Mathematics Unit Planning in a PLC at Work®, High School by Solution Tree

E V E R Y S T U D E N T C A N L E A R N M AT H E M AT I C S

“With clarity and common sense, Mathematics Unit Planning in a PLC at Work, High School guides teachers through the collaborative process of designing aligned and focused units of

members get their hands on it!” —Kim Bailey, Author and Educational Consultant

“This book is a great tool for teacher teams and instructional leaders.

S C H OO L

The detailed protocols provide necessary guidance for unit planning. I will highly recommend this book for our high schools.” —Maria Everett, Coordinator of Secondary Mathematics, Baltimore County Public Schools, Maryland

“In this resource packed with tools and real examples, Sarah Schuhl and her coauthors support teacher teams in seeing mathematics as a connection of ideas, not just a group of

“A critically important tool for high school teacher teams. I also

Unit Planning

H I G H

M AT H E M AT I C S

study in mathematics. This practical and reader-friendly tool will quickly fill with sticky notes and earmarked pages once team

think it will be very valuable for solo educators in alternative

—Sharon Rendon, Coaching Coordinator, College Preparatory

settings and smaller schools and districts.” —Nick Resnick, K–12 Education Specialist

Mathematics Educational Program, California

in a PLC at Work ®

unconnected units.”

Mathematics Unit Planning in a PLC at Work®, High School provides high school teachers with a seven-step framework for collectively planning units of study. Authors Sarah Schuhl, Timothy D. Kanold, Bill Barnes, Darshan of each unit and how teachers can build student self-efficacy. They advocate using the Professional Learning learning experiences. The authors share tools and protocols for effectively performing collaborative tasks, such as unwrapping standards, generating unit calendars, determining academic vocabulary and rigorous lessons, utilizing

S C H O O L

Community at Work (PLC) process to increase mathematics achievement and give students more equitable

H I G H

M. Jain, Matthew R. Larson, and Brittany Mozingo help teams identify what students need to know by the end

and sharing self-reflections, and designing robust units of instruction. This book provides practical insights into collaborative planning and detailed, inspiring models of this work in action. Mathematics teams will:

• Find protocols for unit planning and reproducible templates • Understand how teams can successfully incorporate each unit-planning element in their unit designs • Examine three model units on transformations on the coordinate plane for algebra 1, geometry, and algebra 2 • Review the role of the PLC at Work process in enhancing student learning and teacher collaboration Visit go.SolutionTree.com/MathematicsatWork to download the free reproducibles in this book.

ISBN 978-1-951075-29-3 90000

SolutionTree.com NCTM Stock ID: 16068

9 781951 075293

Schuhl • Kanold Barnes • Jain • Larson • Mozingo

• Learn how to build a shared understanding of the content students need to know in each course by using seven planning elements

Copyright © 2021 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@SolutionTree.com SolutionTree.com Visit go.SolutionTree.com/MathematicsatWork to download the free reproducibles in this book. Printed in the United States of America

Library of Congress Cataloging-in-Publication Data Names: Schuhl, Sarah, author. Title: Mathematics unit planning in a PLC at work. High school / Sarah Schuhl, Timothy D. Kanold, Bill Barnes, Darshan M. Jain, Matthew R. Larson, Brittany Mozingo. Description: Bloomington, IN : Solution Tree Press, 2020. | Series: Every student can learn mathematics series | Includes bibliographical references and index. Identifiers: LCCN 2020014765 (print) | LCCN 2020014766 (ebook) | ISBN 9781951075293 (paperback) | ISBN 9781951075309 (ebook) Subjects: LCSH: Mathematics--Study and teaching (Secondary)--United States. | Professional learning communities--United States. Classification: LCC QA13 .S3646 2020 (print) | LCC QA13 (ebook) | DDC 510.71/273--dc23 LC record available at https://lccn.loc.gov/2020014765 LC ebook record available at https://lccn.loc.gov/2020014766 Solution Tree Jeffrey C. Jones, CEO Edmund M. Ackerman, President Solution Tree Press President and Publisher: Douglas M. Rife Associate Publisher: Sarah Payne-Mills Art Director: Rian Anderson Managing Production Editor: Kendra Slayton Senior Production Editor: Suzanne Kraszewski Content Development Specialist: Amy Rubenstein Copy Editor: Mark Hain Proofreader: Elisabeth Abrams Text and Cover Designer: Kelsey Hergül Editorial Assistants: Sarah Ludwig and Elijah Oates

Table of Contents

About the Authors . Introduction .

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The Purpose of This Book The Parts of This Book . . A Final Thought . . . . .

3 4 . 4

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Part 1 Mathematics Unit Planning and Design Elements

1 2

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Planning for Student Learning of Mathematics in High School Guaranteed and Viable Curriculum . . . . . . . . . . . . . . . Mathematics Unit Planner . . . . . . . . . . . . . . . . . . . Mathematics Concepts and Skills in High School . . . . . . . . Connections Between Mathematics Content and Unit Planning .

Unit Planning as a Collaborative Mathematics Team .

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9 . 10 . 10 . 13

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Mathematics Unit Planning as a Team . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Part 2 Transformations on the Coordinate Plane Unit Examples for Algebra 1, Geometry, and Algebra 2 . . . . . . . . . . . . . . . . . . . . . . . . . .

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MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ÂŽ , HIGH SCHOOL

Algebra 1 Unit: Graphs of Quadratic Functions .

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Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Algebra 1 Graphs of Quadratic Functions Unit Planner . . . . . . . . . . . . . . . . . . . . 57 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Geometry Unit: Transformations and Congruence

. . . . . . . . . . . . . . . . . . . . .

Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Geometry Transformations and Congruence Unit Planner . . . . . . . . . . . . . . . . . . . 81 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Algebra 2 Unit: Graphs of Trigonometric Functions .

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Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Algebra 2 Graphs of Trigonometric Functions Unit Planner . . . . . . . . . . . . . . . . . . 110 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Epilogue: Mathematics Team Organization Final Thoughts .

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117

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118

Appendix A: Create a Proficiency Map .

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Appendix B: Team Checklist and Questions for Mathematics Unit Planning . Generate Essential Learning Standards . . . . . . . . . . . Create a Unit Calendar . . . . . . . . . . . . . . . . . . . Identify Prior Knowledge . . . . . . . . . . . . . . . . . . Determine Vocabulary and Notations . . . . . . . . . . . . Identify Resources and Activities . . . . . . . . . . . . . . Agree on Tools and Technology . . . . . . . . . . . . . . . Record Reflection and Notes . . . . . . . . . . . . . . . . Team Questions to Generate Essential Learning Standards .

121

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123

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123 123 124 124 124 124 124 125

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Table of C ontent s

Team Questions to Create a Unit Calendar . . . . . . . . Team Questions to Identify Prior Knowledge . . . . . . . Team Questions to Determine Vocabulary and Notations . Team Questions to Identify Resources and Activities . . . Team Questions to Agree on Tools and Technology . . . . Team Questions to Record Reflection and Notes . . . . .

References and Resources . Index .

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125 125 125 125 126 126

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127

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vii

About the Authors

Sarah Schuhl, MS, is an educational coach and consultant specializing in mathematics, professional learning communities (PLCs), common formative and summative assessments, school improvement, and response to intervention (RTI). She has worked in schools as a secondary mathematics teacher, high school instructional coach, and K–12 mathematics specialist. Schuhl was instrumental in the creation of a PLC in the Centennial School District in Oregon, helping teachers make large gains in student achievement. She earned the Centennial School District Triple C Award in 2012. Schuhl designs meaningful professional development in districts throughout the United States. Her work focuses on strengthening the teaching and learning of mathematics, having teachers learn from one another when working effectively as collaborative teams in a PLC at Work , and striving to ensure the learning of each and every student through assessment practices and intervention. Her practical approach includes working with teachers and administrators to implement assessments for learning, analyze data, collectively respond to student learning, and map standards.

Since 2015, Schuhl has coauthored the books Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy With K–5 Learners and School Improvement for All: A How-To Guide for Doing the

Right Work. She is a coauthor (with Timothy D. Kanold) of the Every Student Can Learn Mathematics series and the Mathematics at Work™ Plan Book. Previously, Schuhl served as a member and chair of the National Council of Teachers of Mathematics (NCTM) editorial panel for the journal Mathematics Teacher and is currently serving as secretary of the National Council of Supervisors of Mathematics (NCSM). Her work with the Oregon Department of Education includes designing mathematics assessment items, test specifications and blueprints, and rubrics for achievement-level descriptors. She has also contributed as a writer to a middle school mathematics series and an elementary mathematics intervention program. Schuhl earned a bachelor of science in mathematics from Eastern Oregon University and a master of science in mathematics education from Portland State University. To learn more about Sarah Schuhl’s work, follow @SSchuhl on Twitter. Timothy D. Kanold, PhD, is an award-winning educator, author, and consultant and national thought leader in mathematics. He is former director of mathematics and science and served as superintendent of Adlai E. Stevenson High School District 125, a model PLC district in Lincolnshire, Illinois.

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , HIGH SCHOOL

Dr. Kanold is committed to equity and excellence for 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 greater equity for students through the faculty and administrators’ effective delivery of the PLC process.

Barnes is passionate about ensuring equity and access in mathematics for students, families, and staff. His experiences drive his advocacy efforts as he works to ensure opportunity and access to underserved and underperforming populations. He fosters partnership among schools, families, and community resources in an effort to eliminate traditional educational barriers.

He is a past president of NCSM and coauthor of several best-selling mathematics textbooks over several decades. Dr. Kanold has authored or coauthored sixteen books on K–12 mathematics and school leadership since 2011, including the best-selling book HEART! Fully Forming Your Professional Life as a Teacher and Leader. He also has served on writing commissions for NCTM and has authored numerous articles and chapters on school leadership and development for education publications since 2006.

A past president of the Maryland Council of Teachers of Mathematics, Barnes has served as the affiliate service committee eastern region 2 representative for the NCTM and first vice president for NCSM.

Dr. Kanold received the 2017 Ross Taylor / Glenn Gilbert National Leadership Award from NCSM, the international 2010 Damen Award for outstanding contributions to the leadership field of education from Loyola University Chicago, the 1986 Presidential Awards for Excellence in Mathematics and Science Teaching, and the 1994 Outstanding Administrator Award from the Illinois State Board of Education. He serves as an adjunct faculty member for the graduate school at Loyola University Chicago. Dr. Kanold earned a bachelor’s degree in education and a master’s degree in mathematics from Illinois State University. He also completed a master’s degree 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 Timothy D. Kanold’s work, follow @tkanold on Twitter. Bill Barnes, MEd, is the chief academic officer for the Howard County Public School System in Maryland. He is also the historian of NCSM and has served as an adjunct professor for Johns Hopkins University; the University of Maryland, Baltimore County; McDaniel College; and Towson University.

Barnes is the recipient of the 2003 Maryland Presidential Award for Excellence in Mathematics and Science Teaching. He was named Outstanding Middle School Math Teacher by the Maryland Council of Teachers of Mathematics (1999) and Maryland Public Television’s National Teacher Training Institute (NTTI) Master Teacher of the Year (1997). Barnes earned a bachelor of science degree in mathematics from Towson University and a master of science degree in mathematics and science education from Johns Hopkins University. To learn more about Bill Barnes’s work, follow him @BillJBarnes on Twitter. Darshan M. Jain, MST, is director of mathematics and computer science at Adlai E. Stevenson High School in Lincolnshire, Illinois. Jain began his professional career as a mechanical engineer in manufacturing and machine design. Jain’s passion and commitment to students’ learning of mathematics were inspired by working with students through the University of Illinois at Chicago’s Hispanic Mathematics, Science, and Engineering Initiative. Supporting students’ learning in mathematics through a discovery-based collaborative approach led to a career change. Jain is the recipient of the 2010 Golden Apple Award for Excellence in Teaching, 2011 National Board for Professional Teaching Standards recognition, and a 2013 Presidential Award for Excellence in Mathematics and Science Teaching. Jain works to transition curricular teams in adopting state standards and developing curriculum centered on

A bout the Author s

effective and high-leverage pedagogy and assessment. He has extensive experience in leading adult professional learning and speaks at local, state, and national mathematics professional organization meetings. Jain has been featured on pbslearningmedia.org for his leadership in supporting problem solving and student perseverance in learning mathematics. Jain earned a bachelor of science degree in mechanical engineering and master of science in teaching secondary mathematics education. He also holds a master of arts in educational leadership and is currently pursuing a doctorate in education policy, organization, and leadership. To learn more about Darshan Jain’s work, follow him @djain2718 on Twitter. Matthew R. Larson, PhD, is an award-winning educator and author who served as the K–12 mathematics curriculum specialist for Lincoln Public Schools in Nebraska for more than twenty years. He served as president of NCTM from 2016 to 2018. Dr. Larson has taught mathematics at the elementary through college levels and has held an honorary appointment as a visiting associate professor of mathematics education at Teachers College, Columbia University. He is coauthor of several mathematics textbooks, professional books, and articles on mathematics education, and was a contributing writer on the influential publications Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014) and Catalyzing Change in High School Mathematics: Initiating Critical Conversations (NCTM, 2018). A frequent keynote

speaker at national meetings, Dr. Larson’s humorous presentations are well known for their application of research findings to practice. Dr. Larson earned a bachelor’s degree and doctorate from the University of Nebraska–Lincoln, where he is an adjunct professor in the department of mathematics. To learn more about Matthew R. Larson’s work, visit @mlarson_math on Twitter. Brittany Mozingo, MEd, has more than a decade of experience as an educator in at-risk schools. She is currently the academic instructional coach at Fern Creek High School, a DuFour Award– winning school in Louisville, Kentucky. Her work is focused on providing coaching and accountability to teacher teams as they work to become members of a highly effective professional learning community. She specializes in cognitive coaching, systems implementation, and mathematics assessment literacy. Mozingo earned a bachelor’s degree in mathematics and secondary education from Wilmington College and a master’s degree in teacher leadership from the University of Louisville. In addition, she has completed thirty postgraduate hours in adult-learning theory. To learn more about Mozingo’s work, follow her @ MsMoLovesMath on Twitter. To book Sarah Schuhl, Timothy D. Kanold, Bill Barnes, Darshan M. Jain, Matt Larson, or Brittany Mozingo for professional development, contact pd@ SolutionTree.com.

Introduction Timothy D. Kanold

t the heart of your work as teachers of mathematics in high school is the development of student self-efficacy. Student self-efficacy references students’ belief in their capability to learn the mathematics we need them to know by the end of each grade.

order; they need to happen in the right place and the right time in the story arc for each grade level. There is an order to the flow of the high school mathematics content story. And, as high school teachers, we need to fully understand the how and the why of the content trajectories across these grades.

But what exactly does a high school mathematics student need to know by the end of each unit of study throughout each course? And, more important, how does the teacher develop his or her personal teacher self-efficacy to adequately plan for and then deliver on the promise of the mathematics standards for those mathematics units of study to students?

I have spent over half of my life trying to get these story arcs correct—trying to create textbooks that would make contextual sense to both the student and the teacher. In a way, I wanted to help students and teachers develop their self-efficacy to learn mathematics.

I have been trying to answer this question my entire professional life. In 1987, I coauthored my first mathematics textbook: a geometry book for high school students who found mathematics a difficult subject. It was my first real experience in taking a wide body of content for the complete school year and breaking the standards down into reasonable chunks for every teacher and student to learn. As I eventually expanded my textbook writing to include K–12 mathematics students and teachers, I realized the time spent teaching these manageable chunks of content could vary in length from twenty to thirty-five days, and they often had names like units or chapters or modules. As you know, mathematics is a vertically connected curriculum, and the units of study at each grade level cannot be taught in a random

Now for more than a decade, and with the help of our incredible team of mathematics thought leaders and professional developers, lead author of this Mathematics Unit Planning in a PLC at Work ® series Sarah Schuhl and I realize every teacher and teacher team of high school mathematics needs to work collectively with their textbook and other resources to own the planning process for each unit of study. Developing collective teacher efficacy is at the heart of the Professional Learning Community (PLC) at Work process, defined as “Social interactions firmly anchored in instructional practice [that] can move teachers beyond contrived collegiality to a culture that can in turn influence a teacher’s sense of efficacy” (Neugebauer, Hopkins, & Spillane, 2019, p. 13). However, Sabina Neugebauer, Megan Hopkins, and James Spillane (2019) point out that social team interactions must be “anchored in actual teaching and assessing episodes” (p. 13). Teachers then place those

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , HIGH SCHOOL

episodes into manageable chunks of content for their teams’ discussion and work. As we enter the 2020s, it is interesting to observe the integrated nature of how the high school mathematics content standards have shifted and changed, and also how the pedagogical process standards have become more integrated. To some extent, these changes are best viewed through the lens of essential learning standards for every high school student, and from the vantage point of each state’s expectations for college and career readiness. Going back to 2000, the National Council of Teachers of Mathematics (NCTM) describes integrated mathematics as a term used in the United States to connect topics or strands of mathematics throughout each year of secondary school. So what has changed in the past twenty years since NCTM’s 2001 observation? Quite a bit, it turns out. The story arc for each year of high school mathematics is more integrated today than ever before, regardless of how we choose to name high school mathematics courses. In making sense of the high school mathematics standards, we have tended to get a bit too entangled in the details. We think at the “139 standards level” about what we must teach students during the three-year collegereadiness sequence. In most states, students reach proficiency of these standards through defined courses like algebra 1, geometry, and algebra 2. In some cases, there might be other names for these courses, such as math I, math II, and math III, or perhaps year 1, year 2, and year 3. No matter what we call these high school courses, it is best to look at the approximately 139 learning targets through the lens of an ongoing set of integrated standards built around the primary essential learning standards developed throughout every unit in each high school mathematics course. These essential learning standards are built around five domains, which also each include modeling: number, algebra, functions, geometry and measurement, and statistics and probability. Notice that each of these six essential categories are not courses per se; rather, they represent an integrated set of standards that teacher teams package into a progression of topics to create a scope and sequence story arc that makes sense as students pass through each

high school course. For example, whether we call the courses year 1, math I, or algebra 1, a first-year high school mathematics course will integrate authentic mathematical modeling standards and mathematical tasks, statistics that integrate bivariate data sets to linear function connections, and algebraic equations as an equality of two functions. Thus, teams integrate high school students’ learning experiences through a f low of mathematical tasks that are both higher level and lower level in cognitive demand, for almost every daily learning target we teach. There is a significant difference between asking high school mathematics year 1 students to determine f(x) = 500(1.015)x at x = 0 versus asking students to compare and contrast the graphs of f(x) = 500(1.015)x and g(x) = 500(1.021)x . Both of these mathematical tasks are important skills, yet they require varying demonstration levels of understanding. The verbs most states’ standards use, such as create, understand, build, compare, describe, and justify, indicate we are to present our high school students with an integrated learning experience as well. Thus, there is a pedagogical learning expectation that sometimes students learn by observing teacher thinking, taking notes, and answering teacher questions, also known as whole-group discourse, during the mathematics lesson. Yet sometimes students are to experience learning a particular standard through small-group discourse tasks, working with peers as the teacher provides formative feedback, and following prompts for perseverance on carefully chosen mathematical tasks. The expectation is that both whole-group and small-group discourse are integrated into every unit of mathematics. For students to fully learn and demonstrate understanding of mathematics in each of the high school courses, a balanced use of technology is expected. It will be very difficult in the 2020s not to integrate technology into the high school mathematics curriculum. As students learn to explore mathematical models and apply various statistics and functions, including transformations, into this integrated curriculum, they will need proficiency in the use of appropriate graphing, exploration, and statistical tools for the primary college- and career-readiness mathematics courses in your department.

Introduc tion

We have chosen in this book to reference the high school mathematics curriculum within the more widely used and understood algebra 1, geometry, and algebra 2 course names. These course names do not sound like they promote an integrated mathematics program. However, in this era, the high school mathematics curriculum is integrated, no matter the name you choose to give each year of study as the standards of functions and statistics permeate all years of study. My coauthors of this Mathematics at Work unit planning guide for high school—Sarah Schuhl, Bill Barnes, Darshan Jain, Matt Larson, and Brittany Mozingo—and I serve or have served in many mathematics teaching and leading roles. One such role is to serve on our Mathematics at Work team of national thought leaders. As we travel around the United States helping high school teachers improve student learning in mathematics, those teachers often ask us, “How do we collectively plan for a unit of study in mathematics at our grade level?” Answering that question is the purpose of this book.

The Purpose of This Book We want to help your grade-level team learn how to work together to perform the following seven collaborative tasks for each unit of mathematics study throughout the year. Generate Essential Learning Standards for Each Unit Unwrap standards into daily learning targets and write those standards in student-friendly language for essential learning standards. Then use those essential learning standards to drive feedback on common mathematics assessments, classwork, independent practice, and intervention as a collaborative team. Create a Team Unit Calendar Decide the number of days needed to teach each essential learning standard, and the start and end dates of the unit. Decide the dates to administer any common mid-unit or end-of-unit assessments. Establish each date the team will analyze data from any common mid-unit and end-of-unit assessments to plan a team response to student learning.

Identify Prior Knowledge Determine and identify the recent prerequisite content knowledge students need to access the grade-level learning in each unit of study. Decide which mathematical activities (tasks or prompts) to use for students to connect the prior knowledge at the start of each lesson throughout the unit. Use these activities to discern student readiness and entry points into each lesson. Determine Vocabulary and Notations Identify the academic vocabulary students will be reading and using during discourse throughout the unit. Identify any mathematical notation students will need to read, write, and speak during the unit. Identify Resources and Activities Determine which lessons in the team’s current basal curriculum materials align to the essential learning standards in the unit. Determine examples of higher-level and lower-level tasks students must engage in to fully learn each essential learning standard. Agree on Tools and Technology Determine any manipulatives or technology needed to help students master the essential learning standards of the unit. Identify whether the tools or technology needed in the unit will support student learning of the essential learning standards with a focus on conceptual understanding, application, or procedural fluency. Identify which tools and technology, if any, will be part of instruction or available as a resource for common assessments. Record Reflection and Notes When planning the unit, record notes of things to remember when teaching. Do so by answering, for example, these questions: When should students use technology to foster learning? How will students develop and express their understanding of transformations on a coordinate plane, functions, proofs, statistics, and equations? What are the expectations for quality student work? Which mathematical strategies should teachers use throughout the unit? After the unit, reflect on instruction and assessments to keep or change for next year, and record ideas to use when planning the unit for next year.

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , HIGH SCHOOL

The Parts of This Book Part 1 provides detailed insight into how your mathematics team can effectively enact these seven planning tasks for the essential standards you expect students to learn in grades 9, 10, and 11. Part 2 provides three detailed model units and describes a function-transformation story arc for a unit in algebra 1, geometry, and algebra 2. The units connect foundational elements of transformation with quadratic functions, geometric functions, and trigonometric functions, as students move through three years of a college-readiness mathematics sequence. We hope part 2 provides an inspiring model for your high school mathematics course-based teams. The epilogue shares an example for how to organize your course-based team’s work on a unit-by-unit basis so your mathematics department can grow and learn from its work in future years. If your collaborative team does not already have a mathematics unit of study yearlong plan with standards, appendix A provides a proficiency map protocol as a way to organize your standards and to determine when students should be proficient with each standard. Finally, appendix B contains a team checklist and questions for your team to answer as you plan each mathematics unit. Appendix B summarizes the elements of unit planning shared in parts 1 and 2 of this book and is intended to be a quick reference to guide the work of your team in your unit planning.

A Final Thought You might wonder, “Why is this book titled Mathematics Unit Planning in a PLC at Work, High School?”

In 1980, my second mathematics teaching job landed me on the doorstep of an educational leader who would later start an education movement in the United States that would spread throughout North America and even worldwide. He, along with Robert Eaker, was the architect of the Professional Learning Communities at Work movement and my principal for many years. Dr. Richard DuFour expected every grade-level or coursebased team in our school district to answer four critical questions for each unit of study in mathematics (DuFour, DuFour, Eaker, Many, & Mattos, 2016). 1. What do we want all students to know and be able to do? (essential learning standards) 2. How will we know if they know it? (lesson design elements, assessments, and tasks used) 3. How will we respond if they don’t know it? (formative assessment processes) 4. How will we respond if they do know it? (formative assessment processes) As your collaborative team pursues this deep work, remember it all begins with a robust and well-planned response to PLC critical question one (What do we want all students to know and be able to do?). That is the focus of our high school unit planning book. We want to help you plan for and answer the first question for each high school mathematics unit, for each course (regardless of the names you choose for these courses), and for every student. We wish you the best in your mathematics teaching and learning journey, together.

CHAPTER 1

Mathematics is a conceptual domain. It is not, as many people think, a list of facts and methods to be remembered. —Jo Boaler

he first critical question of a PLC is, What do we expect all students to know and be able to do? (DuFour et al., 2016). As your collaborative team successfully answers this question for each unit of study, you build a common understanding of the mathematics students will learn in your course. What is the mathematics story that unfolds as student learning progresses from one mathematics unit to the next? How do the units fit together and build upon one another within and across the first three years of high school mathematics?

Guaranteed and Viable Curriculum Your high school team effectively backward plans the school year by grouping essential mathematics standards into units to create the guaranteed and viable

mathematics curriculum students must learn (Marzano, Warrick, Rains, & DuFour, 2018; Wilkins, 1997). The order you teach the units provides the framework for your course’s mathematics story. Within each unit, your daily lessons create the beginning, middle, and end for that part of the story. Thus, evidence of your team’s guaranteed and viable curriculum includes (1) semester- and yearlong pacing plans of standards (proficiency maps or pacing guides), (2) unit plans, and (3) daily mathematics lessons. The graphic in figure 1.1 illustrates these areas of team planning for a mathematics guaranteed and viable curriculum. The thick black vertical line down the middle shows the end of semester 1 and the start of semester 2 for your team (keep in mind the number of units in

District Yearlong Plan Mathematics

Mathematics

State

Standards in

Unit 1

Unit 2

Unit 3

Unit 4

Unit 7

Unit 8

Unit 9

Unit 10

Unit 8 Plan

Unit 9 Plan

Unit 10 Plan

Mathematics

Mathematics Team Unit Plans Unit 1 Plan

Unit 2 Plan

Unit 3 Plan

Unit 4 Plan

Unit 7 Plan

Mathematics Teacher Daily Lessons

Figure 1.1: Mathematics guaranteed and viable curriculum plan.

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , HIGH SCHOOL

each of your semesters may vary from that shown in figure 1.1). Together, the mathematics units of study tell the story about the course standards students are expected to learn throughout the semester and the academic year, and from one school year to the next.

If your collaborative team does not have a yearlong plan with standards in clearly defined units, see appendix A (page 121), “Create a Proficiency Map,” for additional support. Helping each teacher on your team become comfortable with the progression of mathematics units throughout the school year will support your students’ understanding of the mathematics story arc for various standards.

Mathematics Unit Planner Once your high school course team determines the mathematics units for your course (detailing the standards and time line for each unit), your team next plans for student learning on a unit-by-unit basis (see figure 1.2; Kanold & Schuhl, 2020). The Mathematics Unit Planner in figure 1.2 provides a template for your course-based team to use as you develop a shared understanding of what students are expected to learn in each unit of study. The numbered sections in the Mathematics Unit Planner correspond with the seven elements of unit planning described in the introduction (page 1). In the upcoming chapters, you will see headings that correspond with these seven areas. You will also find completed examples of unit planners for algebra 1 in figure 3.11 (page 59), geometry in figure 4.10 (page 83), and algebra 2 in figure 5.11 (page 113). In Principles to Actions, researchers for the National Council of Teachers of Mathematics (NCTM; 2014) note, “Effective mathematics teaching begins with a

Mathematics Concepts and Skills in High School Students in high school deepen their understanding of number, algebra, functions, geometry and measurement, and statistics and probability situations using content from these domains in order to develop a conceptual understanding of mathematics. Mathematical modeling should permeate the content students learn and support teaching mathematical process standards that develop the necessary habits of mind for successful student mathematical reasoning. These mathematical process standards for learning are an important aspect of instruction, a component of higher-level tasks, and embedded in the strategies and tools students use to problem solve. Students develop a conceptual understanding of high school mathematics from prior knowledge learned in middle school related to ratio and proportional reasoning, linear equations, expressions and equations with rational numbers, geometry, and statistics and probability. Connections from one course to the next throughout high school prepare students to be college and career ready by the time they graduate. Students in your courses learn higher-order reasoning skills and how to apply learning in novel and new situations through the use of higher-level cognitive tasks and modeling. Most current high school state mathematics standards were generated from the recommendations of NCTM’s Principles and Standards for School Mathematics (2000), the Governors' task force for Common Core State Standards (NGA & CCSSO, 2010), and recommendations from Catalyzing Change in High School Mathematics (NCTM, 2018). Table 1.1 (page 12) shows some of the key mathematics concepts students are expected to learn in high school over the

As figure 1.1 (page 9) shows, a district yearlong pacing guide or proficiency map (showing a timeline for student proficiency with each mathematics standard) first defines your course-based team’s guaranteed and viable curriculum. Your team then determines a time frame appropriate for each mathematics unit of study, typically two to four weeks in duration for high school courses. This process eliminates the potential risk of running out of time and skipping end-of-the-year units that address essential standards.

shared understanding among teachers of the mathematics that students are learning and how this mathematics develops along learning progressions” (p. 12). Therefore, as your mathematics department and course-based teams develop unit plans for the year, be sure to make sense of the mathematical content standards students are learning in each semester in your course. Additionally, make sense of the mathematical content trajectories (progressions) students are learning throughout your high school courses.

Planning for Student Learning of Mathematic s in High School

Unit: Start Date:

End Date:

Total Number of Days:

Unit Planning List the essential learning standards for this unit. Essential Learning Standards

Prior Knowledge

List the mathematical academic vocabulary and notations for this unit. Vocabulary and Notations

Possible Resources or Activities

List the possible resources or activities to use when teaching the essential learning standards.

List the essential tools, manipulatives, and technology needed for this unit. Tools and Technology

After the unit, reflect and list what to do again, revise, or change. Reflection and Notes

Unit Calendar Monday

Tuesday

Wednesday

Thursday

Week 1 Week 2 Week 3 Week 4 Week 5

Source: Adapted from Kanold & Schuhl, 2020, p. 30. Figure 1.2: Mathematics Unit Planner.

Visit go.SolutionTree.com/MathematicsatWork for a free reproducible version of this figure.

Friday

List standards from a previous unit or grade students will access in this unit.

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , HIGH SCHOOL

Table 1.1: Mathematics Concepts and Skills in High School High School: Algebra 1, Geometry, and Algebra 2 Number

• Use units to understand problems and model with mathematics. • Extend an understanding of number to include irrational numbers and rational numbers. • Extend the properties of exponents to rational exponents. • Perform arithmetic operations involving complex numbers.

Algebra

• Interpret the structure of expressions and use structure to rewrite equations in different forms. • Understand the process of solving equations and inequalities, including systems of equations, and explain the reasoning used for each. • Understand operations (addition, subtraction, multiplication, division) and apply them to polynomials. • Analyze and make predictions using expressions, equations, and inequalities.

Functions

• Understand the concept of a function and use function notation. • Understand that functions within the same family have distinguishing attributes. • Interpret key features of functions and relate the domain and range to context, if applicable (linear, quadratic, absolute value, exponential, square root, cube root, piecewise, step, polynomial, logarithmic, trigonometric). • Analyze and compare functions represented symbolically, graphically, numerically in tables, or verbally in descriptions. • Build a new function that represents a relationship between two quantities or one from existing functions using transformations. • Interpret expressions for functions based on the situations they model. • Find inverse functions. • Model periodic phenomena with trigonometric functions.

Geometry and Measurement

• Define, represent, describe, and draw transformations that preserve distance and angles and those that do not (rotation, reflection, translation, dilation). • Create geometric proofs related to congruence by first understanding congruence as the result of rigid motions. • Prove geometric theorems algebraically using coordinates. • Prove theorems about similarity by first understanding similarity in terms of similarity transformations. • Solve problems involving right triangles using trigonometric ratios and the Pythagorean theorem. • Understand and apply theorems about circles, including the equation for circles. • Explain volume formulas and use them to solve problems. • Apply geometric concepts in modeling situations.

Statistics and Probability

• Summarize, represent, and interpret data about a single variable (line plot, box plot, histogram, normal curve, measures of center, measures of spread). • Summarize, represent, and interpret data about two categorical or quantitative variables (two-way frequency tables, scatter plots, lines and functions of best fit). • Understand independence and conditional probability and use them to interpret data and solve problems. • Compute probabilities of compound events in terms of a uniform probability model. • Understand and evaluate statistical experiments used to make inferences. • Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Source: Adapted from NCTM, 2018.

• Create equations and inequalities and use them to solve problems.

Planning for Student Learning of Mathematic s in High School

course of three years, regardless of how your school or district determines and organizes the courses in which students learn.

mathematics curriculum. Together, you and your team use the Mathematics Unit Planner template in figure 1.2 (page 11) to record answers to the following questions.

Your students in high school extend learning from middle school to understand, analyze, interpret, and create equations and new functions with key features, grow numbers to include irrational numbers, deepen geometric reasoning with proofs, analyze and interpret statistical data and probability, and create and explain mathematical models. Learning is focused on an understanding of each concept and between concepts in order to develop critical reasoning and proof within a consistent system of mathematics and for application within each course and future experiences.

• What exactly do students need to know and be able to do in this unit?

With so much mathematics content to learn, your mathematics department’s and course-based team’s unit planning helps ensure a guaranteed and viable mathematics curriculum within your course and across your high school course sequence. Planning the units together to learn your own course content in greater depth and its importance in the vertical high school trajectory builds your teacher team self-efficacy.

Connections Between Mathematics Content and Unit Planning For each unit in your high school course, you support your team’s progress toward better understanding the standards that support the guaranteed and viable

• Which mathematics standards should we commonly assess? When? • How does the mathematics learning in this unit connect to the standards students must learn in previous or future units? • Which academic mathematics vocabulary and notations must students learn to read, write, and speak in the unit to be proficient with the standards? • What are examples of higher- and lower-levelcognitive-demand mathematical tasks students should demonstrate proficiency with if they have learned the standards? • Which mathematical tools or technology should students learn or utilize to demonstrate an understanding of the standards in the unit? Answering these questions as a team creates more equitable student learning experiences from one teacher to the next. Teachers come to a consensus on what needs to be taught and how to teach it, ensuring all students, regardless of instructor, receive the same mathematics education. Additionally, developing teacher efficacy strengthens your instructional practices. Consequently, student learning improves because your entire team is working to ensure each student learns the organized mathematics content from one unit to the next. Chapter 2 (page 15) provides tools and protocols that help your high school course-based mathematics team unwrap unit standards. These tools and protocols will also assist you in intentionally addressing each unit-planning element as your mathematics story arc develops for the school year.

Your course-based team and mathematics department may want to explore mathematics learning progressions as defined by your state standards or reference mathematics learning progression documents online, such as those developed by the Common Core Standards Writing Team (2013) at www.math.arizona.edu/~ime/ progressions, or Student Achievement Partners’ (n.d.) coherence map. Your team may also want to engage in a book study, perhaps referencing NCTM resources related to understanding the essential content and skills needed in high school.

E V E R Y S T U D E N T C A N L E A R N M AT H E M AT I C S

“With clarity and common sense, Mathematics Unit Planning in a PLC at Work, High School guides teachers through the collaborative process of designing aligned and focused units of

members get their hands on it!” —Kim Bailey, Author and Educational Consultant

“This book is a great tool for teacher teams and instructional leaders.

S C H OO L

“In this resource packed with tools and real examples, Sarah Schuhl and her coauthors support teacher teams in seeing mathematics as a connection of ideas, not just a group of

“A critically important tool for high school teacher teams. I also

Unit Planning

H I G H

M AT H E M AT I C S

study in mathematics. This practical and reader-friendly tool will quickly fill with sticky notes and earmarked pages once team

think it will be very valuable for solo educators in alternative

—Sharon Rendon, Coaching Coordinator, College Preparatory

settings and smaller schools and districts.” —Nick Resnick, K–12 Education Specialist

Mathematics Educational Program, California

in a PLC at Work ®

unconnected units.”

S C H O O L

Community at Work (PLC) process to increase mathematics achievement and give students more equitable

H I G H

M. Jain, Matthew R. Larson, and Brittany Mozingo help teams identify what students need to know by the end

ISBN 978-1-951075-29-3 90000

SolutionTree.com NCTM Stock ID: 16068

9 781951 075293

Schuhl • Kanold Barnes • Jain • Larson • Mozingo

• Learn how to build a shared understanding of the content students need to know in each course by using seven planning elements

Mathematics Unit Planning in a PLC at Work®, High School

Articles inside

Unit Planning as a Collaborative Mathematics Team

Planning for Student Learning of Mathematics in High School