Preface 

ix  


1  (12) 


1  (3) 

Polya's Four ProblemSolving Steps 



Why Do I Need to Solve Problems Myself When the Teacher Could Just Tell Me How? 




4  (9) 

What Is an Explanation in Mathematics? 



Writing Good Explanations 



Sample Problems and Solutions 




9  (1) 

Answers to Practice Problems 


9  (2) 


11  (2) 

Numbers and the Decimal System 


13  (44) 

Introduction to the Number Systems 


15  (7) 



The Whole Numbers and the Integers 







Relationships among the Number Systems 




21  (1) 

Answers to Practice Problems 


22  (1) 

The Decimal System and Place Value 


22  (8) 

The Origins of the Decimal System 





Values of Places in Decimals and Powers of Ten 



Saying Decimal Numbers and Writing Decimal Numbers with Words 




29  (1) 

Answers to Practice Problems 


29  (1) 


30  (1) 

Representing Decimal Numbers 


30  (15) 

Representing Decimal Numbers with Physical Objects 



Representing Decimal Numbers on Number Lines 




40  (1) 

Answers to Practice Problems 


41  (2) 


43  (2) 

Comparing Sizes of Decimal Numbers 


45  (7) 

Comparing Numbers by Viewing Them as Amounts 



Using Number Lines to Compare Decimal Numbers 



Comparing Negative Decimal Numbers by Viewing Them as Owed Amounts 




50  (1) 

Answers to Practice Problems 


50  (1) 


51  (1) 


52  (5) 



Working with Decimal Numbers That Represent Actual Quantities 




55  (1) 

Answers to Practice Problems 


55  (1) 


56  (1) 


57  (48) 


57  (11) 

A Fraction Is Associated with a Whole 





Understanding Improper Fractions 




61  (1) 

Answers to Practice Problems 


62  (3) 


65  (3) 


68  (9) 

Why Every Fraction Is Equal to Many Other Fractions 





The Simplest Form of a Fraction 




73  (1) 

Answers to Practice Problems 


74  (1) 


75  (2) 


77  (6) 

Fractions on Number Lines 



Decimal Representations of Fractions 




79  (1) 

Answers to Practice Problems 


80  (2) 


82  (1) 

Comparing Sizes of Fractions 


83  (11) 

Comparing Fractions by Converting to Decimal Numbers 



Comparing Fractions by Using Common Denominators 



Comparing Fractions by CrossMultiplying 



Using Other Reasoning to Compare Fractions 




89  (1) 

Answers to Practice Problems 


90  (2) 


92  (2) 


94  (11) 



Percents, Fractions, and Pictures 



Three Types of Percent Problems 




98  (1) 

Answers to Practice Problems 


99  (3) 


102  (3) 


105  (52) 

Interpretations of Addition and Subtraction 


105  (5) 

Addition and Subtraction as Combining and Taking Away 



Addition and Subtraction on Number Lines 



Relating Addition and Subtraction 



Subtraction as Comparison 




110  (1) 

Answers to Practice Problems 


110  (1) 


110  (1) 

Why the Standard Algorithms for Adding and Subtracting Decimal Numbers Work 


110  (11) 





The Subtraction Algorithm 




118  (1) 

Answers to Practice Problems 


118  (1) 


119  (2) 

Adding and Subtracting Fractions 


121  (10) 

Adding and Subtracting Fractions with Like Denominators 



Adding and Subtracting Fractions with Unlike Denominators by Finding Common Denominators 



Writing Mixed Numbers as Improper Fractions 



Writing Finite Decimals as Fractions 



When Is Combining Not Adding? 




127  (1) 

Answers to Practice Problems 


127  (2) 


129  (2) 

When Do We Add Percentages? 


131  (2) 


131  (1) 

Answers to Practice Problems 


132  (1) 


132  (1) 

Percent Increase and Percent Decrease 


133  (9) 

Two Methods for Calculating Percent Increase or Decrease 



Two Methods for Calculating Amounts when the Percent Increase or Decrease Is Given 



The Importance of the Reference Amount 




138  (1) 

Answers to Practice Problems 


139  (1) 


140  (2) 

The Commutative and Associative Properties of Addition and Mental Math 


142  (15) 

Parentheses in Expressions with Three or More Terms 



The Associative Property of Addition 



The Commutative Property of Addition 



Writing Equations That Correspond to a Method of Calculation 



Using the Associative and Commutative Properties to Solve Addition Problems 



The Commutative and Associative Properties of Addition in Algebra 



Other Mental Methods of Addition and Subtraction 



Why Should We Be Able to Use Reasoning to Add and Subtract? 




152  (1) 

Answers to Practice Problems 


153  (1) 


154  (3) 


157  (62) 

The Meaning of Multiplication and Ways to Show Multiplication 


157  (9) 

The Meaning of Multiplication 



Showing Multiplicative Structure by Grouping 



Showing Multiplicative Structure with Organized Lists 



Showing Multiplicative Structure with Array Diagrams 



Showing Multiplicative Structure with Tree Diagrams 



Using Organized Lists, Array Diagrams, and Tree Diagrams 




162  (1) 

Answers to Practice Problems 


163  (2) 


165  (1) 

Why Multiplying Decimal Numbers by 10 Is Easy 


166  (3) 


168  (1) 

Answers to Practice Problems 


168  (1) 


169  (1) 

Multiplication and Areas of Rectangles 


169  (4) 


171  (1) 

Answers to Practice Problems 


171  (1) 


172  (1) 

The Commutative Property of Multiplication 


173  (3) 


175  (1) 

Answers to Practice Problems 


175  (1) 


176  (1) 

Multiplication and Volumes of Boxes 


176  (5) 


179  (1) 

Answers to Practice Problems 


179  (1) 


179  (2) 

The Associative Property of Multiplication 


181  (6) 

Why Does the Associative Property of Multiplication Make Sense? 



Using the Associative and Commutative Properties of Multiplication 




184  (1) 

Answers to Practice Problems 


185  (1) 


185  (2) 

The Distributive Property 


187  (10) 

Expressions Involving Both Multiplication and Addition 



Why Does the Distributive Property Make Sense? 



Variations on the Distributive Property 



FOIL Using the Distributive Property 




193  (1) 

Answers to Practice Problems 


194  (1) 


195  (2) 

Mental Math, Properties of Arithmetic, and Algebra 


197  (11) 

Writing Equations that Correspond to a Mental Calculation Strategy 



Equations and Properties of Arithmetic Are Stepping Stones to Algebra 




201  (1) 

Answers to Practice Problems 


202  (2) 


204  (4) 

Why the Procedure for Multiplying Whole Numbers Works 


208  (11) 

The PartialProducts Multiplication Algorithm 



Relating the Standard and Partial Products Algorithms 



Why We Place Extra Zeros on Some Lines in the Standard Algorithm 



Why the Algorithms Produce Correct Answers 




213  (1) 

Answers to Practice Problems 


213  (3) 


216  (3) 

Multiplication of Fractions, Decimals, and Negative Numbers 


219  (26) 


219  (8) 

The Meaning of Multiplication for Fractions 



The Procedure for Multiplying Fractions 



Explaining Why the Procedure for Multiplying Fractions Gives Correct Answers 



Why Do We Need to Know This? 




224  (1) 

Answers to Practice Problems 


225  (1) 


225  (2) 


227  (2) 

Powers of Numbers Other Than 10 




229  (1) 

Answer to Practice Problem 


229  (1) 


229  (1) 


229  (6) 


232  (1) 

Answers to Practice Problems 


232  (2) 


234  (1) 

Multiplying Negative Numbers 


235  (3) 


237  (1) 

Answers to Practice Problems 


237  (1) 


237  (1) 


238  (7) 


241  (1) 

Answers to Practice Problems 


241  (1) 


242  (3) 


245  (72) 


245  (11) 

The Two Interpretations of Division 



Relating Division and Multiplication 



Answers with or without Remainders 



Division with Negative Numbers 




250  (1) 

Answers to Practice Problems 


251  (2) 


253  (3) 

Understanding Long Division 


256  (9) 

Solving Division Problems without the Standard Longhand Procedure 



Standard Long Division and the Scaffold Method 



Interpreting Long Division from the ``How Many in Each Group?'' Viewpoint 




261  (1) 

Answers to Practice Problems 


262  (1) 


263  (2) 


265  (13) 

Explaining Why Fractions Can Be Expressed in Terms of Division 



Mixed Number Answers to Whole Number Division Problems 



Using Division to Convert Improper Fractions to Mixed Numbers 



Using Long Division to Calculate Decimal Answers to Whole Number Division Problems 



Using Long Division to Calculate Decimal Representations of Fractions 



Using Diagrams to Determine Decimal Representations of Fractions 



Fractions with Negative Numerators or Denominators 




275  (1) 

Answers to Practice Problems 


275  (2) 


277  (1) 


278  (13) 

The Two Interpretations of Division for Fractions 



Dividing by 1/2 Versus Dividing in 1/2 



The ``Invert and Multiply'' Procedure for Fraction Division 



Explaining Why ``Invert and Multiply'' Is Valid by Relating Division to Multiplication 



Using the ``How Many Groups?'' Interpretation to Explain Why ``Invert And Multiply'' Is Valid 



Using the ``How Many in One Group?'' Interpretation to Explain Why ``Invert And Multiply'' Is Valid 




285  (1) 

Answers to Practice Problems 


286  (2) 


288  (3) 


291  (8) 

The Two Interpretations of Division for Decimals 



Explaining the Shifting of Decimal Points by Multiplying and Dividing by the Same Power of 10 



Explaining the Shifting of Decimal Points by Changing from Dollars to Cents 



Explaining the Shifting of Decimal Points by Changing the Unit 



Deciding Where to Put the Decimal Point by Estimating the Size of the Answer 



Dividing Numbers in the Millions, Billions, and Trillions 




297  (1) 

Answers to Practice Problems 


297  (1) 


298  (1) 


299  (18) 





Solving Proportions with Multiplication, Division, and Simple Logical Reasoning 



The Logic Behind Solving Proportions by CrossMultiplying Fractions 



Using Proportions: The Consumer Price Index 




308  (1) 

Answers to Practice Problems 


309  (3) 


312  (5) 


317  (68) 


318  (10) 


321  (2) 

Answers to Practice Problems 


323  (3) 


326  (2) 


328  (17) 

Angles and Reflected Light 




334  (2) 

Answers to Practice Problems 


336  (3) 


339  (6) 


345  (5) 

Definitions of Circle and Sphere 



When Circles or Spheres Meet 




348  (1) 

Answers to Practice Problems 


349  (1) 


350  (1) 


350  (7) 

The Sum of the Angles in a Triangle 




352  (2) 

Answers to Practice Problems 


354  (1) 


355  (2) 

Quadrilaterals and Other Polygons 


357  (10) 

Showing Relationships with Venn Diagrams 



Definitions of Shapes versus Additional Properties the Shapes Have 




362  (1) 

Answers to Practice Problems 


363  (1) 


364  (3) 

Constructions with Straightedge and Compass 


367  (5) 

Dividing a Line Segment in Half and Constructing a Perpendicular Line 



Dividing an Angle in Half 




370  (1) 

Answers to Practice Problems 


370  (1) 


371  (1) 

Polyhedra and Other Solid Shapes 


372  (13) 

Prisms, Cylinders, Pyramids, and Cones 






378  (1) 

Answers to Practice Problems 


379  (2) 


381  (4) 

Geometry of Motion and Change 


385  (48) 

Reflections, Translations, and Rotations 


385  (6) 


388  (1) 

Answers to Practice Problems 


389  (1) 


390  (1) 


391  (17) 








402  (2) 

Answers to Practice Problems 


404  (2) 


406  (2) 


408  (5) 


412  (1) 

Answers to Practice Problems 


412  (1) 


412  (1) 


413  (20) 

When Are Two Shapes Similar? 



Using Similar Triangles to Determine Distances 




421  (2) 

Answers to Practice Problems 


423  (4) 


427  (6) 


433  (38) 

The Concept of Measurement 


434  (8) 




441  (1) 

Answers to Practice Problems 


441  (1) 


442  (1) 

Error and Accuracy in Measurements 


442  (6) 

Interpreting Reported Measurements 



Working with Measurements 




446  (1) 

Answers to Practice Problems 


446  (1) 


447  (1) 

Length, Area, Volume, and Dimension 


448  (5) 


452  (1) 

Answers to Practice Problems 


452  (1) 


453  (1) 

Calculating Perimeters of Polygons, Areas of Rectangles, and Volumes of Boxes 


453  (6) 

Calculating Perimeters of Polygons 



Calculating Areas of Rectangles 



Calculating Volumes of Boxes 




457  (1) 

Answer to Practice Problem 


457  (1) 


457  (2) 

Comparing Sizes of Objects 


459  (2) 


460  (1) 

Answer to Practice Problem 


460  (1) 


460  (1) 

Converting from One Unit of Measurement to Another 


461  (10) 



Area and Volume Conversions 



Approximate Conversions and Checking Your Work 




466  (1) 

Answers to Practice Problems 


467  (1) 


468  (3) 

More About Area and Volume 


471  (76) 

The Moving and Combining Principles about Area 


471  (17) 


474  (2) 

Answers to Practice Problems 


476  (5) 


481  (7) 

Using the Moving and Combining Principles to Prove the Pythagorean Theorem 


488  (6) 


490  (1) 

Answers to Practice Problems 


491  (2) 


493  (1) 

Approximating Areas of Irregular Shapes 


494  (4) 


496  (1) 

Answers to Practice Problems 


497  (1) 


498  (1) 

Cavalieri's Principle about Shearing and Area 


498  (6) 


500  (1) 

Answers to Practice Problems 


501  (1) 


501  (3) 


504  (11) 

Base and Height for Triangles 



The Formula for the Area of a Triangle 



Why Is the Area Formula for Triangles Valid? 




508  (1) 

Answers to Practice Problems 


509  (2) 


511  (4) 


515  (4) 

An Area Formula for Parallelograms 




517  (1) 

Answers to Practice Problems 


517  (1) 


518  (1) 

Areas of Circles and the Number Pi 


519  (10) 


522  (1) 

Answers to Practice Problems 


523  (2) 


525  (4) 

Relating the Perimeter and Area of a Shape 


529  (4) 


531  (1) 

Answer to Practice Problem 


531  (1) 


532  (1) 

Principles for Determining Volumes 


533  (3) 

The Moving and Combining Principles about Volumes 



Cavalieri's Principle about Shearing and Volumes 



Submersing and Volume, Floating and Weight 




535  (1) 

Answers to Practice Problems 


536  (1) 


536  (1) 

Volumes of Prisms, Cylinders, Pyramids, and Cones 


536  (6) 

Volume Formulas for Prisms and Cylinders 



Volume Formulas for Pyramids and Cones 




538  (1) 

Answers to Practice Problems 


539  (1) 


540  (2) 

Areas, Volumes, and Scaling 


542  (5) 


543  (1) 

Answers to Practice Problems 


544  (2) 


546  (1) 


547  (36) 


547  (4) 


549  (1) 

Answers to Practice Problems 


550  (1) 


551  (1) 

Greatest Common Factor and Least Common Multiple 


551  (4) 

Using GCFs and LCMs with Fractions 




554  (1) 

Answers to Practice Problems 


554  (1) 


555  (1) 


555  (7) 

The Sieve of Eratosthenes 



The Trial Division Method for Determining Whether a Number Is Prime 



Factoring into Products of Prime Numbers 



How Many Prime Numbers Are There? 




561  (1) 

Answers to Practice Problems 


561  (1) 


561  (1) 


562  (3) 


563  (1) 

Answers to Practice Problems 


564  (1) 


564  (1) 


565  (5) 


568  (1) 

Answers to Practice Problems 


568  (1) 


568  (2) 

Rational and Irrational Numbers 


570  (13) 

Decimal Representations of Fractions Either Terminate or Repeat 



Writing Repeating and Terminating Decimals as Fractions 



Another Method for Writing Repeating Decimals as Fractions 



The Surprising Fact That 0.99999 . . . = 1 



The Irrationality of the Square Root of 2 and Other Square Roots 




578  (1) 

Answers to Practice Problems 


578  (2) 


580  (3) 


583  (64) 

Mathematical Expressions, Formulas, and Equations 


584  (10) 

Variables, Expressions, and Formulas 





Evaluating Expressions and Formulas 




588  (2) 

Answers to Practice Problems 


590  (2) 


592  (2) 

Solving Equations with Pictures and with Algebra 


594  (14) 



Solving Equations Using Algebra 



Solving Algebra Story Problems with SingaporeStyle Strip Diagrams 




602  (1) 

Answers to Practice Problems 


602  (4) 


606  (2) 


608  (11) 








613  (1) 

Answers to Practice Problems 


614  (2) 


616  (3) 


619  (7) 






623  (1) 

Answers to Practice Problems 


624  (1) 


625  (1) 


626  (13) 

Representing Functions with Tables 



Coordinate Planes and Displaying Functions with Graphs 






636  (1) 

Answers to Practice Problems 


636  (1) 


637  (2) 


639  (8) 


642  (1) 

Answers to Practice Problems 


643  (1) 


644  (3) 


647  (32) 

Designing Investigations and Gathering Data 


648  (4) 






650  (1) 

Answers to Practice Problems 


650  (1) 


651  (1) 

Displaying Data and Interpreting Data Displays 


652  (12) 



Creating and Interpreting Data Displays ``Reading'' Graphs at Different Levels 




660  (1) 

Answers to Practice Problems 


661  (1) 


662  (2) 

The Center of Data: Mean, Median, and Mode 


664  (6) 


666  (1) 

Answers to Practice Problems 


667  (1) 


668  (2) 

The Spread of Data: Percentiles 


670  (9) 


671  (2) 

Answers to Practice Problems 


673  (2) 


675  (4) 


679  (18) 

Basic Principles and Calculation Methods of Probability 


680  (11) 

Principles for Determining Probability 





Probabilities of Equally Likely Outcomes 



The Probability That One of Several Possible Outcomes Occurs 



Independent and Dependent Events 



Using Organized Lists, Tree Diagrams, and Arrays to Calculate Probabilities 




687  (1) 

Answers to Practice Problems 


687  (1) 


688  (3) 

Calculating Probabilities by Considering the Ideal Outcome in a Large Number of Trials 


691  (6) 

Using Fraction Multiplication to Calculate Probabilities 



The Case of Spot, the DrugSniffing Dog 




693  (1) 

Answers to Practice Problems 


693  (1) 


694  (3) 
Bibliography 

697  (4) 
Index 

701  (10) 
Cutouts for Exercises and Problems 

711  