4th Edition
Teaching Fractions and Ratios for Understanding Essential Content Knowledge and Instructional Strategies for Teachers
Written in a user-friendly, conversational style, the fourth edition of this groundbreaking text helps pre-service and in-service mathematics teachers build the comfort and confidence they need to begin talking to children about fractions and ratios, distilling complex ideas and translating research into usable ideas for the classroom.
For two decades, Teaching Fractions and Ratios for Understanding has pushed readers beyond the limits of their current understanding of fractions and rational numbers, challenging them to refine and explain their thinking without falling back on rules and procedures they have relied on throughout their lives. All of the material offered in the book has been used with students, and is presented so that readers can see the brilliance of their insights as well as the issues that challenge their understanding. Each chapter includes children’s strategies and samples of student work for teacher analysis, as well as activities for practicing each thinking strategy, designed to be solved without rules or algorithms, using reasoning alone.
The fourth edition of this popular text has been updated throughout and includes new examples of student work, updated artwork, and more.
As with previous editions, an equally valuable component of this text is the companion book MORE! Teaching Fractions and Ratios for Understanding (2012), a supplement that is not merely an answer key but a resource that provides the scaffolding for the groundbreaking approach to fraction and ratio instruction explored here. MORE! includes in-depth discussions of selected problems in the main text, supplementary activities, Praxis preparation questions, more student work, and templates for key manipulatives.
Preface
1. Proportional Reasoning: An Overview
Student Strategies
Introduction
The Constant of Proportionality
Reasoning: Beyond Mechanization
Invariance and Covariance
Solving Proportions Using K
Multiplicative Thinking
Critical Components of Powerful Reasoning
Getting Started
Analyzing Children’s Thinking
Activities
2. Fractions and Rational Numbers
Student Strategies
New Units and a New Notational System
The Psychology of Units
New Operations and Quantities
Interference of Whole Number Ideas
Problems with Terminology
Development of Sets of Numbers
Kinds of Fractions
What are Fractions?
Rational Numbers
Fractions as Numbers
Fractions, Ratios, and Rates
Many Sources of Meaning
Multiple Interpretations of the Fraction 3/4
Activities
3. Relative Thinking and Measurement
Student Strategies
Two Perspectives on Change
Relative Thinking and Understanding Fractions
Encouraging Multiplicative Thinking
Two Meanings for "More"
The Importance of Measurement
The Compensatory Principle
The Approximation Principle
Recursive Partitioning Principle
Measuring More Abstract Qualities
Other Strategies
Activities
4. Quantities and Covariation
Student Strategies
Building on Children’s Informal Knowledge
Quantities Unquantified
Quantifiable Characteristics
Discussing Proportional Relationships in Pictures
Visualizing, Verbalizing, and Symbolizing Changing Relationships
Covariation and Invariance
Cuisenaire Strips
Scale Factors
Areas and Volumes of Scaled Figures
Similarity
Indirect Measurement
Testing for Similarity
Mockups and Pudgy People
Activities
5. Proportional Reasoning
Student Strategies
The Unit
Units Defined Implicitly
Using Units of Various Types
Reasoning Up and Down
Units and Unitizing
Unitizing Notation
Flexibility in Unitizing
Children’s Thinking
Classroom Activities to Encourage Unitizing
Visual Activities
Reasoning with Ratio Tables
Problem Types
Ratio Tables
Increasing the Difficulty
Analyzing Relationships
Percentage
Percentages as an Instructional Task
Reasoning with Percentages
Activities
6. Reasoning with Fractions
Student Strategies
Visualizing Operations
Equivalent Fractions and Unitizing
Comparing Fractions
Fractions in Between
Activities
7. Fractions as Part–Whole Comparisons
Student Strategies
Part–Whole Fractions: The Big Ideas
Unitizing and Equivalence
Problems in Current Instruction
Fraction Models
Fraction Strips
Comparing Part–Whole Fractions
Discrete Units
Multiplication
Partitive and Quotative Division
Division
Other Rational Number Interpretations
Activities
8. Fractions as Quotients
Student Strategies
Quotients
Partitioning as Fair Sharing
Partitioning Activities
Children’s Partitioning
Equivalence
Simplifying Fractions
Understanding Fractions as Quotients
More Advanced Reasoning
Sharing Different Pizzas
Activities
9. Fractions as Operators
Student Strategies
Operators
Exchange Models
Composition
Area Model for Multiplication
Area Model for Division
Compositions and Paper Folding
Understanding Operators
Activities
10. Fractions as Measures
Student Strategies
Measures of Distance
Static and Dynamic Measurement
The Goals of Successive Partitioning
Understanding Fractions as Measures
Units, Equivalent Fractions, and Comparisons
Fraction Operations
Activities
11. Ratios and Rates
Student Strategies
What is a Ratio?
Notation and Terminology
Equivalence and Comparison of Ratios
Ratios as an Instructional Task
What is a Rate?
Operations with Rates and Ratios
Linear Graphs
Comparing Ratios and Rates Graphically
Speed: The Most Important Rate
Characteristics of Speed
Students’ Misconceptions About Speed
Average Speed
Distance– Speed–Time and Graphs
Activities
12. Changing Instruction
Student Comments
Why Change?
A Summary of Fraction Interpretations
Central Structures
Characteristics of Proportional Thinkers
Obstacles to Change
Sequencing Topics
Directions for Change
Challenging Problems
Biography
Susan J. Lamon is Professor Emerita of Mathematics, Statistics, and Computer Science at Marquette University.
