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This activities manul includes activities designed to be done in class or outside of class. These activities promote critical thinking and discussion and give students a depth of understanding and perspective on the concepts presented in the text.

Sybilla Beckmann earned an undergraduate degree in mathematics from Brown University and a PhD in mathematics from the University of Pennsylvania. She taught and did research in mathematics at Yale University for two years. Since then, she has been at the University of Georgia. When she had children, she became very interested in helping prospective teachers understand and appreciate the mathematics they will teach. This interest led to her book. She enjoys playing the piano, weaving, attending classical music concerts, and traveling with her family.

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 180æ 8J Using the Parallel Postulate to Prove that the Angles in a Triangle Add to 180æ 8K Describing Routes, Using Distances and Angles 8L Explaining Why the Angles in a Triangle Add to 180æ 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

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

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## More About This Textbook

## Overview

This activities manul includes activities designed to be done in class or outside of class. These activities promote critical thinking and discussion and give students a depth of understanding and perspective on the concepts presented in the text.

## Product Details

## Related Subjects

## Meet the Author

Sybilla Beckmann earned an undergraduate degree in mathematics from Brown University and a PhD in mathematics from the University of Pennsylvania. She taught and did research in mathematics at Yale University for two years. Since then, she has been at the University of Georgia. When she had children, she became very interested in helping prospective teachers understand and appreciate the mathematics they will teach. This interest led to her book. She enjoys playing the piano, weaving, attending classical music concerts, and traveling with her family.

## Table of Contents

Chapter 1: PROBLEM SOLVING

1.1 Solving Problems1A A Clinking Glasses Problem 1B Problems about Triangular Numbers 1C What Is a Fair Way to Split the Cost?

1.2 Explaining Solutions1D Who Says You Can’t Do Rocket Science?

Chapter 2:NUMBERS AND THE DECIMAL SYSTEM

2.1 Overview of the Number Systems2.2 The Decimal System and Place Value2A How Many Are There? 2B Showing Powers of Ten

2.3 Representing Decimal Numbers2C Representing Decimal Numbers with Bundled Objects 2D Zooming In and Zooming Out on Number Lines 2E Representing Decimals as Lengths

2.4 Comparing Decimal Numbers2F 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 Numbers2L 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 Fractions3A 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 Numbers3E Counting along Number Lines 3F Fractions on Number Lines, Part 1

3.3 Equivalent Fractions3G 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 Fractions3N 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 Percent3R 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 Subtraction4A 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 Work4E 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 Fractions4L 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 Decrease4S 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 Math4W 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 Multiplication5A Showing Multiplicative Structure

5.2 Why Multiplying Decimal Numbers by 10 Is Easy5B 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 Rectangles5E 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 Boxes5I 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 Property5N 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 Algebra5T 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 Works5Y 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 Fractions6A 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 Decimals6G 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 Numbers6J 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 Notation6N 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 Division7A 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 Division7G 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 Division7N Relating Fractions and Division 7O Mixed-Number Answers to Division Problems 7P Using Division to Calculate Decimal Representations of Fractions

7.4 Dividing Fractions7Q “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 Decimals7V Quick Tricks for Some Decimal Division Problems 7W Decimal Division

7.6 Ratio and Proportion7X 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 Visualization8A 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 Angles8G Angle Explorers 8H Angles Formed by Two Lines 8I Seeing that the Angles in a Triangle Add to 180æ 8J Using the Parallel Postulate to Prove that the Angles in a Triangle Add to 180æ 8K Describing Routes, Using Distances and Angles 8L Explaining Why the Angles in a Triangle Add to 180æ 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 Spheres8S 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 Polygons8W 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 Compass8DD Relating the Constructions to Properties of Rhombuses 8EE Constructing a Square and an Octagon with Straightedge and Compass

8.6 Polyhedra and Other Solid Shapes8FF 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 Rotations9A 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 Symmetry9G 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 Congruence9N 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 Similarity9R 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 Measurement10A 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 Dimension10D Dimension and Size

10.3 Calculating Perimeters of Polygons, Areas of Rectangles, and Volumes of Boxes10E 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 Measurements10J Reporting and Interpreting Measurements

10.5 Converting from One Unit of Measurement to Another10K 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 Area11A 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 Theorem11D 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 Triangles11G Choosing the Base and Height of Triangles 11H Explaining Why the Area Formula for Triangles Is Valid 11I Determining Areas

11.4 Areas of Parallelograms11J 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 Area11L 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 Pi11P 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 Shapes11T Determining the Area of an Irregular Shape

11.8 Relating the Perimeter and Area of a Shape11U How Are Perimeter and Area Related? 11V Can We Determine Area by Measuring Perimeter?

11.9 Principles for Determining Volumes11W 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 Cones11Z 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 Scaling11HH 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 Multiples12A 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 Multiple12E 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 Numbers12K The Sieve of Eratosthenes 12L The Trial Division Method for Determining whether a Number Is Prime

12.4 Even and Odd12M 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 Tests12P The Divisibility Test for 3

12.6 Rational and Irrational Numbers12Q 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 Equations13A 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 Algebra13J 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 Sequences13P 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 Series13W 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 Functions13CC Interpreting Graphs of Functions 13DD Are These Graphs Correct?

13.6 Linear Functions13EE 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 Data14A 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 Displays14F 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 Mode14M 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 Data14Q 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 Probability15A 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 Probabilities15H Using the Meaning of Fraction Multiplication to Calculate a Probability 15I Using Fraction Multiplication and Addition to Calculate a Probability