Eyes on Math

Preface ?ix Acknowledgments ?xiii 1. ?Background ?1 Visualization in Math ?1 Focusing on the Important Math ?3 Building Mathematical Communication ?4 2. ?Grades K-2 ?7 Notice and Wonder Strawberries ?8 Egg Cartons ?10 Bracelets ?12 Daisies ?14 Donuts ?16 Explaining Concepts Counting Up by 1s ?18 Counting Back by 1s ?20 Counting Up by 2s ?22 Counting Back by 2s ?24 Counting Up by 5s ?26 Counting Back by 5s ?28 Counting Up by 10s ?30 Counting Back by 10s ?32 Comparing Numbers by Matching ?34 Benchmark Numbers: All About 5 ?36 Benchmark Numbers: All About 10 ?38 Ordinal Numbers ?40 Addition as Combining ?42 Adding 0 and Adding 1 ?44 Addition: Commutativity ?46 Addition: Changing Addends, But Not the Sum ?48 Adding or Subtracting 10 ?50 Subtraction as Taking Away ?52 Subtraction to Compare ?54 Relating Addition and Subtraction ?56 Naming Two-Digit Numbers ?58 Naming Three-Digit Numbers ?60 Place Value: Grouping in Tens ?62 Place Value: Positions ?64 Comparing Sizes of Numbers ?66 Fractions: Halves ?68 Fractions: Quarters ?70 Measurement: Meaning of Length ?72 Measurement: Effect of Unit Size ?74 Measurement: Standard Units of Length ?76 2-D Shapes Versus 3-D Shapes ?78 Comparing 2-D Shapes ?80 Comparing 3-D Shapes ?82 Composing Shapes ?84 Shape Puzzles ?86 Student Assessment: Draw or Model to Explain Rearranging Addends ?89 Subitizing 8 ?90 Ordinal Numbers: Points of View ?90 Subtracting by Adding ?91 Nonstandard Measurement ?92 3. ?Grades 3-5 ?95 Notice and Wonder Quilts ?96 Design ?98 Juice ?100 Tennis ?102 Pastries ?104 Explaining Concepts Multiplication: Equal Groups ?106 Multiplication: Commutativity ?108 Multiplication: The Distributive Principle ?110 Multiplication: 2-Digit by 2-Digit ?112 Division as Equal Groups or Sharing ?114 Division: Remainders ?116 Rounding Numbers ?118 Place Value: Multiplying and Dividing by Powers of 10 ?120 Place Value: Renaming Numbers ?122 Factors: What They Are ?124 Factors Come in Pairs ?126 Fractions: Representing ?128 Fractions: Equivalence ?130 Fractions: Comparing ?132 Fractions: Mixed Number/Improper Fraction Relationship ?134 Fractions: Common Denominators ?136 Adding Fractions ?138 Multiplying Fractions ?140 Fractions: Multiplying as Resizing ?144 Fractions as Division ?146 Decimals: Relating Hundredths to Tenths ?148 Decimals: Equivalence ?150 Decimals: Adding and Subtracting ?152 Measurement: Time Intervals ?154 Measurement: Area of Rectangles ?156 Perimeter Versus Area ?158 Measurement Conversions ?160 Graphs With Scales ?162 Coordinate Grids ?164 Classification of Shapes ?166 Parallel and Perpendicular Lines ?168 Lines of Symmetry ?170 Patterns Versus Nonpatterns ?172 Algebraic Thinking: Growing Additively ?174 Algebraic Thinking: Shrinking Additively ?176 Algebraic Thinking: Growing Multiplicatively ?178 Student Assessment: Draw or Model to Explain Fractions of a Set ?180 Fraction as Division ?181 Relating Quotients ?182 Comparing Products ?183 Factors ?184 4. ?Grades 6-8 ?187 Notice and Wonder Roller Coasters ?188 A Giant Burger ?190 Going to a Concert ?192 Basketball ?194 A Solar Eclipse ?196 Explaining Concepts Common Factors ?198 Common Multiples ?200 Square Roots ?202 Fraction Division ?204 Ratios: Multiple Ratios Describe Any Situation ?206 Equivalent Ratios ?208 Equivalent Rates ?210 Solving Rate Problems ?212 Describing Percent ?214 Uses of Integers ?216 The Zero Principle ?218 Subtraction of Integers as a Directed Distance ?220 Multiplication and Division of Integers ?222 Area of a Parallelogram ?224 Area of a Triangle ?226 The Pythagorean Theorem ?230 Pi ? ?232 How Measurements Are and Are Not Related ?234 Mean ?236 Variability ?238 Sampling ?240 Probability: What It Means ?242 Unpredictability ?244 Rotations, Reflections, and Translations ?246 Scale Drawings ?248 Dilations ?250 Angles With Parallel Lines ?252 Equivalent Expressions ?254 Equation as a Balance ?256 Different Types of Equations ?258 What Is Linear? ?260 Role of the Slope in the Equation of a Line ?262 Systems of Equations ?264 Function Rules ?266 Student Assessment: Draw or Model to Explain Subtracting Negatives ?268 Relating Percents ?269 Square Root Estimates ?271 Dividing Fractions ?272 Common Factors ?273 Appendix: Connections Between Topics and Common Core Standards ?275 References ?279 Index ?281 About the Authors ?285