Teaching fractions and ratios for understanding : essential content knowledge and instructional strategies for teachers

281 Pages·2012·3.974 MB·English

by  Susan J Lamon

THIRD EDITION
TEACHING
FRACTIONS
AND
RATIOS FOR
UNDERSTANDING
1
Essential Content Knowledge and
Instructional Strategies for Teachers
2
1 1 (cid:51) 12.2
4
1
2 4 2 14.4
4
SUSAN J. LAMON
Teaching Fractions and Ratios
for Understanding
Third Edition

Teaching Fractions and
Ratios for Understanding
Essential Content
Knowledge and Instructional
Strategies for Teachers
Third Edition
Susan J. Lamon
Firstpublished2012
byRoutledge
711ThirdAvenue,NewYork,NY10017
SimultaneouslypublishedintheUK
byRoutledge
2ParkSquare,MiltonPark,Abingdon,OxonOX144RN
RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness
(cid:1)2012Taylor&Francis
TherightofSusanJ.Lamontobeidentifiedasauthorofthisworkhasbeenassertedbyhim/her
inaccordancewithsections77and78oftheCopyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyform
orbyanyelectronic,mechanical,orothermeans,nowknownorhereafterinvented,including
photocopyingandrecording,orinanyinformationstorageorretrievalsystem,withoutpermission
inwritingfromthepublishers.
Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andare
usedonlyforidentificationandexplanationwithoutintenttoinfringe.
LibraryofCongressCataloginginPublicationData
Lamon,SusanJ.,1949-
Teaching fractions and ratios for understanding : essential content knowledge and instructional strategies for
teachers/SusanJ.Lamon.–3rded.
p.cm.
Includesindex.
1.Fractions–Studyandteaching(Elementary)2.Ratioandproportion–Studyandteaching(Elementary)I.Title.
QA137.L362011
372.7'2–dc23
2011019888
ISBN:978-0-415-88612-3(pbk)
ISBN:978-0-203-80316-5(ebk)
TypesetinAldine401BT
byExeterPremediaServicesPrivateLimited
PrintedandboundintheUnitedStatesofAmericaonacid-freepaperbyEdwardsBrothers,Inc.
For Sydney, Kylie and Grant

Contents
Preface xi
1. Proportional Reasoning: An Overview 1
Student Strategies 1
Introduction 2
The Constant of Proportionality 3
Reasoning: Beyond Mechanization 4
Invariance and Covariance 5
Solving Proportions Using k 7
Multiplicative Thinking 9
Critical Components of Powerful Reasoning 9
Getting Started 10
Analyzing Children’s Thinking 13
Activities 14
2. Fractions and Rational Numbers 20
Student Strategies 20
New Units and a New Notational System 21
The Psychology of Units 22
New Operations and Quantities 23
Interference of Whole Number Ideas 24
Problems with Terminology 26
Development of Sets of Numbers 26
Kinds of Fractions 28
What are Fractions? 29
Rational Numbers 29
Fractions as Numbers 30
Fractions, Ratios and Rates 31
Many Sources of Meaning 32
Multiple Interpretations of the Fraction 3 33
4
Activities 35
3. Relative Thinking and Measurement 39
Student Strategies 39
Two Perspectives on Change 40
Relative Thinking and Understanding Fractions 42
viii CONTENTS
Encouraging Multiplicative Thinking 43
Two Meanings for “More” 45
The Importance of Measurement 46
The Compensatory Principle 47
The Approximation Principle 48
Recursive Partitioning Principle 49
Measuring More Abstract Qualities 50
Other Strategies 52
Activities 55
4. Quantities and Covariation 63
Student Strategies 63
Building on Children’s Informal Knowledge 64
Quantities Unquantified 64
Quantifiable Characteristics 66
Discussing Proportional Relationships in Pictures 68
Visualizing, Verbalizing, and Symbolizing Changing Relationships 70
Covariation and Invariance 71
Cuisenaire Strips 73
Scale Factors 75
Similarity 76
Indirect Measurement 80
Testing for Similarity 81
Mockups and Pudgy People 83
Activities 85
5. Proportional Reasoning 96
Student Strategies 96
The Unit 97
Units Defined Implicitly 99
Using Units of Various Types 100
Reasoning Up and Down 102
Units and Unitizing 103
Unitizing Notation 105
Flexibility in Unitizing 105
Children’s Thinking 107
Classroom Activities to Encourage Unitizing 108
Visual Activities 109
Reasoning with Ratio Tables 110
Problem Types 112
Ratio Tables 114
Increasing the Difficulty 116
Analyzing Relationships 117
Percent 119
CONTENTS ix
Percents as an Instructional Task 120
Reasoning with Percents 120
Activities 121
6. Reasoning with Fractions 133
Student Strategies 133
Visualizing Operations 133
Equivalent Fractions and Unitizing 135
Comparing Fractions 136
Fractions in Between 140
Activities 142
7. Fractions as Part–Whole Comparisons 144
Student Strategies 144
Part–Whole Fractions: The Big Ideas 145
Unitizing and Equivalence 146
Problems in Current Instruction 148
Fraction Models 149
Fraction Strips 150
Comparing Part–Whole Fractions 151
Discrete Units 152
Multiplication 155
Partitive and Quotative Division 156
Division 157
Other Rational Number Interpretations 160
Activities 161
8. Fractions as Quotients 170
Student Strategies 170
Quotients 171
Partitioning as Fair Sharing 172
Partitioning Activities 173
Children’s Partitioning 174
Equivalence 177
Should We Reduce? 178
Understanding Fractions as Quotients 179
More Advanced Reasoning 180
Sharing Different Pizzas 181
Activities 184
9. Fractions as Operators 190
Student Strategies 190
Operators 191
Exchange Models 194
Composition 196