4th Edition

# Teaching Fractions and Ratios for Understanding Essential Content Knowledge and Instructional Strategies for Teachers

By Susan J. Lamon Copyright 2020
294 Pages
by Routledge

294 Pages
by Routledge

294 Pages
by Routledge

Also available as eBook on:

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

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

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.