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Math151AFirstSections(1)

Math151AFirstSections(1) - Mathematics of the Secondary...

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Mathematics of the Secondary School Curriculum, I (Math 151) H. Wu January 18, 2012 c Hung-Hsi Wu, 2011

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Contents Chapters Not in This Book v General Introduction vii Suggestions on How to Read This Book ix Prerequisites xi Some Conventions xiii 1 Fractions 1 1.1 Definition of a fraction . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Equivalent fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3 Adding and subtracting fractions . . . . . . . . . . . . . . . . . . . . 34 1.4 Multiplying fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.5 Dividing fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1.6 Complex fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1.7 Percent, ratio, and rate problems . . . . . . . . . . . . . . . . . . . . 72 1.8 Appendix: The basic laws . . . . . . . . . . . . . . . . . . . . . . . . 85 2 Rational Numbers 89 2.1 The rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.2 Adding rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.3 The vectorial representation of addition . . . . . . . . . . . . . . . . . 102 2.4 Multiplying rational numbers . . . . . . . . . . . . . . . . . . . . . . 108 2.5 Dividing rational numbers . . . . . . . . . . . . . . . . . . . . . . . . 117 iii

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iv CONTENTS 2.6 Comparing rational numbers . . . . . . . . . . . . . . . . . . . . . . . 126 3 The Euclidean Algorithm 141 3.1 The reduced form of a fraction . . . . . . . . . . . . . . . . . . . . . . 141 3.2 The Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . 155 4 Basic Isometries and Congruence 163 4.1 The basic vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2 Transformations of the plane . . . . . . . . . . . . . . . . . . . . . . . 190 4.3 The basic isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.4 Congruence, SAS, and ASA . . . . . . . . . . . . . . . . . . . . . . . 224 4.5 A brief pedagogical discussion . . . . . . . . . . . . . . . . . . . . . . 234 5 Dilation and Similarity 237 5.1 The Fundamental Theorem of Similarity . . . . . . . . . . . . . . . . 238 5.2 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 5.3 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 5.4 Appendix: The symmetry of similarity . . . . . . . . . . . . . . . . . 281 6 Symbolic Notation and Linear Equations 283 6.1 Symbolic expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6.2 Solving linear equations in one variable . . . . . . . . . . . . . . . . . 311 6.3 The graphs of linear equations in two variables . . . . . . . . . . . . . 318 6.4 Parallelism and perpendicularity . . . . . . . . . . . . . . . . . . . . . 341 6.5 Simultaneous linear equations . . . . . . . . . . . . . . . . . . . . . . 349 Bibliography 369
Chapters Not in This Book Mathematics of the Secondary School Curriculum, II and III Chapter 7: Linear Functions Chapter 8: Quadratic Functions and Equations Chapter 9: Polynomial and Rational Functions Chapter 10: Exponential and Logarithmic Functions Chapter 11: Polynomial Forms and Complex Numbers Chapter 12: Basic Theorems of Plane Geometry Chapter 13: Ruler and Compass Constructions Chapter 14: Axiomatic Systems Chapter 15: Trigonometry Chapter 16: The Concept of Limit Chapter 17: The Decimal Expansion of a Number Chapter 18: Length and Area Chapter 19: 3-Dimensional Geometry and Volume Chapter 20: Derivative and Integral Chapter 21: Exponents and Logarithms, Revisited v

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Structure of the chapters Chapter 6 Chapter 5 Chapter 2 Chapter 1 Chapter 4 Chapter 3 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 20 Chapter 21 Chapter 12 Chapter 13 Chapter 14 Chapter 11 Chapter 19 
General Introduction This book is a textbook for the mathematics curriculum of grades 6–12 written for teach- ers.

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Math151AFirstSections(1) - Mathematics of the Secondary...

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