دانلود کتاب Mathematical Reasoning for Elementary Teachers

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Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 14
1 Thinking Critically......Page 28
1.1 An Introduction to Problem Solving......Page 30
1.2 Pólya’s Problem-Solving Principles and the Standards for Mathematical Practice of the Common Core State Standards for Mathematics......Page 35
Guess and Check......Page 37
Make an Orderly List......Page 40
Draw a Diagram......Page 41
Pólya’s Problem-Solving Principles and the Standards for Mathematical Practice of the Common Core Standards for Mathematics......Page 43
Look for a Pattern......Page 46
Make a Table......Page 48
Use a Variable......Page 50
Pascal’s Triangle......Page 51
Use a Variable......Page 58
Use Two Variables......Page 63
Working Backward......Page 69
Eliminate Possibilities......Page 70
The Pigeonhole Principle......Page 72
Inductive Reasoning......Page 77
Representational Reasoning......Page 79
Mathematical Statements......Page 81
Deductive Reasoning......Page 83
Chapter 1 Summary......Page 88
Chapter 1 Review Exercises......Page 91
2 Sets and Whole Numbers......Page 94
2.1 Sets and Operations on Sets......Page 96
Venn Diagrams......Page 97
Relationships and Operations on Sets......Page 98
Using Sets for Problem Solving......Page 100
2.2 Sets, Counting, and the Whole Numbers......Page 105
One-to-One Correspondence and Equivalent Sets......Page 106
The Whole Numbers......Page 107
Representingthe Whole Numbers Pictorially and with Manipulatives......Page 108
Ordering the Whole Numbers......Page 109
Problem Solving with Whole Numbers and Venn Diagrams......Page 110
The Set Model of Whole-Number Addition......Page 116
Properties of Whole-Number Addition......Page 118
Subtraction of Whole Numbers......Page 121
Number-Line Model......Page 122
Multiplication as Repeated Addition......Page 128
The Array Model for Multiplication......Page 129
The Cartesian Product Model of Multiplication......Page 130
Properties of Whole-Number Multiplication......Page 132
The Repeated-Subtraction Model of Division......Page 134
The Missing-Factor Model of Division......Page 135
Division by Zero Is Undefined......Page 136
Division with Remainders......Page 137
Why Does Quotient with Remainder (the Division Algorithm) Work?......Page 138
Exponents and the Power Operation......Page 139
Chapter 2 Summary......Page 146
Chapter 2 Review Exercises......Page 150
3 Numeration and Computation......Page 152
The Egyptian System......Page 154
The Roman System......Page 155
The Babylonian System......Page 156
The Mayan System......Page 157
The Indo-Arabic System......Page 159
Physical Models for Positional Systems......Page 160
3.2 Algorithms for Addition and Subtraction of Whole Numbers......Page 164
The Addition Algorithm......Page 165
The Subtraction Algorithm......Page 169
3.3 Algorithms for Multiplication and Division of Whole Numbers......Page 174
Multiplication Algorithms......Page 175
Division Algorithms......Page 178
Easy Combinations......Page 184
Adjustment......Page 185
Working from Left to Right......Page 186
Estimation......Page 187
Rounding......Page 189
Approximating by Rounding......Page 190
Base-Five and Base-Six Place Value Including Conversion to the Decimal System and Operations......Page 194
Addition, Subtraction, and Multiplication in Base Six......Page 199
Chapter 3 Summary......Page 204
Chapter 3 Review Exercises......Page 207
4 Number Theory......Page 210
Divides, Divisors, Factors, Multiples......Page 212
Prime and Composite Numbers......Page 214
The Divisors of a Natural Number......Page 217
There Are Infinitely Many Primes......Page 218
Determining Whether a Given Natural Number Is Prime......Page 219
Divisibility by 2, 5, and 10......Page 224
Divisibilityby 4, 8, and Other Powers of 2......Page 225
Combining Divisibility Tests......Page 226
Applications of Divisibility......Page 228
Illustrating Factors and Divisibility with a Manipulative......Page 229
4.3 Greatest Common Divisors and Least Common Multiples......Page 232
GCD Method 2:Greatest Common Divisor from Prime Factorizations......Page 233
GCD Method 3: Greatest Common Divisor from the Euclidean Algorithm......Page 234
An Application of the Greatest Common Factor......Page 235
LCM Method 1: Least Common Multiples by Intersection of Sets......Page 236
LCM Method 2: Least Common Multiples from Prime Factorizations......Page 237
LCM Method 3:Least Common Multiples by Using the Euclidean Algorithm......Page 238
An Application of the LCM......Page 239
Chapter 4 Summary......Page 244
Chapter 4 Review Exercises......Page 246
5 Integers......Page 248
5.1 Representations of Integers......Page 250
Criteria for the Representation of the Integers......Page 251
Representing Integers with Colored Counters......Page 252
The Addition-by-0 Property with Colored Counters......Page 253
Mail-Time Representations of Integers......Page 254
Number-Line Representations of Integers......Page 255
Addition of Integers by Using Sets of Colored Counters......Page 259
Addition of Integers by Using Mail-Time Stories......Page 262
Addition of Integers by Usinga Number Line......Page 263
Subtraction of Integers with Colored Counters......Page 264
The Equivalence of Subtraction with Addition of the Opposite......Page 266
Subtraction of Integers by Using Mail-Time Stories......Page 267
Subtraction of Integers by Using the Number Line......Page 268
Ordering the Set of Integers......Page 270
Multiplication of Integers......Page 274
Multiplication of Integers by Using Loops of Colored Counters......Page 275
Multiplication of Integers by Using Mail-Time Stories......Page 276
Multiplicationof Integers by Using a Number Line......Page 277
Division of Integers......Page 278
Multiplication and Division with Colored-Counter Arrays......Page 280
Chapter 5 Summary......Page 285
Chapter 5 Review Exercises......Page 287
6 Fractions and Rational Numbers......Page 290
6.1 The Basic Concepts of Fractions and Rational Numbers......Page 292
Fraction Models......Page 293
Equivalent Fractions......Page 295
Fractions in Simplest Form......Page 297
Common Denominators......Page 298
Rational Numbers......Page 299
Ordering Fractions and Rational Numbers......Page 300
Addition of Fractions......Page 308
Proper Fractions and Mixed Numbers......Page 311
Subtraction of Fractions......Page 313
Multiplication of a Fraction by an Integer......Page 318
Multiplication of an Integer by a Fraction......Page 319
Multiplication of a Fraction by a Fraction......Page 320
Division of Fractions......Page 323
Algorithms for Calculating the Division of Fractions......Page 326
Reciprocals as Multiplicative Inverses in the Rational Numbers......Page 327
Properties of Addition and Subtraction......Page 333
Properties of Multiplication and Division......Page 335
Properties of the Order Relation......Page 336
The Density Property of Rational Numbers......Page 337
Estimations......Page 338
Mental Arithmetic......Page 339
Chapter 6 Summary......Page 345
Chapter 6 Review Exercises......Page 349
7 Decimals, Real Numbers, and Proportional Reasoning......Page 350
7.1 Decimals and Real Numbers......Page 352
Representations of Decimals......Page 353
Multiplying and Dividing Decimals by Powers of 10......Page 355
Terminating Decimals as Fractions......Page 356
Repeating Decimals and Rational Numbers......Page 358
The Set of Real Numbers......Page 361
Real Numbers and the Number Line......Page 363
Adding and Subtracting Decimals......Page 367
Multiplying Decimals......Page 369
Dividing Decimals......Page 371
Ratio......Page 375
Proportion......Page 378
Applications of Proportional Reasoning......Page 380
Percent......Page 388
Solving the Three Basic Types of Percent Problems......Page 391
Compound Interest......Page 393
The Mathematics of Growth......Page 395
Chapter 7 Summary......Page 400
Chapter 7 Review Exercises......Page 402
8 Algebraic Reasoning, Graphing, and Connections with Geometry......Page 404
Variables......Page 407
Algebraic Expressions......Page 409
Solving Equations......Page 410
Defining and Visualizing Functions......Page 412
Describing and Visualizing Functions......Page 413
The Cartesian Coordinate Plane......Page 420
The Distance Formula......Page 422
Slope......Page 423
Equations of Lines......Page 426
Nonlinear Functions......Page 429
8.3 Connections between Algebra and Geometry......Page 435
Parallel and Perpendicular Lines......Page 436
Circles......Page 442
Chapter 8 Summary......Page 446
Chapter 8 Review Exercises......Page 449
9 Geometric Figures......Page 452
Points and Lines......Page 458
Line Segments and the Distance between Points......Page 460
Rays, Angles,and Angle Measure......Page 461
Pairs of Angles and the Corresponding-Angles Theorem......Page 463
The Measure of Angles in Triangles......Page 465
Directed Angles......Page 466
Curves and Regions......Page 472
Polygonal Curves and Polygons......Page 475
Triangles......Page 480
Quadrilaterals......Page 481
Regular Polygons......Page 483
9.3 Figures in Space......Page 491
Curves, Surfaces, and Solids......Page 492
Polyhedra......Page 494
Regular Polyhedra......Page 496
Euler’s Formula for Polyhedra......Page 497
Cones and Cylinders......Page 501
Chapter 9 Summary......Page 508
Chapter 9 Review Exercises......Page 511
10 Measurement: Length, Area, and Volume......Page 514
10.1 The Measurement Process......Page 517
The U.S. Customary, or “English,” System of Measures......Page 520
Metric Units: The International System......Page 522
Length......Page 523
Volume and Capacity......Page 525
Weight and Mass......Page 526
Temperature......Page 527
Unit Analysis......Page 528
Measurements in Nonstandard Units......Page 532
The Congruence and Addition Properties of Area......Page 533
Areas of Polygons: A Conceptual Understanding......Page 536
Length of a Curve......Page 539
The Area of a Circle......Page 542
Proving the Pythagorean Theorem......Page 554
Applications of the Pythagorean Theorem......Page 555
The Converse of the Pythagorean Theorem......Page 556
10.4 Volume......Page 562
Volumes of Right Prisms and Right Cylinders......Page 563
Volumes of Pyramids and Cones......Page 564
Volume of a Sphere......Page 566
10.5 Surface Area......Page 569
Surface Area of Right Prisms and Cylinders......Page 570
Surface Area of Pyramids......Page 572
Surface Area of Right Circular Cones......Page 573
The Surface Area of a Sphere......Page 574
Comparing Measurements of Similar Figures......Page 575
Chapter 10 Summary......Page 580
Chapter 10 Review Exercises......Page 583
11 Transformations, Symmetries, and Tilings......Page 586
11.1 Rigid Motions and Similarity Transformations......Page 588
Translations......Page 589
Rotations......Page 590
Reflections......Page 591
Glide–Reflections......Page 593
The Net Outcome of Two Successive Reflections......Page 594
Classification of General Rigid Motions......Page 597
Dilations and Similarity Motions......Page 600
Reflection Symmetry......Page 609
Rotation Symmetry......Page 610
Point Symmetry......Page 611
Border Patterns and Their Classification......Page 612
Wallpaper Patterns......Page 615
11.3 Tilings and Escher-like Designs......Page 620
Regular Tilings of the Plane......Page 622
Semiregular Tilings of the Plane......Page 623
Tilings with Irregular Polygons......Page 624
Escher-like Designs......Page 626
Chapter 11 Summary......Page 634
Chapter 11 Review Exercises......Page 636
12 Congruence, Constructions, and Similarity......Page 640
Congruent Line Segments and Their Construction......Page 642
Corresponding Parts and the Congruence of Triangles......Page 643
The Side–Side–Side (SSS) Property......Page 644
The Triangle Inequality......Page 647
The Side–Angle–Side (SAS) Property......Page 648
The Angle–Side–Angle (ASA) Property......Page 651
The Angle–Angle–Side (AAS) Property......Page 652
Are There SSA and AAA Congruence Properties?......Page 653
Constructing Parallel and Perpendicular Lines......Page 659
Constructing the Midpoint and Perpendicular Bisector of a Line Segment......Page 661
Constructing the Angle Bisector......Page 664
Constructing Regular Polygons......Page 666
Mira™ and Paper-Folding Constructions......Page 668
Constructions with Geometry Software......Page 669
12.3 Similar Triangles......Page 675
The Angle–Angle–Angle (AAA) and Angle–Angle (AA) Similarity Properties......Page 676
The Side–Side–Side (SSS) Similarity Property......Page 677
The Side–Angle–Side (SAS) Similarity Property......Page 678
Geometric Problem Solving with Similar Triangles......Page 679
Chapter 12 Summary......Page 687
Chapter 12 Review Exercises......Page 690
13 Statistics: The Interpretation of Data......Page 694
13.1 Organizing and Representing Data......Page 696
Stem-and-Leaf Plots......Page 697
Histograms......Page 698
Line Graphs......Page 699
Bar Graphs......Page 701
Pie Charts......Page 702
Pictographs......Page 703
Choosing Good Visualizations......Page 704
Measures of Central Tendency......Page 712
The Mean......Page 713
The Mode......Page 715
Measures of Variability......Page 717
Box Plots......Page 720
The Standard Deviation......Page 721
13.3 Statistical Inference......Page 728
Populations and Samples......Page 729
Population Means and Standard Deviations......Page 730
Estimating Population Means and Standard Deviations......Page 731
Distributions......Page 732
z Scores and Percentiles......Page 734
Chapter 13 Summary......Page 740
Chapter 13 Review Exercises......Page 742
14 Probability......Page 746
The Sample Space, Events, and Probability Functions......Page 748
Experimental Probability......Page 750
Theoretical Probability......Page 751
Mutually Exclusive Events......Page 752
Complementary Events......Page 754
The Addition Principle of Counting......Page 760
Factorials and Rearrangements of Ordered Lists......Page 761
The Multiplication Principle of Counting......Page 762
Conditional Probability......Page 765
Independent Events......Page 767
14.3 Permutations and Combinations......Page 771
Formulas for the Number of r-Permutations......Page 773
Formulas for the Number of r-Combinations......Page 774
Solving Problems with Permutations and Combinations......Page 775
Odds......Page 784
Expected Value......Page 785
Geometric Probability......Page 787
Simulation......Page 788
Chapter 14 Summary......Page 794
Chapter 14 Review Exercises......Page 796
Answers to Odd-Numbered Problems......Page 798
Mathematical Lexicon......Page 849
Credits......Page 850
B......Page 852
C......Page 853
D......Page 854
F......Page 855
I......Page 856
M......Page 857
O......Page 858
P......Page 859
R......Page 860
S......Page 861
T......Page 862
Z......Page 863