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
iii
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
iv
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
v
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
vi
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