The Mathematic That Every Secondary School Math Teacher Needs to Know
Buku ini diterbitkan tahun 2011 oleh Routledge, Taylor & Francis. Adalah buku edisi Pertama.
Judul: The Mathematic That Every Secondary School Math Teacher Needs to Know
Oleh: Alan Sultan, et al
Penerbit: Routledge, Taylor & Francis
Tahun: 2011
Jumlah Halaman: 761 hal.
Penulis:
Alan Sultan
Queens College of the City University of New York
Alice F. Artzt
Queens College of the City University of New York
Lingkup Pembahasan:
Buku ini mengemukakan bahwa untuk lebih spesifik, guru matematika perlu memiliki pengetahuan tentang bagaimana membuat pemahaman dan keterampilan matematika diakses bagi semua siswanya. Misalnya, guru akan memberitahu Anda bahwa tahun demi tahun siswa mereka mengalami kesulitan memahami spesifik topic tertentu dalam kurikulum. Dalam buku ini, kita membahas bagian-bagian kesulitan yang khas dan kesalahpahaman umum yang dialami siswa. Kami meneliti mengapa kesulitan-kesulitan ini ada dan pendekatan matematika guru dapat digunakan dalam menjelaskan konsep dan prosedur. Dengan demikian, buku ini menekankan penggunaan beberapa cara untuk mewakili dan memecahkan masalah sehingga Anda akan dapat memenuhi kebutuhan siswa yang paling pasti akan memiliki gaya belajar dan kemampuan yang beragam. Penggunaan teknologi digabungkan untuk menambah beberapa cara masalah dapat direpresentasikan. Selain itu, kemampuannya secara dinamis merupakan konsep dan mensimulasikan masalah, teknologi dapat digunakan untuk membantu siswa memecahkan masalah melalui penemuan, pengenalan pola, dan penalaran induktif.
Sepanjang buku ini mengemukakan kajian kekuatan dan kelemahan dari teknologi yang berbeda mewakili ide-ide matematika, serta memberikan referensi ke beberapa informasi dan website terkadang interaktif yang akan berguna bagi Anda dan siswa Anda.
Buku ini juga mengemukakan bahwa selain meningkatkan kompetensi siswa dan pemahaman matematika, bagian penting lain dari pekerjaan guru adalah untuk dapat menarik siswa dalam materi pelajaran.
Daftar Isi:
Preface/Introduction xvii
Notes to the Reader/Professor xxi
Acknowledgments xxvii
CHAPTER 1 Intuition and Proof 1
1.1 Introduction 1
1.2 Can Intuition Really Lead Us Astray? 1
1.3 Some Fundamental Methods of Proof 8
1.3.1 Direct Proof 9
1.3.2 Proof by Contradiction 10
1.3.3 Proof by Counterexample 12
1.3.4 The Finality of Proof 13
CHAPTER 2 Basics of Number Theory 17
2.1 Introduction 17
2.2 Odd, Even, and Divisibility Relationships 17
2.3 The Divisibility Rules 25
2.4 Facts about Prime Numbers 31
2.4.1 The Prime Number Theorem 36
2.5 The Division Algorithm 38
2.6 The Greatest Common Divisor (GCD) and the Euclidean Algorithm 42
2.7 The Division Algorithm for Polynomials 48
2.8 Different Base Number Systems 51
2.9 Modular Arithmetic 56
2.9.1 Application: RSA Encryption 59
2.10 Diophantine Analysis 61
CHAPTER 3 Theory of Equations 69
3.1 Introduction 69
3.2 Polynomials: Modeling, Basic Rules, and the Factor Theorem 70
3.3 Synthetic Division 75
3.4 The Fundamental Theorem of Algebra 80
3.5 The Rational Root Theorem and Some Consequences 85
3.6 The Quadratic Formula 90
3.7 Solving Higher Order Polynomials 95
3.7.1 The Cubic Equation 95
3.7.2 Cardan’s Contribution 100
3.7.3 The Fourth Degree and Higher Equations 101
3.8 The Role of the Graphing Calculator in Solving Equations 103
3.8.1 The Newton–Raphson Method 104
3.8.2 The Bisection Method–Unraveling the Workings of the Calculator 108
CHAPTER 4 Measurement: Area and Volume 113
4.1 Introduction 113
4.2 Areas of Simple Figures and Some Surprising Consequences 113
4.3 The Circle 125
4.3.1 An Informal Proof of the Area of a Circle 126
4.3.2 Archimedes’ Proof of the Area of a Circle 127
4.3.3 Limits And Areas of Circles 130
4.3.4 Using Technology to Find the Area of a Circle 131
4.3.5 Computation of π 133
4.3.6 Finding Areas of Irregular Shapes 139
4.4 Volume 146
4.4.1 Introduction to Volume 146
4.4.2 A Special Case: Volumes of Solids of Revolution 150
4.4.3 Cavalieri’s Principle 153
4.4.4 Final Remarks 154
CHAPTER 5 The Triangle: Its Study and Consequences 159
5.1 Introduction 159
5.2 The Law of Cosines and Surprising Consequences 159
5.2.1 Congruence 161
5.3 The Law of Sines 165
5.5 Sin(A+ B) 175
5.6 The Circle Revisited 179
5.6.1 Inscribed and Central Angles 180
5.6.2 Secants and Tangents 183
5.6.3 Ptolemy’s Theorem 185
5.7 Technical Issues 191
5.8 Ceva’s Theorem 195
5.9 Pythagorean Triples 200
5.10 Other Interesting Results about Areas 204
5.10.1 Heron’s Theorem 205
5.10.2 Pick’s Theorem 206
CHAPTER 6 Building the Real Number System 215
6.1 Introduction 215
6.2 Part 1: The Beginning Laws: An Intuitive Approach 216
6.3 Negative Numbers and Their Properties: An Intuitive Approach 220
6.4 The First Rules for Fractions 224
6.5 Rational and Irrational Numbers: Going Deeper 231
6.6 The Teacher’s Level 234
6.7 The Laws of Exponents 242
6.7.1 Integral Exponents 242
6.8 Radical and Fractional Exponents 245
6.8.1 Radicals 245
6.8.2 Fractional Exponents 247
6.8.3 Irrational Exponents 250
6.9 Working with Inequalities 253
6.10 Logarithms 259
6.10.1 Rules for Logarithms 263
6.11 Solving Equations 267
6.11.1 Some Issues 268
6.11.2 Logic Behind Solving Equations 269
6.11.3 Equivalent Equations 272
6.12 Part 2: Review of Geometric Series: Preparation for Decimal Representation 277
6.13 Decimal Expansion 280
6.14 Decimal Periodicity 289
6.15 Decimals: Uniqueness of Representation 293
6.16 Countable and Uncountable Sets 297
6.16.1 Algebraic and Transcendental Numbers Revisited 303
CHAPTER 7 Building the Complex Numbers 307
7.1 Introduction 307
7.2 The Basics 307
7.2.1 Operating on the Complex Numbers 308
7.3 Picturing Complex Numbers and Connections to Transformation Geometry 315
7.3.1 An Interesting Problem 319
7.3.2 The Magnitude of a Complex Number 323
7.4 The Polar Form of Complex Numbers and De Moivre’s Theorem 325
7.4.1 Roots of Complex Numbers 330
7.5 A Closer Look at the Geometry of Complex Numbers 334
7.6 Some Connections to Roots of Polynomials 340
7.7 Euler’s Amazing Identity and the Irrationality of e 343
7.8 Fractal Images 347
7.8.1 Other Ways to Generate Fractal Images 351
7.9 Logarithms of Complex Numbers and Complex Powers 352
CHAPTER 8 Induction, Recursion, and Fractal Dimension 357
8.1 Introduction 357
8.2 Recursive Relations 357
8.2.1 Solving Recursive Relations 361
8.3 Induction 372
8.3.1 Taking Induction to a Higher Level 376
8.3.2 Other Forms of Induction 378
8.4 Fractals Revisited and Fractal Dimension 387
8.4.1 The Chaos Game 389
8.4.2 Fractal Dimension 390
CHAPTER 9 Functions and Modeling 397
9.1 Introduction 397
9.2 Functions 397
9.2.1 The Historical Notion of Function 398
9.2.2 Functions Today 398
9.2.3 Functions – The More General Notion 400
9.2.4 Ways of Representing Functions 401
9.3 Modeling with Functions 406
9.3.1 Some Types of Models 407
9.3.2 Which Model Should We Use? 412
9.4 What Does Best Fit Mean? 420
9.4.1 What is Behind Finding the Line of Best Fit? 421
9.4.2 How Well Does a Function Fit the Data? 425
9.5 Finding Exponential and Power Functions That Fit Curves 427
9.5.1 How Calculators Find Exponential and Power Regressions 428
9.5.2 Things to Watch Out for in Curve Fitting 431
9.6 Fitting Data Exactly With Polynomials 433
9.7 1–1 Functions 439
9.7.1 The Rudiments 439
9.7.2 Why Are 1–1 Functions Important? 442
9.7.3 Inverse Functions in More Depth 442
9.7.4 Finding the Inverse Function 444
9.7.5 Graphing the Inverse Function 446
CHAPTER 10 Geometric Transformations 451
10.1 Introduction 451
10.2 Transformations: The Secondary School Level 452
10.2.1 Basic Ideas 453
10.3 Bringing in the Main Tool – Functions 458
10.4 The Matrix Approach – a Higher Level 466
10.4.1 Reflections, Rotations, and Dilations 466
10.4.2 Compositions of Transformations 469
10.4.3 Reflecting about Arbitrary Lines 474
10.5 Matrix Transformations 482
10.5.1 The Basics 482
10.5.2 Matrix Transformations in More Detail – A Technical Point 485
10.6 Transforming Areas 492
10.7 Connections to Fractals 497
10.7.1 Translations 498
10.8 Transformations in Three dimensions 504
10.9 Reflecting on Reflections 507
CHAPTER 11 Trigonometry 513
11.1 Introduction 513
11.2 Typical Applications Using Angles and Basic Trigonometric Functions 514
11.2.1 Engineering and Astronomy 514
11.2.2 Forces Acting on a Body 516
11.3 Extending Notions of Trigonometric Functions 523
11.3.1 Trigonometric Functions of Angles More than 90 Degrees 524
11.3.2 Some Useful Trigonometric Relationships 528
11.4 Radian Measure 537
11.4.1 Conversion 537
11.4.2 Areas and Arc Length in Terms of Radians 539
11.5 Graphing Trigonometric Curves 544
11.5.1 The Graphs of Sin θ and Cos θ 545
11.5.2 The Graph of y = Tan θ 552
11.6 Modeling with Trigonometric Functions 555
11.7 Inverse Trigonometric Functions 560
11.8 Trigonometric Identities 567
11.9 Solutions of Cubic Equations Using Trigonometry 575
11.10 Lissajous Curves 578
11.11 Vectors 581
11.11.1 Basic Vector Algebra 582
11.11.2 Components of Vectors 587
11.11.3 Using Vectors to Prove Geometric Theorems 592
CHAPTER 12 Data Analysis and Probability 599
12.1 Introduction 599
12.2 Basic Ideas of Probability 600
12.2.1 Different Approaches to Probability 600
12.2.2 Issues with the Approaches to Probability 603
12.3 The Set Theoretic Approach to Probability 605
12.3.1 Some Elementary Results in Probability 608
12.4 Elementary Counting 613
12.5 Conditional Probability and Independence 618
12.5.1 Some Misconceptions in Probability 623
12.6 Bernoulli Trials 626
12.7 The Normal Distribution 629
12.8 Classic Problems: Counterintuitive Results in Probability 635
12.8.1 The Birthday Problem 636
12.8.2 The Monty Hall Problem 636
12.8.3 The Gunfight 637
12.8.4 Simulation 638
12.9 Fair and Unfair Games 641
12.9.1 Games Where No Money is Involved 641
12.9.2 Games Where Money is Involved 643
12.9.3 The General Notion of Expectation 644
12.9.4 The Cereal Box Problem 646
12.10 Geometric Probability 650
12.10.1 Some Surprising Consequences 652
12.10.2 Monte Carlo Revisited 654
12.11 Data Analysis 658
12.11.1 Plotting Data 659
12.11.2 Mean, Median, Mode 664
12.12 Lying with Statistics 669
12.12.1 What Can you Do to Talk Back to Statistics? 673
CHAPTER 13 Introduction to Non-Euclidean Geometry 677
13.1 Introduction 677
13.2 Can We Believe Our Eyes? 678
13.2.1 What Are the Errors in the Proofs? 683
13.3 The Parallel Postulate 685
13.3.1 What Can We Prove with the Parallel Postulate? 686
13.3.2 What Can We Prove Without the Parallel Postulate? 688
13.4 Undefined Terms 690
13.5 Strange Geometries 692
13.5.1 Hyperbolic Geometry 693
13.5.2 Euclid’s Axioms in the Hyperbolic World 695
13.5.3 Area in Hyperbolic Space 700
13.5.4 Spherical Geometry 702
CHAPTER 14 Three Problems of Antiquity 705
14.1 Introduction 705
14.2 Some Basic Constructions 705
14.3 Three Problems of Antiquity and Constructible Numbers 710
14.3.1 Constructible Numbers 711
14.3.2 Geometrically Constructible Numbers 711
14.3.3 The Constructible Plane 713
14.3.4 Solving the Three Problems of Antiquity 716
Bibliography 721
Appendix 723
Index 729
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