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Mathematics for Elementary School Teachers and Activities

ISBN: 9780201725872 | 0201725878
Edition: 1st
Format: Hardcover
Publisher: Addison Wesley
Pub. Date: 1/1/2005

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SummaryTable of Contents
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.
Prefaceix
Problem Solving
1(12)
Solving Problems
1(3)
... MORE
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
74(1)
Problems
75(2)
Fractions as Numbers
77(6)
Fractions on Number Lines
Decimal Representations of Fractions
Practice Problems
79(1)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
99(3)
Problems
102(3)
Addition and Subtraction
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
Practice Problems
110(1)
Answers to Practice Problems
110(1)
Problems
110(1)
Why the Standard Algorithms for Adding and Subtracting Decimal Numbers Work
110(11)
What Is an Algorithm?
The Addition Algorithm
The Subtraction Algorithm
Practice Problems
118(1)
Answers to Practice Problems
118(1)
Problems
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?
Practice Problems
127(1)
Answers to Practice Problems
127(2)
Problems
129(2)
When Do We Add Percentages?
131(2)
Practice Problems
131(1)
Answers to Practice Problems
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)
Answers to Practice Problems
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
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?
Practice Problems
152(1)
Answers to Practice Problems
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)
Answers to Practice Problems
163(2)
Problems
165(1)
Why Multiplying Decimal Numbers by 10 Is Easy
166(3)
Practice Problems
168(1)
Answers to Practice Problems
168(1)
Problems
169(1)
Multiplication and Areas of Rectangles
169(4)
Practice Problems
171(1)
Answers to Practice Problems
171(1)
Problems
172(1)
The Commutative Property of Multiplication
173(3)
Practice Problems
175(1)
Answers to Practice Problems
175(1)
Problems
176(1)
Multiplication and Volumes of Boxes
176(5)
Practice Problems
179(1)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
225(1)
Problems
225(2)
Powers
227(2)
Powers of Numbers Other Than 10
Practice Problem
229(1)
Answer to Practice Problem
229(1)
Problems
229(1)
Multiplying Decimals
229(6)
Practice Problems
232(1)
Answers to Practice Problems
232(2)
Problems
234(1)
Multiplying Negative Numbers
235(3)
Practice Problems
237(1)
Answers to Practice Problems
237(1)
Problems
237(1)
Scientific Notation
238(7)
Practice Problems
241(1)
Answers to Practice Problems
241(1)
Problems
242(3)
Division
245(72)
The Meaning of Division
245(11)
The Two Interpretations of Division
Relating Division and Multiplication
Answers with or without Remainders
Division with Negative Numbers
Practice Problems
250(1)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
309(3)
Problems
312(5)
Geometry
317(68)
Visualization
318(10)
Practice Problems
321(2)
Answers to Practice Problems
323(3)
Problems
326(2)
Angles
328(17)
Angles and Reflected Light
Practice Problems
334(2)
Answers to Practice Problems
336(3)
Problems
339(6)
Circles and Spheres
345(5)
Definitions of Circle and Sphere
When Circles or Spheres Meet
Practice Problems
348(1)
Answers to Practice Problems
349(1)
Problems
350(1)
Triangles
350(7)
The Sum of the Angles in a Triangle
Practice Problems
352(2)
Answers to Practice Problems
354(1)
Problems
355(2)
Quadrilaterals and Other Polygons
357(10)
Showing Relationships with Venn Diagrams
Definitions of Shapes versus Additional Properties the Shapes Have
Practice Problems
362(1)
Answers to Practice Problems
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)
Answers to Practice Problems
370(1)
Problems
371(1)
Polyhedra and Other Solid Shapes
372(13)
Prisms, Cylinders, Pyramids, and Cones
The Platonic Solids
Practice Problems
378(1)
Answers to Practice Problems
379(2)
Problems
381(4)
Geometry of Motion and Change
385(48)
Reflections, Translations, and Rotations
385(6)
Practice Problems
388(1)
Answers to Practice Problems
389(1)
Problems
390(1)
Symmetry
391(17)
Defining Symmetry
Creating Symmetry
Analyzing Designs
Practice Problems
402(2)
Answers to Practice Problems
404(2)
Problems
406(2)
Congruence
408(5)
Practice Problems
412(1)
Answers to Practice Problems
412(1)
Problems
412(1)
Similarity
413(20)
When Are Two Shapes Similar?
Using Similar Triangles to Determine Distances
Practice Problems
421(2)
Answers to Practice Problems
423(4)
Problems
427(6)
Measurement
433(38)
The Concept of Measurement
434(8)
Systems of Measurement
Practice Problems
441(1)
Answers to Practice Problems
441(1)
Problems
442(1)
Error and Accuracy in Measurements
442(6)
Interpreting Reported Measurements
Working with Measurements
Practice Problems
446(1)
Answers to Practice Problems
446(1)
Problems
447(1)
Length, Area, Volume, and Dimension
448(5)
Practice Problems
452(1)
Answers to Practice Problems
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)
Answer to Practice Problem
457(1)
Problems
457(2)
Comparing Sizes of Objects
459(2)
Practice Problem
460(1)
Answer to Practice Problem
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)
Answers to Practice Problems
467(1)
Problems
468(3)
More About Area and Volume
471(76)
The Moving and Combining Principles about Area
471(17)
Practice Problems
474(2)
Answers to Practice Problems
476(5)
Problems
481(7)
Using the Moving and Combining Principles to Prove the Pythagorean Theorem
488(6)
Practice Problems
490(1)
Answers to Practice Problems
491(2)
Problems
493(1)
Approximating Areas of Irregular Shapes
494(4)
Practice Problems
496(1)
Answers to Practice Problems
497(1)
Problems
498(1)
Cavalieri's Principle about Shearing and Area
498(6)
Practice Problems
500(1)
Answers to Practice Problems
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)
Answers to Practice Problems
509(2)
Problems
511(4)
Areas of Parallelograms
515(4)
An Area Formula for Parallelograms
Practice Problems
517(1)
Answers to Practice Problems
517(1)
Problems
518(1)
Areas of Circles and the Number Pi
519(10)
Practice Problems
522(1)
Answers to Practice Problems
523(2)
Problems
525(4)
Relating the Perimeter and Area of a Shape
529(4)
Practice Problem
531(1)
Answer to Practice Problem
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)
Answers to Practice Problems
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)
Answers to Practice Problems
539(1)
Problems
540(2)
Areas, Volumes, and Scaling
542(5)
Practice Problems
543(1)
Answers to Practice Problems
544(2)
Problems
546(1)
Number Theory
547(36)
Factors and Multiples
547(4)
Practice Problems
549(1)
Answers to Practice Problems
550(1)
Problems
551(1)
Greatest Common Factor and Least Common Multiple
551(4)
Using GCFs and LCMs with Fractions
Practice Problems
554(1)
Answers to Practice Problems
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)
Answers to Practice Problems
561(1)
Problems
561(1)
Even and Odd
562(3)
Practice Problems
563(1)
Answers to Practice Problems
564(1)
Problems
564(1)
Divisibility Tests
565(5)
Practice Problems
568(1)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
602(4)
Problems
606(2)
Sequences
608(11)
Arithmetic Sequences
Geometric Sequences
Other Sequences
Practice Problems
613(1)
Answers to Practice Problems
614(2)
Problems
616(3)
Series
619(7)
Arithmetic Series
Geometric Series
Practice Problems
623(1)
Answers to Practice Problems
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)
Answers to Practice Problems
636(1)
Problems
637(2)
Linear Functions
639(8)
Practice Problems
642(1)
Answers to Practice Problems
643(1)
Problems
644(3)
Statistics
647(32)
Designing Investigations and Gathering Data
648(4)
Populations and Samples
Surveys Experiments
Practice Problems
650(1)
Answers to Practice Problems
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)
Answers to Practice Problems
661(1)
Problems
662(2)
The Center of Data: Mean, Median, and Mode
664(6)
Practice Problems
666(1)
Answers to Practice Problems
667(1)
Problems
668(2)
The Spread of Data: Percentiles
670(9)
Practice Problems
671(2)
Answers to Practice Problems
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)
Answers to Practice Problems
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)
Answers to Practice Problems
693(1)
Problems
694(3)
Bibliography697(4)
Index701(10)
Cutouts for Exercises and Problems711

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