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

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of contents
Preface
Chapter 1 Proportional Reasoning: An Overview
	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
Chapter 2 Fractions and Rational Numbers
	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 .
	Activities
Chapter 3 Relative Thinking and Measurement
	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
Chapter 4 Quantities and Covariation
	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
Chapter 5 Proportional Reasoning
	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
Chapter 6 Reasoning with Fractions
	Visualizing Operations
	Equivalent Fractions and Unitizing
	Comparing Fractions
	Fractions in Between
	Activities
Chapter 7 Fractions as Part–Whole Comparisons
	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
Chapter 8 Fractions as Quotients
	Quotients
	Partitioning as Fair Sharing
	Partitioning Activities
	Children’s Partitioning
	Equivalence
	Simplifying Fractions
	Understanding Fractions as Quotients
	More Advanced Reasoning
	Sharing Different Pizzas
	Activities
Chapter 9 Fractions as Operators
	Operators
	Exchange Models
	Composition
	Area Model for Multiplication
	Area Model for Division
	Compositions and Paper Folding
	Understanding Operators
	Activities
Chapter 10 Fractions as Measures
	Measures of Distance
	Static and Dynamic Measurement
	The Goals of Successive Partitioning
	Understanding Fractions as Measures
	Units, Equivalent Fractions, and Comparisons
	Fraction Operations
	Activities
Chapter 11 Ratios and Rates
	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
Chapter 12 Changing Instruction
	Why Change?
	A Summary of Fraction Interpretations
	Central Structures
	Characteristics of Proportional Thinkers
	Obstacles to Change
	Sequencing Topics
	Directions for Change
	Challenging Problems
Index