Table Of ContentTHE MATHEMATICS THAT EVERY SECONDARY SCHOOL
MATH TEACHER NEEDS TO KNOW
What knowledge of mathematics do secondary school math teachers need to facilitate under-
standing, competency, and interest in mathematics for all of their students? This unique text
and resource bridges the gap between the mathematics learned in college and the mathematics
taught in secondary schools. Written in an informal, clear, and interactive learner-centered style,
it is designed to help pre-service and in-service teachers gain the deep mathematical insight they
need to engage their students in learning mathematics in a multifaceted way that is interesting,
developmental,connected,deep,understandable,andoften,surprisingandentertaining.
Launch questions at the beginning of each section capture interest and involve readers in the
learningofthemathematicalconcepts.
Student Learning Opportunities provide chances to practice problems associated with what has
been learned in the chapter, to complete proofs that are mentioned but not proved, and to apply
whathasbeenlearnedtosolvingreal-lifeproblems.
Questions from the Classroom are featured in every chapter and in the Student Learning
Opportunities, such as the deep “why” conceptual questions that middle or secondary school
students are curious about, questions requiring analysis and correction of typical student errors
and misconceptions, questions focused on counterintuitive results, and questions that contain
activitiesand/ortaskssuitableforusewithsecondaryschoolstudents.
Highlightedthemesthroughoutthechaptersaidreadersinbecomingteacherswhohavegreat
“MATH-N-SIGHT”
M MultipleApproaches/Representations
A ApplicationstoRealLife
T Technology
H History
N NatureofMathematics:ReasoningandProof
S SolvingProblems
I InterlinkingConcepts:Connections
G GradeLevels
H HoningofMathematicalUnderstandingandSkills
T TypicalErrors
ThistextisalignedwiththerecentlyreleasedCommonCoreStateStandards,andisideallysuited
for a capstone mathematics course in a secondary mathematics certification program. It is also
appropriate for any methods or mathematics course for pre- or in-service secondary mathematics
teachers,andisavaluableresourceforclassroomteachers.
AlanSultanisProfessor,Mathematics,QueensCollegeoftheCityUniversityofNewYork.
Alice F. Artzt is Professor, Secondary Mathematics Education, Queens College of the City
UniversityofNewYork.
STUDIES IN MATHEMATICAL THINKING AND LEARNING
Alan H. Schoenfeld, Series Editor
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Baroody/Dowker (Eds.) The Development of Arithmetic Concepts and Skills: Constructing Adaptive
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Boaler Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and Their
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Chazan/Callis/Lehman(Eds.) EmbracingReason:EgalitarianIdealsandtheTeachingofHighSchool
Mathematics
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Cobb/Bauersfeld(Eds.) TheEmergenceofMathematicalMeaning:InteractioninClassroomCultures
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Cohen Teachers’ Professional Development and the Elementary Mathematics Classroom: Bringing
UnderstandingstoLight
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Clements/Sarama/DiBiase (Eds.) Engaging Young Children in Mathematics: Standards for Early
ChildhoodMathematicsEducation
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English(Ed.) MathematicalandAnalogicalReasoningofYoungLearners
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English(Ed.) MathematicalReasoning:Analogies,Metaphors,andImages
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Fennema/Romberg(Eds.) MathematicsClassroomsThatPromoteUnderstanding
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Fernandez/Yoshida Lesson Study: A Japanese Approach to Improving Mathematics Teaching and
Learning
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Greer/Mukhopadhyay/Powell/Nelson-Barber(Eds.) CulturallyResponsiveMathematicsEducation
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Kaput/Carraher/Blanton(Eds.) AlgebraintheEarlyGrades
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Lajoie ReflectionsonStatistics:Learning,Teaching,andAssessmentinGradesK-12
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Lehrer/Chazan (Eds.) Designing Learning Environments for Developing Understanding of Geometry
andSpace
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Ma Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathe-
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Martin Mathematics Success and Failure Among African-American Youth: The Roles of Sociohistorical
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Romberg/Fennema/Carpenter (Eds.) Integrating Research on the Graphical Representation of
Functions
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Romberg/Carpenter/Dremock(Eds.) UnderstandingMathematicsandScienceMatters
(cid:2)
Romberg/Shafer The Impact of Reform Instruction on Mathematics Achievement: An Example of a
SummativeEvaluationofaStandards-BasedCurriculum
(cid:2)
Sarama/Clements Early Childhood Mathematics Education Research: Learning Trajectories for Young
Children
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Schliemann/Carraher/Brizuela (Eds.) Bringing Out the Algebraic Character of Arithmetic: From
Children’sIdeastoClassroomPractice
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Schoenfeld(Ed.) MathematicalThinkingandProblemSolving
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Senk/Thompson (Eds.) Standards-Based School Mathematics Curricula: What Are They? What Do
StudentsLearn?
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Solomon MathematicalLiteracy:DevelopingIdentitiesofInclusion
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Sophian TheOriginsofMathematicalKnowledgeinChildhood
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Sternberg/Ben-Zeev(Eds.) TheNatureofMathematicalThinking
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Stylianou/Blanton/Knuth(Eds.) TeachingandLearningProofAcrosstheGrades:AK-16Perspective
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Wilcox/Lanier (Eds.) Using Assessment to Reshape Mathematics Teaching: A Casebook for Teachers
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For additional information on titles in the Studies in Mathematical Thinking and Learning Series
visitwww.routledge.com/education
THE MATHEMATICS THAT EVERY
SECONDARY SCHOOL MATH
TEACHER NEEDS TO KNOW
Alan Sultan
Queens College of the City University of New York
Alice F. Artzt
Queens College of the City University of New York
Firstpublished2011
byRoutledge
270MadisonAvenue,NewYork,NY10016
SimultaneouslypublishedintheUK
byRoutledge
2ParkSquare,MiltonPark,Abingdon,OxonOX144RN
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This edition published in the Taylor & Francis e-Library, 2010.
To purchase your own copy of this or any of Taylor & Francis or Routledge’s
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(cid:2)c2011 Routledge, Taylor & Francis
Allrightsreserved.Nopartofthisbookmaybereprintedorreproduced
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ISBN 0-203-85753-4 Master e-book ISBN
ISBN13:978-0-415-99413-2(pbk)
ISBN13:978-0-203-85753-3(ebk)
BRIEF CONTENTS
Preface/Introduction xvii
NotestotheReader/Professor xxi
Acknowledgments xxvii
CHAPTER 1 IntuitionandProof 1
CHAPTER 2 BasicsofNumberTheory 17
CHAPTER 3 TheoryofEquations 69
CHAPTER 4 Measurement:AreaandVolume 113
CHAPTER 5 TheTriangle:ItsStudyandConsequences 159
CHAPTER 6 BuildingtheRealNumberSystem 215
CHAPTER 7 BuildingtheComplexNumbers 307
CHAPTER 8 Induction,Recursion,andFractalDimension 357
CHAPTER 9 FunctionsandModeling 397
CHAPTER10 GeometricTransformations 451
CHAPTER11 Trigonometry 513
CHAPTER12 DataAnalysisandProbability 599
CHAPTER13 IntroductiontoNon-EuclideanGeometry 677
CHAPTER14 ThreeProblemsofAntiquity 705
Bibliography 721
Appendix 723
Index 729
CONTENTS
Preface/Introduction xvii
NotestotheReader/Professor xxi
Acknowledgments xxvii
(cid:2) CHAPTER 1 Intuition and Proof 1
1.1 Introduction 1
1.2 CanIntuitionReallyLeadUsAstray? 1
1.3 SomeFundamentalMethodsofProof 8
1.3.1 DirectProof 9
1.3.2 ProofbyContradiction 10
1.3.3 ProofbyCounterexample 12
1.3.4 TheFinalityofProof 13
(cid:2) CHAPTER 2 Basics of Number Theory 17
2.1 Introduction 17
2.2 Odd,Even,andDivisibilityRelationships 17
2.3 TheDivisibilityRules 25
2.4 FactsaboutPrimeNumbers 31
2.4.1 ThePrimeNumberTheorem 36
2.5 TheDivisionAlgorithm 38
2.6 TheGreatestCommonDivisor(GCD)andtheEuclideanAlgorithm 42
2.7 TheDivisionAlgorithmforPolynomials 48
2.8 DifferentBaseNumberSystems 51
2.9 ModularArithmetic 56
2.9.1 Application:RSAEncryption 59
2.10 DiophantineAnalysis 61
