Elementary School Mathematics For Parents And Teachers - Volume 2

by Raz Kupferman

Overview

'The exposition is exceptionally clear, and keeps its audience in mind: in the end, this is "˜a book for adults on mathematics for children'. "˜Adults' here includes parents as well as teachers; in the author's words, "˜Every parent is automatically an educator!' There is considerable merit in including parents in the elementary mathematics conversation; it is a pleasure to see books like Kupferman's that explain elementary math to adults. If our students, in their future role as teachers, can enlist parents as allies, everyone will benefit. These books are one good way to begin addressing that challenge.'
MAA ReviewsThis book covers the elementary school mathematics curriculum common in most parts of the world. Its aim is to serve educators (teachers and parents) as a guide for teaching mathematics at elementary school level. The book focuses both on content knowledge and on pedagogical content knowledge. It bridges the gap between fundamental mathematical principles and good teaching practices. It also offers the reader a glimpse on how mathematicians perceive elementary mathematics and presents ideas for specific mathematical activities.Volume 2 focuses on content taught in the higher grades of elementary school. It covers the following topics: multiplication and division of multi-digit numbers, divisibility and primality, divisibility signs, sequences, fractions and their representations, and fraction arithmetic.The author is also a co-founder of Matific, an adaptive game-based teaching and learning tool for primary school mathematics. Independent studies have shown Matific to improve test scores, reduce maths anxiety, and increase motivation. Matific is available in 26 languages and aligned to mathematics curricula in 46 countries. Awards include Best Mathematics Instructional Solution, Best Game-Based Curriculum Solution and Best Educational App. For a trial, visit https://www.matific.com.

Product Details

ISBN-13:	9789813108936
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/23/2017
Pages:	304
Product dimensions:	7.00(w) x 9.90(h) x 0.60(d)

Table of Contents

Preface v

Acknowledgments xi

19 Multiplication of Multi-Digit Numbers 1

19.1 Repeated addition 2

19.1 Products of one-significant-digit numbers 3

19.2 Products of a single-digit number and a multi-digit number 6

19.3.1 Using the distributive property 6

19.3.2 Multiplication without regrouping 10

19.3.3 Multiplication with regrouping 12

19.4 Products of multi-digit numbers 14

19.4.1 Using the distributive property 14

19.4.2 Multiplication with regrouping 16

19.5 Mental calculations 19

19.6 Other multiplication algorithms 19

19.6.1 Napier bones 20

19.6.2 The binary algorithm 22

19.6.3 The peasants' algorithm 23

20 Division of Multi-Digit Numbers 29

20.1 Division without regrouping 32

20.1.1 Using the distributive property 32

20.1.2 Algorithm 33

20.2 Division with regrouping 34

20.2.1 First regroup and then evaluate 34

20.2.2 Illustrating the division process 36

20.2.3 Long division: Step-by-step 40

20.3 Division by a multi-digit number 43

20.4 Chunking 45

21 The Order of Operations 49

21.1 Problem statement 50

21.2 From computation networks to arithmetic expressions 52

21.3 Parentheses 53

21.4 Precedence of operations 55

21.4.1 Deleting redundant parentheses 57

22 Division with a Remainder 61

22.1 Remainders 62

22.1.1 Notation 64

22.1.2 The range of the remainder 65

22.1.3 Inverse multiplication equation 66

22.2 Word problems 68

22.3 Evaluating division with a remainder 70

22.4 Arithmetic of remainders 72

22.4.1 Odd and even numbers 72

22.4.2 Modulo-3 classification of integers 74

22.4.3 Modulo-3 addition 76

22.4.4 Modulo-3 multiplication 78

23 Prime and Composite Numbers 85

23.1 Multiples and divisors 85

23.1.1 Properties of multiples and divisors 87

23.1.2 Enumerating the divisors of a given number 88

23.2 Prime numbers 89

23.2.1 Prime and composite numbers 89

23.2.2 The fundamental theorem of arithmetic 90

23.2.3 Why is 1 not a prime number? 94

23.2.4 Prime number factorization: The fingerprints of natural numbers 95

23.3 Enumerating the divisors of a given number 96

23.4 Prime numbers and rectangular arrays 97

23.5 More on prime numbers 99

23.5.1 The sieve of Eratosthenes 99

23.5.2 There are infinitely many prime numbers 101

23.5.3 Large prime numbers 103

23.5.4 The density of prime numbers 103

23.5.5 Twin prime numbers 104

23.5.6 Goldbach's conjecture 105

24 Common Multiples and Common Divisors 109

24.1 Least common multiple 109

24.1.1 Common multiples 109

24.1.2 Evaluation of the least common multiple 110

24.2 Greatest common divisor 113

24.2.1 Common divisors 113

24.2.2 Evaluation of the greatest common divisor 114

24.3 Relation between least common multiple and greatest common divisor 120

25 Divisibility Rules 123

25.1 The principles underlying divisibility rules 124

25.1.1 Remainder arithmetic 124

25.1.2 Subtraction of a multiple does not change the remainder 125

25.1.3 Prime number factorization 126

25.2 Divisibility rules 126

25.2.1 Divisibility by 10 126

25.2.2 Divisibility by 5 127

25.2.3 Divisibility by 4 128

25.2.4 Divisibility by 8 130

25.2.5 Divisibility by 9 131

25.2.6 Divisibility by 3 132

25.2.7 Divisibility by 7 133

25.2.8 Divisibility by 11 135

25.3 Divisibility by composite numbers 137

25.3.1 Divisibility by 6 137

25.3.2 Divisibility by 4 138

25.3.3 Divisibility by 12 138

25.4 Validation using divisibility tests 139

26 Sequences 143

26.1 Some examples 144

26.1 Arithmetic progressions 145

26.2.1 Finding the element at a given location 145

26.2.2 Arithmetic series 146

26.3 Geometric progressions 149

26.3.1 Exponential growth 150

26.3.1 Exponential decay 150

26.3.1 Geometric progressions in daily life 150

26.4 Square numbers 151

26.4 Triangular numbers 152

26.4 The Fibonacci sequence 153

26.6.1 Fibonacci numbers in daily life 154

26.6.1 The golden ratio 156

26.7 The isoperimetric problem 157

26.8 The sequence of factorials 159

27 Fractions 163

27.1 Natural numbers and fractions 164

27.1.1 Natural numbers as quantifiers 164

27.1.2 Fractions as quantifiers 166

27.2 Unit fractions 168

27.2.1 Definition 168

27.2.2 Quantifying with unit fractions 169

27.3 General fractions 173

27.3.1 Definition 173

27.3.2 Quantifying with general fractions 174

27.3.3 The rational numbers 176

27.4 Fractions in early history 177

27.5 Fractions as part of a quantity 178

28 Quotients of Natural Numbers 181

28.1 Division of natural numbers 182

28.2 Equivalent fractions 184

28.2.1 Equivalent fractions and multiplicative scaling 185

28.2.2 Infinitude of equivalent fractions 186

28.2.3 Natural numbers as fractions 187

28.2.4 Expansion and reduction 189

28.3 Improper fractions and mixed numbers 191

28.3.1 Improper fractions 191

28.3.2 Mixed numbers 191

28.3.3 From improper fractions to mixed numbers 192

28.3.4 From mixed numbers to improper fractions 193

29 Fraction Comparison 195

29.1 Order relations and inclusion 196

29.2 Fraction comparison 197

29.2.1 Comparison using concrete models 198

29.2.2 Fractions with like denominators 201

29.2.3 Fractions with like numerators 202

29.2.4 Finding a common denominator 203

29.2.5 Finding a common numerator 206

29.2.6 Comparison using an intermediary 207

29.3 Factions on the number line 208

29.4 Density of the rational numbers 209

29.5 McKay's theorem 210

30 Fraction Addition and Subtraction 215

30.1 The meaning of fraction addition 216

30.2 The meaning of fraction subtraction 218

30.3 Evaluation of fraction addition and subtraction 220

30.3.1 Using concrete models 220

30.3.2 Fractions with like denominators 223

30.3.3 Fractions with unlike denominators 225

30.4 Fractions in ancient Egypt 227

31 Fraction Multiplication 231

31.1 Numbers and multiplication revisited 232

31.1.1 The number concept revisited 232

31.1.2 Multiplication revisited 234

31.2 Evaluating products 239

31.2.1 Fractions multiplied by natural numbers 240

31.2.2 Natural numbers multiplied by fractions 241

31.2.3 Fractions multiplied by fractions 245

31.3 Multiplicative inverses 250

31.4 Is multiplication a magnifying operation? 252

32 Fraction Division 257

32.1 Division revisited 258

32.1.1 Quotients of natural numbers 259

32.1.2 Quotients of fractions 263

32.1.3 Division is inverse to multiplication 264

32.2 Evaluating quotients 266

32.2.1 Fractions divided by whole numbers 266

32.2.2 Whole number divided by fractions 268

32.2.3 Fractions divided by fractions 271

32.3 Misconceptions 275

32.3.1 Divided by two, divided by hall 275

32.3.2 Is division a reducing operation? 275

Index 279

Common Core Index 283