Preface |
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xx | |
Acknowledgments |
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xxi | |
About the Author |
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xxi | |
Chapter 1 How Students Learn Math and What Math We Want Them to Learn |
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1 | (14) |
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1 | (1) |
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1 | (1) |
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Research on Mathematical Learning |
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2 | (3) |
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The Importance of Conceptual Understanding |
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2 | (1) |
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A Constructivist Approach |
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3 | (1) |
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4 | (1) |
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5 | (1) |
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What Mathematics We Want Students to Learn |
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5 | (5) |
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Differing Perspectives on What Mathematics Is |
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5 | (2) |
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Differing Perspectives on What Mathematics Is Valued and at What Grade Levels |
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7 | (2) |
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9 | (1) |
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Research on Mathematics Teaching |
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10 | (2) |
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Importance of a Teacher's Pedagogical Content Knowledge |
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10 | (1) |
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Research About Classroom Environment |
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11 | (1) |
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Applying What You've Learned |
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12 | (1) |
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13 | (2) |
Chapter 2 Focusing Instruction on Big Ideas and Mathematical Processes |
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15 | (20) |
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15 | (1) |
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15 | (1) |
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Organizing Content Around Big Ideas |
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16 | (5) |
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Different Approaches to Big Ideas |
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16 | (1) |
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Big Ideas Applying to Many Mathematical Domains |
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17 | (3) |
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Examples of More Domain-Based Big Ideas |
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20 | (1) |
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Focusing Instruction on Proportional Reasoning, Algebraic Reasoning, and Spatial Reasoning |
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21 | (4) |
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21 | (2) |
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23 | (1) |
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23 | (2) |
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Focusing Instruction on Math Processes and Practices |
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25 | (7) |
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Standards for Mathematical Practice |
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25 | (1) |
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25 | (2) |
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27 | (2) |
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Tools and Representations |
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29 | (2) |
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Communication and Discourse |
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31 | (1) |
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Assessing Student Understanding |
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32 | (1) |
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Applying What You've Learned |
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32 | (1) |
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32 | (3) |
Chapter 3 Assessment and Evaluation |
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35 | (28) |
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35 | (1) |
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36 | (1) |
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36 | (2) |
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37 | (1) |
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37 | (1) |
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37 | (1) |
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How Assessment Has Changed |
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38 | (1) |
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Characteristics of Good Assessment |
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38 | (2) |
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38 | (2) |
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Sources of Assessment Data |
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40 | (11) |
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40 | (1) |
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Conversations and Interviews |
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41 | (1) |
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42 | (2) |
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44 | (1) |
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44 | (1) |
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45 | (1) |
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46 | (3) |
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49 | (2) |
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51 | (1) |
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Assessing Practices and Processes |
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51 | (3) |
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Assessing Problem Solving |
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51 | (1) |
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52 | (2) |
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54 | (1) |
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55 | (1) |
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Using Technology in Assessment |
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55 | (1) |
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Interpreting Assessment Results |
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56 | (1) |
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56 | (2) |
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Anecdotal Versus Letter Grades or Percents |
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57 | (1) |
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57 | (1) |
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Pitfalls of Percent Grades |
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57 | (1) |
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Including Group Evaluations |
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58 | (1) |
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Evaluations Based on Observation Assessment Data |
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58 | (1) |
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58 | (2) |
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Preparing Students for Large-Scale Assessment |
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59 | (1) |
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Applying What You've Learned |
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60 | (1) |
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61 | (2) |
Chapter 4 Planning Instruction |
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63 | (26) |
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63 | (1) |
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63 | (1) |
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Math Anxiety: A Special Challenge in Teaching Mathematics |
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64 | (1) |
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64 | (1) |
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Diminishing Anxiety in Students |
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65 | (1) |
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Diminishing Anxiety in Teachers |
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65 | (1) |
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Unit Planning and Lesson Planning |
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65 | (2) |
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65 | (1) |
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66 | (1) |
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67 | (1) |
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67 | (3) |
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The Importance of Varied Lesson Styles |
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67 | (1) |
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68 | (1) |
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Specific Lesson Strategies |
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69 | (1) |
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70 | (3) |
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Heterogeneous and Homogeneous Grouping |
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70 | (1) |
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71 | (1) |
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71 | (1) |
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Suitable Tasks for Group Work |
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71 | (1) |
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71 | (1) |
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71 | (1) |
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Balancing Whole-Class, Small-Group, and Individual Instruction |
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72 | (1) |
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73 | (1) |
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Planning Opportunities for Practice |
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73 | (2) |
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73 | (1) |
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What Should Be Practiced? |
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73 | (1) |
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How Much Practice Is Needed? |
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74 | (1) |
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What Should Practice Look Like? |
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74 | (1) |
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Differentiating Instruction: Supporting Individual Differences |
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75 | (10) |
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Moving from the Concrete to the Symbolic |
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75 | (2) |
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77 | (2) |
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79 | (1) |
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Students with Learning Disabilities |
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79 | (2) |
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81 | (3) |
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For Those Seeking Even More |
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84 | (1) |
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85 | (1) |
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85 | (1) |
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Making Contact with Parents |
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86 | (1) |
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Applying What You've Learned |
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86 | (1) |
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87 | (2) |
Chapter 5 Early Number |
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89 | (20) |
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89 | (1) |
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89 | (1) |
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Numbers in the Child's World |
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90 | (1) |
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90 | (1) |
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91 | (1) |
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91 | (3) |
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92 | (1) |
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93 | (1) |
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93 | (1) |
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94 | (4) |
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Comparing and Relating Numbers |
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94 | (1) |
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One-to-One Correspondence |
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94 | (1) |
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Position in the Counting Sequence |
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95 | (1) |
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95 | (1) |
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95 | (1) |
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96 | (2) |
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98 | (1) |
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98 | (1) |
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Reading and Writing Numerals |
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99 | (2) |
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Common Errors and Misconceptions |
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101 | (1) |
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Appropriate Manipulatives |
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102 | (3) |
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Appropriate Children's Books |
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105 | (1) |
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Assessing Student Understanding |
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106 | (1) |
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Applying What You've Learned |
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107 | (1) |
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107 | (2) |
Chapter 6 Early Operations |
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109 | (28) |
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109 | (1) |
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109 | (1) |
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How the Four Operations Are Related |
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110 | (1) |
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110 | (6) |
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110 | (1) |
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111 | (1) |
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Understanding Addition and Subtraction Situations |
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112 | (1) |
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Solving Addition and Subtraction Number Stories |
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113 | (1) |
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Relating Addition and Subtraction |
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113 | (1) |
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Addition and Subtraction Principles |
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114 | (2) |
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Common Errors and Misconceptions |
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116 | (1) |
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Appropriate Manipulatives |
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117 | (1) |
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Multiplication and Division |
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118 | (13) |
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Meanings of Multiplication |
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119 | (2) |
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121 | (2) |
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Understanding Multiplication and Division Situations |
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123 | (1) |
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Multiplication and Division Situations |
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124 | (1) |
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Solving Multiplication and Division Problems |
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124 | (1) |
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Relating Multiplication and Division |
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125 | (1) |
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Multiplication and Division Principles |
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125 | (6) |
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Common Errors and Misconceptions |
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131 | (1) |
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Appropriate Manipulatives |
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132 | (1) |
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Appropriate Children's Books |
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133 | (1) |
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Assessing Student Understanding |
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133 | (1) |
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Applying What You've Learned |
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133 | (1) |
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134 | (3) |
Chapter 7 Developing Fact Fluency |
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137 | (12) |
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137 | (1) |
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137 | (1) |
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138 | (1) |
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Why Is It Important to Learn the Facts? |
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138 | (1) |
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What Does Fact Fluency Mean? |
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138 | (1) |
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Learning Addition and Subtraction Facts |
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139 | (3) |
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Typical Sequence for Acquiring Addition and Subtraction Facts |
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141 | (1) |
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Learning Multiplication and Division Facts |
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142 | (3) |
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Typical Sequence for Acquiring Multiplication and Division Facts |
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145 | (1) |
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What Kind of Practice Is Useful? |
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145 | (1) |
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How Should Facts Be Assessed? |
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How Should They Not Be Assessed? |
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145 | (1) |
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Common Errors and Misconceptions |
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146 | (1) |
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Appropriate Manipulatives |
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147 | (1) |
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Appropriate Children's Books |
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147 | (1) |
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Assessing Student Understanding |
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147 | (1) |
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Applying What You've Learned |
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148 | (1) |
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148 | (1) |
Chapter 8 Representing Larger Whole Numbers |
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149 | (24) |
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149 | (1) |
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149 | (1) |
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Larger Numbers in the Students' World |
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150 | (1) |
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150 | (6) |
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Counting Based on Place Value |
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156 | (1) |
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156 | (2) |
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157 | (1) |
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157 | (1) |
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158 | (1) |
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Common Errors and Misconceptions |
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158 | (1) |
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Appropriate Manipulatives |
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159 | (1) |
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160 | (8) |
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160 | (1) |
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161 | (1) |
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162 | (4) |
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Prime and Composite Numbers |
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166 | (2) |
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Common Errors and Misconceptions |
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168 | (1) |
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Appropriate Manipulatives |
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168 | (1) |
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Appropriate Children's Books |
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169 | (1) |
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Assessing Student Understanding |
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169 | (1) |
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Applying What You've Learned |
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170 | (1) |
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170 | (3) |
Chapter 9 Estimation and Calculation Strategies with Larger Whole Numbers |
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173 | (36) |
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173 | (1) |
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173 | (1) |
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174 | (12) |
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Estimating Sums and Differences |
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174 | (2) |
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Varied Approaches to Addition and Subtraction |
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176 | (9) |
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Communicating About Adding and Subtracting |
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185 | (1) |
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Common Errors and Misconceptions |
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186 | (1) |
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Appropriate Manipulatives |
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187 | (2) |
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Multiplication and Division |
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189 | (13) |
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Multiplying and Dividing Using Powers of 10 |
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189 | (2) |
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Estimating Products and Quotients |
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191 | (2) |
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Algorithms for Multiplication and Division |
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193 | (8) |
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Communicating About Multiplying and Dividing |
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201 | (1) |
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Common Errors and Misconceptions |
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202 | (2) |
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Appropriate Manipulatives |
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204 | (1) |
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Appropriate Children's Books |
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205 | (1) |
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Assessing Student Understanding |
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205 | (2) |
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Applying What You've Learned |
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207 | (1) |
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208 | (1) |
Chapter 10 Fractions |
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209 | (38) |
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209 | (1) |
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209 | (1) |
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Representing and Comparing Fractions |
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210 | (12) |
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Fractions in the Child's World |
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210 | (1) |
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How the Fraction Meanings Are Equivalent |
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211 | (1) |
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Fraction Definitions and Principles |
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212 | (4) |
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216 | (1) |
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Mixed Numbers and Fractions |
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217 | (1) |
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Visualizing with Fractions |
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218 | (1) |
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What Students Might Learn When Comparing Fractions |
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218 | (4) |
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Relating Fractions to Decimals |
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222 | (1) |
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Common Errors and Misconceptions |
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222 | (2) |
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Appropriate Manipulatives |
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224 | (3) |
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227 | (13) |
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Adding and Subtracting Fractions |
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227 | (6) |
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Multiplying and Dividing Fractions |
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233 | (7) |
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Common Errors and Misconceptions |
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240 | (1) |
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Appropriate Manipulatives |
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241 | (1) |
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Appropriate Children's Books |
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242 | (1) |
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Assessing Student Understanding |
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242 | (1) |
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Applying What You've Learned |
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243 | (1) |
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244 | (3) |
Chapter 11 Decimals |
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247 | (22) |
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247 | (1) |
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247 | (1) |
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248 | (9) |
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249 | (1) |
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249 | (3) |
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252 | (1) |
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Equivalent Decimals and Precision |
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252 | (1) |
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Equivalent Fractions and Decimals |
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252 | (2) |
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Rounding and Estimating Decimals |
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254 | (1) |
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Reading and Writing Decimals |
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255 | (1) |
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255 | (2) |
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257 | (5) |
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Adding and Subtracting Decimals |
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257 | (2) |
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Multiplying and Dividing Decimals |
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259 | (3) |
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Common Errors and Misconceptions |
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262 | (1) |
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Appropriate Manipulatives |
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263 | (3) |
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Appropriate Children's Books |
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266 | (1) |
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Assessing Student Understanding |
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266 | (1) |
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Applying What You've Learned |
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267 | (1) |
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268 | (1) |
Chapter 12 Ratio and Proportion |
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269 | (24) |
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269 | (1) |
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269 | (1) |
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270 | (1) |
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270 | (7) |
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271 | (3) |
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274 | (3) |
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277 | (2) |
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278 | (1) |
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278 | (1) |
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Role of Proportional Reasoning in Other Strands |
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279 | (2) |
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279 | (1) |
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280 | (1) |
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280 | (1) |
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280 | (1) |
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281 | (1) |
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281 | (1) |
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281 | (6) |
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281 | (6) |
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Common Errors and Misconceptions |
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287 | (1) |
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Appropriate Manipulatives |
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288 | (1) |
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Appropriate Children's Books |
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289 | (1) |
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Assessing Student Understanding |
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289 | (1) |
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Applying What You've Learned |
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290 | (1) |
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291 | (2) |
Chapter 13 Extending the Number System to Negative and Irrational Numbers |
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293 | (22) |
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293 | (1) |
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293 | (1) |
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294 | (1) |
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294 | (1) |
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Reading and Writing Integers |
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294 | (1) |
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295 | (1) |
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295 | (1) |
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296 | (2) |
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298 | (2) |
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300 | (3) |
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303 | (2) |
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305 | (1) |
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305 | (3) |
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Using Powers of 10 to Give a Sense of Number Size |
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306 | (1) |
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Order of Operations Including Exponents |
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307 | (1) |
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Rational Versus Irrational Numbers |
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308 | (1) |
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Common Errors and Misconceptions |
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309 | (1) |
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Appropriate Manipulatives |
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310 | (2) |
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Appropriate Children's Books |
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312 | (1) |
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Assessing Student Understanding |
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312 | (1) |
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Applying What You've Learned |
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313 | (1) |
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314 | (1) |
Chapter 14 Patterns and Algebra |
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315 | (38) |
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315 | (1) |
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315 | (1) |
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316 | (8) |
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316 | (3) |
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Describing and Extending Patterns |
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319 | (3) |
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322 | (1) |
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Mathematical Situations Rich in Patterns |
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322 | (1) |
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Using Patterns to Develop Mathematical Concepts |
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323 | (1) |
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Common Errors and Misconceptions |
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324 | (1) |
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Appropriate Manipulatives |
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325 | (2) |
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327 | (12) |
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Moving from Pattern to Algebra |
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327 | (2) |
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Algebra as Generalizing Number Relationships |
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329 | (1) |
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330 | (3) |
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333 | (1) |
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Using Graphs to Describe Relationships |
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334 | (5) |
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339 | (6) |
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Common Errors and Misconceptions |
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345 | (2) |
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Appropriate Manipulatives |
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347 | (1) |
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Appropriate Children's Books |
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348 | (1) |
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Assessing Student Understanding |
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349 | (1) |
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Applying What You've Learned |
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349 | (1) |
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350 | (3) |
Chapter 15 3-D and 2-D Shapes |
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353 | (48) |
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353 | (1) |
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354 | (1) |
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Fundamental Aspects of Geometry |
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354 | (1) |
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Development of Geometric Thinking |
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355 | (2) |
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Identifying and Classifying Shapes |
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357 | (1) |
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Identification of 2-D and 3-D Shapes |
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357 | (1) |
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Geometric Attributes and Properties |
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358 | (3) |
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358 | (1) |
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Sorting and Patterning with Shapes |
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359 | (2) |
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361 | (2) |
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Polyhedrons, Spheres, Cones, and Cylinders |
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361 | (1) |
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362 | (1) |
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Using Properties to Classify 2-D Shapes |
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363 | (4) |
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363 | (1) |
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363 | (4) |
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367 | (1) |
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367 | (1) |
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Lines, Segments, and Rays |
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368 | (1) |
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369 | (4) |
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Reflection Symmetry of 2-D and 3-D Shapes |
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369 | (2) |
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Rotational Symmetry of 2-D and 3-D Shapes |
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371 | (2) |
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373 | (7) |
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373 | (4) |
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377 | (1) |
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378 | (2) |
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Decomposing and Composing Shapes |
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380 | (5) |
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381 | (2) |
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383 | (2) |
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Congruence and Similarity |
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385 | (3) |
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386 | (2) |
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388 | (1) |
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388 | (3) |
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Determining Similarity of 2-D Shapes |
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389 | (1) |
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Constructing Similar 2-D Shapes |
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|
389 | (2) |
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Common Errors and Misconceptions |
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|
391 | (3) |
|
Appropriate Manipulatives |
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|
394 | (3) |
|
Appropriate Children's Books |
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|
397 | (1) |
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Assessing Student Understanding |
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|
397 | (1) |
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Applying What You've Learned |
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|
398 | (1) |
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|
399 | (2) |
Chapter 16 Location and Movement |
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401 | (26) |
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|
401 | |
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|
40 | (362) |
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402 | (1) |
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Developing Positional Vocabulary |
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|
402 | (2) |
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Using Dance, Song, and Play |
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|
402 | (1) |
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|
402 | (2) |
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Maps and Coordinate Grids |
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|
404 | (2) |
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Drawing and Interpreting Maps |
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404 | (1) |
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404 | (2) |
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406 | (12) |
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Euclidean Transformations-Slides, Flips, and Turns |
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|
406 | (1) |
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Transformations on Simple Grids |
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407 | (1) |
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Transformations on Coordinate Grids |
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|
407 | (1) |
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407 | (2) |
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|
409 | (3) |
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412 | (2) |
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|
414 | (1) |
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Non-Euclidean Transformations-Dilations |
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414 | (2) |
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416 | (1) |
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|
417 | (1) |
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Common Errors and Misconceptions |
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|
418 | (3) |
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Appropriate Manipulatives |
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|
421 | (1) |
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Appropriate Children's Books |
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|
422 | (1) |
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Assessing Student Understanding |
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|
422 | (1) |
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Applying What You've Learned |
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|
423 | (1) |
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|
424 | (3) |
Chapter 17 The Nature of Measurement, with a Focus on Length and Area |
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427 | (50) |
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427 | (1) |
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|
427 | (1) |
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The Nature of Measurement |
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|
428 | (2) |
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Addressing Measurement Principles in the Classroom |
|
|
429 | (1) |
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Stages of Measurement Instruction |
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|
430 | (2) |
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|
432 | (1) |
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The Imperial System of Measurement |
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|
432 | (1) |
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The Metric System of Measurement |
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433 | (1) |
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SI-Derived and Other Units |
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|
434 | (1) |
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|
434 | (1) |
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|
434 | (12) |
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Introducing Length Concepts |
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|
435 | (1) |
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Length: The Definition/Comparison Stage |
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|
435 | (1) |
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Length: The Nonstandard Unit Stage |
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|
436 | (3) |
|
Length: The Standard Unit Stage |
|
|
439 | (5) |
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|
444 | (2) |
|
Common Errors and Misconceptions |
|
|
446 | (3) |
|
Appropriate Manipulatives |
|
|
449 | (2) |
|
|
451 | (17) |
|
Introducing Area Concepts |
|
|
451 | (1) |
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Area: The Definition/Comparison Stage |
|
|
451 | (1) |
|
Area: The Nonstandard Unit Stage |
|
|
452 | (5) |
|
Area: The Standard Unit Stage |
|
|
457 | (3) |
|
Formulas for Calculating Areas of Shapes |
|
|
460 | (7) |
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|
467 | (1) |
|
Common Errors and Misconceptions |
|
|
468 | (3) |
|
Appropriate Manipulatives |
|
|
471 | (2) |
|
Appropriate Children's Books |
|
|
473 | (1) |
|
Assessing Student Understanding |
|
|
474 | (1) |
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Applying What You've Learned |
|
|
474 | (1) |
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|
475 | (2) |
Chapter 18 Volume, Mass, Time, and Angles |
|
477 | (52) |
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|
477 | (1) |
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|
477 | (1) |
|
Measuring Volume and Mass |
|
|
478 | (1) |
|
|
478 | (13) |
|
Volume: The Definition/Comparison Stage |
|
|
479 | (1) |
|
Volume: The Nonstandard Unit Stage |
|
|
480 | (4) |
|
Volume: The Standard Unit Stage |
|
|
484 | (2) |
|
|
486 | (1) |
|
Solving Problems Involving Volume |
|
|
487 | (4) |
|
Introducing Mass Concepts |
|
|
491 | (6) |
|
Mass: The Definition/Comparison Stage |
|
|
492 | (1) |
|
Mass: The Nonstandard Unit Stage |
|
|
493 | (2) |
|
Mass: The Standard Unit Stage |
|
|
495 | (2) |
|
Common Errors and Misconceptions |
|
|
497 | (3) |
|
Appropriate Manipulatives |
|
|
500 | (2) |
|
|
502 | (1) |
|
Introducing Time Concepts |
|
|
503 | (8) |
|
Time: The Definition/Comparison Stage |
|
|
505 | (1) |
|
Time: The Nonstandard Unit Stage |
|
|
505 | (1) |
|
Time: The Standard Unit Stage |
|
|
506 | (4) |
|
Measuring and Calculating Elapsed Time |
|
|
510 | (1) |
|
Common Errors and Misconceptions |
|
|
511 | (2) |
|
Appropriate Manipulatives |
|
|
513 | (1) |
|
|
513 | (2) |
|
Introducing Angle Concepts |
|
|
515 | (6) |
|
Angles: The Definition/Comparison Stage |
|
|
516 | (1) |
|
Angles: The Nonstandard Unit Stage |
|
|
516 | (2) |
|
Angles: The Standard Unit Stage |
|
|
518 | (3) |
|
Common Errors and Misconceptions |
|
|
521 | (2) |
|
Appropriate Manipulatives |
|
|
523 | (2) |
|
Appropriate Children's Books |
|
|
525 | (1) |
|
Assessing Student Understanding |
|
|
525 | (1) |
|
Applying What You've Learned |
|
|
526 | (1) |
|
|
527 | (2) |
Chapter 19 Data |
|
529 | (58) |
|
|
529 | (1) |
|
|
529 | (1) |
|
|
530 | (4) |
|
|
530 | (1) |
|
|
530 | (1) |
|
Describing a Sorting Rule |
|
|
531 | (2) |
|
|
533 | (1) |
|
Common Errors and Misconceptions |
|
|
534 | (1) |
|
Appropriate Manipulatives |
|
|
535 | (1) |
|
|
536 | (3) |
|
|
536 | (1) |
|
Asking Good Survey Questions |
|
|
536 | (1) |
|
Choosing Data Collection Topics |
|
|
537 | (1) |
|
|
538 | (1) |
|
Factors Influencing Data Collection |
|
|
538 | (1) |
|
Common Errors and Misconceptions |
|
|
539 | (1) |
|
Appropriate Manipulatives |
|
|
540 | (1) |
|
|
540 | (20) |
|
|
540 | (17) |
|
Appropriate Use of Data Display Formats |
|
|
557 | (3) |
|
Common Errors and Misconceptions |
|
|
560 | (5) |
|
Appropriate Manipulatives |
|
|
565 | (2) |
|
|
567 | (8) |
|
|
567 | (3) |
|
Drawing Inferences from Graphs |
|
|
570 | (2) |
|
|
572 | (3) |
|
Common Errors and Misconceptions |
|
|
575 | (1) |
|
|
576 | (5) |
|
Measures of Central Tendency |
|
|
576 | (3) |
|
|
579 | (1) |
|
|
580 | (1) |
|
Common Errors and Misconceptions |
|
|
581 | (1) |
|
Appropriate Manipulatives |
|
|
582 | (1) |
|
Appropriate Children's Books |
|
|
582 | (1) |
|
Assessing Student Understanding |
|
|
582 | (1) |
|
Applying What You've Learned |
|
|
583 | (1) |
|
|
584 | (3) |
Chapter 20 Probability |
|
587 | (20) |
|
|
587 | (1) |
|
|
587 | (1) |
|
Fundamental Notions About Probability |
|
|
588 | (1) |
|
Probability Misconceptions |
|
|
588 | (1) |
|
Early Work with Probability |
|
|
589 | (1) |
|
|
589 | (2) |
|
Outcomes, Events, and Sample Space |
|
|
591 | (1) |
|
Simple and Compound Events |
|
|
591 | (1) |
|
Independent and Dependent Compound Events |
|
|
591 | (1) |
|
Probabilities as Ratios and Fractions |
|
|
592 | (1) |
|
|
592 | (3) |
|
|
593 | (1) |
|
Making Predictions Based on Experimental Results |
|
|
594 | (1) |
|
|
595 | (4) |
|
The Transition from Experimental to Theoretical for Compound Events |
|
|
595 | (2) |
|
Reconciling Theoretical and Experimental Probability |
|
|
597 | (1) |
|
Determining Theoretical Probability |
|
|
597 | (2) |
|
Common Errors and Misconceptions |
|
|
599 | (2) |
|
Appropriate Manipulatives |
|
|
601 | (2) |
|
Appropriate Children's Books |
|
|
603 | (1) |
|
Assessing Student Understanding |
|
|
603 | (1) |
|
Applying What You've Learned |
|
|
604 | (1) |
|
|
605 | (2) |
Index |
|
607 | |