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| Preface | |
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| Acknowledgments | |
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| Background | |
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| Visualization in Math | |
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| Focusing on the Important Math | |
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| Building Mathematical Communication | |
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| Grades K-2 | |
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| Counting Up by 1s | |
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| Counting Back by 1s | |
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| Counting Up by 2s | |
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| Counting Back by 2s | |
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| Counting Up by 5s | |
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| Counting Back by 5s | |
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| Counting Up by 10s | |
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| Counting Back by 10s | |
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| Comparing Numbers by Matching | |
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| Benchmark Numbers: All About 5 | |
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| Benchmark Numbers: All About 10 | |
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| Ordinal Numbers | |
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| Addition as Combining | |
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| Addition to Describe Part-Part-Whole Situations | |
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| Adding 0 and Adding 1 | |
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| Addition: Commutativity | |
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| Addition: Changing Addends, But Not the Sum | |
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| Adding or Subtracting 10 | |
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| Subtraction as Taking Away | |
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| Subtraction to Compare | |
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| Relating Addition and Subtraction | |
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| Naming Two-Digit Numbers | |
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| Naming Three-Digit Numbers | |
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| Place Value: Grouping in Tens | |
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| Place Value: Positions | |
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| Comparing Sizes of Numbers | |
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| Fractions: Halves | |
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| Fractions: Quarters | |
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| Measurement: Meaning of Length | |
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| Measurement: Effect of Unit Size | |
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| Measurement: Standard Units of Length | |
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| 2-D Shapes Versus 3-D Shapes | |
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| Comparing 2-D Shapes | |
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| Comparing 3-D Shapes | |
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| Composing Shapes | |
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| Shape Puzzles | |
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| Grades 3-5 | |
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| Multiplication: Equal Groups | |
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| Multiplication: Commutativity | |
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| Multiplication: The Distributive Principle | |
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| Multiplication: 2-Digit by 2-Digit | |
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| Division as Equal Groups or Sharing | |
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| Division: Remainders | |
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| Rounding Numbers | |
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| Place Value: Multiplying and Dividing by Powers of 10 | |
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| Place Value: Renaming Numbers | |
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| Factors: What They Are | |
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| Factors: Come in Pairs | |
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| Fractions: Representing | |
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| Fractions: Equivalence | |
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| Fractions: Comparing | |
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| Fractions: Mixed Number/Improper Fraction Relationship | |
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| Fractions: Common Denominators | |
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| Adding Fractions | |
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| Multiplying Fractions | |
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| Fractions: Multiplying as Resizing | |
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| Fractions as Division | |
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| Decimals: Relating Hundredths to Tenths | |
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| Decimals: Equivalence | |
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| Decimals: Adding and Subtracting | |
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| Measurement: Time Intervals | |
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| Measurement: Area of Rectangles | |
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| Perimeter versus Area | |
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| Measurement Conversions | |
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| Graphs with Scales | |
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| Coordinate Grids | |
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| Classification of Shapes | |
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| Parallel and Perpendicular Lines | |
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| Lines of Symmetry | |
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| Patterns versus Non-patterns | |
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| Algebraic Thinking: Growing Additively | |
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| Algebraic Thinking: Shrinking Additively | |
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| Algebraic Thinking: Growing Multiplicatively | |
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| Grades 6-8 | |
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| Common Factors | |
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| Common Multiples | |
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| Square Roots | |
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| Fraction Division | |
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| Ratios: Multiple Ratios Describe Any Situation | |
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| Equivalent Ratios | |
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| Equivalent Rates | |
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| Solving Rate Problems | |
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| Describing Percent | |
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| Uses of Integers | |
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| The Zero Principle | |
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| Subtraction of Integers as a Directed Distance | |
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| Multiplication and Division of Integers | |
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| Area of a Parallelogram | |
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| Area of a Triangle | |
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| The Pythagorean Theorem | |
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| Pi | |
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| How Measurements Are and Are Not Related | |
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| Mean | |
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| Variability | |
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| Sampling | |
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| Probability: What It Means | |
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| Unpredictability | |
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| Rotations, Reflections, and Translations | |
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| Scale Drawings | |
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| Dilatations | |
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| Angles with Parallel Lines | |
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| Equivalent Expressions | |
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| Equation as a Balance | |
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| Different Types of Equations | |
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| What Is Linear? | |
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| Role of the Slope in the Equation of a Line | |
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| Systems of Equations | |
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| Function Rules | |
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| Appendix: Connections Between Topics and Common Core Standards | |
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| References | |
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| Index | |
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| About the Authors | |