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

Description

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