaningful
 
14. Responding to the Angle-Trisection Question
 
Probability and Statistics Ideas
 
1. Introduction of a Sample Space
 
2. Using Sample Spaces to Solve Tricky Probability Problems
 
3. Introducing Probability Through Counting (or Probability as Relative Frequency)
 
4. In Probability You Cannot Always Rely on Your Intuition
 
5. When “Averages” Are Not Averages: Introducing Weighted Averages
 
6. The Monty Hall Problem: “Let’s Make a Deal”
 
7. Conditional Probability in Geometry
 
8. Introducing the Pascal Triangle
 
9. Comparing Means Algebraically
 
10. Comparing Means Geometrically
 
11. Gambling Can Be Deceptive
 
Other Topics Ideas
 
1. Asking the Right Questions
 
2. Making Arithmetic Means Meaningful
 
3. Using Place Value to Strengthen Reasoning Ability
 
4. Prime Numbers
 
5. Introducing the Concept of Relativity
 
6. Introduction to Number Theory
 
7. Extracting a Square Root
 
8. Introducing Indirect Proof
 
9. Keeping Differentiation Meaningful
 
10. Irrationality of the Square Root of an Integer That Is Not a Perfect Square
 
11. Introduction to the Factorial Function x!
 
12. Introduction to the Function x to the (n) Power
 
13. Introduction to the Two Binomial Theorems
 
14. Factorial Function Revisited
 
15. Extension of the Factorial Function r! to the Case Where r Is Rational
 
16. Prime Numbers Revisited
 
17. Perfect Numbers

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.”    

Journal of School Improvement, Volume 3, Issue 2, Fall 2002

"A 'must' for any who wish for more proven classroom practices. From geometry to algebra, teachers will find it packed with ideas."

California Bookwatch, September 2006
Key features
  • 114 Great Ideas for teaching secondary math curricula.
  • Covers algebra, geometry, trigonometry, probability and statistics.
  • Includes topics for advanced learners, including indirect proofs, the binomial theorem, irrationality, relativity, and beyond.
  • By two unique authors: Mathematician, author, professor, university dean, and popular commentator Alfred S. Posamentier and mathematical pioneer Herbert A. Hauptman, recipient of the 1985 Nobel Prize in chemistry.

Sample Materials & Chapters

Preface

Algebra Idea 1


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