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.

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   3H Misconceptions 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

 

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

 

 

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.

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