Uncomplicating ALGEBRA to Meet Common Core Standards in Math, K–8 Uncomplicating AA LLGG EE BB RR AA to Meet Common Core Standards in Math, K–8 MARIAN SMALL Teachers College, Columbia University www.nelson.com New York and London Published by Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027; distributed in Canada by Nelson Education, 1120 Birchmount Road, Toronto, ON, Canada M1K 5G4. Copyright © 2014 by Teachers College, Columbia University All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, or any information storage and retrieval system, without permission from the publisher. Text Design: Lynne Frost Library of Congress Cataloging-in-Publication Data Small, Marian. Uncomplicating algebra to meet common core standards in math, K–8 / Marian Small. pages cm “Teachers College, Columbia University.” Includes bibliographical references and index. ISBN 978-0-8077-5517-4 (pbk. : alk. paper) ISBN 978-0-8077-7309-3 (e-book) 1. Algebra—Study and teaching (Primary)—Standards—United States. 2. Algebra—Study and teaching (Elementary)—Standards—United States. 3. Mathematics—Study and teaching (Primary)—Standards—United States. 4. Mathematics—Study and teaching (Elementary)—Standards—United States. I. Title. QA159.S57 2014 372.7'1021873—dc23 2014000913 ISBN 978-0-8077-5517-4 (paper) ISBN 978-0-8077-7309-3 (ebook) Printed on acid-free paper Manufactured in the United States of America 21 20 19 18 17 16 15 14 8 7 6 5 4 3 2 1 C ONTENTS Preface vii Organization of the Book vii Acknowledgments viii Introduction 1 Increased Focus on Algebra 1 What Is Algebra? 1 General Reasons Students Struggle with Algebra 3 Focusing on the CCSSM Standards for Mathematical Practice 8 Focusing on the CCSSM Standards for Mathematical Content 10 Summary 10 Kindergarten 11 Using Equations to Describe Addition and Subtraction 11 Summary 13 Grade 1 15 Using Equations to Describe Addition and Subtraction 15 Summary 21 Grade 2 23 Using Equations to Represent Problems and Relationships 23 Summary 28 Grade 3 29 Using Equations to Represent Problems 29 Patterns in the Addition and Multiplication Tables 33 Summary 42 Grade 4 43 Arithmetic Generalizations 43 Representing Situations Using Equations 44 Pattern Rules 49 Solving Measurement Problems with Equations 52 Summary 54 v vi Contents Grade 5 55 Analyzing Patterns 55 Using Equations to Represent and Solve Problems 58 Solving Measurement Problems with Equations 60 Using a Coordinate Grid to Visualize Algebraic Relationships 62 Summary 66 Grade 6 67 Observing Proportional Relationships 67 Using a Coordinate Grid to Visualize Algebraic Relationships 70 Variables to Describe Generalizations 73 Equivalent Expressions 79 Equations and Inequalities Involving Rational Numbers 83 Representing Linear Relationships 86 Solving Measurement Problems with Equations 89 Summary 91 Grade 7 93 Recognizing Proportionality 93 Properties of Operations with Rational Numbers 97 Equivalent Expressions 100 Using Algebra to Solve Problems 102 Solving Measurement Problems with Equations 108 Summary 110 Grade 8 111 Exponent Conventions and Properties 111 Equations of Lines 113 Solving Linear Equations 122 Solving Two Equations in Two Unknowns 128 Functions 137 Linear Functions 145 Pythagorean Th eorem 150 Measurement Problems 152 Linear Trends 154 Summary 156 Conclusion 157 Bibliography 159 Index 161 About the Author 165 P REFACE ORGANIZATION OF THE BOOK Th is resource is intended to help teachers improve student success in learning algebra by sharing approaches that will lead to a deeper and richer understanding of the subject. Th e resource is organized by grade level around the Common Core State Stan- dards for Mathematics (CCSSM) that are related to algebraic thinking. Th e grades covered in this resource begin with Kindergarten, where the fi rst relevant standard is found in the Operations and Algebraic Th inking domain, and end with Grade 8, where the focus is on working with linear equations and functions. For each sec- tion, a portion of the relevant standard is presented, followed by a delineation of important underlying ideas associated with that portion of the standard, as well as some Good Questions to Ask to bring those underlying ideas out. Th e discussions of underlying ideas include •• background on the mathematics of the standard, •• suggestions for appropriate representations of the specifi c mathematical ideas, •• suggestions for explaining the ideas to students, and •• cautions about misconceptions or situations to avoid. Following each set of underlying ideas is a group of Good Questions to Ask that can be used for classroom instruction, student practice, or assessment. Among the questions are many open questions, as well as more directed conceptual ques- tions that might be supplemental to what teachers normally are provided in the resources they use. Th e Common Core State Standards for Mathematical Practice underlie the content throughout and are explicitly mentioned in a number of instances. For Whom Is Th is Book Useful and Why? Th is resource is designed to aid math teachers of Kindergarten–Grade 5 in build- ing a solid foundation for student work in algebra in the middle grades and to aid vii viii Preface teachers of Grades 6–8 in preparing students for work in algebra in the secondary grades. It is also intended to serve as a resource for math coaches in assisting class- room teachers in their transition to teaching mathematics within the more demand- ing framework of the Common Core State Standards. I expect this book to be help- ful as well to preservice teachers as they prepare themselves to understand and teach math in a way that will foster a deep level of understanding in their students. Considering the Bigger Picture While I would hope that all users would read the entire book, I particularly en- courage this approach for math coaches and preservice teachers. For grade-level or grade-band teachers, I suggest reading the Introduction and the grade-level sections that most directly apply for their particular groups of students, but also becoming acquainted with the mathematics related to algebra taught in grades directly below and above their groups. Because students in any classroom possess diff erent levels of knowledge, in order to diff erentiate instruction appropriately, teachers must be aware of missing prerequisite knowledge, as well as suitable di- rections for moving forward. Lastly, I hope that using this book helps make algebra make more sense both to the readers and to their students. ACKNOWLEDGMENTS I have been fortunate to work with excellent editors in preparing this manuscript. I wish to thank, too, several reviewers whose comments were very helpful in shap- ing the manuscript. Most importantly, I have been fortunate to have interacted with so many teachers who have responded positively to my approach to mathe- matics teaching and learning. Th ese educators have continued to encourage me to write more. I thank these professionals for their personal support, as well as for sharing my work with their colleagues. Uncomplicating ALGEBRA to Meet Common Core Standards in Math, K–8