# Mathematics for Elementary School Teachers and Activities

• ISBN13:

• ISBN10:

## 0201725878

• Edition: 1st
• Format: Hardcover
• Purchase Benefits
• Free Shipping On Orders Over \$35!
Your order must be \$35 or more to qualify for free economy shipping. Bulk sales, PO's, Marketplace items, eBooks and apparel do not qualify for this offer.
• Get Rewarded for Ordering Your Textbooks! Enroll Now
List Price: \$102.20

### Summary

The goal of this text is to provide prospective elementary teachers with a deep understanding of the mathematics they will be called on to teach. Through a careful, mathematically precise development of concepts, this text asks that students go beyond simply knowing how to carry out mathematical procedures. Students must also be able to explain why mathematics works the way it does. Being able to explain why is a vital skill for teachers. Through activities, examples and applications, the author expects students to write and solve problems, make sense of the mathematics, and write clear, logical explanations of the mathematical concepts. The accompanying Activities Manual promotes engagement, exploration, and discussion of the material, rather than passive absorption. Both students and instructors should find this material fun, interesting, and rewarding.

Preface ix
 Problem Solving
1(12)
 Solving Problems
1(3)
 Polya's Four Problem-Solving Steps
 Why Do I Need to Solve Problems Myself When the Teacher Could Just Tell Me How?
 Explaining Solutions
4(9)
 What Is an Explanation in Mathematics?
 Writing Good Explanations
 Sample Problems and Solutions
 Practice Problems
9(1)
9(2)
 Problems
11(2)
 Numbers and the Decimal System
13(44)
 Introduction to the Number Systems
15(7)
 The Counting Numbers
 The Whole Numbers and the Integers
 The Rational Numbers
 The Real Numbers
 Relationships among the Number Systems
 Practice Problems
21(1)
22(1)
 The Decimal System and Place Value
22(8)
 The Origins of the Decimal System
 Place Value
 Values of Places in Decimals and Powers of Ten
 Saying Decimal Numbers and Writing Decimal Numbers with Words
 Practice Problems
29(1)
29(1)
 Problems
30(1)
 Representing Decimal Numbers
30(15)
 Representing Decimal Numbers with Physical Objects
 Representing Decimal Numbers on Number Lines
 Practice Problems
40(1)
41(2)
 Problems
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
 Practice Problems
50(1)
50(1)
 Problems
51(1)
 Rounding Decimal Numbers
52(5)
 How to Round
 Working with Decimal Numbers That Represent Actual Quantities
 Practice Problems
55(1)
55(1)
 Problems
56(1)
 Fractions
57(48)
 The Meaning of Fractions
57(11)
 A Fraction Is Associated with a Whole
 Equal Parts
 Understanding Improper Fractions
 Practice Problems
61(1)
62(3)
 Problems
65(3)
 Equivalent Fractions
68(9)
 Why Every Fraction Is Equal to Many Other Fractions
 Common Denominators
 The Simplest Form of a Fraction
 Practice Problems
73(1)
74(1)
 Problems
75(2)
 Fractions as Numbers
77(6)
 Fractions on Number Lines
 Decimal Representations of Fractions
 Practice Problems
79(1)
80(2)
 Problems
82(1)
 Comparing Sizes of Fractions
83(11)
 Comparing Fractions by Converting to Decimal Numbers
 Comparing Fractions by Using Common Denominators
 Comparing Fractions by Cross-Multiplying
 Using Other Reasoning to Compare Fractions
 Practice Problems
89(1)
90(2)
 Problems
92(2)
 Percent
94(11)
 The Meaning of Percent
 Percents, Fractions, and Pictures
 Three Types of Percent Problems
 Practice Problems
98(1)
99(3)
 Problems
102(3)
105(52)
105(5)
 Addition and Subtraction as Combining and Taking Away
 Addition and Subtraction on Number Lines
 Subtraction as Comparison
 Practice Problems
110(1)
110(1)
 Problems
110(1)
 Why the Standard Algorithms for Adding and Subtracting Decimal Numbers Work
110(11)
 What Is an Algorithm?
 The Subtraction Algorithm
 Practice Problems
118(1)
118(1)
 Problems
119(2)
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
 Practice Problems
127(1)
127(2)
 Problems
129(2)
131(2)
 Practice Problems
131(1)
132(1)
 Problems
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
 Practice Problems
138(1)
139(1)
 Problems
140(2)
 The Commutative and Associative Properties of Addition and Mental Math
142(15)
 Parentheses in Expressions with Three or More Terms
 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?
 Practice Problems
152(1)
153(1)
 Problems
154(3)
 Multiplication
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
 Practice Problems
162(1)
163(2)
 Problems
165(1)
 Why Multiplying Decimal Numbers by 10 Is Easy
166(3)
 Practice Problems
168(1)
168(1)
 Problems
169(1)
 Multiplication and Areas of Rectangles
169(4)
 Practice Problems
171(1)
171(1)
 Problems
172(1)
 The Commutative Property of Multiplication
173(3)
 Practice Problems
175(1)
175(1)
 Problems
176(1)
 Multiplication and Volumes of Boxes
176(5)
 Practice Problems
179(1)
179(1)
 Problems
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
 Practice Problems
184(1)
185(1)
 Problems
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
 Practice Problems
193(1)
194(1)
 Problems
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
 Practice Problems
201(1)
202(2)
 Problems
204(4)
 Why the Procedure for Multiplying Whole Numbers Works
208(11)
 The Partial-Products 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
 Practice Problems
213(1)
213(3)
 Problems
216(3)
 Multiplication of Fractions, Decimals, and Negative Numbers
219(26)
 Multiplying Fractions
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?
 Practice Problems
224(1)
225(1)
 Problems
225(2)
 Powers
227(2)
 Powers of Numbers Other Than 10
 Practice Problem
229(1)
229(1)
 Problems
229(1)
 Multiplying Decimals
229(6)
 Practice Problems
232(1)
232(2)
 Problems
234(1)
 Multiplying Negative Numbers
235(3)
 Practice Problems
237(1)
237(1)
 Problems
237(1)
 Scientific Notation
238(7)
 Practice Problems
241(1)
241(1)
 Problems
242(3)
 Division
245(72)
 The Meaning of Division
245(11)
 The Two Interpretations of Division
 Relating Division and Multiplication
 Division with Negative Numbers
 Practice Problems
250(1)
251(2)
 Problems
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
 Practice Problems
261(1)
262(1)
 Problems
263(2)
 Fractions and Division
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
 Practice Problems
275(1)
275(2)
 Problems
277(1)
 Dividing Fractions
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
 Practice Problems
285(1)
286(2)
 Problems
288(3)
 Dividing Decimals
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
 Practice Problems
297(1)
297(1)
 Problems
298(1)
 Ratio and Proportion
299(18)
 Ratios and Fractions
 Equivalent Ratios
 Solving Proportions with Multiplication, Division, and Simple Logical Reasoning
 The Logic Behind Solving Proportions by Cross-Multiplying Fractions
 Using Proportions: The Consumer Price Index
 Practice Problems
308(1)
309(3)
 Problems
312(5)
 Geometry
317(68)
 Visualization
318(10)
 Practice Problems
321(2)
323(3)
 Problems
326(2)
 Angles
328(17)
 Angles and Reflected Light
 Practice Problems
334(2)
336(3)
 Problems
339(6)
 Circles and Spheres
345(5)
 Definitions of Circle and Sphere
 When Circles or Spheres Meet
 Practice Problems
348(1)
349(1)
 Problems
350(1)
 Triangles
350(7)
 The Sum of the Angles in a Triangle
 Practice Problems
352(2)
354(1)
 Problems
355(2)
357(10)
 Showing Relationships with Venn Diagrams
 Definitions of Shapes versus Additional Properties the Shapes Have
 Practice Problems
362(1)
363(1)
 Problems
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
 Practice Problems
370(1)
370(1)
 Problems
371(1)
 Polyhedra and Other Solid Shapes
372(13)
 Prisms, Cylinders, Pyramids, and Cones
 The Platonic Solids
 Practice Problems
378(1)
379(2)
 Problems
381(4)
 Geometry of Motion and Change
385(48)
 Reflections, Translations, and Rotations
385(6)
 Practice Problems
388(1)
389(1)
 Problems
390(1)
 Symmetry
391(17)
 Defining Symmetry
 Creating Symmetry
 Analyzing Designs
 Practice Problems
402(2)
404(2)
 Problems
406(2)
 Congruence
408(5)
 Practice Problems
412(1)
412(1)
 Problems
412(1)
 Similarity
413(20)
 When Are Two Shapes Similar?
 Using Similar Triangles to Determine Distances
 Practice Problems
421(2)
423(4)
 Problems
427(6)
 Measurement
433(38)
 The Concept of Measurement
434(8)
 Systems of Measurement
 Practice Problems
441(1)
441(1)
 Problems
442(1)
 Error and Accuracy in Measurements
442(6)
 Interpreting Reported Measurements
 Working with Measurements
 Practice Problems
446(1)
446(1)
 Problems
447(1)
 Length, Area, Volume, and Dimension
448(5)
 Practice Problems
452(1)
452(1)
 Problems
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
 Practice Problem
457(1)
457(1)
 Problems
457(2)
 Comparing Sizes of Objects
459(2)
 Practice Problem
460(1)
460(1)
 Problems
460(1)
 Converting from One Unit of Measurement to Another
461(10)
 Dimensional Analysis
 Area and Volume Conversions
 Approximate Conversions and Checking Your Work
 Practice Problems
466(1)
467(1)
 Problems
468(3)
471(76)
 The Moving and Combining Principles about Area
471(17)
 Practice Problems
474(2)
476(5)
 Problems
481(7)
 Using the Moving and Combining Principles to Prove the Pythagorean Theorem
488(6)
 Practice Problems
490(1)
491(2)
 Problems
493(1)
 Approximating Areas of Irregular Shapes
494(4)
 Practice Problems
496(1)
497(1)
 Problems
498(1)
 Cavalieri's Principle about Shearing and Area
498(6)
 Practice Problems
500(1)
501(1)
 Problems
501(3)
 Areas of Triangles
504(11)
 Base and Height for Triangles
 The Formula for the Area of a Triangle
 Why Is the Area Formula for Triangles Valid?
 Practice Problems
508(1)
509(2)
 Problems
511(4)
 Areas of Parallelograms
515(4)
 An Area Formula for Parallelograms
 Practice Problems
517(1)
517(1)
 Problems
518(1)
 Areas of Circles and the Number Pi
519(10)
 Practice Problems
522(1)
523(2)
 Problems
525(4)
 Relating the Perimeter and Area of a Shape
529(4)
 Practice Problem
531(1)
531(1)
 Problems
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
 Practice Problems
535(1)
536(1)
 Problems
536(1)
 Volumes of Prisms, Cylinders, Pyramids, and Cones
536(6)
 Volume Formulas for Prisms and Cylinders
 Volume Formulas for Pyramids and Cones
 Practice Problems
538(1)
539(1)
 Problems
540(2)
 Areas, Volumes, and Scaling
542(5)
 Practice Problems
543(1)
544(2)
 Problems
546(1)
 Number Theory
547(36)
 Factors and Multiples
547(4)
 Practice Problems
549(1)
550(1)
 Problems
551(1)
 Greatest Common Factor and Least Common Multiple
551(4)
 Using GCFs and LCMs with Fractions
 Practice Problems
554(1)
554(1)
 Problems
555(1)
 Prime Numbers
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?
 Practice Problems
561(1)
561(1)
 Problems
561(1)
 Even and Odd
562(3)
 Practice Problems
563(1)
564(1)
 Problems
564(1)
 Divisibility Tests
565(5)
 Practice Problems
568(1)
568(1)
 Problems
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
 Practice Problems
578(1)
578(2)
 Problems
580(3)
 Functions and Algebra
583(64)
 Mathematical Expressions, Formulas, and Equations
584(10)
 Variables, Expressions, and Formulas
 Equations
 Evaluating Expressions and Formulas
 Practice Problems
588(2)
590(2)
 Problems
592(2)
 Solving Equations with Pictures and with Algebra
594(14)
 Solutions of Equations
 Solving Equations Using Algebra
 Solving Algebra Story Problems with Singapore-Style Strip Diagrams
 Practice Problems
602(1)
602(4)
 Problems
606(2)
 Sequences
608(11)
 Arithmetic Sequences
 Geometric Sequences
 Other Sequences
 Practice Problems
613(1)
614(2)
 Problems
616(3)
 Series
619(7)
 Arithmetic Series
 Geometric Series
 Practice Problems
623(1)
624(1)
 Problems
625(1)
 Functions
626(13)
 Representing Functions with Tables
 Coordinate Planes and Displaying Functions with Graphs
 Formulas for Functions
 Practice Problems
636(1)
636(1)
 Problems
637(2)
 Linear Functions
639(8)
 Practice Problems
642(1)
643(1)
 Problems
644(3)
 Statistics
647(32)
 Designing Investigations and Gathering Data
648(4)
 Populations and Samples
 Surveys Experiments
 Practice Problems
650(1)
650(1)
 Problems
651(1)
 Displaying Data and Interpreting Data Displays
652(12)
 Common Data Displays
 Creating and Interpreting Data Displays ``Reading'' Graphs at Different Levels
 Practice Problems
660(1)
661(1)
 Problems
662(2)
 The Center of Data: Mean, Median, and Mode
664(6)
 Practice Problems
666(1)
667(1)
 Problems
668(2)
670(9)
 Practice Problems
671(2)
673(2)
 Problems
675(4)
 Probability
679(18)
 Basic Principles and Calculation Methods of Probability
680(11)
 Principles for Determining Probability
 Experimental 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
 Practice Problems
687(1)
687(1)
 Problems
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 Drug-Sniffing Dog
 Practice Problems
693(1)