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9780130449412

Mathematics for High School Teachers- An Advanced Perspective

by ; ; ;
  • ISBN13:

    9780130449412

  • ISBN10:

    0130449415

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2002-11-25
  • Publisher: Pearson

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Supplemental Materials

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Summary

This book gives readers a comprehensive look at the most important concepts in the mathematics taught in grades 9-12. Real numbers, functions, congruence, similarity, area and volume, trigonometry and more. For high school mathematics teachers, mathematics supervisors, mathematics coordinators, mathematicians, and users of the University of Chicago School Mathematics Project materials for grades 7-12 who want a comprehensive reference book to use throughout their careers or anyone who wants a better understanding of mathematics.

Table of Contents

Preface ix
What is Meant By ``An Advanced Perspective''?
1(16)
An Example of Concept Analysis: Parallelism
1(4)
An Example of Problem Analysis: Matching an Average
5(5)
An Example of Mathematical Connections: +and.
10(7)
PART I ALGEBRA AND ANALYSIS WITH CONNECTIONS TO GEOMETRY
Real Numbers and Complex Numbers
17(50)
The Real Numbers
18(29)
Rational numbers and irrational numbers
20(5)
The number line and decimal representation of real numbers
25(9)
Periods of periodic decimals
34(6)
The distributions of various types of real numbers
40(7)
The Complex Numbers
47(20)
The complex numbers and the complex plane
48(6)
The geometry of complex number arithmetic
54(9)
Chapter Projects
63(3)
Bibliography
66(1)
Functions
67(68)
The Definitions. Historical Evolution, and Basic Machinery of Functions
68(18)
What is a function?
68(8)
Problem analysis: from equations to functions
76(4)
Some types of functions
80(6)
Properties of Real Functions
86(27)
Analyzing real functions
87(8)
Composition and inverse functions
95(6)
Monotone real functions
101(4)
Limit behavior of real functions
105(8)
Problems Involving Real Functions
113(22)
Fitting linear and exponential function to data
113(8)
Fitting polynomial functions to data
121(4)
An extended analysis of the box problem
125(5)
Chapter Projects
130(3)
Bibliography
133(2)
Equations
135(44)
The Concept of Equation
136(10)
Equality, equivalence, and isomorphism
136(4)
Solving equations
140(6)
Algebraic Structures and Solving Equations
146(14)
Solving equations of the form a * x = b
146(3)
Solving equations of the form ax + b = cx + d
149(4)
Quadratic and other polynomial equations
153(7)
The Solving Process
160(19)
Genealized addition and multiplication properties of equality
160(4)
Applying the same function to both sides of an equation
164(3)
Solving inequalities
167(6)
Extended analysis: averages of speeds
173(5)
Chapter Projects
178(1)
Bibliography
178(1)
Integers and Polynomials
179(66)
Natural Numbers, Induction, and Recursion
180(24)
Recursion and proof by mathematical induction
180(6)
Mathematical induction
186(7)
More applications of mathematical induction
193(4)
An extended analysis of an induction situation
197(7)
Divisibility Properties of the Integers
204(25)
The Division Algorithm
204(4)
Divisibility of integers
208(4)
Solving linear Diophantine equations
212(6)
The Fundamental Theorem of Arithmetic
218(6)
Base representation of positive integers
224(5)
Divisibility Properties of Polynomials
229(16)
The Division Algorithm for polynomials
229(7)
The Euclidean Algorithm and prime factorization for polynomials
236(5)
Chapter Projects
241(2)
Bibliography
243(2)
Numbers System Structures
245(30)
The Systems of Modular Arithmetic
246(15)
Integer congruence
246(6)
Applications of integer congruence to calendars and cryptology
252(4)
The Chinese Remainder Theorem
256(5)
Number Fields
261(14)
Ordered fields
262(5)
Archimedean and complete ordered fields
267(4)
The structure of the complex number system
271(2)
Chapter Projects
273(1)
Bibliography
274(1)
PART II GEOMETRY WITH CONNECTIONS TO ALGEBRA AND ANALYSIS
Congruence
275(86)
Euclid and Congruence
275(27)
Euclid's Elements
275(8)
Deduction and proof
283(7)
General properties of definitions
290(5)
Definitions of congruence from Euclid to modern times
295(7)
The Congruence Transformations
302(31)
Translations
302(4)
Rotations
306(7)
Reflections
313(8)
Glide reflections
321(3)
Are there other congruence transformations?
324(4)
Congruent graphs
328(5)
Symmetry
333(11)
Reflection symmetry
333(6)
Other congruence transformation symmetries
339(5)
Traditional Congruence Revisited
344(17)
Sufficient conditions for congruence
344(7)
Concept analysis: analyzing a geometric figure
351(2)
General theorems about congruence
353(4)
Chapter Projects
357(1)
Bibliography
358(3)
Distance and Similarity
361(70)
Distance
362(21)
What is distance?
362(6)
Minimum distance problems
368(7)
Extended analysis: locus problems
375(4)
Distance on the surface of a sphere
379(4)
Similar Figures
383(31)
When are two figures similar?
384(7)
Similarity of graphs
391(5)
Similar polygons
396(4)
Similar acrs
400(4)
When many theorems become one
404(4)
Types of similarity transformations
408(6)
Distances within Figures
414(17)
Geometric means
415(4)
Similarity and parallel lines
419(9)
Chapter Projects
428(1)
Bibliography
429(2)
Trigonometry
431(46)
Angle Measure and the Trigonometric Ratios
432(12)
Angle measure and arc length
432(2)
The trigonometric ratios
434(6)
Extended analysis: indirect measurement problems
440(4)
The Trigonometric Functions and Their Connections
444(15)
The trigonometric functions
444(6)
Modeling with trigonometric functions
450(6)
The historical and conceptual evolution of trigonometry
456(3)
Properties of the Sine and Cosine Functions
459(18)
Algebraic properties of the trigonometric functions
460(3)
Geometric properties of the sine and cosine functions
463(6)
Analytical properties of the sine and cosine functions
469(5)
Chapter Projects
474(2)
Bibliography
476(1)
Area and Volume
477(70)
Area
478(33)
What is area?
478(8)
Area formulas for triangles
486(6)
Extended analysis: the line through a given point minimizing area
492(8)
From polygons to regions bounded by curves
500(6)
The problem of quadrature
506(2)
Area as representing probability
508(3)
Volume
511(17)
What is volume?
512(3)
From cubes to polyhedra
515(8)
From polyhedra to spheres
523(5)
Relationships among Area, Volume, and Dimension
528(19)
Surface area
528(4)
The Isoperimetric Inequalities
532(4)
The Fundamental Theorem of Similarity
536(5)
Fractional dimension
541(3)
Chapter Projects
544(2)
Bibliography
546(1)
Axiomatics and Euclidean Geometry
547(36)
Constructing Euclidean Geometry
548(23)
Axioms for incidence
548(4)
Axioms for betweenness
552(4)
Congruence and the basic figures
556(8)
Geometry without the Parallel Postulate
564(4)
Euclid's Fifth Postulate
568(3)
The Cartesian Model for Euclidean Geometry
571(12)
The Cartesian coordinate system
571(5)
Verifying the definition of Euclidean geometry: the relationship between a mathemetical theory and its models
576(4)
Chapter Projects
580(2)
Bibliography
582(1)
Index 583

Supplemental Materials

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The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

Excerpts

Mathematics for High School Teachers-An Advanced Perspectiveis intended as a text for mathematics courses for prospective or experienced secondary school mathematics teachers and all others who wish to examine high school mathematics from a higher point of view. Preliminary versions of the book have been used in a variety of ways, ranging from junior and senior (capstone) or graduate mathematics courses for pre-service secondary mathematics education majors to graduate professional development courses for teachers. Some courses included both undergraduate and graduate students and practicing teachers with good success. There is enough material in this book for at least a full year (two semesters) of study under normal conditions, even if only about half of the problems are assigned. With a few exceptions, the chapters are relatively independent and an instructor may choose from them. However, some chapters contain more sophisticated content than others. Here are four possible sequences for a full semester''s work: Algebra emphasis: Chapters 1-6 Geometry emphasis: Chapters 1, 7-11 Introductory emphasis: Chapters 1, 3, 4, 7, 8,10 More advanced emphasis: Chapters 1, 2, 5, 6, 9,11. In each sequence we suggest beginning with Chapter 1 so that students are aware of the features of this book and of some of the differences between it and other mathematics texts they may have used. More information and suggestions in this regard can be found in the Instructor''s Notes. Additional instructional resources are also at the web site http://www.prenhall.com/usiskin. The presentation assumes the student has had at least one year of calculus and a post-calculus mathematics course (such as real analysis, linear algebra, or abstract algebra) in which proofs were required and algebraic structures were discussed. The term "from an advanced standpoint" is taken to mean that the text examines high school mathematical ideas from a perspective appropriate for college mathematics majors, and makes use of the kind of mathematical knowledge and sophistication the student is gaining or has gained in other courses. Two basic characteristics of Mathematics for High School Teachers-An Advanced Perspective,taken together, distinguish courses taught from this book from many current courses. First, the material is rooted in the core mathematical content and problems of high school mathematics courses before calculus. Specifically, the development emanates from the major concepts found in high school mathematics: numbers, algebra, geometry, and functions. Second, the concepts and problems are treated from a mathematically advanced standpoint, and differ considerably from materials designed for high school students. The authors feel that the mathematical content in this book lies in an area of mathematics that is of great benefit to all those interested in mathematics at the secondary school level, but is rarely seen by them. Specifically, we have endeavored to include: analyses of alternate definitions, language, and approaches to mathematical ideas extensions and generalizations of familiar theorems discussions of the historical contexts in which concepts arose and have changed over time applications of the mathematics in a wide range of settings analyses of common problems of high school mathematics from a deeper mathematical level demonstrations of alternate ways of approaching problems, including ways with and without calculator and computer technology connections between ideas that may have been studied separately in different courses relationships of ideas studied in school to ideas students may encounter in later study. There are many reasons why we believe a teacher or other person interested in high school mathematics should have this knowledge. Mere are a few. Knowing alternate approaches helps in making decisions regarding curriculum, selection of materials, and lesson plans. Being able to connect, extend, and relate mathematical ideas to each other and to the mathematics a student may take later helps in designing courses and responding to student questions. Having a sense of history and the stories behind the mathematics can make lessons more interesting and engaging for both teacher and student. Encountering the richness of the mathematics that is studied at the high school level helps us to understand why some students are turned on by that mathematics, while others have difficulty with it.

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