Chapter 1: PROBLEM SOLVING

__1.1 Solving Problems __

� 1A A Clinking Glasses Problem � 1B Problems about Triangular Numbers � 1C What Is a Fair Way to Split the Cost?

__1.2 Explaining Solutions __

� 1D Who Says You Can’t Do Rocket Science?

Chapter 2:NUMBERS AND THE DECIMAL SYSTEM

2.1 Overview of the Number Systems

2.2 The Decimal System and Place Value

� 2A How Many Are There? � 2B Showing Powers of Ten

__2.3 Representing Decimal Numbers __

� 2C Representing Decimal Numbers with Bundled Objects � 2D Zooming In and Zooming Out on Number Lines � 2E Representing Decimals as Lengths

__2.4 Comparing Decimal Numbers __

� 2F Places of Larger Value Count More than Lower Places Combined � 2G Misconceptions in Comparing Decimal Numbers � 2H Finding Smaller and Smaller Decimal Numbers � 2I Finding Decimals between Decimals � 2J Decimals between Decimals on Number Lines � 2K “Greater Than” and “Less Than” with Negative Decimal Numbers

__2.5 Rounding Decimal Numbers __

� 2L Why Do We Round? � 2M Explaining Rounding � 2N Can We Round This Way? � 2O Can We Round This Way?

Chapter 3: FRACTIONS

__3.1 The Meaning of Fractions __

� 3A Fractions of Objects � 3B The Whole Associated with a Fraction � 3C Is the Meaning of Equal Parts Always Clear? � 3D Improper Fractions

__3.2 Fractions as Numbers __

� 3E Counting along Number Lines � 3F Fractions on Number Lines, Part 1

__3.3 Equivalent Fractions __

� 3G Equivalent Fractions � 3HMisconceptions about Fraction Equivalence � 3I Common Denominators � 3J Solving Problems by Changing Denominators � 3K Fractions on Number Lines, Part 2 � 3L Simplifying Fractions � 3M When Can We “Cancel” to Get an Equivalent Fraction?

__3.4 Comparing Fractions __

� 3N Can We Compare Fractions this Way? � 3O What Is Another Way to Compare these Fractions? � 3P Comparing Fractions by Reasoning � 3Q Can We Reason this Way?

__3.5 Percent __

� 3R Pictures, Percentages, and Fractions � 3S Calculating Percents of Quantities by Using Benchmark Fractions � 3T Calculating Percentages � 3U Calculating Percentages with Pictures and Percent Diagrams � 3V Calculating Percentages by Going through 1 � 3W Calculating a Quantity from a Percentage of It

Chapter 4: ADDITION AND SUBTRACTION

__4.1 Interpretations of Addition and Subtraction __

� 4A Addition and Subtraction Story Problems � 4B Solving Addition and Subtraction Story Problems � 4C The Shopkeeper’s Method of Making Change � 4D Addition and Subtraction Story Problems with Negative

Numbers

__4.2 Why the Common Algorithms for Adding and Subtracting Decimal Numbers Work __

� 4E Adding and Subtracting with Ten-Structured Pictures � 4F Understanding the Common Addition Algorithm � 4G Understanding the Common Subtraction Algorithm � 4H Subtracting across Zeros � 4I Regrouping with Dozens and Dozens of Dozens � 4J Regrouping with Seconds, Minutes, and

Hours � 4K A Third Grader’s Method of Subtraction

__4.3 Adding and Subtracting Fractions __

� 4L Fraction Addition and Subtraction � 4M Mixed Numbers and Improper Fractions � 4N Adding and Subtracting Mixed Numbers � 4O Are These Story Problems for ½ - 1/3 � 4Q What Fraction Is Shaded?

__4.4 When Do We Add Percentages? __

� 4R Should We Add These Percentages?

__4.5 Percent Increase and Percent Decrease __

� 4S Calculating Percent Increase and Decrease � 4T Calculating Amounts from a Percent Increase or Decrease � 4U Percent of versus Percent Increase or Decrease � 4V Percent Problem Solving

__4.6 The Commutative and Associative Properties of Addition and Mental Math __

� 4W Mental Math � 4X Using Properties of Addition in Mental Math � 4Y Using Properties of Addition to Aid Learning of Basic Addition Facts � 4Z Writing Correct Equations � 4AA Writing Equations That Correspond to a Method of Calculation � 4BB Other Ways to Add and Subtract

Chapter 5: MULTIPLICATION

__5.1 The Meaning of Multiplication and Ways to Show Multiplication __

� 5A Showing Multiplicative Structure

__5.2 Why Multiplying Decimal Numbers by 10 Is Easy __

� 5B Multiplying by 10 � 5C If We Wrote Numbers Differently, Multiplying by 10 Might Not Be So Easy � 5D Multiplying by Powers of 10 Explains the Cycling of Decimal Representations of Fractions

__5.3 The Commutative Property of Multiplication and Areas of Rectangles __

� 5E Multiplication, Areas of Rectangles, and the Commutative Property � 5F Explaining the Commutative Property of Multiplication � 5G Using the Commutative Property of Multiplication � 5H Using Multiplication to Estimate How Many

__5.4 The Associative Property of Multiplication and Volumes of Boxes __

� 5I Ways to Describe the Volume of a Box with Multiplication � 5J Explaining the Associative Property � 5K Using the Associative and Commutative Properties of Multiplication � 5L Different Ways to Calculate the Total Number of Objects � 5M How Many Gumdrops?

__5.5 The Distributive Property __

� 5N Order of Operations � 5O Explaining the Distributive Property � 5P The Distributive Property and FOIL

� 5Q Using the Distributive Property � 5R Why Isn’t 23 × 23 Equal to 20 × 20 + 3 × 3? � 5S Squares and Products Near Squares

__5.6 Mental Math, Properties of Arithmetic, and Algebra __

� 5T Using Properties of Arithmetic to Aid the Learning of Basic Multiplication Facts � 5U Solving Arithmetic Problems Mentally � 5V Which Properties of Arithmetic Do These Calculations Use? � 5W Writing Equations That Correspond to a Method of Calculation � 5X Showing the Algebra in Mental Math

__5.7 Why the Procedure for Multiplying Whole Numbers Works __

� 5Y The Standard Versus the Partial-Products Multiplication Algorithm � 5Z Why the Multiplication Algorithms Give Correct Answers, Part 1 � 5AA Why the Multiplication Algorithms Give Correct Answers, Part 2 � 5BB The Standard Multiplication Algorithm Right Side Up and Upside Down

Chapter 6: MULTIPLICATION OF FRACTIONIS, DECIMALS, AND NEGATIVE NUMBERS

__6.1 Multiplying Fractions __

� 6A Writing and Solving Fraction Multiplication Story Problems � 6B Misconceptions with Fraction Multiplication � 6C Explaining Why the Procedure for Multiplying Fractions Gives Correct Answers � 6D When Do We Multiply Fractions? � 6E Multiplying Mixed Numbers � 6F What Fraction Is Shaded?

__6.2 Multiplying Decimals __

� 6G Multiplying Decimals � 6H Explaining Why We Place the Decimal Point Where We Do When We Multiply Decimals � 6I Decimal Multiplication and Areas of Rectangles

__6.3 Multiplying Negative Numbers __

� 6J Patterns with Multiplication and Negative Numbers � 6K Explaining Multiplication with Negative Numbers (and 0) � 6L Using Checks and Bills to Interpret Multiplication with Negative Numbers � 6M Does Multiplication Always Make Larger?

__6.4 Scientific Notation __

� 6N Scientific Notation versus Ordinary Decimal Notation � 6O Multiplying Powers of 10 � 6P How Many Digits Are in a Product of Counting Numbers?� 6Q Explaining the Pattern in the Number of Digits in Products

Chapter 7: DIVISION

__7.1 The Meaning of Division __

� 7A The Two Interpretations of Division � 7B Why Can’t We Divide by Zero? � 7C Division Story Problems � 7D Can We Use Properties of Arithmetic to Divide? � 7E Reasoning about Division � 7F Rounding to Estimate Solutions to Division Problems

__7.2 Understanding Long Division __

� 7G Dividing without Using a Calculator or Long Division � 7H Understanding the Scaffold Method of Long Division � 7I Using the Scaffold Method � 7J Interpreting Standard Long Division from the “How Many in Each Group?” Viewpoint � 7K Zeros in Long Division � 7L Using Long Division to Calculate Decimal Number Answers to Whole Number Division Problems � 7M Errors in Decimal Answers to Division

Problems

__7.3 Fractions and Division__

� 7N Relating Fractions and Division � 7O Mixed-Number Answers to Division Problems � 7P Using Division to Calculate Decimal Representations of Fractions

__7.4 Dividing Fractions__

� 7Q “How Many Groups?” Fraction Division Problems � 7R “How Many in One Group?” Fraction Division Problems � 7S Using “Double Number Lines” to Solve “How Many in One Group?” Division Problems � 7T Explaining “Invert and Multiply” by Relating Division to Multiplication � 7U Are These Division Problems?

__7.5 Dividing Decimals __

� 7V Quick Tricks for Some Decimal Division Problems � 7W Decimal Division

__7.6 Ratio and Proportion __

� 7X Comparing Mixtures � 7Y Using Ratio Tables � 7Z Using Strip Diagrams to Solve Ratio Problems � 7AA Using Simple Reasoning to Find Equivalent Ratios and Rates � 7BB Solving Proportions with Multiplication and Division � 7CC Ratios, Fractions, and Division � 7DD Solving Proportions by Cross-

Multiplying Fractions � 7EE Can You Always Use a Proportion? � 7FF The Consumer Price Index

Chapter 8: GEOMETRY

__8.1 Visualization __

� 8A What Shapes Do These Patterns Make? � 8B Parts of a Pyramid � 8C Slicing through a Board � 8D Visualizing Lines and Planes � 8E The Rotation of the Earth and Time Zones � 8F Explaining the Phases of the Moon

__8.2 Angles __

� 8G Angle Explorers � 8H Angles Formed by Two Lines � 8I Seeing that the Angles in a Triangle Add to 180� � 8J Using the Parallel Postulate to Prove that the Angles in a Triangle Add to 180� � 8K Describing Routes, Using Distances and Angles � 8L Explaining Why the Angles in a Triangle Add to 180� by Walking and Turning � 8M Angles and Shapes Inside Shapes � 8N Angles of Sun Rays � 8O How the Tilt of the Earth

Causes Seasons � 8P How Big Is the Reflection of Your Face in a Mirror? � 8Q Why Do Spoons Reflect Upside Down? � 8R The Special Shape of Satellite Dishes

__8.3 Circles and Spheres __

� 8S Points That Are a Fixed Distance from a Given Point � 8T Using Circles � 8U The Global Positioning System (GPS) � 8V Circle Curiosities

__8.4 Triangles, Quadrilaterals, and Other Polygons __

� 8W Using a Compass to Draw Triangles and Quadrilaterals � 8X Making Shapes by Folding Paper � 8Y Constructing Quadrilaterals with Geometer’s Sketchpad � 8Z Relating the Kinds of Quadrilaterals � 8AA Venn Diagrams Relating Quadrilaterals � 8BB Investigating Diagonals of Quadrilaterals with Geometer’s Sketchpad � 8CC Investigating Diagonals of Quadrilaterals (Alternate)

__8.5 Constructions with Straightedge and Compass __

� 8DD Relating the Constructions to Properties of Rhombuses � 8EE Constructing a Square and an Octagon with Straightedge and Compass

__8.6 Polyhedra and Other Solid Shapes __

� 8FF Patterns for Prisms, Cylinders, Pyramids, and Cones � 8GG Making Prisms and Pyramids � 8HH Analyzing Prisms and Pyramids � 8II What’s Inside the Magic 8 Ball? � 8JJ Making Platonic Solids with Toothpicks and Marshmallows � 8KK Why Are There No Other Platonic Solids? � 8LL Relating the

Numbers of Faces, Edges, and Vertices of Polyhedra

Chapter 9: GEOMETRY OF MOTION AND CHANGE

__9.1 Reflections, Translations, and Rotations __

� 9A Exploring Rotations � 9B Exploring Reflections � 9C Exploring Reflections with Geometer’s Sketchpad

� 9D Exploring Translations with Geometer’s Sketchpad � 9E Exploring Rotations with Geometer’s Sketchpad � 9F Reflections, Rotations, and Translations in a Coordinate Plane

__9.2 Symmetry __

� 9G Checking for Symmetry � 9H Frieze Patterns � 9I Traditional Quilt Designs � 9J Creating Symmetrical Designs with Geometer’s Sketchpad � 9K Creating Symmetrical Designs (Alternate) � 9L Creating Escher-Type Designs with Geometer’s Sketchpad (for Fun) � 9M Analyzing Designs

__9.3 Congruence __

� 9N Triangles and Quadrilaterals of Specified Side Lengths � 9O Describing a Triangle � 9P Triangles with an Angle, a Side, and an Angle Specified � 9Q Using Triangle Congruence Criteria

__9.4 Similarity __

� 9R A First Look at Solving Scaling Problems � 9S Using the “Scale Factor,” “Relative Sizes,” and “Set up a Proportion” Methods � 9T A Common Misconception about Scaling � 9U Using Scaling to Understand Astronomical Distances � 9V More Scaling Problems � 9W Measuring Distances by “Sighting” � 9X Using Shadows to Determine the Height of a Tree

Chapter 10: MEASUREMENT

__10.1 Fundamentals of Measurement __

� 10A The Biggest Tree in the World � 10B What Do “6 Square Inches” and “6 Cubic Inches” Mean? � 10C Using a Ruler

__10.2 Length, Area, Volume, and Dimension __

� 10D Dimension and Size

__10.3 Calculating Perimeters of Polygons, Areas of Rectangles, and Volumes of Boxes__

� 10E Explaining Why We Add to Calculate Perimeters of Polygons � 10F Perimeter Misconceptions � 10G Explaining Why We Multiply to Determine Areas of Rectangles � 10H Explaining Why We Multiply to Determine Volumes of Boxes � 10I Who Can Make the Biggest Box?

__10.4 Error and Accuracy in Measurements __

� 10J Reporting and Interpreting Measurements

__10.5 Converting from One Unit of Measurement to Another __

� 10K Conversions: When Do We Multiply? When Do We Divide? � 10L Conversion Problems � 10M Converting Measurements with and without Dimensional Analysis � 10N Areas of Rectangles in Square Yards and Square Feet � 10O Volumes of Boxes in Cubic Yards and Cubic Feet � 10P Area and Volume Conversions: Which Are Correct and Which Are Not?

Chapter 11: MORE ABOUT AREA AND VOLUME

__11.1 The Moving and Additivity Principles about Area __

� 11A Different Shapes with the Same Area � 11B Using the Moving and Additivity Principles � 11C Using the Moving and Additivity Principles to Determine Surface Area

__11.2 Using the Moving and Additivity Principles to Prove the Pythagorean Theorem __

� 11D Using the Pythagorean Theorem � 11E Can We Prove the Pythagorean Theorem by Checking Examples? � 11F A Proof of the Pythagorean Theorem

__11.3 Areas of Triangles __

� 11G Choosing the Base and Height of Triangles � 11H Explaining Why the Area Formula for Triangles Is Valid � 11I Determining Areas

__11.4 Areas of Parallelograms __

� 11J Do Side Lengths Determine the Area of a Parallelogram? � 11K Explaining Why the Area Formula for Parallelograms Is

Valid

__11.5 Cavalieri’s Principle about Shearing and Area __

� 11L Shearing a Toothpick Rectangle to Make a Parallelogram � 11M Is This Shearing? � 11N Shearing Parallelograms � 11O Shearing Triangles

__11.6 Areas of Circles and the Number Pi __

� 11P How Big Is the Number π? � 11Q Over- and Underestimates for the Area of a Circle � 11R Why the Area Formula for Circles Makes Sense � 11S Using the Circle Circumference and Area Formulas to Find Areas and Surface Areas

__11.7 Approximating Areas of Irregular Shapes __

� 11T Determining the Area of an Irregular Shape

__11.8 Relating the Perimeter and Area of a Shape __

� 11U How Are Perimeter and Area Related? � 11V Can We Determine Area by Measuring Perimeter?

__11.9 Principles for Determining Volumes __

� 11W Using the Moving and Additivity Principles to Determine Volumes � 11X Determining Volumes by Submersing in Water � 11Y Floating Versus Sinking: Archimedes’s Principle

__11.10 Volumes of Prisms, Cylinders, Pyramids, and Cones __

� 11Z Why the Volume Formula for Prisms and Cylinders Makes Sense � 11AA Filling Boxes and Jars � 11BB Comparing the Volume of a Pyramid with the Volume of a Rectangular Prism � 11CC The 13

in the Volume Formula for Pyramids and Cones � 11DD Using Volume Formulas with Real Objects

� 11EE Volume and Surface Area Contests � 11FF Volume Problems � 11GG The Volume of a Rhombic Dodecahedron

__11.11 Areas, Volumes, and Scaling __

� 11HH Areas and Volumes of Similar Boxes � 11II Areas and Volumes of Similar Cylinders � 11JJ Determining Areas and Volumes of Scaled Objects � 11KK A Scaling Proof of the Pythagorean Theorem

Chapter 12: NUMBER THEORY

__12.1 Factors and Multiples __

� 12A Factors, Multiples, and Rectangles � 12B Problems about Factors and Multiples � 12C Finding All Factors � 12D Do Factors Always Come in Pairs?

__12.2 Greatest Common Factor and Least Common Multiple __

� 12E Finding Commonality � 12F The “Slide Method” � 12G Problems Involving Greatest Common Factors and Least Common Multiples � 12H Flower Designs � 12I Relationships between the GCF and the LCM and Explaining the Flower Designs � 12J Using GCFs and LCMs with Fractions

__12.3 Prime Numbers __

� 12K The Sieve of Eratosthenes � 12L The Trial Division Method for Determining whether a Number Is Prime

__12.4 Even and Odd __

� 12M Why Can We Check the Ones Digit to Determine whether a Number Is Even or Odd? � 12N Questions about Even and Odd Numbers � 12O Extending the Definitions of Even and Odd

__12.5 Divisibility Tests __

� 12P The Divisibility Test for 3

__12.6 Rational and Irrational Numbers __

� 12Q Decimal Representations of Fractions � 12R Writing Terminating and Repeating Decimals as Fractions � 12S What Is 0.9999 ...? � 12T The Square Root of 2 � 12U Pattern Tiles and the Irrationality of the Square Root of 3

Chapter 13: FUNCTIONS AND ALGEBRA

__13.1 Mathematical Expressions, Formulas, and Equations __

� 13A Writing Expressions and a Formula for a Flower Pattern � 13B Expressions in Geometric Settings � 13C Expressions in 3D Geometric Settings � 13D Equations Arising from Rectangular Designs � 13E Expressions with Fractions � 13F Evaluating Expressions with Fractions Efficiently and Correctly � 13G Expressions for Story Problems � 13H Writing Equations for Story Situations � 13I Writing Story Problems for

Equations

__13.2 Solving Equations Using Number Sense, Strip Diagrams, and Algebra __

� 13J Solving Equations Using Number Sense � 13K Solving Equations Algebraically and with a Pan Balance � 13L How Many Pencils Were There? � 13M Solving Story Problems with Strip Diagrams and with Equations � 13N Modifying Problems � 13O Solving Story Problems

__13.3 Sequences __

� 13P Arithmetic Sequences of Numbers Corresponding to Sequences of Figures � 13Q Deriving Formulas for Arithmetic Sequences � 13R Sequences and Formulas � 13S Geometric Sequences � 13T Repeating Patterns � 13U The Fibonacci Sequence in Nature and Art � 13V What’s the Rule?

__13.4 Series __

� 13W Sums of Counting Numbers � 13X Sums of Odd Numbers � 13Y Sums of Squares � 13Z Sums of Powers of Two � 13AA An Infinite Geometric Series � 13BB Making Payments into an Account

__13.5 Functions __

� 13CC Interpreting Graphs of Functions � 13DD Are These Graphs Correct?

__13.6 Linear Functions __

� 13EE A Function Arising from Proportions � 13FF Arithmetic Sequences as Functions � 13GG Analyzing the Way Functions Change � 13HH Story Problems for Linear Functions � 13II Deriving the Formula for Temperature in Degrees Fahrenheit in Terms of Degrees Celsius

Chapter 14: STATISTICS

__14.1 Formulating Questions, Designing Investigations, and Gathering Data __

� 14A Challenges in Formulating Survey Questions � 14B Choosing a Sample � 14C Using Random Samples

� 14D Using Random Samples to Estimate Population Size by Marking (Capture—Recapture) � 14E Which Experiment Is Better?

__14.2 Displaying Data and Interpreting Data Displays __

� 14F What Do You Learn from the Display? � 14G Display These Data about Pets � 14H What Is Wrong with These Displays? � 14I Three Levels of Questions about Graphs � 14J The Length of a Pendulum and the Time It Takes to Swing � 14K Investigating Small Bags of Candies � 14L Balancing a

Mobile

__14.3 The Center of Data: Mean, Median, and Mode __

� 14M The Average as “Making Even” or “Leveling Out” � 14N The Average as “Balance Point” � 14O Same Median, Different Average � 14P Can More Than Half Be above Average?

__14.4 Percentiles and the Distribution of Data __

� 14Q Determining Percentiles � 14R Percentiles versus Percent Correct � 14S Box-and-Whisker Plots � 14T How Percentiles Inform You about the Distribution of Data: The Case of Household Income � 14U Distributions of Random Samples

Chapter 15: PROBABILITY

__15.1 Basic Principles and Calculation Methods of Probability __

� 15A Comparing Probabilities � 15B Experimental versus Theoretical Probability: Picking Cubes from a Bag � 15C If You Flip 10 Pennies, Should Half Come Up Heads? � 15D Number Cube Rolling Game � 15E Picking Two Marbles from a Bag of 1 Black and 3 Red Marbles � 15F Applying Probability � 15G Some Probability Misconceptions

__15.2 Using Fraction Arithmetic to Calculate Probabilities __

� 15H Using the Meaning of Fraction Multiplication to Calculate a Probability � 15I Using Fraction Multiplication and Addition to Calculate a Probability