Bringing the Common Core Math Standards to Life As high school math teachers shift to the Common Core State Standards, the question remains: What do the standards actually look like in the classroom? This book answers that question by taking you inside of real Common Core classrooms across the country. You’ll see how exemplary teachers are meeting the new requirements and engaging students in math. Through these detailed examples of effective instruction, you will uncover how to bring the standards to life in your own classroom! Special features: ◆ A clear explanation of the big shifts happening in the classroom as a result of the Common Core State Standards ◆ Real examples of how exemplary teachers are using engaging strategies and tasks to teach algebra, geometry, trigonometry, statistics, mathematics across the curriculum, and more ◆ A detailed analysis of each example to help you understand why it is effective and how you can try it with your own students ◆ Practical, ready-to-use tools you can take back to your classroom, including unit plans and classroom handouts Yvelyne Germain-McCarthy is Professor Emerita of mathematics education at the Uni- versity of New Orleans. She is a frequent speaker at conferences and a consultant to school districts. Ivan P. Gill is Associate Professor and Science Education Coordinator at the University of New Orleans, where he has taught elementary and secondary science methods courses. Other Eye On Education Books Available from Routledge (www.routledge.com/eyeoneducation) Bringing the Common Core Math Standards to Life: Exemplary Practices from Middle Schools Yvelyne Germain-McCarthy Strategies for Common Core Mathematics: Implementing the Standards for Mathematical Practice, 6–8 Leslie Texas and Tammy Jones Guided Math in Action: Building Each Student’s Mathematical Profi ciency with Small-Group Instruction Nicki Newton Using Formative Assessment to Drive Math Instruction in Grades 3–5 Jennifer Taylor-Cox and Christine Oberdorf Math in Plain English: Literacy Strategies for the Mathematics Classroom Amy Benjamin Using Children’s Literature to Teach Problem Solving in Math: Addressing the Common Core in K–2 Jeanne White Flipping Your English Class to Reach All Learners: Strategies and Lesson Plans Troy Cockrum Authentic Learning Experiences: A Real-World Approach to Project-Based Learning Dayna Laur Teaching Students to Dig Deeper: The Common Core in Action Ben Johnson Common Core Literacy Lesson Plans: Ready-to-Use Resources, 6–8 Edited by Lauren Davis Big Skills for the Common Core: Literacy Strategies for the 6–12 Classroom Amy Benjamin and Michael Hugelmeyer Assessing Critical Thinking in Middle and High Schools: Meeting the Common Core Rebecca Stobaugh Bringing the Common Core Math Standards to Life Exemplary Practices from High Schools Yvelyne Germain-McCarthy and Ivan P. Gill Second edition published 2015 by Routledge 7 11 Third Avenue, New York, NY 10017 a nd by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN R outledge is an imprint of the Taylor & Francis Group, an informa business © 2015 Taylor & Francis T he right of Yvelyne Germain-McCarthy and Ivan P. Gill to be identifi ed as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Author’s photograph courtesy of her son, Germain McCarthy, New Orleans, LA A ll rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice :Product or corporate names may be trademarks or registered trademarks, and are used only for identifi cation and explanation without intent to infringe. First edition published by Eye On Education (as Bringing the NCTM Standards to Life: Exemplary Practices for High Schools) 2001 L ibrary of Congress Cataloging-in-Publication Data G ermain-McCarthy, Yvelyne, 1948– author. Bringing the common core math standards to life : exemplary practices from high schools / Yvelyne Germain-McCarthy and Ivan Gill. pages cm 1. Mathematics—Study and teaching (Secondary)—Standards—Evaluation. I. Gill, Ivan, author. II. Title. Q181.G36 2015 510.71’273—dc23 2014015068 I SBN: 978-0-415-73474-5 (hbk) I SBN: 978-0-415-73342-7 (pbk) I SBN: 978-1-315-81976-1 (ebk) T ypeset in Palatino b y Apex CoVantage, LLC Contents Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Trends and Issues Leading to the Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Curriculum and Evaluation Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Professional Standards for Teaching Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 The Assessment Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 The Third International Mathematics and Science Study . . . . . . . . . . . . . . . . . .4 Trends in International Mathematics and Science Study . . . . . . . . . . . . . . . . . .5 Program for International Student Assessment . . . . . . . . . . . . . . . . . . . . . . . . . .5 Constructivism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 Skills for Citizens of the Twenty-fi rst Century . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Productive Struggle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Principles and Standards for School Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . .10 NCTM’s Quest for a Coherent Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Curriculum Focal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Focus in High School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 The Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 The Common Core State Standards for Mathematics (CCSSM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 Principles to Actions: Ensuring Mathematical Success for All . . . . . . . . . . . . . . . . . .19 2 Exemplary Practice: What Does It Look Like? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 Envisioning a Reform-based Classroom Environment . . . . . . . . . . . . . . . . . . . . . . . .26 Example One: Reformed-based Instruction? . . . . . . . . . . . . . . . . . . . . . . . . . . .26 Example Two: Traditional Instruction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 Exemplary Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 3 Michael Lehman: Group Performance Assessments . . . . . . . . . . . . . . . . . . . . . . . .39 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 4 Mark Lonergan: The Mathematics of the Theater . . . . . . . . . . . . . . . . . . . . . . . . . .64 Background on the Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 vi ◆ Contents Preparing for the Launch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 The Launch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 5 Henry Kranendonk: Raisin Geometry—Exploring the Balance Point and Its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 6 Virginia Highstone: Equations That Model Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116 7 Craig Jensen: A Dice Game for Engaging Teachers in Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 Warm-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 Launch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 Explore—Part A: Patterns in Products of Polynomials . . . . . . . . . . . . . . . . . .121 Explore—Part B: Patterns in Factors of Polynomials . . . . . . . . . . . . . . . . . . . .122 Explore—Part C: Back to Dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123 Summarize—Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 Summarize—Pedagogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132 8 Ben Preddy: Integrating Mathematics with Literacy . . . . . . . . . . . . . . . . . . . . . .133 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 9 Murray Siegel: Discovering the Central Limit Theorem . . . . . . . . . . . . . . . . . . .144 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 Contents ◆ vii Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153 1 0 Susan Morere, Mary Ann Bates, Nezha Whitecotton, and Angela Guthrie: A Lesson Study on Real-World Applications of the Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154 Lesson Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 University of New Orleans’ Lesson Study Process . . . . . . . . . . . . . . . . . . . . . . . . . .156 Lesson Study across Content Areas and Schools . . . . . . . . . . . . . . . . . . . . . . .156 The Lesson Study Open House . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158 Launching the Lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164 Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169 Lesson Study Colloquium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173 Appendix 10.1: Lesson Study Open House . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182 Appendix 10.2: Refl ection on Lesson Study Implementation at Patrick Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191 1 1 Claudia Carter: How a Rumor Spreads—An Introduction to Logistic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202 1 2 Teachers Adapting Tasks to Closely Align to CCSSM . . . . . . . . . . . . . . . . . . . . .203 Carol Lavery: Constructing an Equilateral Triangle, Fairfax High School, Virginia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211 Anecdotal Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211 Note to Readers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212 Rick Barlow: Composition of Functions, Fremont High School, Sunnyvale, California . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .214 NCTM Process Standard: Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . .214 The Overall Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .216 What I Would Change for Next Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .216 Cheryl Tokarski: Exploring Absolute Value, Chase Collegiate School, Waterbury, Connecticut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217 Holly Erwin-Harding: Calculating Areas of Irregular Figures, Blacksburg New School, Blacksburg, Virginia . . . . . . . . . . . . . . . . . . . . . . . . . . . .219 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221 viii ◆ Contents Andrew Winstead and Uriel Rodriguez: Quadratic Models for Real-Life Data from Hurricane Katrina, New Orleans Math and Science Charter School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222 Original Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222 Adapted Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227 Caroline Delaney: What Would Struggling Students Like to Know about Linear Equations? Warren Easton Charter High School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .228 Original Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .228 Multiple-Choice Assessment with Challenging Tasks . . . . . . . . . . . . . . . . . . .229 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .230 13 Don Crossfi eld: Marching from Algebra to Trigonometry and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231 Discussion between Colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .239 Unit Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .242 1 4 The CCSSM: Making It Happen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243 CCSSM: New Way for Teaching Math? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243 Guidelines and Recommendations for Implementing CCSSM . . . . . . . . . . . . . . . .244 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245 Acknowledgments T he authors would like to express sincere appreciation to the teachers profi led in this book for their hard work and for their commitment to improving the teaching and learning of our children. In particular, we wish to thank them for sharing their ideas and for responding to our many communications. We also wish to express gratitude to the countless teachers, too numerous to mention by name, with whom we have worked, for sharing their passion for learning, their expertise in teaching, and their belief in their students’ success. Y velyne acknowledges Robert Sickles and Lauren Davis for suggestions that sparked improvements in this second edition of the book. She also wants to thank Kerry Davidson, Bryan Jones, and Noreen Lackett for their expert leadership of the Louisiana State Systemic Initiative Program; codirecting grants from this program have challenged her to grow in the areas of developing effective professional development experiences for teachers. She also thanks Craig Jensen, Dongming Wei, Kenneth Hol- laday, Ivan Gill, and Norma Felton, who worked with her on the grants and helped to form the University of New Orleans Mathematics and Science Collaborative for Teaching and Learning, which sparked collaborative enterprises between local schools and the mathematics and education departments. She extends special appreciation to her family: her loving Heavenly Father; her parents, Georges and Eugenie Germain; her brothers, Gerard, Serge, and Claude Ger- main; her nieces, Georgine and Hermiaune Germain; her sister-in-law, Sara Germain, her nephew, Claude Germain; and friend, Lina Botavara for providing a wonderful retreat away from home when deadlines were approaching. Most important, she expresses loving gratitude to her husband, Henry McCarthy; her sons, Julian and Ger- main McCarthy; and her nephew, Carlos Germain, for their love, patience, and unfailing support. Ivan gratefully acknowledges his friends, family, and colleagues who have sup- ported this effort and taken the load off other aspects of his work, most especially his wife, Jeanette, as well as Pat Austin and Abram Himelstein. His friend and dog Moose provided expert commentary. F inally, we wish to dedicate this book to children in impoverished sections of the world where freedom or the means to participate in the types of learning environments described in this book are but a dream. We donate the proceeds from this book to promoting the best education available to children from Rivière Froide, a small town in Yvelyne’s homeland, Haiti.