Marian Small is an international professional developer, conference speaker, bestselling author, former dean of education at the University of New Brunswick, and longtime professor of mathematics education. Visit Marian's website at www.onetwoinfinity.ca for in-person and online professional development. Amy Lin is the director of teaching and learning at Seneca College. Amy has worked as a teacher in elementary, secondary, and postsecondary levels and as a mathematics coach, a curriculum consultant, administrator, and ministry researcher.

• Preface  ixAcknowledgments  xiii.  Background   Visualization in Math   Focusing on the Important Math  3Building Mathematical Communication  42.  Grades K–2  7Notice and WonderStrawberries  8Egg Cartons   Bracelets   2Daisies   4Donuts   Explaining ConceptsCounting Up by s   8Counting Back by s  2 Counting Up by 2s  22Counting Back by 2s  24Counting Up by 5s  2 Counting Back by 5s  28Counting Up by s  3 Counting Back by s  32Comparing Numbers by Matching  34Benchmark Numbers: All About 5  3 Benchmark Numbers: All About  38Ordinal Numbers  4 Addition as Combining  42Adding and Adding  44Addition: Commutativity  4 Addition: Changing Addends, But Not the Sum  48Adding or Subtracting  5 Subtraction as Taking Away  52Subtraction to Compare  54Relating Addition and Subtraction  5 Naming Two-Digit Numbers  58Naming Three-Digit Numbers   Place Value: Grouping in Tens   2Place Value: Positions   4Comparing Sizes of Numbers   Fractions: Halves   8Fractions: Quarters  7 Measurement: Meaning of Length  72Measurement: Effect of Unit Size  74Measurement: Standard Units of Length  7 2-D Shapes Versus 3-D Shapes  78Comparing 2-D Shapes  8 Comparing 3-D Shapes  82Composing Shapes  84Shape Puzzles  8 Student Assessment: Draw or Model to ExplainRearranging Addends  89Subitizing 8  9 Ordinal Numbers: Points of View  9 Subtracting by Adding  9 Nonstandard Measurement  923.  Grades 3–5  95Notice and WonderQuilts  9 Design  98Juice   Tennis   2Pastries   4Explaining ConceptsMultiplication: Equal Groups   Multiplication: Commutativity   8Multiplication: The Distributive Principle   Multiplication: 2-Digit by 2-Digit   2Division as Equal Groups or Sharing   4Division: Remainders   Rounding Numbers   8Place Value: Multiplying and Dividing by Powers of   2 Place Value: Renaming Numbers   22Factors: What They Are   24Factors Come in Pairs   2 Fractions: Representing   28Fractions: Equivalence   3 Fractions: Comparing   32Fractions: Mixed Number/Improper Fraction Relationship   34Fractions: Common Denominators   3 Adding Fractions   38Multiplying Fractions   4 Fractions: Multiplying as Resizing   44Fractions as Division   4 Decimals: Relating Hundredths to Tenths   48Decimals: Equivalence   5 Decimals: Adding and Subtracting   52Measurement: Time Intervals   54Measurement: Area of Rectangles   5 Perimeter Versus Area   58Measurement Conversions   Graphs With Scales   2Coordinate Grids   4Classification of Shapes   Parallel and Perpendicular Lines   8Lines of Symmetry   7 Patterns Versus Nonpatterns   72Algebraic Thinking: Growing Additively   74Algebraic Thinking: Shrinking Additively   7 Algebraic Thinking: Growing Multiplicatively   78Student Assessment: Draw or Model to ExplainFractions of a Set   8 Fraction as Division   8 Relating Quotients   82Comparing Products   83Factors   844.  Grades –8   87Notice and WonderRoller Coasters   88A Giant Burger   9 Going to a Concert   92Basketball   94A Solar Eclipse   9 Explaining ConceptsCommon Factors   98Common Multiples  2 Square Roots  2 2Fraction Division  2 4Ratios: Multiple Ratios Describe Any Situation  2 Equivalent Ratios  2 8Equivalent Rates  2 Solving Rate Problems  2 2Describing Percent  2 4Uses of Integers  2 The Zero Principle  2 8Subtraction of Integers as a Directed Distance  22 Multiplication and Division of Integers  222Area of a Parallelogram  224Area of a Triangle  22 The Pythagorean Theorem  23 Pi    232How Measurements Are and Are Not Related  234Mean  23 Variability  238Sampling  24 Probability: What It Means  242Unpredictability  244Rotations, Reflections, and Translations  24 Scale Drawings  248Dilations  25 Angles With Parallel Lines  252Equivalent Expressions  254Equation as a Balance  25 Different Types of Equations  258What Is Linear?  2 Role of the Slope in the Equation of a Line  2 2Systems of Equations  2 4Function Rules  2 Student Assessment: Draw or Model to ExplainSubtracting Negatives  2 8Relating Percents  2 9Square Root Estimates  27 Dividing Fractions  272Common Factors  273Appendix: Connections Between Topics and Common Core Standards  275References  279Index  28 About the Authors  285

Marian Small is an international professional developer, conference speaker, bestselling author, former dean of education at the University of New Brunswick, and longtime professor of mathematics education. Visit Marian's website at www.onetwoinfinity.ca for in-person and online professional development. Amy Lin is the director of teaching and learning at Seneca College. Amy has worked as a teacher in elementary, secondary, and postsecondary levels and as a mathematics coach, a curriculum consultant, administrator, and ministry researcher.

• Preface  ixAcknowledgments  xiii.  Background   Visualization in Math   Focusing on the Important Math  3Building Mathematical Communication  42.  Grades K–2  7Notice and WonderStrawberries  8Egg Cartons   Bracelets   2Daisies   4Donuts   Explaining ConceptsCounting Up by s   8Counting Back by s  2 Counting Up by 2s  22Counting Back by 2s  24Counting Up by 5s  2 Counting Back by 5s  28Counting Up by s  3 Counting Back by s  32Comparing Numbers by Matching  34Benchmark Numbers: All About 5  3 Benchmark Numbers: All About  38Ordinal Numbers  4 Addition as Combining  42Adding and Adding  44Addition: Commutativity  4 Addition: Changing Addends, But Not the Sum  48Adding or Subtracting  5 Subtraction as Taking Away  52Subtraction to Compare  54Relating Addition and Subtraction  5 Naming Two-Digit Numbers  58Naming Three-Digit Numbers   Place Value: Grouping in Tens   2Place Value: Positions   4Comparing Sizes of Numbers   Fractions: Halves   8Fractions: Quarters  7 Measurement: Meaning of Length  72Measurement: Effect of Unit Size  74Measurement: Standard Units of Length  7 2-D Shapes Versus 3-D Shapes  78Comparing 2-D Shapes  8 Comparing 3-D Shapes  82Composing Shapes  84Shape Puzzles  8 Student Assessment: Draw or Model to ExplainRearranging Addends  89Subitizing 8  9 Ordinal Numbers: Points of View  9 Subtracting by Adding  9 Nonstandard Measurement  923.  Grades 3–5  95Notice and WonderQuilts  9 Design  98Juice   Tennis   2Pastries   4Explaining ConceptsMultiplication: Equal Groups   Multiplication: Commutativity   8Multiplication: The Distributive Principle   Multiplication: 2-Digit by 2-Digit   2Division as Equal Groups or Sharing   4Division: Remainders   Rounding Numbers   8Place Value: Multiplying and Dividing by Powers of   2 Place Value: Renaming Numbers   22Factors: What They Are   24Factors Come in Pairs   2 Fractions: Representing   28Fractions: Equivalence   3 Fractions: Comparing   32Fractions: Mixed Number/Improper Fraction Relationship   34Fractions: Common Denominators   3 Adding Fractions   38Multiplying Fractions   4 Fractions: Multiplying as Resizing   44Fractions as Division   4 Decimals: Relating Hundredths to Tenths   48Decimals: Equivalence   5 Decimals: Adding and Subtracting   52Measurement: Time Intervals   54Measurement: Area of Rectangles   5 Perimeter Versus Area   58Measurement Conversions   Graphs With Scales   2Coordinate Grids   4Classification of Shapes   Parallel and Perpendicular Lines   8Lines of Symmetry   7 Patterns Versus Nonpatterns   72Algebraic Thinking: Growing Additively   74Algebraic Thinking: Shrinking Additively   7 Algebraic Thinking: Growing Multiplicatively   78Student Assessment: Draw or Model to ExplainFractions of a Set   8 Fraction as Division   8 Relating Quotients   82Comparing Products   83Factors   844.  Grades –8   87Notice and WonderRoller Coasters   88A Giant Burger   9 Going to a Concert   92Basketball   94A Solar Eclipse   9 Explaining ConceptsCommon Factors   98Common Multiples  2 Square Roots  2 2Fraction Division  2 4Ratios: Multiple Ratios Describe Any Situation  2 Equivalent Ratios  2 8Equivalent Rates  2 Solving Rate Problems  2 2Describing Percent  2 4Uses of Integers  2 The Zero Principle  2 8Subtraction of Integers as a Directed Distance  22 Multiplication and Division of Integers  222Area of a Parallelogram  224Area of a Triangle  22 The Pythagorean Theorem  23 Pi    232How Measurements Are and Are Not Related  234Mean  23 Variability  238Sampling  24 Probability: What It Means  242Unpredictability  244Rotations, Reflections, and Translations  24 Scale Drawings  248Dilations  25 Angles With Parallel Lines  252Equivalent Expressions  254Equation as a Balance  25 Different Types of Equations  258What Is Linear?  2 Role of the Slope in the Equation of a Line  2 2Systems of Equations  2 4Function Rules  2 Student Assessment: Draw or Model to ExplainSubtracting Negatives  2 8Relating Percents  2 9Square Root Estimates  27 Dividing Fractions  272Common Factors  273Appendix: Connections Between Topics and Common Core Standards  275References  279Index  28 About the Authors  285

Marian Small is an international professional developer, conference speaker, bestselling author, former dean of education at the University of New Brunswick, and longtime professor of mathematics education. Visit Marian's website at www.onetwoinfinity.ca for in-person and online professional development. Amy Lin is the director of teaching and learning at Seneca College. Amy has worked as a teacher in elementary, secondary, and postsecondary levels and as a mathematics coach, a curriculum consultant, administrator, and ministry researcher.

• Preface  ixAcknowledgments  xiii.  Background   Visualization in Math   Focusing on the Important Math  3Building Mathematical Communication  42.  Grades K–2  7Notice and WonderStrawberries  8Egg Cartons   Bracelets   2Daisies   4Donuts   Explaining ConceptsCounting Up by s   8Counting Back by s  2 Counting Up by 2s  22Counting Back by 2s  24Counting Up by 5s  2 Counting Back by 5s  28Counting Up by s  3 Counting Back by s  32Comparing Numbers by Matching  34Benchmark Numbers: All About 5  3 Benchmark Numbers: All About  38Ordinal Numbers  4 Addition as Combining  42Adding and Adding  44Addition: Commutativity  4 Addition: Changing Addends, But Not the Sum  48Adding or Subtracting  5 Subtraction as Taking Away  52Subtraction to Compare  54Relating Addition and Subtraction  5 Naming Two-Digit Numbers  58Naming Three-Digit Numbers   Place Value: Grouping in Tens   2Place Value: Positions   4Comparing Sizes of Numbers   Fractions: Halves   8Fractions: Quarters  7 Measurement: Meaning of Length  72Measurement: Effect of Unit Size  74Measurement: Standard Units of Length  7 2-D Shapes Versus 3-D Shapes  78Comparing 2-D Shapes  8 Comparing 3-D Shapes  82Composing Shapes  84Shape Puzzles  8 Student Assessment: Draw or Model to ExplainRearranging Addends  89Subitizing 8  9 Ordinal Numbers: Points of View  9 Subtracting by Adding  9 Nonstandard Measurement  923.  Grades 3–5  95Notice and WonderQuilts  9 Design  98Juice   Tennis   2Pastries   4Explaining ConceptsMultiplication: Equal Groups   Multiplication: Commutativity   8Multiplication: The Distributive Principle   Multiplication: 2-Digit by 2-Digit   2Division as Equal Groups or Sharing   4Division: Remainders   Rounding Numbers   8Place Value: Multiplying and Dividing by Powers of   2 Place Value: Renaming Numbers   22Factors: What They Are   24Factors Come in Pairs   2 Fractions: Representing   28Fractions: Equivalence   3 Fractions: Comparing   32Fractions: Mixed Number/Improper Fraction Relationship   34Fractions: Common Denominators   3 Adding Fractions   38Multiplying Fractions   4 Fractions: Multiplying as Resizing   44Fractions as Division   4 Decimals: Relating Hundredths to Tenths   48Decimals: Equivalence   5 Decimals: Adding and Subtracting   52Measurement: Time Intervals   54Measurement: Area of Rectangles   5 Perimeter Versus Area   58Measurement Conversions   Graphs With Scales   2Coordinate Grids   4Classification of Shapes   Parallel and Perpendicular Lines   8Lines of Symmetry   7 Patterns Versus Nonpatterns   72Algebraic Thinking: Growing Additively   74Algebraic Thinking: Shrinking Additively   7 Algebraic Thinking: Growing Multiplicatively   78Student Assessment: Draw or Model to ExplainFractions of a Set   8 Fraction as Division   8 Relating Quotients   82Comparing Products   83Factors   844.  Grades –8   87Notice and WonderRoller Coasters   88A Giant Burger   9 Going to a Concert   92Basketball   94A Solar Eclipse   9 Explaining ConceptsCommon Factors   98Common Multiples  2 Square Roots  2 2Fraction Division  2 4Ratios: Multiple Ratios Describe Any Situation  2 Equivalent Ratios  2 8Equivalent Rates  2 Solving Rate Problems  2 2Describing Percent  2 4Uses of Integers  2 The Zero Principle  2 8Subtraction of Integers as a Directed Distance  22 Multiplication and Division of Integers  222Area of a Parallelogram  224Area of a Triangle  22 The Pythagorean Theorem  23 Pi    232How Measurements Are and Are Not Related  234Mean  23 Variability  238Sampling  24 Probability: What It Means  242Unpredictability  244Rotations, Reflections, and Translations  24 Scale Drawings  248Dilations  25 Angles With Parallel Lines  252Equivalent Expressions  254Equation as a Balance  25 Different Types of Equations  258What Is Linear?  2 Role of the Slope in the Equation of a Line  2 2Systems of Equations  2 4Function Rules  2 Student Assessment: Draw or Model to ExplainSubtracting Negatives  2 8Relating Percents  2 9Square Root Estimates  27 Dividing Fractions  272Common Factors  273Appendix: Connections Between Topics and Common Core Standards  275References  279Index  28 About the Authors  285