Table of Contents
Preface v
Acknowledgments xiii
1 Counting and Natural Numbers 1
1.1 The natural numbers 1
1.1.1 Development of the number concept 1
1.1.2 Matching between sets 2
1.1.3 Counting 3
1.1.4 The number zero 5
1.2 Order relations between numbers 6
1.2.1 Comparison between sets 6
1.2.2 Comparison between numbers 7
1.2.3 Successor and predecessor 8
1.2.4 There are infinitely many natural numbers 9
1.2.5 The number line diagram 9
1.3 Natural numbers as determining order 10
1.4 Estimation 10
1.5 More on numbers and sets 11
1.5.1 What is three? 11
1.5.2 Comparison of infinite sets 12
2 The Decimal Representation System 15
2.1 Development of the decimal system 15
2.1.1 The legend of the shepherd 15
2.1.2 Ancient decimal representation 16
2.1.3 Place-value notation 18
2.1.4 Zero as a place holder 18
2.2 Early age learning of the decimal system 20
2.2.1 The hundred table 21
2.3 Numerals in other cultures 22
2.3.1 Ancient Egypt 22
2.3.2 China 23
2.3.3 Ancient Rome 23
3 The Four Operations of Arithmetic 25
3.1 Arithmetic operations 25
3.1.1 Operators and operands 26
3.1.2 Concatenating operations 26
3.2 Arithmetic expressions 27
3.3 Teaching the operations of arithmetic 28
4 Addition 31
4.1 What is addition? 31
4.1.1 Addition as a model for joining together 31
4.1.2 Addition as a model for appending 32
4.1.3 The addition operation 32
4.2 Word problems 34
4.2.1 Modeling with addition 34
4.2.2 Interpretation 34
4.3 The properties of addition 34
4.3.1 The commutative property 35
4.3.2 The associative property 36
4.3.3 The identity property of zero 39
4.3.4 Laws of variation 40
4.4 Evaluating addition 41
4.4.1 If you can count, you can add 41
4.4.2 Addends whose sum is 10 43
4.4.3 Addition within 20 43
4.4.4 Using the properties of addition 45
4.4.5 The addition table 45
4.4.6 Addition on the number line 46
5 Subtraction 49
5.1 What is subtraction? 49
5.1.1 Subtraction as a model for taking away 50
5.1.2 Subtraction as a model for taking apart 51
5.1.3 Subtraction as a model for comparison 52
5.1.4 Subtraction as a model for complementation 52
5.1.5 The subtraction operation 53
5.1.6 Subtraction on the number line 54
5.2 Word problems 55
5.2.1 Modeling with subtraction 55
5.2.2 Interpretation 56
5.3 The properties of subtraction 57
5.3.1 Subtraction is not commutative 57
5.3.2 Subtraction is not associative 58
5.3.3 Laws of variation 58
5.3.4 Adjoint subtraction equation 62
5.3.5 Subtraction of zero 63
5.4 Evaluating subtraction 63
5.4.1 If you can count, you can subtract 63
5.4.2 Backward and forward counting 63
5.4.3 Subtraction within 20 65
5.4.4 Validation 66
5.5 Negative numbers 66
6 Even and Odd Numbers 69
6.1 Definition of parity 69
6.2 Arithmetic of parity 72
6.3 Determining the parity of a number 74
6.4 More on parity 75
6.4.1 Why don't we define parity for fractions? 75
6.4.2 Parity in error control 76
7 Multiplication 79
7.1 What is multiplication? 79
7.1.1 Multiplication as a model for repeated addition 79
7.1.2 Multiplication as a model for proportional comparison 80
7.1.3 The multiplication operation 80
7.1.4 Multiplication and counting 81
7.1.5 Rectangular arrays 82
7.2 Properties of multiplication 82
7.2.1 The commutative property 82
7.2.2 The associative property 84
7.2.3 The distributive property 86
7.2.4 Laws of variation 87
7.2.5 The identity property of one 87
7.2.6 Multiplication by zero 87
7.3 Word problems 88
7.3.1 Repeated addition 88
7.3.2 Combinatorial problems 88
7.4 Evaluating multiplication 91
7.4.1 If you can count, you can multiply 91
7.4.2 The multiplication table 92
7.4.3 Various evaluation strategies 92
7.4.4 Multiplication by 10 94
8 Division 97
8.1 What is division? 97
8.1.1 Division as a model for sharing 97
8.1.2 Division as a model for rationing 98
8.1.3 The division operation 99
8.2 Word problems 100
8.2.1 Modeling with division 100
8.2.2 Interpretation 101
8.2.3 Sneak preview: Fraction division 102
8.3 Properties of division 103
8.3.1 Division is not commutative 103
8.3.2 Division is not associative 103
8.3.3 The distributive property 104
8.3.4 Laws of variation 106
8.3.5 Interchanging multiplication and division 108
8.3.6 Adjoint division equation 108
8.3.7 Division of zero and division by zero 109
8.3.8 Division by 1 112
8.3.9 Division of a number by itself 112
5.4 Evaluating division 113
8.4.1 If you can count, you can divide 113
8.4.2 Evaluation by repeated addition 113
8.4.3 Evaluation by repeated subtraction 114
8.4.4 Chunking 114
8.4.5 Division by 10 115
8.4.6 Division by 5 116
9 Regrouping 121
9.1 Decimal units 121
9.2 Place-value notation 122
9.2.1 Zero as a place holder 124
9.3 Regrouping 125
9.3.1 Non-standard decimal representations 125
9.4 Numeral systems and complexity 126
10 Addition of Multi-Digit Numbers 129
10.1 Addition without regrouping 129
10.2 Vertical addition without regrouping 130
10.3 Addition with regrouping 132
10.4 Vertical addition with regrouping 134
11 Subtraction of Multi-Digit Numbers 139
11.1 Subtraction without regrouping 139
11.2 Vertical subtraction without regrouping 140
11.3 Vertical subtraction with regrouping 141
11.3.1 First regroup and then evaluate 142
11.3.2 The standard algorithm 143
11.3.3 Multiple regroupings 143
11.4 The French algorithm 146
12 Give Me Five! 151
12.1 Quinary numeral system 151
12.1.1 The shepherd's tale revisited 151
12.1.2 A new numeral system 152
12.1.3 Place-value notation 154
12.2 Addition 155
12.2.1 Addition of multi-digit numbers without regrouping 156
12.2.2 Addition of multi-digit numbers with regrouping 157
12.3 Subtraction 158
12.4 Parity 158
12.5 Multiplication 159
12.6 Division 160
13 Introduction to Geometry 163
13.1 Euclidean geometry 163
13.2 Elementary school geometry 164
13.3 Set-theoretic concepts 165
13.4 Three-dimensional space 165
13.5 Geometric figures 166
13.6 Congruence 167
13.7 Measurements 168
14 Planes and Lines 171
14.1 Planes 171
14.1.1 Plane geometry 172
14.2 Lines 172
14.3 Postulates and theorem 172
14.3.1 Three Euclidean postulates 172
14.3.2 A sample theorem 175
14.3.3 Ordering of points on a line 176
14.4 Segments and rays 177
14.4.1 Line segments 177
14.4.2 Rays 177
14.1.1 Broken lines 178
14.5 Segment arithmetic 179
14.5.1 Segment comparison 179
14.5.2 Segment addition 179
14.5.3 Segment subtraction 180
14.5.4 Segment multiplication 181
14.5.5 Segment division 181
15 Length 183
15.1 Length comparison 183
15.1.1 Comparison by juxtaposition 183
15.1.2 Comparison by transitivity 184
15.1.3 Comparison by concatenation 184
15.2 The length of a segment 185
15.2.1 Standard measuring units 186
15.2.2 The meter 186
15.2.3 Using multiple measuring units 187
15.2.4 Systems of measuring units 187
15.2.5 Length-measuring instruments 188
15.3 The length of curves 189
15.3.1 The length of a broken line 189
15.3.2 The length of more complicated curves 190
15.3.3 Infinitely long curves 191
16 Angles 193
16.1 What is an angle? 193
16.2 Angle arithmetic 195
16.2.1 Angle comparison 195
16.2.2 Angle addition 197
16.2.3 Angle subtraction 197
16.2.4 Angle multiplication 198
16.2.5 Angle division 198
16.3 Angle measurement 200
16.3.1 The degree 200
16.3.2 Types of angles 201
16.3.3 The protractor 201
16.4 Parallel lines 202
17 Polygons 207
17.1 What is a polygon? 207
17.2 Triangles 210
17.2.1 The sum of the angles in a triangle 210
17.2.2 Classification of triangles according to their sides 211
17.2.3 Classification of triangles according to their angles 212
17.2.4 Inclusion relations between types of triangles 213
17.2.5 Congruent triangles and constructions 215
17.3 Quadrilaterals 218
17.3.1 The sum of the angles in a quadrilateral 218
17.3.2 Squares 219
17.3.3 Rectangles 219
17.3.4 Rhombuses 220
17.3.5 Parallelograms 220
17.3.6 Trapezoids 221
17.3.7 Kites 221
17.4 General polygons 222
17.4.1 The sum of the angles 222
17.4.2 The number of diagonals 223
17.4.3 Regular polygons 224
18 Area 227
18.1 The area concept 227
18.2 Area comparison 228
18.3 Area measurement 231
18.4 The area of polygons 232
18.4.1 Rectangles 232
18.4.2 Triangles 233
18.4.3 General polygons 235
18.5 Area and scaling 235
Index 239
Common Core Index 245
