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

by Alfred S. Posamentier

• ISBN13: 9781412927062
• ISBN10: 1412927064
• Edition: 2nd
• Format: Paperback
• Copyright: 2006-05-12
• Publisher: Corwin Pr

Summary

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 actual equations 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.

Table of Contents

Preface	ix
About the Authors	xi
Introductory Idea	1	(2)
Coming to Terms With Mathematical Terms	3	(4)
Algebra Ideas	7	(72)
Introducing the Product of Two Negatives	9	(2)
Multiplying Polynomials by Monomials (Introducing Algebra Tiles)	11	(2)
Multiplying Binomials (Using Algebra Tiles)	13	(3)
Factoring Trinomials (Using Algebra Tiles)	16	(4)
Multiplying Binomials (Geometrically)	20	(2)
Factoring Trinomials (Geometrically)	22	(2)
Trinomial Factoring	24	(1)
How Algebra Can Be Helpful	25	(1)
Automatic Factoring of a Trinomial	26	(2)
Reasoning Through Algebra	28	(1)
Pattern Recognition Cautions	29	(1)
Caution With Patterns	30	(2)
Using a Parabola as a Calculator	32	(3)
Introducing Literal Equations: Simple Algebra to Investigate an Arithmetic Phenomenon	35	(4)
Introducing Nonpositive Integer Exponents	39	(2)
Importance of Definitions in Mathematics (Algebra)	41	(3)
Introduction to Functions	44	(1)
When Algebra Explains Arithmetic	45	(1)
Sum of an Arithmetic Progression	46	(2)
Averaging Rates	48	(2)
Using Triangular Numbers to Generate Interesting Relationships	50	(2)
Introducing the Solution of Quadratic Equations Through Factoring	52	(1)
Rationalizing the Denominator	53	(1)
Paper Folding to Generate a Parabola	54	(1)
Paper Folding to Generate an Ellipse	55	(2)
Paper Folding to Generate a Hyperbola	57	(2)
Using Concentric Circles to Generate a Parabola	59	(1)
Using Concentric Circles to Generate an Ellipse	60	(1)
Using Concentric Circles to Generate a Hyperbola	61	(1)
Summing a Series of Powers	62	(3)
Sum of Limits	65	(2)
Linear Equations With Two Variables	67	(3)
Introducing Compound Interest Using the ``Rule of 72''	70	(2)
Generating Pythagorean Triples	72	(2)
Finding Sums of Finite Series Geometry Ideas	74	(5)
Geometry Ideas	79	(108)
Sum of the Measures of the Angles of a Triangle	81	(1)
Introducing the Sum of the Measures of the Interior Angles of a Polygon	82	(2)
Sum of the Measures of the Exterior Angles of a Polygon: I	84	(2)
Sum of the Measures of the Exterior Angles of a Polygon: II	86	(2)
Triangle Inequality	88	(1)
Don't Necessarily Trust Your Geometric Intuition	89	(4)
Importance of Definitions in Mathematics (Geometry)	93	(4)
Proving Quadrilaterals to Be Parallelograms	97	(1)
Demonstrating the Need to Consider All Information Given	98	(2)
Midlines of a Triangle	100	(2)
Length of the Median of a Trapezoid	102	(2)
Pythagorean Theorem	104	(3)
Simple Proofs of the Pythagorean Theorem	107	(3)
Angle Measurement With a Circle by Moving the Circle	110	(4)
Angle Measurement With a Circle	114	(2)
Introducing and Motivating the Measure of an Angle Formed by Two Chords	116	(2)
Using the Property of the Opposite Angles of an Inscribed Quadrilateral	118	(1)
Introducing the Concept of Slope	119	(3)
Introducing Concurrency Through Paper Folding	122	(2)
Introducing the Centroid of a Triangle	124	(1)
Introducing the Centroid of a Triangle Via a Property	125	(2)
Introducing Regular Polygons	127	(2)
Introducing π	129	(3)
The Lunes and the Triangle	132	(3)
The Area of a Circle	135	(1)
Comparing Areas of Similar Polygons	136	(2)
Relating Circles	138	(1)
Invariants in Geometry	139	(3)
Dynamic Geometry to Find an Optimum Situation	142	(2)
Construction-Restricted Circles	144	(2)
Avoiding Mistakes in Geometric Proofs	146	(4)
Systematic Order in Successive Geometric Moves: Patterns!	150	(3)
Introducing the Construction of a Regular Pentagon	153	(5)
Euclidean Constructions and the Parabola	158	(7)
Euclidean Constructions and the Ellipse	165	(5)
Euclidean Constructions and the Hyperbola	170	(7)
Constructing Tangents to a Parabola From an External Point P	177	(3)
Constructing Tangents to an Ellipse	180	(2)
Constructing Tangents to a Hyperbola	182	(5)
Trigonometry Ideas	187	(30)
Derivation of the Law of Sines: I	189	(1)
Derivation of the Law of Sines: II	190	(1)
Derivation of the Law of Sines: III	191	(1)
A Simple Derivation for the Sine of the Sum of Two Angles	192	(2)
Introductory Excursion to Enable an Alternate Approach to Trigonometry Relationships	194	(3)
Using Ptolemy's Theorem to Develop Trigonometric Identities for Sums and Differences of Angles	197	(5)
Introducing the Law of Cosines: I (Using Ptolemy's Theorem)	202	(2)
Introducing the Law of Cosines: II	204	(2)
Introducing the Law of Cosines: III	206	(1)
Alternate Approach to Introducing Trigonometric Identities	207	(3)
Converting to Sines and Cosines	210	(1)
Using the Double Angle Formula for the Sine Function	211	(1)
Making the Angle Sum Function Meaningful	212	(2)
Responding to the Angle-Trisection Question	214	(3)
Probability and Statistics Ideas	217	(20)
Introduction of a Sample Space	219	(1)
Using Sample Spaces to Solve Tricky Probability Problems	220	(2)
Introducing Probability Through Counting (or Probability as Relative Frequency)	222	(1)
In Probability You Cannot Always Rely on Your Intuition	223	(2)
When ``Averages'' Are Not Averages: Introducing Weighted Averages	225	(1)
The Monty Hall Problem: ``Let's Make a Deal''	226	(3)
Conditional Probability in Geometry	229	(1)
Introducing the Pascal Triangle	230	(3)
Comparing Means Algebraically	233	(1)
Comparing Means Geometrically	234	(2)
Gambling Can Be Deceptive	236	(1)
Other Topics Ideas	237	(41)
Asking the Right Questions	239	(1)
Making Arithmetic Means Meaningful	240	(1)
Using Place Value to Strengthen Reasoning Ability	241	(1)
Prime Numbers	242	(3)
Introducing the Concept of Relativity	245	(1)
Introduction to Number Theory	246	(1)
Extracting a Square Root	247	(1)
Introducing Indirect Proof	248	(1)
Keeping Differentiation Meaningful	248	(2)
Irrationality of √m	250	(2)
Introduction to the Factorial Function x!	252	(2)
Introduction to the Function x(n)	254	(3)
Introduction to the Two Binomial Theorems	257	(2)
Factorial Function Revisited	259	(3)
Extension of the Factorial Function r! to the Case Where r Is Rational	262	(14)
Prime Numbers Revisited	276	(2)
Perfect Numbers	278