List of Figures |
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xv | |
List of Tables |
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xix | |
About the Authors |
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xxi | |
Acknowledgments |
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xxiii | |
Series |
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xxv | |
Introduction |
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1 | (2) |
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1 | (1) |
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The Structure of the Book |
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1 | (2) |
1 Professional Learning and Three Grand Organizers for Arithmetic Instruction |
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3 | (12) |
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Scenario 1: School-Based Mathematics Leader |
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3 | (1) |
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Scenario 2: District-Wide Math Recovery® Leader |
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4 | (1) |
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Scenario 3: Mathematics Coach |
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4 | (1) |
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Three Grand Organizers for Arithmetic Instruction |
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5 | (10) |
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Guiding Principles for Instruction |
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5 | (3) |
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Domains of Arithmetic Knowledge |
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8 | (3) |
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Dimensions of Mathematizing |
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11 | (4) |
2 From Whole Numbers to Fractions |
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15 | (12) |
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The Reorganization Hypothesis |
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15 | (1) |
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16 | (1) |
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Fragmenting and Partitioning |
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17 | (1) |
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18 | (1) |
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19 | (1) |
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20 | (2) |
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22 | (2) |
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Introduction to Content Chapters |
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24 | (1) |
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24 | (1) |
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25 | (1) |
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25 | (2) |
3 Assessing Stages of Units Coordination |
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27 | (14) |
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Tasks for Assessing Units Coordination |
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27 | (6) |
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Rubric for Assessing Units Coordination |
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33 | (2) |
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35 | (6) |
4 Teaching Students at Stage 1: Fragmenting |
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41 | (16) |
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41 | (1) |
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Fragmenting and Partitioning |
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42 | (2) |
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Testing Equality of Shares |
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44 | (1) |
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Working with Students at the First Three Levels of Fragmenting |
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45 | (5) |
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Fragmenting Problem Type 1: Equal Sharing |
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46 | (1) |
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Fragmenting Problem Type 2: Drawing Unit Fractions |
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46 | (3) |
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Fragmenting Problem Type 3: Making Wholes from Parts |
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49 | (1) |
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50 | (3) |
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A4.1 Equal Sharing of Single Items |
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50 | (1) |
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A4.2 Making One Equal Share |
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50 | (1) |
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A4.3 Drawing a Unit Fraction of One Bar and Justifying the Amount |
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51 | (1) |
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A4.4 Making Connected Numbers |
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51 | (1) |
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A4.5 Drawing a Unit Fraction of a Segmented Bar and Justifying the Amount |
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52 | (1) |
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53 | (4) |
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IA4.1 Share an Energy Bar Fairly |
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53 | (1) |
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IA4.2 Make Only Your Own Share |
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54 | (1) |
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IA4.3 Slicing Cakes in a Bakery |
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55 | (1) |
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IA4.4 Working with Connected Numbers |
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55 | (2) |
5 Transitioning to Stage 2: Part-Whole Reasoning |
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57 | (14) |
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57 | (1) |
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Fractions as Parts Within Wholes |
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58 | (1) |
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Fractions as Parts Out of Wholes |
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59 | (1) |
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Working with Students with Parts Within Wholes Fraction Schemes |
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60 | (4) |
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64 | (2) |
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A5.1 Parts Within the Whole |
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64 | (1) |
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A5.2 Parts Out of the Whole |
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64 | (1) |
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A5.3 Finding an Equal Share |
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65 | (1) |
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65 | (1) |
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66 | (5) |
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66 | (1) |
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IA5.2 Tiny Creature Journeys |
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67 | (1) |
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68 | (1) |
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IA5.4 Comparing Unit Fractions |
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68 | (1) |
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IA5.5 Finding Unit Fractions of Chocolate Bars |
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69 | (2) |
6 Teaching Students at Stage 2: Measuring with |
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71 | (1) |
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71 | (1) |
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Unit Fractions and the Whole |
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72 | (1) |
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The Measurement Scheme for Unit Fractions |
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73 | (1) |
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73 | (1) |
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Reciprocal Relationship Between the Unit Fraction and the Whole |
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74 | (1) |
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Measuring Non-Unit Fractions |
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75 | (1) |
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Comparing Fractions as Quantities |
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75 | (2) |
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77 | (1) |
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Assessing Measurement Conceptions of Unit Fractions |
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77 | (1) |
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Instruction on Fractions as Measures |
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78 | (1) |
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78 | (2) |
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A6.1 Naming Unit Fractions |
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78 | (1) |
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A6.2 Producing Unit Fractions from the Whole |
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79 | (1) |
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A6.3 Producing the Whole from Unit Fractions |
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79 | (1) |
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A6.4 Producing Non-Unit Fractions from Unit Fractions |
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80 | (1) |
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80 | (5) |
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IA6.1 Comparing Unit Fractions |
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80 | (1) |
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IA6.2 Guess My Fraction (Fractions as Measures) |
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81 | (1) |
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IA6.3 Left, Right, or Just Right (Placing Fractions on a Number Line) |
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82 | (1) |
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1A6.4 Different but the Same (Commensurate Fractions) |
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82 | (3) |
7 Teaching Students at Stage 2: Reversible Reasoning |
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85 | (12) |
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85 | (1) |
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86 | (1) |
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87 | (1) |
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Reversible Reasoning with Proper Fractions |
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87 | (1) |
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Coordinating Three Levels of Units in Activity |
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88 | (1) |
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Assessing Reversible Reasoning |
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88 | (1) |
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Instruction on Reversible Reasoning |
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89 | (1) |
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89 | (3) |
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90 | (1) |
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A7.2 Reversing Proper Fractions with Rods |
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90 | (1) |
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A7.3 Reversing Proper Fractions with Strips |
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91 | (1) |
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A7.4 Reversible Multiplicative Reasoning |
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91 | (1) |
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92 | (5) |
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92 | (1) |
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IA7.2 My Number, Your Number |
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93 | (1) |
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IA7.3 My Fraction, Your Fraction |
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93 | (1) |
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94 | (1) |
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IA7.5 Reversing Multiplicative Reasoning |
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95 | (2) |
8 Teaching Students at Stage 3: Fractions as Numbers |
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97 | (14) |
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97 | (1) |
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The Iterative Fraction Scheme: Fractions as Numbers |
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98 | (1) |
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Assessing an Iterative Fraction Scheme |
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98 | (1) |
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Promoting an Iterative Fraction Scheme: Magic Candy Bars |
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99 | (2) |
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Promoting an Iterative Fraction Scheme: Fraction Comparisons and Drawing Longer Bars |
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101 | (1) |
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Tasks to Solidify Conceiving of Fractions as Numbers in Their Own Right |
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102 | (1) |
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103 | (3) |
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A8.1 Drawing 'Large' Fractions |
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103 | (1) |
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A8.2 Drawing the Whole Given a Fraction Larger than the Whole |
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104 | (1) |
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A8.3 Drawing a Mixed Number |
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104 | (1) |
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105 | (1) |
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A8.5 Drawing a Longer Fruit Strip |
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105 | (1) |
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106 | (1) |
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106 | (5) |
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106 | (1) |
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IA8.2 Fraction Comparison Game |
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107 | (1) |
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IA8.3 Drawing a Longer Fruit Strip |
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108 | (1) |
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IA8.4 Making a Fraction Number Sequence |
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108 | (3) |
9 Teaching Students at Stages 2 and 3: Equal Sharing of Multiple Items |
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111 | (14) |
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111 | (1) |
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Equally Sharing Multiple Items and the. Fifth Level of Fragmenting (Stage 3 Students) |
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112 | (2) |
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Equally Sharing Multiple Items: Using Repeated Halving |
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114 | (1) |
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Equally Sharing Multiple Items: Sharing Whole Bars First, and Then Partitioning a Single Bar or Using Repeated Halving |
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115 | (1) |
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Equally Sharing Multiple Items: Bricolage Solutions for Harder Problems |
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116 | (2) |
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Equally Sharing Multiple Items: Solutions Involving Multiplication and Division |
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118 | (1) |
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118 | (5) |
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A9.1 Equally Sharing 3 Cookie Dough Logs among 4 Bakers |
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119 | (1) |
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A9.2 Equally Sharing 6 Fruit Bars among 5 Friends |
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120 | (1) |
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A9.3 Equally Sharing 5 Sub Sandwiches among 7 People |
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121 | (1) |
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A9.4 Equally Sharing 3 Different Cakes among 5 People |
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122 | (1) |
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123 | (2) |
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IA9.1 Equally Sharing Multiple Items of the Same Size |
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123 | (1) |
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IA9.2 Equally Sharing Multiple Items of Different Sizes |
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124 | (1) |
10 Teaching Students at Stages 2 and 3: Multiplying Fractions |
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125 | (18) |
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125 | (1) |
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Taking Unit Fractions of Unit Fractions |
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125 | (3) |
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Taking Non-Unit Fractions of Unit Fractions |
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128 | (1) |
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Taking Unit Fractions of Non-Unit Fractions |
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129 | (1) |
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Taking Fractions of Fractions, Generally |
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130 | (3) |
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133 | (3) |
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133 | (1) |
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A10.2 Taking Fractions of Unit Fractions |
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134 | (1) |
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A10.3 Taking Unit Fractions of Fractions |
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134 | (2) |
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A10.4 Taking Fractions of Fractions |
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136 | (1) |
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136 | (7) |
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136 | (1) |
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IA10.2 Taking Fractions of Fractions I |
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137 | (1) |
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IA10.3 Making 1/4 of a Recipe |
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138 | (1) |
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IA10.4 Taking Fractions of Fractions II |
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139 | (1) |
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IA10.5 Notating and Articulating Patterns in Reasoning |
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140 | (3) |
11 Teaching Students at Stages 2 and 3: Adding and Subtracting Fractions |
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143 | (18) |
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143 | (2) |
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Transforming Bars to Show Multiple Fractions |
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145 | (1) |
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Pulling Two Fractions Out of the Same Bar with a Co-Measurement |
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146 | (2) |
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Commensurate Fractions More Generally |
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148 | (3) |
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151 | (4) |
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152 | (1) |
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A11.2 Pulling Tim Fractions from the Same Bar |
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153 | (1) |
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154 | (1) |
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A11.4: Naming Fractions the Same Size |
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154 | (1) |
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155 | (6) |
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IA11.1 Transforming Bars I |
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155 | (1) |
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IA11.2 nansforming Bars II |
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156 | (1) |
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IA11.3 Add and/or Subtract |
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157 | (1) |
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IA11.4 Making Same-Size Fractions with Smaller Parts |
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158 | (1) |
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IA11.5 Making Same-Size Fractions with Larger Parts |
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159 | (2) |
12 Teaching Students at Stages 2 and 3: Dividing Fractions |
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161 | (14) |
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161 | (1) |
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Fraction Division Problems with a Measurement Meaning |
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161 | (2) |
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A Multiplicative Solution to Fraction Division Problems with a Measurement Meaning |
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163 | (2) |
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A Quantitative Meaning for a Reciprocal |
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165 | (1) |
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Problems to Elicit Multiplicative Solutions of Fraction Measurement Division Problems |
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166 | (1) |
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167 | (3) |
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A12.1 Drawing Pictures of Whole Number Division |
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167 | (1) |
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A12.2 Meanings of Division |
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167 | (1) |
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A12.3 Writing Division Stories |
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168 | (1) |
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169 | (1) |
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170 | (5) |
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IA12.1 Meanings of Division |
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170 | (1) |
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IA12.2 Writing Division Stories |
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171 | (1) |
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IA12.3 Solving Measurement Division Problems I |
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171 | (1) |
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172 | (1) |
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IA12.5 Solving Measurement Division Problems II |
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173 | (2) |
13 From Fractions to Algebra |
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175 | (10) |
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Fractions and Algebra: Similar Ways of Thinking |
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175 | (1) |
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176 | (1) |
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Multiplicative Reasoning with Quantitative Unknowns |
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177 | (4) |
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The 'Other' Fraction Division |
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181 | (1) |
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Working on the Algebraic Meaning of Fraction Division |
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182 | (3) |
Glossary |
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185 | (4) |
Appendix |
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189 | (2) |
References |
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191 | (4) |
Index |
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195 | |