Mathematics for Elementary School Teachers and Activities

by Beckmann, Sybilla

• ISBN13: 9780201725872
• ISBN10: 0201725878
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-01-01
• Publisher: Addison Wesley

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.

Table of Contents

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)
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)
Bibliography	697	(4)
Index	701	(10)
Cutouts for Exercises and Problems	711