Mathematics Unit Planning in a PLC at Work®, Grades 3–5 by Solution Tree

“This book gives collaborative teams a strategic and systematic approach to designing dynamic mathematics units of study that build and connect over the course of a school year. Educators to answer the first critical question of learning (What do we want students to know and be able to do?) and come away with a collective set of valuable tools for planning high-quality mathematics instruction.”

G R A D E S

—Tracey Hulen, Mathematics Specialist,

3–5

T.H. Educational Solutions, Fairfax, Virginia

“Teachers frequently lament that they need to see real examples they can use as models to jumpstart their collaborative work.

collaborative process of designing aligned and focused units

This book includes exactly that! With a well-developed planning

of study in mathematics. This practical and reader-friendly

process along with templates and completed examples to use

tool will quickly fill with sticky notes and earmarked pages

as models, teams will quickly be able to get started writing, or

once team members get their hands on it!”

revising, effective unit plans.”

—Kim Bailey, Author and Educational Consultant

—Chris Jakicic, Author and Educational Consultant

in a PLC at Work ®

in a PLC at Work, Grades 3–5 guides teachers through the

Unit Planning

“With clarity and common sense, Mathematics Unit Planning

M AT H E M AT I C S

who read this book will have an opportunity to dig in deeply

Mathematics Unit Planning in a PLC at Work®, Grades 3–5 provides third- to fifth-grade mathematics teachers with a seven-step framework for collectively planning units of study. Authors Sarah Schuhl, Timothy D. Kanold, end of each unit and how teachers can build student self-efficacy. They advocate using the Professional Learning Community at Work (PLC) process to increase mathematics achievement and give students more equitable as unwrapping standards, generating unit calendars, determining academic vocabulary and rigorous lessons,

3–5

learning experiences. The authors share tools and protocols for effectively performing collaborative tasks, such

G R A D E S

Jennifer Deinhart, Matthew R. Larson, and Mona Toncheff help teams identify what students need to know by the

utilizing and sharing self-reflections, and designing robust fraction units. This book provides practical insights into collaborative planning and detailed, inspiring models of this work in action. Mathematics teams will: • Learn how to build a shared understanding of the content students need to know in each grade level by using seven planning elements

• Understand how teams can successfully incorporate each unit-planning element in their unit designs • Examine three model units on fractions, one for each grade level • 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.

SolutionTree.com NCTM Stock ID: 16019

Schuhl • Kanold Deinhart • Larson • Toncheff

• Find protocols for unit planning and reproducible templates

G R ADES

3–5

Copyright © 2020 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. | Kanold, Timothy D., author. | Deinhart, Jennifer, author. | Larson, Matthew R., author. | Toncheff, Mona, author. Title: Mathematics unit planning in a PLC at work. Grades 3-5 / Sarah Schuhl, Timothy D. Kanold, Jennifer Deinhart, Matthew R. Larson, Mona Toncheff. Description: Bloomington, IN : Solution Tree Press, [2020] | Includes bibliographical references and index. Identifiers: LCCN 2019051280 (print) | LCCN 2019051281 (ebook) | ISBN 9781951075255 (paperback) | ISBN 9781951075262 (ebook) Subjects: LCSH: Mathematics--Study and teaching (Elementary)--United States. | Professional learning communities. Classification: LCC QA13 .S364 2020 (print) | LCC QA13 (ebook) | DDC 372.7/043--dc23 LC record available at https://lccn.loc.gov/2019051280 LC ebook record available at https://lccn.loc.gov/2019051281 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: Evie Madsen Proofreader: Mark Hain Cover Designer: Kelsey Hergül Editorial Assistant: Sarah Ludwig

Table of Contents

About the Authors . Introduction .

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

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

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

2 3 . 3

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

Part 1 Mathematics Unit Planning and Design Elements .

1 2

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

Planning for Student Learning of Mathematics in Grades 3–5 . Guaranteed and Viable Curriculum . . . . . . . . . . . . . . . Mathematics Unit Planner . . . . . . . . . . . . . . . . . . . Mathematics Concepts and Skills for Grades 3–5 . . . . . . . . Connections Between Mathematics Content and Unit Planning .

Unit Planning as a Collaborative Mathematics Team .

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

9 . 9 . 10 . 13

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

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

Mathematics Unit Planning as a Team . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Part 2 Fraction Unit Examples, Grades 3–5 .

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

Grade 3 Unit: Fraction Understanding . Essential Learning Standards

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

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

37 38

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 3 – 5

Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Grade 3 Fraction Understanding Unit Planner . . . . . . . . . . . . . . . . . . . . . . . . . 53 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Grade 4 Unit: Fraction Equivalence, Addition, and Subtraction .

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

Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Grade 4 Fraction Equivalence, Addition, and Subtraction Unit Planner . . . . . . . . . . . . . 73 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Grade 5 Unit: Fraction Addition and Subtraction .

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

Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Grade 5 Fraction Addition and Subtraction Unit Planner . . . . . . . . . . . . . . . . . . . . 92 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Epilogue: Mathematics Team Organization . Final Thoughts .

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Appendix A: Create a Proficiency Map .

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

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 . Team Questions to Create a Unit Calendar . . . . . . . . . Team Questions to Identify Prior Knowledge . . . . . . . .

. . . . . . . . . .

101

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

101 101 102 102 102 102 102 103 103 103

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

Table of C ontent s

Team Questions to Determine Vocabulary and Notations . Team Questions to Identify Resources and Activities . . . Team Questions to Determine Tools and Technology . . . Team Questions to Record Reflection and Notes . . . . .

References and Resources . Index .

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

103 103 104 104

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

105

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

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

109

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 Kâ&#x20AC;&#x201C;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. TM

Since 2015, Schuhl has coauthored the books Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy With Kâ&#x20AC;&#x201C;5 Learners and School Improvement for All: A Guide to 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 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 Schuhlâ&#x20AC;&#x2122;s work, follow her @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. Dr. Kanold is committed to equity and excellence for students, faculty, and school administrators. He

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 3 – 5

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. He is a past president of the National Council of Supervisors of Mathematics (NCSM) and coauthor of many 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 and IPPY 2018 Gold Medal Award–winning book HEART! He also has served on several writing commissions for the NCTM and has authored numerous articles and chapters on school leadership and development for education publications since 2006. Dr. Kanold received the 2017 Ross Taylor/Glen Gilbert Mathematics Education Leadership Award from the NCSM, the international 2010 Damen Award for outstanding contributions to the leadership field of education from Loyola University Chicago, 1986 Presidential Awards for Excellence in Mathematics and Science Teaching, and 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 him @tkanold on Twitter. Jennifer Deinhart, MEd, is an educational consultant and K–8 mathematics specialist. Deinhart is currently working as a mathematics instructional coach at Rose Hill Elementary, part of Fairfax County Public Schools. During her time at Mason Crest Elementary in Annandale, Virginia, the school was recognized as the first national model PLC school to receive the DuFour Award. A passionate educator with more than twenty years of experience working with

diverse populations within Title I schools, she works collaboratively with teams of teachers to provide quality mathematics instruction. Deinhart has been part of multiple leadership teams and now also support schools around the nation in learning and implementing the PLC at Work process. She has worked with others to develop meaningful collaborative team structures that focus on student learning, reflecting on results, and designing instruction that meets the needs of all learners. Deinhart coauthored an article that appeared in the Journal of Mathematics and Science: Collaborative Explorations and has been a leader on several curriculum projects for Fairfax County Public Schools. She received a bachelor’s degree from Buffalo State College, State University of New York, and a master’s of education degree specializing in K–8 mathematics leadership from George Mason University. To learn more about Jennifer Deinhart’s work, follow her at @jenn_deinhart 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, where he currently serves as associate superintendent for instruction. He served as president of the NCTM from 2016–2018. 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, 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.

A bout the Author s

To learn more about Matthew R. Larson’s work, visit @mlarson_math on Twitter.

ensure all students receive high-quality mathematics instruction.

Mona Toncheff, MEd, an educational consultant and author, worked as both a mathematics teacher and as a mathematics specialist for the Phoenix Union High School District in Arizona. In the latter role, she coached and provided professional development to high school teachers and administrators related to quality mathematics teaching and learning and working in effective collaborative teams. She currently serves as a supervisor teacher for the University of Arizona Teach Program.

Toncheff is currently an active member of the National Council of Supervisors of Mathematics (NCSM) board and has served NCSM in the roles of secretary (2007–2008), director of Western region 1 (2012–2015), second vice-president (2015–2016), first vice-president (2016–2017), marketing and e-news editor (2017–2018), president-elect (2018–2019), and president (2019–2021). In addition to her work with NCSM, she has served as the president of Arizona Mathematics Leaders (2016–2018) and is the current past-president. She was named 2009 Phoenix Union High School District Teacher of the Year and in 2014, she received the Copper Apple Award for leadership in mathematics from the Arizona Association of Teachers of Mathematics.

Toncheff has supervised the culture change from teacher isolation to PLCs, creating articulated standards and relevant district common assessments and providing ongoing professional development on best practices, equity and access, technology, response to intervention, high-quality grading practices, and assessment for learning. As a writer and consultant, Toncheff works with educators and leaders nationwide to build collaborative teams, empowering them with effective strategies for aligning curriculum, instruction, and assessment to

Toncheff earned a bachelor of science degree from Arizona State University and a master of education degree in educational leadership from Northern Arizona University. To learn more about Mona Toncheff’s work, follow her @toncheff5 on Twitter. To book Sarah Schuhl, Timothy D. Kanold, Jennifer Deinhart, Matthew R. Larson, or Mona Toncheff for professional development, contact pd@SolutionTree.com.

Introduction By Timothy D. Kanold

t the heart of your work as teachers of mathematics for grades 3–5 is developing student self-efficacy. Student self-efficacy references a student’s belief in his or her capability to learn the mathematics you need students to know by the end of each grade. But what exactly does a grade 3–5 mathematics student need to know by the end of each unit of study throughout the school year? And, more important, how does a teacher develop his or her personal selfefficacy to adequately plan for and then deliver those mathematics units of study to students? I have been trying to answer this question my entire professional life. In 1987 I coauthored my first mathematics textbook (a geometry book for 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 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 from twenty to thirty-five days, and often had names like units or chapters or modules. As you know, mathematics is a vertically connected curriculum, and units of study at each grade level cannot be taught in random order; the units must exist in the right sequence of the story for each grade level. There is an order to the flow of the third-, fourth-,

and fifth-grade mathematics content story. And, as third-, fourth-, and fifth-grade teachers, you need to fully understand the how and why of the content trajectories across these grades. I have spent over half of my life trying to get these story arcs correct—trying to create textbooks that make contextual sense to both the students and the teacher. In a way, I wanted to help students and teachers develop their self-efficacy to learn mathematics. Sarah Schuhl, lead author of these by grade-band books, and I realize every grades 3–5 mathematics teacher and teacher team needs to work collaboratively with his or her textbook and other resources to in order to collectively own the planning process for each unit of study. Developing collective teacher efficacy is at the heart of the Professional Learning Communities (PLC) at Work process: “Social interactions firmly anchored in instructional practice can move teachers beyond contrived collegiality to a culture that can in turn influence a teachers’ sense of efficacy” (Neugebauer, Hopkins, & Spillane, 2019, p. 13). However, educators and coauthors Sabina Neugebauer, Megan Hopkins, and James Spillane (2019) point out that social team interactions must be “anchored in actual teaching and assessing episodes.” Teachers then place those episodes into manageable chunks of content for the team’s discussion and work. In 2019, when Solution Tree asked Sarah and I to develop the Mathematics at Work Plan Book (Kanold

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 3 – 5

& Schuhl, 2020), we jumped at the chance to provide a book that would help you organize your mathematics work and story arc for the entire school year. The weekly planners we created for the plan book provide helpful organizational tools and these may be completely sufficient for your team. However, we also realize you might need more specific direction with the elements of planning we ask you to prepare for each unit of mathematics study. The coauthors of this Mathematics at Work unit planning guide for grades 3–5 (Sarah Schuhl, Jennifer Deinhart, Matthew R. Larson, Mona Toncheff, 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 elementary school teachers improve student learning in mathematics, we often hear the question, How do we collectively plan for a unit of study in mathematics at our grade level? And that 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. And 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 for 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 their prior knowledge at the start of each lesson throughout the unit. Use these activities to discern student readiness and entry points for each lesson. Determine Vocabulary and Notations Identify the academic vocabulary students will be reading and using during discourse throughout the unit. Identify any mathematical notations 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 higherand 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 and technology needed for 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 (by answering, for example, these questions: When should students use manipulatives? How will students write fractions? What are the expectations for 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.

Introduc tion

The Parts of This Book Part 1 provides detailed insight into how your mathematics team can effectively respond to these seven planning tasks for the essential standards you expect students to learn in grades 3, 4, and 5. Part 2 provides three detailed model units for fractions (one for each grade level) and describes the fractions story arc for grades 3, 4, and 5 related to equivalence and addition and subtraction of fractions. We hope part 2 provides an inspiring model for your grade-level team.

A Final Thought You might wonder, “Why is this book titled Mathematics Unit Planning in a PLC, Grades 3–5?” 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 was the architect of the Professional Learning Communities at Work movement (along with Robert Eaker) and my principal for many years. Dr. Richard DuFour expected every grade-level or

course-based 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 this guide book for grades 3–5. We want to help you plan for and answer the first question for each mathematics unit, grade level, and student. We wish you the best in your mathematics teaching and learning journey, together.

PA R T 1 Mathematics Unit Planning and Design Elements

Creating a guaranteed, viable curriculum is the number-one factor for increased levels of learning. â&#x20AC;&#x201D;Robert J. Marzano

PA R T 1 As your third-, fourth-, or fifth-grade team develops clarity about what mathematics students will learn at each grade level, it brings a laser-like focus to the content and processes students must learn in each unit throughout the year. Your team clarifies the depth of learning required for students to become proficient on the mathematics standards, and team members build a shared understanding of the content students must learn in each unit of study. Together, you determine the mathematics your team must teach and assess throughout each unit.

students must learn. PLC experts Richard DuFour, Rebecca DuFour, Robert Eaker, Thomas W. Many, and Mike Mattos (2016) note that this practice will:

The action of intentional planning as a team for student learning of mathematics on a unit-by-unit basis develops your individual and team collective teacher efficacy.

• Create ownership among all teachers required to teach and support the intended curriculum

Working together with your colleagues as a collaborative mathematics team, you erase the inequities in student learning expectations that otherwise could exist across a grade level or course. Together, you and your teammates determine what students must know and be able to do. Then, your team does the work to ensure every student learns through agreed-on, high-quality instruction, common assessments, and formative assessment processes. Your team recognizes the many challenges inherent in students learning robust mathematics standards and takes collective responsibility to close gaps and extend learning as needed. To erase inequities and ensure grade-level learning of mathematics for each student, your third-, fourth-, or fifth-grade team begins with an agreed-on guaranteed and viable curriculum. Your grade-level team, or vertical grades 3–5 team in smaller schools, works collaboratively to ensure students learn identified essential mathematics standards within the school year. On a unit-by-unit basis, your team builds a shared understanding of the essential mathematics standards

• Promote clarity among your colleagues • Ensure consistent curricular priorities among colleagues • Ensure common pacing required for effective common assessments • Ensure the curriculum is viable (that you can teach it in the allotted calendared time)

It might be surprising, but in a PLC at Work, teacher teams build mathematics units from the standards, not from the chapters in a textbook. Too often, textbooks include more learning than your state or province may require, or the textbooks may be missing content that you need to supplement to better match the standards and local curriculum expectations. Thus, your team starts with making sense of the standards students must learn in each unit of study, and then utilizes the most effective resources for teaching and learning. Part 1 consists of two chapters. Chapter 1 (page 9) describes the mathematics content and skills students must learn in grades 3–5. Teams start by understanding what mathematics students must learn in third, fourth, and fifth grades. Chapter 2 (page 15) provides protocols and tools your grade-level collaborative team can use to plan for the student learning each mathematics unit requires. Your team’s understanding of the mathematics content students must learn and your framework for each mathematics unit allow your team to use a backward design approach to ensuring every student learns mathematics.

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, members build a common understanding of the mathematics students learn at your grade level. 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 on one another within and across third, fourth, and fifth grades?

Guaranteed and Viable Curriculum Your third-, fourth-, or fifth-grade team effectively backward plans the year by grouping essential mathematics standards into units to create the guaranteed and viable mathematics curriculum students must learn. The order you teach the units provides the framework for your grade-level 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) a yearlong standards pacing plan (proficiency map or pacing guide), (2) unit plans, and (3) daily lessons. The graphic in figure 1.1 (page 10) illustrates these areas of team planning for a mathematics guaranteed and viable curriculum. Together, the mathematics units of study tell the story of the grade-level standards teachers expect students to learn throughout the year and from one year to the next. As figure 1.1 (page 10) shows, a district yearlong pacing guide or proficiency map (showing a time line for student proficiency with each mathematics

standard) first defines your grade-level team’s guaranteed and viable curriculum. Your team then determines a time frame appropriate for each mathematics unit, typically two to four weeks for grades 3–5. This process eliminates the potential risk of running out of time and skipping units or essential standards that fall at the end of the year. If your collaborative team does not have a yearlong plan with standards in clearly defined units, see appendix A (page 99), “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 team determines the mathematics units for your grade level (detailing the standards and time line for each unit) for the year, the team next plans for student learning on a unit-by-unit basis (see figure 1.2, page 11; Kanold & Schuhl, 2020, p. 30). The Mathematics Unit Planner in figure 1.2 (page 11) provides a template your team can use as it develops 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. Throughout this book, you will see numbered headings that correspond with these seven areas. (Find completed examples of unit planners for third grade in figure 3.11 [page 55], fourth grade in figure 4.11 [page 75], and fifth grade in figure 5.10 [page 93].)

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 3 – 5

District Yearlong Plan Mathematics

Mathematics

State Standards

in Unit 1

in Unit 2

in Unit 3

in Unit 4

in Unit 5

in Unit 6

in Unit 7

in Unit 8

Unit 6 Plan

Unit 7 Plan

Unit 8 Plan

Mathematics Team Unit Plans Unit 1 Plan

Unit 2 Plan

Unit 3 Plan

Unit 4 Plan

Unit 5 Plan

= lesson

Figure 1.1: Mathematics guaranteed and viable curriculum plan.

In Principles to Action, researchers for the National Council of Teachers of Mathematics (NCTM; 2014) note, “Effective mathematics teaching begins with a shared understanding among teachers of the mathematics that students are learning and how this mathematics develops along learning progressions” (p. 12). Therefore, before diving into each individual unit plan for the year, as a team, first consider the mathematical content students are learning in your grade. Additionally, make sense of the mathematical content trajectories (progressions) students are learning across the grades 3–5 band.

Mathematics Concepts and Skills for Grades 3–5 Students in grades 3–5 deepen their understanding of number, place value, addition and subtraction of whole numbers, and geometry and measurement learned in grades preK–2. Throughout grades 3–5, teachers expect students to grow their mathematical understanding of number to include larger whole number values as well as fractions and decimals, learn multiplication and division computations, and to more critically reason with geometry, measurement, and data. Table 1.1 (page 12) shows some of the key mathematics concepts teachers expect students to learn in grades 3–5, both by grade and as a vertical trajectory the National Council of Teachers of Mathematics’s (2006) Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics first defines. Grades 3–5 students are still developing flexibility with number to develop number sense and mathematical

reasoning. Third and fourth graders work to develop procedural fluency with addition and subtraction of whole numbers. Throughout grades 3–5, students first develop a conceptual understanding of multiplication and division, and in fifth grade, demonstrate procedural fluency for multiplication of whole numbers. In sixth grade, students develop procedural fluency for whole-number division. Students develop fraction understanding, application, and procedural fluency throughout grades 3–5 as they make sense of fractions; generate equivalent fractions; compare fractions; and add, subtract, multiply, and divide fractions. Grade-level teams expect students to simultaneously develop an understanding of decimals and connect their work with fractions and whole numbers to work with decimals. In geometry and measurement, students learn to find areas of rectangles, perimeters of polygons, and volumes of rectangular prisms. They deepen their ability to classify two-dimensional figures and then teachers introduce plotting points on a coordinate plane. Students also solve measurement problems involving conversions. Your team may want to explore mathematics learning progressions as defined in your state standards or reference online mathematics learning progression documents, such as those of the Common Core Standards Writing Team (2013), or Student Achievement Partners’s (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 for mathematics in grades 3–5.

Mathematics Teacher Daily Lessons

Planning for Student Learning of Mathematic s in Grades 3 – 5

Unit: Start Date:

End Date:

Total Number of Days:

Unit Planning Essential Learning Standards

List the essential learning standards for this unit.

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

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

Prior Knowledge

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 3 – 5

Table 1.1: Key Mathematics Concepts and Skills for Grades 3–5 Grade 3 Number and Operations

Geometry and Measurement

Grade 5

• Understand multiplication and division strategies for multiplication and division within 100. • Understand and develop fluency with multidigit addition and subtraction based on place value.

• Understand and develop fluency with multidigit multiplication and understand finding quotients of whole numbers involving multidigit dividends. • Fluently add and subtract using the standard algorithm.

• Fluently multiply multidigit whole numbers using the standard algorithm. • Understand division strategies with two-digit divisors. • Understand and develop fluency with operations involving decimals to the hundredths place.

• Understand fractions, especially unit fractions with specified denominators. • Understand fraction equivalence and comparison.

• Understand fraction equivalence and comparison. • Understand fraction addition and subtraction with like denominators. • Understand fraction multiplication by whole numbers. • Understand decimal fractions.

• Develop fluency with addition and subtraction of fractions. • Understand fraction multiplication. • Understand fraction division involving a whole number and unit fraction.

• Solve measurement problems involving mass and time. • Understand arrays and the area of a rectangle. • Describe and analyze twodimensional shapes.

• Solve measurement problems involving conversions. • Analyze and classify twodimensional figures using properties. • Measure and classify angles.

• Solve measurement problems involving conversions. • Understand volume of rectangular prisms. • Classify two-dimensional shapes.

Source: Adapted from NCTM, 2006.

With so much mathematics content to learn, your team’s collaborative unit planning helps ensure a guaranteed and viable mathematics curriculum at your grade level and across grades 3–5. Planning the units together to more deeply learn your own grade-level content and its importance in the grade 3–5 trajectory builds teacher team self-efficacy.

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

Connections Between Mathematics Content and Unit Planning

• Which academic mathematics vocabulary and notations must students learn to read, write, and speak to be proficient on the unit standards?What are examples of higher- and lower-level-cognitive-demand mathematical tasks students should demonstrate proficiency with if they have learned the standards?

For each unit at your grade level, you support your teams’ progress toward better understanding the standards that support the guaranteed and viable 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.

• 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 mathematical tools or technology should students learn or utilize to demonstrate an understanding the unit standards?

Fractions

Grade 4

Planning for Student Learning of Mathematic s in Grades 3 â&#x20AC;&#x201C; 5

Answering these questions as a team creates more equitable student learning experiences from one teacher to the next. 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 third-, fourth-, or fifth-grade mathematics team unpack unit standards and learn how to intentionally address each unit-planning element as your mathematics story arc develops for the school year.

G R A D E S

—Tracey Hulen, Mathematics Specialist,

3–5

T.H. Educational Solutions, Fairfax, Virginia

“Teachers frequently lament that they need to see real examples they can use as models to jumpstart their collaborative work.

collaborative process of designing aligned and focused units

This book includes exactly that! With a well-developed planning

of study in mathematics. This practical and reader-friendly

process along with templates and completed examples to use

tool will quickly fill with sticky notes and earmarked pages

as models, teams will quickly be able to get started writing, or

once team members get their hands on it!”

revising, effective unit plans.”

—Kim Bailey, Author and Educational Consultant

—Chris Jakicic, Author and Educational Consultant

in a PLC at Work ®

in a PLC at Work, Grades 3–5 guides teachers through the

Unit Planning

“With clarity and common sense, Mathematics Unit Planning

M AT H E M AT I C S

who read this book will have an opportunity to dig in deeply

3–5

learning experiences. The authors share tools and protocols for effectively performing collaborative tasks, such

G R A D E S

Jennifer Deinhart, Matthew R. Larson, and Mona Toncheff help teams identify what students need to know by the

SolutionTree.com NCTM Stock ID: 16019

Schuhl • Kanold Deinhart • Larson • Toncheff

• Find protocols for unit planning and reproducible templates

G R ADES

3–5