Table Of ContentUncomplicating
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
