ARITHMETIC
1 Teaching Middle School Mathematics: Standards, Recommendations and Teacher Candidates’ Perspectives 1 11 Introduction 1 12 The Importance of Content Knowledge 2 13 The Importance of Pedagogical Content Knowledge 6 14 The Need for Teachers with ‘Deep Understanding’ of Mathematics 8 15 Learning to Address Misconceptions Through Mathematical Connections 10 16 Making Conceptual Connections 12 17 Using Technology in the Classroom 14 18 Conclusion 16 2 Modeling Mathematics in the Digital Era 17 21 Introduction 17 22 Modeling Mathematics and Mathematical Modeling 18 221 Modeling Mathematics: Selecting a Referent or Object to Act upon 22 23 Modeling Mathematics 30 231 Modeling Equivalent Surface Area Functions— Stamping Functions 32 232 Modeling Correlations—Burning the Candle 36 233 Modeling Systems of Equations—The Flagpole Factory 37 234 Data-Tables Revisited 41 24 The Problem Solving Board 44 25 Conclusion 49 26 Activity Set 51 3 Reasoning and Proof 53 31 Introduction 53 32 Elementary Reasoning at the Concrete Operational Stage 54 33 Supporting Plausible Mathematical Reasoning and Formal Proofs with CARE 62 331 Commonplaces 63 332 Authority 64 333 Reason 66 334 Experience 68 34 CARE and Rotating Squares 68 35 Basic Elements of Plausible Reasoning 71 36 Basic Elements of Propositional Logic 76 361 Conjunction, Disjunction and Implication 76 362 Application of the Logical Conjunction Truth Values in Spreadsheet Modeling 76 363 Application of the Logical Disjunction Truth Values in Spreadsheet Programming 77 364 De Morgan Rules 78 365 Converse, Inverse and Contrapositive Statements 80 366 Modus Ponens: [(p ) q) ^ p] ) q 80 367 Modus Tollens: [(p ) q) ^ :q] ) :p 80 37 Conclusion 81 38 Activity Set 82 4 Modeling Mathematics with Fractions 85 41 Introduction 85 42 Transition from Arithmetic of Integers to Arithmetic of Fractions 88 43 The Meaning of Multiplying and Dividing Fractions 89 431 Multiplying Two Fractions 89 432 Dividing Fractions 94 44 Conceptual Meaning of the Invert and Multiply Rule: From Integers to Fractions 101 441 The Case of Dividing Integers 102 442 The Case of Dividing Fractions 102 443 The Invert and Multiply Rule as a Change of Unit 103 45 Unit Fractions 105 451 Unit Fractions as Benchmark Fractions 105 452 Representation of a Unit Fraction as a Sum of Two Like Fractions 106 453 Representation of 1/2 as a Sum of Three Different Unit Fractions 109 454 Egyptian Fractions and the Greedy Algorithm 109 46 From Long Division to Decimal Representation of Fractions 112 47 Rational Numbers in Non-Decimal Bases 114 471 Conversion of Integers into Non-Decimal Bases 114 472 Conversion of Common Fractions into Non-Decimal Bases 116 473 Visual Representation of Conversion of Common Fractions in Different Bases 117 48 Integer Sequences as Sources of Fractions 119 49 Continued Fractions 120 491 Euclidean Algorithm 120 492 Continued Fraction Representation of Common Fractions 122 410 Connecting Fractions to Quadratic Equations 123 411 Conclusion 126 412 Activity Set 126 5 Decimal and Percent Representation of Rational Numbers 129 51 Introduction 129 52 Developing Numerical Fluency 130 53 Transitioning from Numerical Fluency to the Base-Ten Notational System 138 54 Decimals and Decimal Fractions 141 55 Decimals and Common Fractions 142 56 Decimal and Percent 142 57 Repeating Decimals 145 58 From Representation to Operations 149 581 Comparison 150 582 Addition and Subtraction 151 583 Multiplication and Division 153 59 Conclusion 153 510 Activity Set 155 6 Ratio and Proportion 157 61 Introduction 157 62 Ratios 157 621 Part-Part Ratios 158 622 Part-Whole Ratios 159 623 Reflecting Quotients as Ratios 161 624 Reflecting Rates as Ratios 161 63 Proportions 163 631 Scaling 171 632 Proportions and Conversions 174 633 The Golden Ratio 175 634 The Silver Ratio 177 635 Metallic Means 178 64 Conclusion 178 65 Activity Set 180 7 Geometry and Measurement 183 71 Introduction 183 72 Geometry 184 721 Geometric Thinking and Van Hiele Levels 185 722 Developing Geometric Vocabulary 189 723 Deductive Proofs 190 724 Transformations 192 725 Tessellations 197 726 Parallel Lines 202 727 Pythagorean Theorem, Pythagorean Triples, and the Distance Formula 205 728 Triangles and Introductory Trigonometry 208 729 Quadrilaterals 209 73 Measurement 209 731 Application of Scaling Factors in Measurement 212 732 Circumference, Area of a Circle, and P 213 74 Conclusion 219 75 Activity Set 221 8 Combinatorics 223 81 Introduction 223 82 The Rules of Sum and Product 226 83 Tree Diagrams 226 84 Permutation of Letters in a Word 228 85 Combinations Without Repetition 231 851 Developing Counting Techniques 231 852 Counting Rectangles on a Checkerboard 236 853 Why Are There 256 Representations of 9 Through a Sum of Ordered Integers? 238 86 Combinations with Repetition 240 87 The Sum of Powers of Integers as a Combinatorial Tool 244 88 Conclusion 246 89 Activity Set 247 9 Conceptual Approach to the Ideas of Middle School Algebra 249 91 Introduction 249 92 From Early Algebra to Graphing and Problem Posing with Technology 250 921 Three Problems from Different Grade Levels 250 922 Problem Reformulation 251 923 Solution of Problem 95 Using Kid Pix 252 924 Opening a Window to Traditional School Algebra 254 93 Different Types of Generalization in School Algebra 255 931 Algebraic Generalization of the First Kind 256 932 Algebraic Generalization of the Second Kind 259 933 From Summation of Odd Numbers to Summation of Squares 260 934 The Sum of Cubes in the Multiplication Table 262 94 Conceptual Generalizations of the First Kind 264 941 Patterns of Elimination Leading to Polygonal Numbers 265 942 Developing Subsequences of Fibonacci Numbers 266 95 Parameterization as Conceptual Generalization of the Second Kind 267 96 Insufficiency of Generalization by Empirical Induction 268 97 Numeric Tables as a Context for Developing Algebraic Skills 271 971 Finding the Sum of All Numbers in the n n Addition Table 272 972 Finding the Sum of All Numbers in the n n Multiplication Table 276 973 Exploring Divisibility of Numbers in Addition and Multiplication Tables 279 98 Solving Algebraic Word Problems Through Conceptual Strategies/Shortcuts 283 99 Inequalities 288 991 Algebraic Inequalities as Tools of Digital Fabrication 289 992 Digitally Fabricating Points, Segments and Arcs 290 993 Arithmetic Mean-Geometric Mean Inequality 294 910 Algebraic Recreations 297 911 Conclusion 300 912 Activity Set 300 10 Patterns and Functions 303 101 Introduction 303 102 From Patterns to Functions 306 103 From Visual to Symbolic 309 104 Patterns Structured by Colors 312 1041 From Colors to Functions 313 1042 From Empirical Conjectures to Mathematical Induction Proof 315 1043 Two-Color Pattern Guided by Consecutive Odd Numbers 317 1044 More Patterns Defined by an Arithmetic Sequence 319 105 Finding Patterns Formed by Functions 323 1051 Developing Formula for f 1 ðn;mÞ 323 1052 The Cases of the Position of the First R and the First/Last B 324 1053 Alternative Verification of the Obtained Results 325 106 The Case of Three Colors 326 107 The Case of p Colors 328 1071 Pattern Guided by Arithmetic Sequence with Difference One 328 1072 Pattern Guided by Arithmetic Sequence with Difference Two 329 1073 Pattern Guided by Arithmetic Sequence with Difference Three 330 1074 Pattern Guided by Arithmetic Sequence with Difference Four 331 1075 Pattern Guided by Arithmetic Sequence with Difference m 332 108 From Generaization to Computerization 333 109 Conclusion 337 1010 Activity Set 337 11 Financial Literacy and Blockchain 341 111 Introduction 341 112 Financial Literacy 341 1121 Earning 343 1122 Budgeting 344 1123 Checking 347 1124 Interest 349 1125 Credit and Borrowing 352 1126 Mortgages 353 1127 Savings 354 1128 Credit Reports 355 1129 Planning and Paying for College 355 11210 Insurance 357 11211 Investments 359 11212 Common Financial Pitfalls 360 113 Blockchain 361 1131 Blockchain Basics 361 1132 Blockchain Security Concerns 362 1133 Consensus Mechanisms and Incentivation 362 1134 Cryptocurrency and Other Applications 363 1135 Advantages of Blockchain 363 1136 Disadvantages of Blockchain 364 114 Conclusion 364 115 Activity Set 365 12 Probability and Statistical Data Analysis 367 121 Introduction 367 122 Experiments with Equally Likely Outcomes 368 123 Randomness and Sample Space 371 124 Different Representations of a Sample Space 372 125 Fractions as Tools in Measuring Chances 375 126 Explorations with Addition and Multiplication Tables 376 1261 Computational Experiments with Pairs of Random Integers 377 1262 Selecting Even Numbers in Addition Tables 379 1263 Selecting Even Numbers in Multiplication Tables 379 1264 Selecting Multiples of Three in Addition Tables 380 1265 Selecting Multiples of Three in Multiplication Tables 381 1266 Theoretical Probabilities and Their Monotone Convergence 381 127 Bernoulli Trials and the Law of Large Numbers 382 128 Classic Problems that Motivated Theoretical Development 384 129 A Modification of the Problem of De Méré 386 1210 Experimental Probability Requires a Long Series of Observations 387 1211 Probability Experiments with Spinners and Palindromes 388 1212 Monty Hall Dilemma as a Paradox in the Theory of Probability 390 1213 Transition from Probability Theory to Statistical Data Analysis 392 1214 Graphic Representations of Numeric Data 393 1215 Measures of Central Tendency 395 1216 Measures of Dispersion 396 1217 The Binomial Distribution and the Normal Curve 398 1218 The z-Score and the Standard Deviation 401 1219 Bivariate Analysis 402 1220 Conclusion 405 1221 Activity Set 406