Beyond the Common Core: A Handbook for Mathematics in a PLC at Work, High School by Solution Tree

H A N D B O O K

F O R

Mathematics in a PLC at WorkTM

—WILLIAM BARNES, COORDINATOR OF SECONDARY MATHEMATICS, HOWARD COUNTY PUBLIC SCHOOLS, ELLICOTT CITY, MARYLAND

H IGH SCHO O L

“No one in America is more qualified than Tim Kanold to address how the PLC at Work process can help educators provide the mathematics curriculum and powerful instruction that are essential to helping more students learn mathematics at dramatically higher levels. Kanold and Toncheff draw on their expertise in the PLC process and their leadership in mathematics education to provide educators with an invaluable resource. If you can only read one book to improve student achievement in mathematics, this is the book!” —RICHARD DUFOUR, EDUCATIONAL AUTHOR AND CONSULTANT

Beyond the Common Core: A Handbook for Mathematics in a PLC at Work™, High School helps mathematics teacher teams within professional learning communities (PLCs) focus their curriculum and vision on the elements of instruction and assessment necessary for student achievement. Using the practical guidelines in this handbook, high school mathematics teachers and administrators will go beyond state or local standards to create the essential conditions to ensure success in mathematics for all students.

1. Make sense of the agreed-on essential learning standards (content and practices) and pacing 2. Identify higher-level-cognitive-demand mathematical tasks 4. Develop scoring rubrics and proficiency expectations for the common assessment instruments 5. Plan and use common homework assignments

7. Use in-class formative assessment processes effectively 8. Use a lesson-design process for lesson planning and collective team inquiry 9. Ensure evidence-based student goal setting and action for the next unit of study 10. Ensure evidence-based adult goal setting and action for the next unit of study

Visit go.solution-tree.com/mathematicsatwork to download the reproducibles in this book. solution-tree.com

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T I M OT H Y D. KAN O LD

3. Develop common assessment instruments

6. Use higher-level-cognitive-demand mathematical tasks effectively

TIMOTHY D. KANOLD Series Editor

BEYOND THE COMMON CORE

H A N D B O O K

F O R

Mathematics in a PLC at Work

HIGH SC HO O L

Series editor and author Timothy D. Kanold and author Mona Toncheff present ten high-leverage team actions (HLTAs) to assist teams in narrowing their vision for instruction and assessment and increasing student learning while implementing the Common Core State Standards or district and local standards. These HLTAs are steps taken before, during, and after a unit of instruction—to prepare for instruction, to use formative assessment during instruction, and to reflect on the results of student assessment and prepare for the next unit of instruction. The authors describe how to:

BEYOND THE COMMON CORE A HANDBOOK FOR MATHEMATICS IN A PLC AT WORKTM

“This book will serve as an invaluable resource. It provides a roadmap for team improvement with specific actions to support the development of highly functioning mathematics-teaching teams and empowers the experts in the classroom with the processes, skills, and knowledge needed to support differentiated growth.”

BEYOND THE COMMON CORE

HI GH SCHOOL

Mona Toncheff Timothy D. Kanold

Copyright © 2015 by Solution Tree Press Materials appearing here are copyrighted. With one exception, all rights are reserved. Readers may reproduce only those pages marked “Reproducible.” Otherwise, no part of this book may be reproduced or transmitted in any form or by any means (electronic, photocopying, recording, or otherwise) without prior written permission of the publisher. 555 North Morton Street Bloomington, IN 47404 800.733.6786 (toll free) / 812.336.7700 FAX: 812.336.7790 email: info@solution-tree.com solution-tree.com Visit go.solution-tree.com/mathematicsatwork to download the reproducibles in this book. Printed in the United States of America 18 17 16 15 14

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Library of Congress Cataloging-in-Publication Data

Toncheff, Mona, author. Beyond the common core : a handbook for mathematics in a PLC at work. High school / Mona Toncheff, Timothy D. Kanold ; Timothy D. Kanold (editor). pages cm Includes bibliographical references and index. ISBN 978-1-936763-50-4 (perfect bound) 1. Mathematics--Study and teaching (Secondary)--Standards--United States. 2. Professional learning communities. I. Kanold, Timothy D., author. II. Title. QA13.T66 2015 510.71’2--dc23 2014029226 Solution Tree Jeffrey C. Jones, CEO Edmund M. Ackerman, President Solution Tree Press President: Douglas M. Rife Associate Acquisitions Editor: Kari Gillesse Editorial Director: Lesley Bolton Managing Production Editor: Caroline Weiss Senior Production Editor: Suzanne Kraszewski Copy Editor: Sarah Payne-Mills Proofreader: Elisabeth Abrams Text and Cover Designer: Laura Kagemann Text Compositor: Rachel Smith

Table of Contents About the Series Editor .

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About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The Grain Size of Change Is the Teacher Team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Knowing Your Vision for Mathematics Instruction and Assessment .

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A Cycle for Analysis and Learning: The Instructional Unit . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Before the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 During the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 After the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 CHAPTER 1

Before the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 HLTA 1: Making Sense of the Agreed-On Essential Learning Standards (Content and Practices) and Pacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

HLTA 2: Identifying Higher-Level-Cognitive-Demand Mathematical Tasks .

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The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

HLTA 3: Developing Common Assessment Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

HLTA 4: Developing Scoring Rubrics and Proficiency Expectations for the Common Assessment Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

HLTA 5: Planning and Using Common Homework Assignments .

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The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Setting Your Before-the-Unit Priorities for Team Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

CHAPTER 2

During the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 HLTA 6: Using Higher-Level-Cognitive-Demand Mathematical Tasks Effectively .

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The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

HLTA 7: Using In-Class Formative Assessment Processes Effectively . . . . . . . . . . . . . . . . . . . 90 The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

HLTA 8: Using a Lesson-Design Process for Lesson Planning and Collective Team Inquiry . . 110 The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Setting Your During-the-Unit Priorities for Team Action .

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127

CHAPTER 3

After the Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 HLTA 9: Ensuring Evidence-Based Student Goal Setting and Action for the Next Unit of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

HLTA 10: Ensuring Evidence-Based Adult Goal Setting and Action for the Next Unit of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 The What . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 The How . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Your Team’s Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Setting Your After-the-Unit Priorities for Team Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 EPILOGUE

Taking Your Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 APPENDIX A

Standards for Mathematical Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 APPENDIX B

Standards for Mathematical Practice Evidence Tool . . . . . . . . . . . . . . . . . . . . . . . . . 167 Mathematical Practice 1: “Make Sense of Problems and Persevere in Solving Them” . . . . . . . 167 Mathematical Practice 2: “Reason Abstractly and Quantitatively” .

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Table of Contents

Mathematical Practice 3: “Construct Viable Arguments and Critique the Reasoning of Others” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Practice 4: “Model With Mathematics” .

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168

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168

Mathematical Practice 5: “Use Appropriate Tools Strategically” . . . . . . . . . . . . . . . . . . . . . 168 Mathematical Practice 6: “Attend to Precision” .

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Mathematical Practice 7: “Look for and Make Use of Structure” .

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169 169

Mathematical Practice 8: “Look for and Express Regularity in Repeated Reasoning” . . . . . . . 170 APPENDIX C

Cognitive-Demand-Level Task-Analysis Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 APPENDIX D

Sources for Higher-Level-Cognitive-Demand Tasks . . . . . . . . . . . . . . . . . . . . . . . . . 173 APPENDIX E

How the Mathematics at Work High-Leverage Team Actions Support the NCTM Principles to Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Resources .

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175

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179

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

About the Series Editor Timothy D. Kanold, PhD, is an award-winning educator, author, and consultant. He is former director of mathematics and science and served as superintendent of Adlai E. Stevenson High School District 125, a model professional learning community district in Lincolnshire, Illinois. He serves as an adjunct faculty member for the graduate school at Loyola University Chicago. Dr. Kanold is committed to a vision for Mathematics at Work™, a process of learning and working together that builds knowledge sharing, equity, and excellence for all students, faculty, and school administrators. He conducts highly motivational professional development leadership seminars worldwide with a focus on turning school vision into realized action that creates increased learning opportunities for students through the effective delivery of professional learning communities for faculty and administrators. He is a past president of the National Council of Supervisors of Mathematics and coauthor of several best-selling mathematics textbooks. He has served on writing commissions for the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics. He has authored numerous articles and chapters on school mathematics, leadership, and professional development for education publications. In 2010, Dr. Kanold received the prestigious international Damen Award for outstanding contributions to the leadership field of education from Loyola University Chicago. He also received the Outstanding Administrator Award from the Illinois State Board of Education in 1994 and the Presidential Award for Excellence in Mathematics and Science Teaching in 1986. Dr. Kanold earned a bachelor’s degree in education and a master’s degree in applied mathematics from Illinois State University. He completed a master’s in educational administration at the University of Illinois and received a doctorate in educational leadership and counseling psychology from Loyola University Chicago. To learn more about Dr. Kanold’s work, visit his blog Turning Vision Into Action at http://tkanold .blogspot.com, or follow @tkanold on Twitter. To book Dr. Kanold for professional development, contact pd@solution-tree.com.

About the Authors Mona Toncheff, MEd, is the mathematics content specialist for Phoenix Union High School District in Arizona. She oversees the development and implementation of mathematics curriculum, instruction, and assessment. Mona also coordinates and instructs district professional development activities for teachers designed to improve mathematics instruction and student achievement. She has served in public education for over twenty years. Mona has supervised the culture change from teacher isolation to professional learning communitiesâ&#x20AC;&#x201D;creating articulated standards and relevant district common assessments and providing ongoing professional development for over two hundred high school mathematics teachers regarding best practices, equity and access, technology, response to intervention, and assessment for learning. Mona served as the National Council of Supervisors of Mathematics secretary (2007â&#x20AC;&#x201C;2008) and currently serves the National Council of Supervisors of Mathematics board as the Western Region 1 director. In 2009, Mona was selected as the Phoenix Union High School District Teacher of the Year where she also received the Copper Apple Award for leadership in mathematics from the Arizona Association of Teachers of Mathematics in 2014. She earned a bachelor of science from Arizona State University and a master of education in educational leadership from Northern Arizona University. To learn more about Mona Toncheffâ&#x20AC;&#x2122;s work, visit her blog at puhsdmath.blogspot.com or follow @ toncheff5 on Twitter. To book Mona Toncheff for professional development, contact pd@solution-tree.com. In addition to being the series editor, Timothy D. Kanold, PhD, is a coauthor of this book.

xiii

Introduction You have high impact on the front lines as you snag children in the river of life. —Tracy Kidder Your work as a high school mathematics teacher is one of the most important, and at the same time, one of the most difficult jobs to do well in education. Since the release of our 2012 Solution Tree Press series Common Core Mathematics in a PLC at Work™, our authors, reviewers, school leaders, and consultants from the Mathematics at Work™ team have had the opportunity to work with thousands of high school teachers and teacher teams from across the United States who are just like you: educators trying to urgently and consistently seek deeper and more meaningful solutions to a sustained effort for meeting the challenge of improved student learning. From California to Virginia, Utah to Florida, Oregon to New York, Wisconsin to Texas, and beyond, we have discovered a thirst for implementation of K–12 mathematics programs that will sustain student success over time. This focus on high school is a significant component of the K–12 effort toward improved student learning and preparation for college and career. Certainly the Common Core State Standards (CCSS) have served as a catalyst for much of the national focus and conversation about improving student learning. However, your essential work as a high school mathematics teacher and as part of a collaborative team in your local school and district takes you well beyond your states’ standards—whatever they may be. As the authors of the National Council of Teachers of Mathematics (NCTM, 2014) publication Principles to Actions: Ensuring Mathematical Success for All argue, standards in and of themselves do not describe the essential conditions necessary to ensure mathematics learning for all students. You, as the classroom teacher, are the most important ingredient to student success. Thus, this high school mathematics teaching and assessing handbook is designed to take you beyond the product of standards themselves by providing you and your collaborative team with the guidance, support, and process tools necessary to achieve mathematics program and department greatness within the context of higher levels of demonstrated student learning and performance. Whether you are from a state that is participating in one of the CCSS assessment consortia, or from a state that uses a unique mathematics assessment designed only for that state, it is our hope that this handbook provides a continual process that allows you to move toward a local program of great mathematics teaching and learning for you and your students. Your daily work in mathematics begins by understanding that what does make a significant difference (in terms of high levels of student achievement) are the thousands of instructional and assessment decisions you and your collaborative team will make every year—every day and in every unit.

The Grain Size of Change Is the Teacher Team We believe that the best strategy to achieve the expectations of CCSS-type state standards is to create schools and districts that operate as professional learning communities (PLCs), and, more specifically, within a PLC at Work™ culture as outlined by Richard DuFour, Rebecca DuFour, Robert Eaker, and

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

Thomas Many (2010). We believe that the PLC process supports a grain size of change that is just right— not too small (the individual teacher) and not too big (the district office)—for impacting deep change. The adult knowledge capacity development and growth necessary to deliver on the promise of standards that expect student demonstrations of understanding reside in the engine that drives the PLC school culture: you and your teacher team. There is a never-ending aspect to your professional journey and the high-leverage teacher and teacher team actions that measure your impact on student learning. This idea is at the very heart of your work. As John Hattie (2012) states in Visible Learning for Teachers: Maximizing Impact on Learning: My role as a teacher is to evaluate the effect I have on my students. It is to “know thy impact,” it is to understand this impact, and it is to act on this knowing and understanding. This requires that teachers gather defensible and dependable evidence from many sources, and hold collaborative discussions with colleagues and students about this evidence, thus making the effect of their teaching visible to themselves and to others. (p. 19)

Knowing Your Vision for Mathematics Instruction and Assessment Quick—you have thirty seconds: turn to a colleague and declare your vision for mathematics instruction and assessment in your mathematics department and in your school. What exactly will you say? More importantly, on a scale of 1 (low) to 6 (high), what would be the degree of coherence between your and your colleagues’ visions for instruction and assessment? We have asked these vision questions to more than 10,000 mathematics teachers across the United States since 2011, and the answers have been consistent: wide variance on mathematics instruction and assessment coherence from teacher to teacher (low scores of 1, 2, or 3 mostly) and general agreement that the idea of some type of a formative assessment process is supposed to be in a vision for mathematics instruction and assessment. A favorite team exercise we use to capture the vision for instruction and assessment is to ask a team of three to five teachers to draw a circle in the middle of a sheet of poster paper. We ask each team member to write a list (outside of the circle) of three or four vital adult behaviors that reflect his or her vision for instruction and assessment. After brainstorming, the team will have twelve to fifteen vital teacher behaviors. We then ask the team to prepare its vision for mathematics instruction and assessment inside the circle. The vision must represent the vital behaviors each team member has listed in eighteen words or less. We indicate, too, that the vision should describe a “compelling picture of the school’s future that produces energy, passion, and action in yourself and others” (Kanold, 2011, p. 12). Team members are allowed to use pictures, phrases, or complete sentences, but all together the vision cannot be more than eighteen words. In almost every case, in all of our workshops, professional development events, conferences, institutes, and onsite work, we have been asked a simple, yet complex question: How? How do you begin to make decisions and do your work in ways that will advance your vision for mathematics instruction and assessment in your middle school? How do you honor what is inside your circle? And how do you know that your circle, your defined vision for mathematics instruction and assessment, represents the “right things” to pursue that are worthy of your best energy and effort?

Introduction

In Common Core Mathematics in a PLC at Work, High School (Kanold, Zimmermann, Carter, Kanold, & Toncheff, 2012), we explain how understanding formative assessment as a research-affirmed process for student and adult learning serves as a catalyst for successful CCSS mathematics content implementation. In the series, we establish the pursuit of assessment as a process of formative feedback and learning for the students and the adults as a highly effective practice to pursue (see chapter 4 of the series). In this handbook, we provide tools for how to achieve that collaborative pursuit: how to engage in ten high-leverage team actions (HLTAs) steeped in a commitment to a vision for mathematics instruction and assessment that will result in greater student learning than ever before.

A Cycle for Analysis and Learning: The Instructional Unit The mathematics unit or chapter of content creates a natural cycle of manageable time for a teacher’s and team’s work throughout the year. What is a unit? For the purposes of your work in this handbook, we define a unit as a chunk of mathematics content. It might be a chapter from your textbook or other materials for the course, a part of a chapter or set of materials, or a combination of various short chapters or content materials. A unit generally lasts no less than two to three weeks and no more than four to five weeks. As DuFour, DuFour, and Eaker (2008), the architects of the PLC at Work process, advise, there are four critical questions every collaborative team in a PLC at Work culture asks and answers on a unit-byunit basis: 1. What do we want all students to know and be able to do? (The essential learning standards) 2. How will we know if they know it? (The assessment instruments and tasks teams use) 3. How will we respond if they don’t know it? (Formative assessment processes for intervention) 4. How will we respond if they do know it? (Formative assessment processes for extension and enrichment)

The unit or chapter of content, then, becomes a natural cycle of time that is not too small (such as one week) and not too big (such as nine weeks) for meaningful analysis, reflection, and action by you and your teacher team throughout the year as you seek to answer the four critical questions of a PLC. A unit should be analyzed based on content-standard clusters—that is, three to five essential standards (or sometimes a cluster of standards) for the unit. Thus, a teacher team, an administrative team, or a district office team, does this type of analysis about eight to ten times per year. This Mathematics at Work handbook consists of three chapters that fit the natural rhythm of your ongoing work as a teacher of mathematics and as part of a teacher team. The chapters bring a focus to ten high-leverage team actions (HLTAs) your team takes before, during, and in the immediate aftermath of a unit of instruction as you respond to the four critical questions of a PLC throughout the year, as highlighted in figure I.1 (page 4). Figure I.1 lists the ten high-leverage team actions within their timeframe in relation to the unit of instruction (before, during, or after) and then links the actions to the critical questions of a PLC that they address.

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

High-Leverage Team Actions

1. What do we want all students to know and be able to do?

2. How will we know if they know it?

3. How will we respond if they donâ&#x20AC;&#x2122;t know it?

4. How will we respond if they do know it?

Before-the-Unit Team ActionsÂ HLTA 1. Making sense of the agreed-on essential learning standards (content and practices) and pacing HLTA 2. Identifying higher-levelcognitive-demand mathematical tasks HLTA 3. Developing common assessment instruments HLTA 4. Developing scoring rubrics and proficiency expectations for the common assessment instruments HLTA 5. Planning and using common homework assignments During-the-Unit Team Actions HLTA 6. Using higher-level-cognitivedemand mathematical tasks effectively HLTA 7. Using in-class formative assessment processes effectively HLTA 8. Using a lesson-design process for lesson planning and collective team inquiry After-the-Unit Team Actions HLTA 9. Ensuring evidence-based student goal setting and action for the next unit of study HLTA 10. Ensuring evidence-based adult goal setting and action for the next unit of study = Fully addressed with high-leverage team action = Partially addressed with high-leverage team action

Figure I.1: High-leverage team actions aligned to the four critical questions of a PLC. Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.

Introduction

Before the Unit In chapter 1, we provide insight into the work of your collaborative team before the unit begins, along with the tools you will need in this phase. Your collaborative team expectation should be (as best you can) to complete this teaching and assessing work in preparation for the unit. There are five before-the-unit high-leverage team actions (HLTAs) for collaborative team agreement on a unit-by-unit basis. HLTA 1. Making sense of the agreed-on essential learning standards (content and practices) and pacing HLTA 2. Identifying higher-level-cognitive-demand mathematical tasks HLTA 3. Developing common assessment instruments HLTA 4. Developing scoring rubrics and proficiency expectations for the common assessment instruments HLTA 5. Planning and using common homework assignments

Once your team has taken these action steps, the mathematics unit begins.

During the Unit In chapter 2, we provide the tools for and insight into the formative assessment work of your collaborative team during the unit. This chapter teaches deeper understanding of content, discussing the Common Core Mathematical Practices and processes and using higher-level-cognitive-demand mathematical tasks effectively. It helps your team with daily lesson design and study ideas as ongoing in-class student assessment becomes part of a teacher-led formative process. This chapter introduces three during-the-unit high-leverage team actions your team works through on a unit-by-unit basis. HLTA 6. Using higher-level-cognitive-demand mathematical tasks effectively HLTA 7. Using in-class formative assessment processes effectively HLTA 8. Using a lesson-design process for lesson planning and collective team inquiry

The end of each unit results in some type of student assessment. You pass back the assessments scored and with feedback. Then what? What are students to do? What are you to do?

After the Unit In chapter 3, we provide tools for and insight into the formative work your collaborative team does after the unit is over. After students have taken the common assessment, they are expected to reflect on the results of their work and be willing to use the common unit assessment instrument for formative feedback purposes.

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

In addition, there is another primary formative purpose to using a common end-of-unit assessment, which Hattie (2012) describes in Visible Learning for Teachers: “This [teachers collaborating] is not critical reflection, but critical reflection in light of evidence about their teaching” (p. 19, emphasis added). From a practical point of view, an end-of-unit analysis of the common assessment focuses your team’s next steps for teaching and assessing for the next unit. Thus, there are two end-of-unit high-leverage team actions your team works through on a unit-by-unit basis. HLTA 9. Ensuring evidence-based student goal setting and action for the next unit of study HLTA 10. Ensuring evidence-based adult goal setting and action for the next unit of study

In Principles to Actions: Ensuring Mathematical Success for All, NCTM (2014) presents a modern-day view of professional development for mathematics teachers: building the knowledge capacity of every teacher. More importantly, however, you and your colleagues should intentionally act on that knowledge and transfer what you learn into daily classroom practice through the ten high-leverage teacher team actions presented in this handbook. For more information on the connection between these two documents, see appendix E on page 175. Although given less attention, the difficult work of collective inquiry and action orientation has a more direct impact on student learning than when you work in isolation (Hattie, 2009). Through your team commitment (the engine that drives the PLC at Work culture and process of collective inquiry and action research), you will find meaning in the collaborative work with your colleagues. In Great by Choice, Jim Collins (Collins & Hansen, 2011) asks, “Do we really believe that our actions count for little, that those who create something great are merely lucky, that our circumstances imprison us?” He then answers, “Our research stands firmly against this view. Greatness is not primarily a matter of circumstance; greatness is first and foremost a matter of conscious choice and discipline” (p. 181). We hope this handbook helps you focus your time, energy, choices, and pursuit of a great teaching journey.

CHAPTER 1

Before the Unit Teacher: Know thy impact. â&#x20AC;&#x201D;John Hattie

In conjunction with the scope and sequence your district mathematics curriculum provides, your collaborative team prepares a roadmap that describes the knowledge students will know and be able to demonstrate at the conclusion of the unit. To create this roadmap, your collaborative team prepares and organizes your work around five before-the-unit-begins high-leverage team actions (HTLAs). HLTA 1. Making sense of the agreed-on essential learning standards (content and practices) and pacing HLTA 2. Identifying higher-level-cognitive-demand mathematical tasks HLTA 3. Developing common assessment instruments HLTA 4. Developing scoring rubrics and proficiency expectations for the common assessment instruments HLTA 5. Planning and using common homework assignments

These five team pursuits are based on step one of the PLC teaching-assessing-learning cycle (Kanold, Kanold, & Larson, 2012) shown in figure 1.1 (page 8). This cycle drives your pursuit of a meaningful formative assessment and learning process for your team and for your students throughout the unit and the year. In this chapter, we describe each of the five before-the-unit-begins high-leverage team actions in more detail (the what) along with suggestions for how to achieve these pursuits (the how). Each HLTA section ends with an opportunity for you to evaluate your current reality (your teamâ&#x20AC;&#x2122;s progress). The chapter ends with time for reflection and action (setting your Mathematics at Work priorities for team action).

The ultimate outcome of planning before the unit is for you and your team members to gain a clear understanding of the impact of your expectations for student learning and demonstrations of understanding during the unit.

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

Step One Collaborative teams identify learning standards and design common unit tasks and assessment instruments.

Before the Unit

Collaborative teams use ongoing assessment feedback to improve instruction.

Step Two Teachers implement formative assessment classroom strategies.

The PLC TeachingAssessing-Learning Cycle

Step Four Students use assessment instruments from step one for motivation, reflection, and action.

Step Three Students take action on in-class formative assessment feedback.

Source: Kanold, Kanold, & Larson, 2012. Figure 1.1: Step one of the PLC teaching-assessing-learning cycle.

Step Five

Before the Unit

HLTA 1: Making Sense of the Agreed-On Essential Learning Standards (Content and Practices) and Pacing An excellent mathematics program includes curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world. —National Council of Teachers of Mathematics

Recall there are four critical questions every collaborative team in a PLC asks and answers on an ongoing unit-by-unit basis. 1. What do we want all students to know and be able to do? (The essential learning standards) 2. How will we know if they know it? (The assessment instruments and tasks teams use) 3. How will we respond if they don’t know it? (Formative assessment processes for intervention) 4. How will we respond if they do know it? (Formative assessment processes for extension and enrichment)

High-Leverage Team Action

1. What do we want all students to know and be able to do?

2. How will we know if they know it?

Before-the-Unit Action HLTA 1. Making sense of the agreed-on essential learning standards (content and practices) and pacing = Fully addressed with high-leverage team action

3. How will we respond if they don’t know it?

4. How will we respond if they do know it?

In most high school mathematics courses, there will be ten to twelve mathematics units (or chapters) during the school year. These units may also consist of several learning modules depending on how your high school curriculum and courses are structured. An ongoing challenge is for you and your team to determine how to best make sense of and develop understanding for each of the agreed-on essential learning standards within the mathematics unit.

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

The What

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. (NGA & CCSSO, 2010, pp. 6–8)

While schools and districts use many names for learning standards—learning goals, learning objectives, and so on—this handbook references the broad mathematical concepts and understandings for the entire unit as essential learning standards. For more specific lesson-by-lesson daily outcomes, we use daily learning objectives or the daily essential questions. We use the terms learning goals or learning targets to reference the outcome for student proficiency in each standard. The daily learning objectives and the tasks and activities representing those objectives help students understand the four to five essential learning standards for the unit in order to demonstrate proficiency (outcomes) on those standards. The daily learning objectives must also articulate for students what they are to understand and do that day. A unit of instruction connects topics in mathematics that are naturally grouped together—the essential ideas or content standard clusters. Each of those units should consist of about four to six essential learning standards taught to every student in the course. These essential learning standards may consist of several daily learning objectives that require student understanding. The essential learning standards are framed as overarching questions posed to the class during the unit. It might take three to five days of instruction and two to three daily learning objectives to fully answer the essential questions. The context of the lesson is the driving force for the entire lesson-design process. Each lesson context centers on clarity of the mathematical content and the processes for student learning.

This first high-leverage team action enhances clarity on the first PLC critical question for collaborative team learning: What do we want all students to know and be able to do? In light of the Common Core State Standards for mathematics, the essential learning standards for the unit—the guaranteed and viable mathematics curriculum—include the essential standards students will learn, when they will learn each essential standard (the pacing of the unit), and how they will learn it (via process standards such as the Common Core Standards for Mathematical Practice). The Standards for Mathematical Practice “describe varieties of expertise that mathematic educators at all levels should seek to develop in their students” (NGA & CCSSO, 2010, p. 6). Following are the eight Standards for Mathematical Practice, which we include in full in appendix A (page 163).

Before the Unit

This first high-leverage team action serves NCTMâ&#x20AC;&#x2122;s (2014) Principles to Actions curriculum principle, professionalism principle, and teaching and learning principle as teachers establish goals for student learning. See appendix E (page 175) for how all ten HLTAs support NCTMâ&#x20AC;&#x2122;s guiding practices for school mathematics.

The How As you and your collaborative team unpack Common Core mathematics content standards or your specific state standards (the essential learning standards) for a unit, it is also important to decide which Standards for Mathematical Practice (or processes) will receive focused development throughout the unit of instruction, and what mathematical tasks you will use during the unit to help students learn both the essential content standards and the Mathematical Practices or process standards. Thus, your team identifies, explores, and discusses: 1. The meaning of the essential content learning standards for the unit 2. The intentional Mathematical Practices or processes for student learning and understanding to be developed during the unit 3. The mathematical tasks (higher- and lower-level cognitive demand) to be used during the unit

Unpacking a Learning Standard How can your team explore the general unpacking of content and linking the content to the Mathematical Practices for any unit? For example, consider the high school content standard cluster Construct and compare linear, quadratic, and exponential models and solve problems in the domain Linear, Quadratic, and Exponential Models (F-LE). The essential learning standards for such a unit usually occur during the second semester of either an algebra 1 or integrated mathematics I course (see figure 1.2, page 12).

The crux of any successful mathematics lesson rests on your collaborative team identifying and determining methods of teaching for understanding with the essential learning standards for the unit. Although you might develop daily learning objectives for each lesson as part of curriculum writing or review, your collaborative team should take time during lesson-design discussions to make sense of the essential learning standards for the unit and to consider how the essential learning standards for the unit are connected. This involves unpacking the mathematics content as well as the Mathematical Practices or processes each student will engage in as he or she learns the mathematics of the unit. Unpacking, in this case, means making sense of the mathematics listed in the standard, making sense of how the content connects to content learned in other mathematics courses as well as within the current course, and making sense of how students might develop both conceptual understanding and procedural skill with the mathematics listed in the standard.

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

Content standard cluster: Construct and compare linear, quadratic, and exponential models and solve problems. F-LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions. F-LE.1a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. F-LE.1b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

F-LE.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F-LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F-LE.4: For exponential models, express as a logarithm the solution to abct = d, where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Source for standards: NGA & CCSSO, 2010, pp. 70–71. Figure 1.2: Common Core mathematics essential learning standards for linear, exponential functions unit.

Some of your team members may already have deep knowledge of the relationships between linear and exponential functions (and quadratics, too). Other members may only know about these essential learning standards on a very superficial level. For example, why does F-LE.2 connect arithmetic and geometric sequences to constructing linear and exponential models? Some team members may not yet understand the depth of understanding and the various mathematical representations for students to meet this essential learning standard. You and your collaborative team can use the discussion questions in figure 1.3 to discuss an appropriate learning progression of the essential standards for this type of unit in your algebra or integrated mathematics I course; identify the depth of the essential learning standards for constructing and comparing linear and exponential models (quadratic, too, if applicable); and discuss explicit connections to previous and future units with functions for your course, or for future high school courses. Additionally, you and your collaborative team should plan and discuss how you want students to demonstrate their understanding of the mathematics content and practices used during the unit. What would you expect to see here? Your team conversations should focus on student thinking and solution strategies or pathways for the essential learning standards of the unit. This level of unpacking the meaning of the essential learning standards is crucial before you can plan effective student engagement within the mathematics instruction. For example, once your team identifies the depth of the unit’s essential learning standards, you can then discuss specific solution strategies and learning processes (problem solving, reasoning, precision, and modeling) that you want students to explore during the unit. You and your collaborative team will need to decide which tools (for example, graphing calculators, statistics programs, graph paper, paper and pencil, or physical models) students will use to construct, model, or build the functions. You will need to specify which essential standards students need to explore first to create a deeper meaning for them of the various types of representations: tables, graphs, and function rules.

F-LE.1c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Before the Unit

Directions: Within your collaborative team, answer the following questions that address linear and exponential functions. 1. What does it mean to construct and compare function models?

2. How might students engage in the construction and comparison of function models? What types of mathematical tasks should they do?

4. How might students connect numerical representations (arithmetic and geometric) to the visual (graphical) and analytical (function) representations of linear and exponential models?

5. What is the role of quadratics in this content standard cluster, and how could we connect quadratics to the content from this unit? (See F-LE.3.)

6. Why is it important for students to be able to compare these three types of functions as they progress through the course?

7. What is the natural progression of these topics in your current high school course sequence? (See F-LE.4.) How can you help your students to understand this progression?

Figure 1.3: Sample essential learning standard discussion tool for linear and exponential functions. Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.

For example, once you make sense of how to compare linear and exponential functions and connect them to their various representations, you might decide that students will need to engage in Mathematical Practice 4, “Model with mathematics,” in order to build both linear and exponential functions. You might also determine that students will engage in Mathematical Practice 5, “Use appropriate tools strategically,” as you plan opportunities for them to use technology as a tool for creating linear or exponential functions. (Selecting good tasks will be discussed further in the following section on the second high-leverage team action). Students engaged in these Mathematical Practices will be asked to check for reasonableness of computations to determine if the solution is appropriate given the original problem context. For example, students can use a graphing calculator or a software tool to examine a set of data and determine the best-fitting model from a list of familiar functions. Consider the world population task in figure 1.4 (page 14). Your students could create a scatterplot of the data and then use the regression capabilities of the graphing calculator or computer software to determine the best-fitting model for the data. In this mathematical task, students compare linear, exponential, and quadratic models too.

3. What Mathematical Practices should we highlight during a unit of instruction for linear and exponential comparison content?

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

Directions: The world population (in millions) from 1960 through 2010 is shown in the following data table. Answer each question. Year

1960

1965

1970

1975

1980

1985

1990

1995

2000

2010

Population (millions)

3,039

3,345

3,707

4,707

4,086

4,454

4,851

5,688

6,083

6,902

1. Use a graphing and statistics tool to make a scatter plot of the data. Examine the data tables and use the regression capabilities of the graphing tool to determine the best-fitting models for the data.

Exponential function:

Quadratic function:

Visually, all three models appear to be reasonable fits over the given domain. To decide which model is best to use, consider the context of the problem, especially if the model is going to be used to extrapolate for years outside the given domain.

2. Use the three models from question one to determine a value for the world population in 1950. The actual world population in 1950 was 2,556,000,053. Which of the three model’s 1950 population value comes closest to the actual value?

3. The world population reached 7 billion in 2011, and is expected to reach 8 billion by 2027. If this prediction is true, which model—linear, exponential, or quadratic—would be the best predictor?

4. Is the world population rate of change staying the same, increasing, or slowing down over the next fifty years?

Source: Zimmerman, Carter, Kanold, & Toncheff, 2012. Figure 1.4: Predicting world population: Creating and comparing a linear, quadratic, and exponential function. Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.

Linear function:

Before the Unit

The key element of this first high-leverage team action is your ability to make sense of the essential learning standards and to also plan for student engagement in the Mathematical Practices that support the standard. See Appendix A (page 163) for a complete listing of the Mathematical Practices and Appendix B (page 167) to review evidence for each Mathematical Practice. In general, to begin any discussion of the essential standards with your team, you can use the discussion tool in figure 1.5 for each unit of the course.

1. List the agreed-on essential learning standards for this unit.

2. What is the prerequisite knowledge needed to engage students with the essential learning standards?

3. What is the time frame available to teach this unit, and how will that time be distributed for each essential learning standard?

4. What are the mathematics vocabulary and literacy skills necessary for student success in this unit?

5. What are specific teaching strategies we can use to most effectively teach each essential learning standard for the unit? (See the questions in figure 1.3, page 13.)

6. Which Mathematical Practices or processes should we highlight during the unit in order to better engage students in the process of understanding each essential learning standard?

Figure 1.5: Discussion tool for making sense of the agreed-on essential learning standards for the unit. Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.

After using the discussion tool in figure 1.5, you and your collaborative team can use the result of your conversations to create a transparent map of the unit and to articulate the intent of the unit to all team members. Consider the sample unit plan for an algebra I or an integrated mathematics I or II course presented in figure 1.6 (pages 16–17) and designed to support the first high-leverage team action—Making sense of the agreed-on essential learning standards (content and practices) and pacing—for a unit supporting quadratic functions.

Directions: Discuss the following prompts or questions with your collaborative teams to unpack essential learning standards, prerequisite skills for the unit, associated Mathematical Practices or processes relevant to the current unit of study for your grade level, and pacing decisions for the unit.

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

Unit Name: Quadratic Functions

Unit Number: 7

Time Frame

Purpose

Eighteen fifty-minute class periods (including review and test)

This unit examines the parameters of a quadratic function, graphing with and without a table, identifying key features of a quadratic function, and then comparing all functions learned throughout the year.

Common Core Essential Learning Standards 1. Interpret functions that arise in applications in terms of the context.

2. Analyze functions using different representations. F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F-IF.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima. F-IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F-IF.8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F-IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 3. Build a function that models a relationship between two quantities. F-BF.1: Write a function that describes a relationship between two quantities. F-BF.1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. 4. Build new functions from existing functions. F-BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (Limit to quadratic functions for this unit.) Essential Learning Standards

Enduring Understanding

Student will be able to:

Students can model and interpret real-world situations with quadratic functions.

1. Interpret functions that arise in applications in terms of the context. 2. Analyze functions using different representations. 3. Build a function that models a relationship between two quantities.

Essential Questions 1. How are different representations of quadratic functions used to identify key features of the graph? 2. What are the distinctions of a quadratic function compared to linear and exponential? 3. What is the minimum or maximum of the graph, and how does it behave? 4. How do the a, b, and c values change the function and the domain and range of the function?

F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Before the Unit

Key Mathematics Vocabulary • zeros • extreme values • standard form • factored form • perfect square trinomial • discriminant • radical • maximum and minimum • parabola • vertex • leading coefficient • axis of symmetry • x-intercept • roots (solutions) Prior Knowledge (What Knowledge and Skills Need to Be Spiraled?) Prerequisite understanding for this unit includes: • Domain and range • How to graph a point and make a graph from a table • Rate of change • GCF and like terms • Linear functions, factoring, and solving quadratics

Source for standards: NGA & CCSSO, 2010, pp. 69–70. Figure 1.6: Sample unit plan progression of content for a quadratic functions unit of study. Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.

When unpacking the essential learning standards, your team will also develop a lesson progression that makes sense for student understanding of the essential learning standards for the unit.

Making Sense of the Unit Content Progression Once your team identifies the essential learning standards for the unit, you decide how much time to dedicate to each essential standard as well as a natural progression of lessons during the unit. As with any planning, this is a starting point to understanding the depth of student understanding required for the unit and how to organize the development of mathematical concepts within the unit. Your team should share how each concept is connected to previous standards and upcoming standards in order to make explicit and logical connections for the unit content for students. Figure 1.7 (pages 18–19) provides a sample unit progression for exponential functions across standards for students. An additional sample unit plan is provided at go.solution-tree.com/mathematicsatwork.

• quadratic equation

Day 10 F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases, and using technology for more complicated (exponential only).

Day 9 F-BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Day 8 F-BF.1: Write a function that describes a relationship between two quantities.*

Day 13 F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative.

Day 7 F-BF.1: Write a function that describes a relationship between two quantities.*

Day 12 F-IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Day 6

F-LE.5: Interpret the parameters in a linear or exponential function in terms of a context.*

Day 11

F-IF.8a: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of a function.

A-SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*

F-IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. (Percent rate of change)

Day 15

F-LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

F-LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.*

Day 14

Day 5

Day 4

Day 3

Day 2

Day 1

Unit Seven Plan: Twenty Instructional Days

18 BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

Review for unit seven

S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Review for unit seven

Day 19 Assessment for unit seven

Day 20

Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure and an additional sample.

Figure 1.7: Sample unit progression for exponential functions—across standards.

Source: Adapted from Grossmont Union High School District, El Cajon, California (July 2013). Source for standards: NGA & CCSSO, 2010, pp. 70–71.

*Specific modeling standards appear throughout the high school standards marked with an asterisk. In addition, “model with mathematics” is an expected mathematical practice and process for students.

Statistics are also brought back into this unit through fitting an exponential function to data.

The unit also compares linear and exponential functions in a context and from various representations of those types of functions.

The unit also extends ideas and understandings from unit 3 on linear functions to the parallel ideas for exponential functions with special emphasis on the notion of constant percent rate of change (as compared to constant rate of change in chapter 3), repeated multiplication as the big idea, geometric sequences and recursive definitions, the meaning of the dependent variable, the meaning of the independent variable, parameters and their meanings, ways of measuring amount of growth via the constant difference (linear), and constant ratio (exponential).

The unit provides a deep examination of exponential functions and the laws of exponents, including definition of exponent notation, sum law for exponents, product law for exponents, definition of negative exponent notation, and basic characteristics of exponential functions.

8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. This unit is packed with content, and thus it needs twenty days to unfold the clusters of comparing, building, and interpreting exponential functions.

This unit connects to the students’ prior knowledge on the domain Expressions and Equations (8.EE) under the content standard cluster Work with radicals and integer exponents.

Notes for Unit Seven

Day 18

Day 17

Day 16

Before the Unit 19

BEYOND THE COMMON COR E: A H A NDBOOK FOR M ATHEM ATICS IN A PLC AT WOR K

This first high-leverage team action is both a district and a teacher team responsibility. The district office needs to provide guidance about the proper scope and sequence of the essential learning standards of the unit. At the same time, your course-based team members need to be clear on the intent of the essential standards, the rationale for teaching the standards in a specific order, and the nuances of the meaning and intent of each essential learning standard. At a minimum, your team should take the time to write a set of notes similar to the ones shown at the end of figure 1.7 as you work to better understand the intent of the mathematics content and progressions for the unit.

It is helpful to diagnose your team’s current reality and action prior to launching the unit or chapter. Ask each team member to individually assess your team on the first high-leverage team action using the status check tool in table 1.1. Discuss your perception of your team’s progress on making sense of the agreed-on essential learning standards (content and practices) and pacing. It matters less which stage your team is at and more that you and your team members are committed to working together to focus on understanding the learning standards and the best mathematical tasks, activities, and strategies for increasing student understanding and achievement as your team seeks stage IV—sustaining. Your responses to table 1.1 will help you determine what you and your team are doing well with respect to your focus on essential learning standards and where you might need to place more attention before the unit begins. Once your team unpacks and understands the essential learning standards you are ready to identify and prepare for higher-level-cognitive-demand mathematical tasks related to those essential learning standards. It is necessary to include tasks at varying levels of demand during instruction. The idea is to match the tasks and their cognitive demand to the essential learning standard expectations for the unit. Selecting mathematical tasks together is the topic of the second high-leverage team action, HLTA 2.

Your Team’s Progress

H A N D B O O K

F O R

Mathematics in a PLC at WorkTM

—WILLIAM BARNES, COORDINATOR OF SECONDARY MATHEMATICS, HOWARD COUNTY PUBLIC SCHOOLS, ELLICOTT CITY, MARYLAND

H IGH SCHO O L

Visit go.solution-tree.com/mathematicsatwork to download the reproducibles in this book. solution-tree.com

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T I M OT H Y D. KAN O LD

3. Develop common assessment instruments

6. Use higher-level-cognitive-demand mathematical tasks effectively

TIMOTHY D. KANOLD Series Editor

BEYOND THE COMMON CORE

H A N D B O O K

F O R

Mathematics in a PLC at Work

HIGH SC HO O L

BEYOND THE COMMON CORE A HANDBOOK FOR MATHEMATICS IN A PLC AT WORKTM

BEYOND THE COMMON CORE

HI GH SCHOOL

Mona Toncheff Timothy D. Kanold