Fundamental Mathematics
for
Elementary and Middle School Teachers
Table of Contents
Preface ix
Chapter 1
The Development of the Rational Numbers
1.1 Sets and the Whole Numbers 1
1.2 Patterns and Inductive Reasoning 11
1.3 The Integers 19
1.4 The Rational Numbers 26
1.5 Ratios and the Fundamental Property of Fractions 36
1.6 Ordering Rational Numbers Using Number Sense 44
1.7 Counting and Measuring 55
1.8 Chapter Summary 65
1.9 NCTM Standards Implemented in this Chapter 68
1.10 Practice Test 70
Chapter 2
Deductive Reasoning and the Real Numbers
2.1 Deductive Reasoning: Logic and Proof 75
2.2 More about Sets and Logic 85
2.3 Deductive Reasoning and Irrational Numbers 92
2.4 Imaginary and Complex Numbers (optional) 100
2.5 A Brief Overview of the Basic Operations 105
2.6 Problem Solving Strategies 112
2.7 Chapter Summary 121
2.8 NCTM Standards Implemented in this Chapter 123
2.9 Practice Test 125
Chapter 3
Numeration and Measurement Systems
3.1 Numeration Systems 131
3.2 Numeration System Conversions 141
3.3 Decimal Form: Expanding Place Value to All Rational Numbers 148
3.4 Equivalent Forms of Rational Numbers 158
3.5 Measurement Systems I: The Metric and American Systems 166
3.6 Measurement Systems II: Time, Angle, Area, and Volume 175
3.7 Functions: Examining Relationships Between Quantities 184
3.8 Chapter Summary 191
3.9 NCTM Standards Implemented in this Chapter 193
3.10 Practice Test 195
Chapter 4
Addition and Subtraction
4.1 The Meaning of Addition and Subtraction 201
4.2 Addition and Subtraction with Signed Numbers 209
4.3 Properties of Addition and Subtraction 218
4.4 Units and Their Roles in Addition and Subtraction 225
4.5 Intersecting Sets and Endpoint Problems 236
4.6 Computation I: Mental Arithmetic and Estimation 244
4.7 Computation II: Decimal Arithmetic 251
4.8 Computation III: Fractions 258
4.9 Chapter Summary 265
4.10 NCTM Standards Implemented in this Chapter 267
4.11 Practice Test 269
Chapter 5
Multiplication and Division I: Meaning
5.1 Multiplication as Repeated Addition 275
5.2 Division in the Context of Repeated Addition 285
5.3 Multiplication as a Means of Comparison 299
5.4 Division in the Context of Comparisons 309
5.5 Proportional Reasoning 318
5.6 Dimensional Analysis and Conversions 327
5.7 Chapter Summary and Additional Problem Solving 338
5.8 NCTM Standards Implemented in this Chapter 346
5.9 Practice Test 348
Chapter 6
Multiplication and Division II:
Properties and Computational Methods
6.1 Properties of Multiplication and Division 355
6.2 Multiplication and Division of Integers 365
6.3 Computing Whole Number Products 371
6.4 Computing Whole Number Quotients 380
6.5 Computing Products and Quotients of Rational Numbers 388
6.6 Real Numbers in Decimal Form 399
6.7 Chapter Summary 408
6.8 NCTM Standards Implemented in this Chapter 411
6.9 Practice Test 413
Chapter 7
Number Theory
7.1 Factors and Multiples 417
7.2 Primes and Composites 425
7.3 Common Factors and Multiples 434
7.4 Chapter Summary 441
7.5 NCTM Standards Implemented in this Chapter 443
7.6 Practice Test 444
Chapter 8
Data Analysis
8.1 Organizing and Representing Univariate Data 449
8.2 Organizing and Representing Multivariate Data 462
8.3 Shapes and Centers of Distributions 477
8.4 Measures of Dispersion 493
8.5 Measures of Position Based on Rank 505
8.6 Collecting Data and Designing Studies 517
8.7 Chapter Summary 533
8.8 NCTM Standards Implemented in this Chapter 536
8.9 Practice Test 538
Chapter 9
Probability
9.1 Basic Concepts 543
9.2 Independence and Conditionality 553
9.3 Probabilities of Intersections 561
9.4 Counting, Permutations, and Combinations 568
9.5 Probability and Inferential Statistics 577
9.6 Chapter Summary 584
9.7 NCTM Standards Implemented in this Chapter 587
9.8 Practice Test 589
Chapter 10
Introductory Geometry
10.1 Dimensionality, Lines, and Angles 593
10.2 Classifying Two-Dimensional Figures 606
10.3 Properties of Polygons 617
10.4 Three-Dimensional Figures 626
10.5 Symmetry 638
10.6 Chapter Summary 645
10.7 NCTM Standards Implemented in this Chapter 648
10.8 Practice Test 650
Chapter 11
Measurement
11.1 A Review of the Basic Ideas of Measurement 655
11.2 One-Dimensional Measurements 661
11.3 Proving the Pythagorean Theorem 670
11.4 Areas of Polygons 678
11.5 Areas of Curved Figures 688
11.6 Surface Areas of Three-Dimensional Shapes 696
11.7 Volumes 703
11.8 Chapter Summary 712
11.9 NCTM Standards Implemented in this Chapter 715
11.10 Practice Test 717
Chapter 12
Location and Change
12.1 Location and Rectangular Coordinate Systems 721
12.2 Direction 732
12.3 Additional Coordinate Systems 741
12.4 Transformations 747
12.5 Slope as a Means of Comparison and Measure of Change 753
12.6 Chapter Summary 763
12.7 NCTM Standards Implemented in this Chapter 766
12.8 Practice Test 768
Appendices
Appendix A: Sample Skills Quizzes 773
Appendix B: Grid Paper 777
Index
vii