Mathematics Unit Planning in a PLC at Work®, Grades 6–8

“Mathematics Unit Planning in a PLC at Work, Grades 6–8 answers the question, ‘What must we do during our team planning to make a difference for our students?’ From planning essential learning standards to recording reflections and notes, teams to develop necessary structures for the critical work of planning and implementing common units.”

—John W. Staley, NCSM: Leadership in Mathematics Education President 2015–2017, United States National Commission

G R A D E S

for Mathematics Instruction Chair 2018–2020

6–8

through a thorough unit-planning process that ensures deep “This book, with its emphasis on mathematics and useful examples of tools and templates for PLC teacher teams, is an excellent contribution to the field.”

process is critical for teacher teams who collectively believe that every student can learn challenging mathematics content and collectively commit to addressing the specific learning needs of individual students.”

Education, Western Kentucky University

—Becky Walker, Assistant Superintendent of Academics and Innovation, Howard-Suamico School District, Green Bay, Wisconsin

in a PLC at Work ®

—Janet Tassell, Associate Professor, School of Teacher

grade-level mathematics learning for every student. This

Unit Planning

“This book intentionally guides collaborative teams of teachers

M AT H E M AT I C S

this comprehensive text provides guidance for collaborative

Mathematics Unit Planning in a PLC at Work®, Grades 6–8 provides sixth- to eighth-grade mathematics teachers with a seven-step framework for collectively planning units of study. Authors Sarah Schuhl, Timothy D. Kanold, need to know by the 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 collaborative tasks, such as unwrapping standards, generating unit calendars, determining academic vocabulary

6–8

students more equitable learning experiences. The authors share tools and protocols for effectively performing

G R A D E S

Jessica Kanold-McIntyre, Suyi Chuang, Matthew R. Larson, and Mignon Smith help teams identify what students

and rigorous lessons, utilizing and sharing self-reflections, and designing robust proportional reasoning units. 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 proportional reasoning, 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.

ISBN 978-1-951075-27-9 90000

SolutionTree.com NCTM Stock ID: 16045

9 781951 075279

Schuhl • Kanold Kanold-McIntyre • Chuang • Larson • Smith

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

G R ADES

6–8

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. | Kanold, Timothy D., author. | Kanold-McIntyre, Jessica, author.

Title: Mathematics unit planning in a PLC at work. Grades 6-8 / Sarah Schuhl, Timothy D. Kanold, Jessica Kanold-McIntyre, Suyi Chuang, Matthew R. Larson, Mignon Smith.

Description: Bloomington, IN : Solution Tree Press, [2020] | Series: Every student can learn mathematics | Includes bibliographical references and index.

Identifiers: LCCN 2020008919 (print) | LCCN 2020008920 (ebook) | ISBN 9781951075279 (paperback) | ISBN 9781951075286 (ebook) Subjects: LCSH: Mathematics--Study and teaching (Elementary)--United States. | Professional learning communities.

Classification: LCC QA13 .S3645 2020 (print) | LCC QA13 (ebook) | DDC 372.7--dc23 LC record available at https://lccn.loc.gov/2020008919 LC ebook record available at https://lccn.loc.gov/2020008920 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 Production Editor: Alissa Voss Content Development Specialist: Amy Rubenstein Copy Editor: Jessi Finn Proofreader: Sarah Ludwig Text and Cover Designer: Kelsey Hergül Editorial Assistants: Sarah Ludwig and Elijah Oates

Table of Contents

About the Authors .

Introduction .

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

ix

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

1

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 6–8 .

Guaranteed and Viable Curriculum . . . . . . . . . . . . . . .

Mathematics Unit Planner . . . . . . . . . . . . . . . . . . .

Mathematics Concepts and Skills for Grades 6–8 . . . . . . . .

Connections Between Mathematics Content and Unit Planning .

Unit Planning as a Collaborative Mathematics Team .

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

5

9

9 . 10 . 10 . 13

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

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

15

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

v

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MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK Ž , GR A D ES 6 – 8

Part 2 Proportional Reasoning Unit Examples, Grades 6-8 .

3

4

5

Grade 6 Unit: Ratios and Rates .

. . . . . . . . . . . . . . . . . . . . . . . . 33

39

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

Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Grade 6 Ratios and Rates Unit Planner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Grade 7 Unit: Proportional Reasoning .

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

59

Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Grade 7 Proportional Reasoning Unit Planner . . . . . . . . . . . . . . . . . . . . . . . . . 76 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Grade 8 Unit: Linear Functions and Equations .

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

81

Essential Learning Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Unit Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Prior Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Vocabulary and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Resources and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Tools and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Reflection and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Grade 8 Linear Functions and Equations Unit Planner . . . . . . . . . . . . . . . . . . . . 102 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Epilogue: Mathematics Team Organization .

Final Thoughts .

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

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

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 . . . .

105 106

109

. . . . . . . . . .

111

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

111 111 112 112 112

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

Table of C ontent s

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 . . . . . . . .

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 .

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

112 112 113 113 113 113 113 114 114

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

115

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

117

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, a nd 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.

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MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

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.

He is a past president of 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! 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.

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.

Jessica Kanold-McIntyre is an educational consultant and author committed to supporting teacher implementation of rigorous mathematics curriculum and assessment practices blended with research-informed instructional practices. She works with teachers and schools around the country to meet the needs of their students.

Specifically, she specializes in building and supporting the collaborative teacher culture through the curriculum, assessment, and instruction cycle.

She has served as a middle school principal, assistant principal, and mathematics teacher and leader. As principal of Aptakisic Junior High School in Buffalo Grove, Illinois, she supported her teachers in implementing initiatives, such as the Illinois Learning Standards; the Next Generation Science Standards; and the College, Career, and Civic Life Framework for Social Studies State Standards, while also supporting a one-to-one iPad environment for all students. She focused on teacher instruction through the PLC process, creating learning opportunities around formative assessment practices, data analysis, and student engagement. She previously served as assistant principal at Aptakisic, where she led and supported special education, RTI, and English learner staff through the PLC process.

As a mathematics teacher and leader, KanoldMcIntyre strived to create equitable and rigorous learning opportunities for all students while also providing them with cutting-edge 21st century experiences that engage and challenge them. As a mathematics leader, she developed and implemented a districtwide process for the Common Core State Standards in Illinois and led a collaborative process to create mathematics curriculum guides for K–8 mathematics, algebra 1, and algebra 2. She has also served as a board member for NCSM. Kanold-McIntyre earned a bachelor’s degree in elementary education from Wheaton College and a master’s degree in educational administration from Northern Illinois University.

To learn more about Jessica Kanold-McIntyre’s work, follow @jkanold on Twitter.

Suyi Chuang is a professional learning supervisor for Loudoun C ou nt y P ubl ic S c ho ol s i n Ashburn, Virginia, as well as the district’s certified project-based learning facilitator. Chuang is passionate about ensuring that each and every student is provided opportunities to excel through rich, authentic learning experiences.

A bout the Author s

Previously, Chuang served Loudoun County Public Schools as supervisor of K–12 mathematics. Her educational experiences include designing and implementing sustained professional learning around project-based learning, performance-based assessments, personalized learning, and equity.

Chuang received a certificate in educational leadership and administration from George Washington University and her bachelor’s and master’s degrees in mathematics education from New York University.

To learn more about Suyi Chuang’s work, follow @sugee27 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.

Mignon Smith is a mathematics teacher and consultant in Bethesda, Maryland. In addition to her experience in teaching mathematics at all levels in public and private school environments, she has served as a mathematics instructional specialist, curriculum writer, department chair, and grade-level chair. Mignon is passionate about supporting high-quality instruction in mathematics, while focusing on building and sustaining positive school cultures and working with underrepresented student populations.

Mignon is a member of NCTM and the Association for Supervision and Curriculum Development (ASCD).

In 2010, she was selected as Teacher of the Year at Early College Academy, a PLC in Spring, Texas, where she became immersed in the work of the PLC process. She was a member of Leading Educators’ 2015 cohort and was an intern for Building Excellent Schools Summer Leaders in 2016.

To book Sarah Schuhl, Timothy D. Kanold, Jessica Kanold-McIntyre, Suyi Chuang, Matthew R. Larson, or Mignon Smith for professional development, contact pd@SolutionTree.com.

xi

Introduction

By Timothy D. Kanold

A

t the heart of our work as teachers of mathematics for grades 6–8 is developing student self-efficacy. Student self-efficacy refers to students’ belief in their capability to learn the mathematics you need them to know by the end of each grade and in preparation for high school.

But what exactly do grades 6–8 mathematics students need to know by the end of each unit of study throughout the school year? And, more important, how does a middle school teacher develop his or her personal self-efficacy to confidently plan for those mathematics units of study and then deliver them to students?

I have been trying to answer this question throughout my 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 in length 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 place and the right time in the mathematics story arc for each grade level. There is an order

to the flow of the sixth-, seventh-, and eighth-grade mathematics content story. And, as sixth-, seventh-, and eighth-grade teachers in your mathematics department, you need to fully understand the how and the why of the content trajectories across these grades.

Grades 6–8 mathematics courses are an enigma in some ways, as the sequencing of the courses offered to students is more complex. Throughout my work across the United States, I have observed middle schools that offer as many as nine different mathematics course choices for grades 6–8. That is a lot of unit planning!

My colleagues and I are often asked what the ideal sequencing of courses is for these three grade levels. We base our response on a simple belief: In this decade, the middle school mathematics curriculum is a floor and not a ceiling. Thus, your grade-level mathematics course standards for grades 6, 7, and 8 provide the base level of essential learning for each and every student. This creates the floor (so to speak) of expected learning and mathematics course work for every middle school student by the end of eighth grade. However, your middle school mathematics program can provide access and opportunities to learn beyond the middle school standards as appropriate. In some instances, this may mean two course sequences of learning for your students. Just be sure student movement between these levels can be fluid.

Lead author Sarah Schuhl and I realized that teachers and teacher teams of grades 6–8 mathematics need to collectively work with their textbooks and other

1

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MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

resources to own the planning process for each unit of study.

Generate Essential Learning Standards for Each Unit

Developing collective teacher efficacy is at the heart of the professional learning community (PLC) process: “Social interactions firmly anchored in instructional practice 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, 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” (p.

13). Teachers then place those episodes into manageable chunks of content for the team’s discussion and work.

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.

In 2019, when Solution Tree asked Sarah and me to develop the Mathematics at Work™ Plan Book (Kanold & Schuhl, 2020), we jumped at the chance to offer 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 sufficient for your team. However, we also realized 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 6–8 (Sarah Schuhl, Jessica Kanold-McIntyre, Matthew R. Larson, Suyi Chuang, Mignon Smith, 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 middle school teachers improve student learning in mathematics, teachers often ask us, “How do we collectively plan for a unit of study in mathematics at our grade level?”

And answering that question and providing that direction 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.

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. Choose 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 the recent prerequisite content knowledge students need to access the grade-level learning in the unit of study. Decide which mathematical activities (tasks or prompts) to use for students to connect their prior knowledge at the start of lessons 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. Also identify any mathematical notations students will need to read, write, and speak during the unit.

Identify Resources and Activities Decide which lessons in the team’s current basal curriculum materials align with 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 technolog y needed to help students master the essential learning standards of the unit. Identify whether the tools or technology needed for the unit will support student learning of the essential learning standards with a

Introduc tion

focus on conceptual understanding, application, or procedural f luency. Decide 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 technology to foster learning? How will students develop and express their understanding of equivalent expressions, ratio, and proportion? 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.

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 learning standards you expect students to learn in grades 6, 7, and 8.

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, Grades 6–8?” 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, Dr. Richard DuFour, was the architect of the Professional Learning Communities at Work movement (along with Robert Eaker) and my principal for many years. Dr. 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)

Part 2 provides three detailed model mathematics units for proportional reasoning (one for each grade level) and describes the proportional reasoning story arc for grades 6, 7, and 8 related to ratios and rates, proportional relationships with tables and graphs, and linear functions and equations. We hope part 2 offers an inspiring model for your grade-level team.

2. How will we know if they know it? (lessondesign elements, assessments, and tasks used)

The epilogue shares an example for how to organize your grade-level 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

As your collaborative team pursues this deep work, remember it all begins with a robust and well-planned response to PLC critical question 1 (What do we want all students to know and be able to do?). That is the focus of our grades 6–8 unit planning book.

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)

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.

3

CHAPTER 3

Grade 6 Unit: Ratios and Rates

In fifth grade, students engage in problems that require them to form ordered pairs of corresponding terms from two patterns and then graph the ordered pairs on a coordinate plane. However, building an understanding of ratios and equivalent ratios does not become a priority until sixth grade, the start of the learning arc for proportional relationships. In seventh grade, students will deepen their understanding of ratios and rates to generate proportional relationships using equivalent ratios and solve proportional reasoning problems using multiple representations. Students will identify the unit rate from a graph, table, equation, diagram, or verbal description of a proportional relationship. In eighth grade, students will extend their knowledge of proportional relationships by comparing proportional relationships represented in different ways and exploring linear functions.

As your sixth-grade team plans units related to ratios, start with clarity about what you expect students to know and be able to do related to ratios and unit rates in sixth grade and what is needed in seventh grade for students to be successful.

In sixth grade, students are expected to master the concept of ratio and unit rate. While there is more emphasis on representing and describing ratios, rates, and proportional relationships, students also deepen their understanding of equivalence and start to grapple with representing equivalent ratios in tables and graphs and applying ratio reasoning to percentages. Students

wrestle with the similarities and differences between their work with fractions and their work with ratios.

As a result, your sixth-grade team has a goal to design a sound unit of study that builds the foundation of proportional relationships in a way that develops student confidence and self-efficacy. A solid foundation contributes to student success in seventh and eighth grades as students develop their understanding of proportional relationships and begin to explore functions (specifically, linear functions).

This chapter shares an example of how your sixthgrade mathematics team could plan for a ratios and rates unit. The standards that follow may or may not match your exact state standards, but most likely contain the key ideas for most of your local district standards. The state standards related to ratio understanding used in this unit are the same as those shared in table P2.1 (page 36) and are labeled standards 6.1–6.4.

6.1. Understand the concept of a ratio and use ratio language to describe the relationship between two quantities.

6.2. Understand the concept of a unit rate associated with a ratio, and use rate language in the context of a ratio relationship.

Š 2021 by Solution Tree Press. All rights reserved.

E

xploring the concept of ratios and rates is a new development at the sixth-grade level; however, the skills students need to find equivalent ratios build on multiplication and division concepts students learned in grades 3–5.

6.3. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.4. Solve unit rate problems including those involving unit pricing and constant speed.

39

40

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

The distinction between the two sets of standards in the unit is the bold (need-to-know) standards incorporate the students’ prior learning experiences related to multiplication and division from the primary grades.

Therefore, it is appropriate to expect a sixth grader would achieve mastery by the end of the year.

However, solving unit rate problems and representing them are skills that students will continue to develop in seventh grade, with an expectation for proficiency at that time. During seventh grade, students will have opportunities to sharpen their skills related to unit rate problems and then apply their knowledge of unit rate to the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions. They will further build on this knowledge when they examine linear function representations in eighth grade.

Once your team has clarity about the unit standards and how they fit into the proportional reasoning story arc for sixth-grade students, it is time to begin creating your unit plan. The Mathematics Unit Planner (see figure 1.2, page 11) will guide your team and provide a location to record team agreements. Your team’s work starts with generating essential learning standards in student-friendly language for assessment and reflection.

Essential Learning Standards The essential learning standards are the driver for common assessments and student reflections related to learning. They are written as I can statements and generated from the state standards students must learn in the unit. They form the daily learning targets for lessons. Now that your team knows the unit standards, how will it create the essential learning standards?

What exactly do students have to know and be able to do to be proficient with the standards in the unit? Your team can use the team protocol to unwrap and make sense of mathematics standards; see figure 2.4 (pages 20–21)

for the template with directions. Completed templates for the need-to-know standards appear in figures 3.1, 3.2 (page 42), and 3.3 (page 43).

During the process of unwrapping standard 6.1 in figure 3.1, it might be helpful to consider how students are expected to show proficiency of standard 6.1 on the common end-of-unit assessment or end-of-year state assessment. Having a clear picture of the end goal in mind can support your sixth-grade team as it generates ideas and navigates conversations that prepare members to complete the components of the unwrapping protocol.

Sometimes the wording of mathematics standards leads teammates to varied interpretations. The process of circling verbs and underlining noun phrases helps bring a clarity to and focus on what students must be able to do. Following are questions to consider during your team discussions about standard 6.1.

• What experiences should a sixth grader have with understanding the concept of a ratio?

• How will we engage students with the language of a ratio?

• How will our team ensure that students have a conceptual understanding of a ratio and can reason about ratios?

Standard 6.2, shown in figure 3.2 (page 42), requires students to understand the concept of a unit rate as it relates to a given ratio. It would be difficult for students to recognize a unit rate without first having a strong understanding of the concept of a ratio. So, during the unwrapping of standard 6.2, teams must keep in mind the connections between these two standards.

Standard 6.2 requires students to apply their understanding of ratios to calculate a rate, specifically a unit rate. Students extend their understanding of ratio language from “for every,” “for each,” and “per” to language about the unit rate, such as “for every 1” and “for each 1.” Students use multiplication and division strategies from fifth grade to calculate the unit rate.

Standard 6.3, shown in figure 3.3 (page 43), focuses on building an understanding of how to generate equivalent ratios using strategies such as repeated addition, skip counting, and scalar multiplication or division.

The representations, models, and tools that students use to construct and represent equivalent ratios support their understanding of proportional reasoning and their ability to compare two quantities.

© 2021 by Solution Tree Press. All rights reserved.

The bold standards reflect the need-to-know essential standards in the unit (see figure 2.3, page 19). These are the standards your team uses when creating common mid-unit assessments and uses first when addressing student gaps or extensions in learning. Your team will locate these more critical unit standards in district documents or state, provincial, or national guidelines.

These standards are not, however, the only standards students will learn in the unit. The standards not in bold are the important-to-know standards your team includes on the common end-of-unit assessment.

Grade 6 Unit: Ratios and Rates

41

Unit: Grade 6—Ratios and Rates State Standard in the Unit: 6.1. Understand the concept of a ratio and use ratio language to describe the relationship between two quantities.

Conceptual Understanding What do students need to know?

• Understand a ratio is a relationship between two quantities.

• Describe the relationship between two quantities using ratios.

• Write a ratio using the three different forms: (1) A:B, (2) A to A B, and (3) B .

Academic Vocabulary and Notations Ratio Quantity

Ratio language (for every, for each, per) A Ratio notation: A:B, A to B, B

Essential Learning Standard (In student-friendly language: I can . . .)

• I can write a ratio in different forms and explain the ratio relationship between two quantities.

Proficiency Level of Understanding 4 Advanced

Create a context for a given ratio written in any form, and use ratio language to describe a ratio relationship between the two quantities.

3 Proficient

Accurately write a ratio and use ratio language to describe a ratio relationship between two quantities.

2 Partial 1 Minimal

Write a ratio using the three different forms.

Write a ratio using the three different forms, but may struggle with order, accuracy, or both.

Exemplar Tasks to Meet Standard: Task 1 In Ms. Smith’s class, there are 15 boys and 12 girls.

• What is the ratio of boys to girls? Write your ratio two different ways.

• What is the ratio of girls to the whole class? Write your ratio two different ways.

Task 2 Abby mixes 3 cups of nuts, 2 cups of raisins, and 1 cup of chocolate chips to make a trail mix.

Write a ratio to compare the amount of raisins in the trail mix to the total amount of trail mix.

Draw a picture to model the ratio and explain your thinking.

Figure 3.1: Unwrap and make sense of standard 6.1.

© 2021 by Solution Tree Press. All rights reserved.

• Concept of a ratio • Ratio language • Relationship between two quantities

Procedural Knowledge and Skills What do students need to do?

42

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Unit: Grade 6—Ratios and Rates State Standard in the Unit: 6.2. Understand the concept of a unit rate associated with a ratio, and use rate language in the context of a ratio relationship.

Conceptual Understanding What do students need to know?

• Understand the concept of a unit rate.

• Use rate language in the context of a ratio relationship.

• Find the unit rate for a given ratio.

Academic Vocabulary and Notations

Unit rate Rate language (for each 1, for every 1, per 1 unit) Essential Learning Standard (In student-friendly language: I can . . .)

• I can find the unit rate for a given ratio and explain the meaning of the unit rate.

Proficiency Level of Understanding 4 Advanced

Create a ratio in the context of a problem, find the unit rate, and explain the meaning of the unit rate in the context of the problem.

3 Proficient

Accurately find the unit rate for a given ratio, and explain the unit rate using rate language in the context of the problem.

2 Partial 1 Minimal

Find the unit rate for a given ratio. Some calculation errors may occur.

Identify the unit rate for a given ratio with support.

Exemplar Tasks to Meet Standard:

Task 1 Use the Nutrition Facts label to answer the following questions.

• How many calories are there per cookie?

• How many grams of fat are there per cookie?

Nutrition Facts Serving Size Servings Per Container

4 cookies (30g) 4

Amount Per Serving

Calories 220

Calories from Fat 110 % Daily Value*

Total Fat 12g

18%

Saturated Fat 6g

30%

Task 2

Trans Fat 0.5g

Samara earned $54.00 babysitting for 6 hours. Lucy earned $96.00 babysitting for 12 hours. Who earned more per hour?

Explain how you know. Show your work.

Cholesterol 10mg

2%

Sodium 70mg

4%

Total Carbohydrates 25g

8%

Dietary Fiber 1g

4%

Sugars 20g Protein 3g

Figure 3.2: Unwrap and make sense of standard 6.2.

© 2021 by Solution Tree Press. All rights reserved.

• Unit rates • Ratios • Unit rate associated with a ratio • Rate language • Context of a ratio relationship

Procedural Knowledge and Skills What do students need to do?

Grade 6 Unit: Ratios and Rates

43

Unit: Grade 6—Ratios and Rates State Standard in the Unit: 6.3. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

Conceptual Understanding What do students need to know?

Procedural Knowledge and Skills What do students need to do?

• Make tables of equivalent ratios involving whole-number measurements.

• Find missing values in tables of equivalent ratios.

• Plot ordered pairs from a ratio table on the coordinate plane.

• Compare ratios using ratio tables.

Academic Vocabulary and Notations Equivalent ratios

Ratio table

Essential Learning Standard (In student-friendly language: I can . . .)

• I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

Proficiency Level of Understanding 4 Advanced

Given a ratio table or a graph showing ordered pairs from a ratio table, create a ratio relationship with a ratio greater than or less than the one given OR create ratio tables for ratios involving nonwhole-number measurements.

3 Proficient

Create ratio tables, plot ordered pairs from the ratio tables on the coordinate plane, and compare ratios using ratio tables.

2 Partial 1 Minimal

Create ratio tables and plot the pairs of values from a ratio table on the coordinate plane.

Create ratio tables and plot the pairs of values from a ratio table on the coordinate plane, with prompting and support from the teacher.

Exemplar Tasks to Meet Standard: Task 1

Task 3

The ratio of the number of boys to the number of girls in the sixth-grade class is 3:4. There are 350 students in the sixthgrade class. Using what you know, answer the following.

• How many girls are in the class?

• How many boys are in the class?

Use a ratio table to show at least three ordered pairs that represent the ratio 5:8, showing 5 feet for every 8 steps.

Make a second ratio table to show at least three ordered pairs that represent the ratio 4:6, showing 4 feet for every 6 steps. Compare the ratios. Which one shows a larger rate of feet per step?

Task 2 Complete the following ratio table.

Cost ($) Candy Bars

6 1

3

12

18 7

Plot the points on a coordinate plane. What is the unit rate?

Figure 3.3: Unwrap and make sense of standard 6.3.

© 2021 by Solution Tree Press. All rights reserved.

• Equivalent ratios • Tables of equivalent ratios with whole-number measurements • Missing values in the tables • Ratios as ordered pairs • Coordinate plane • Ratio comparison

44

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Standards 6.1–6.3 are the need-to-know standards in this ratios and rates unit. Your team will similarly build a shared understanding of the remaining important-to-know standard in the unit—standard 6.4. Your sixth-grade team combines both types of standards to generate the ratios and rates unit essential learning standards and daily learning targets (see figure 3.4, page 45). (See figure 2.5, page 22, for the template with directions.)

The I can statements (in the center column of figure 3.4, page 45) show the essential learning standards for the ratios and rates unit. These statements are the drivers for team common assessments and student reflections. Your team writes these statements in the Essential Learning Standards row of the Mathematics Unit Planner (see figure 3.11, pages 56–57).

Sometimes when planning sixth-grade units, teams group together standards that closely align to form one student-friendly essential learning standard. However, grouping standards is not needed in this unit since each essential learning standard is distinct, and together, the number of essential standards already falls within the targeted range of three to six essential learning standards. The daily learning targets are based on the concepts and skills portion of the unwrapping work and share the story of the learning for the unit.

Both the essential learning standards and the daily learning targets provide resources for your team as you

plan lessons and assessments. The next step for team unit planning is to add the dates for common assessments to the calendar and create the learning story with daily learning targets.

Unit Calendar The unit calendar includes common assessment dates and the student-friendly targets that help shape the story of the learning about to take place. This calendar is not meant to be a lockstep approach to teaching among team members. Rather, the goal is to have an idea of how long to spend on each essential learning standard and the general order of when to address learning targets. Your sixth-grade mathematics team might want to include time at the beginning of the unit to explore multiplication, division, and coordinate graphing problems like those in fifth-grade standards.

Intentionally including time in the unit to connect back to prior learning is just one consideration when developing the unit calendar. Your team might also consider adding f lex days to allow for responding to student learning needs after common mid-unit assessments. Determining when to give the common mid-unit assessments for the need-to-know standards not only helps govern the pace of the lessons but also ensures that evidence of student learning is a priority among all team members.

An example of a team-developed calendar for grade 6 appears in figure 3.5 (page 46). The calendar articulates the time line of learning targets, common assessment dates, and important events (such as holidays) so the team has a solid picture of how it will use the instructional time.

If this is the first time your team is making a calendar, start with identifying the dates of your agreed-on common assessments in the unit. Next, work together as a team to plan the flow of the unit, but understand that through the reflection during each unit, the flow and storyline will become stronger and more precise each year. When creating the calendar, your team also identifies the common homework assignments to use as part of your team formative processes (see Mathematics Homework and Grading in a PLC at Work [Kanold, Barnes et al., 2018]).

© 2021 by Solution Tree Press. All rights reserved.

Standard 6.3 expects students to use a variety of representations. Consider incorporating tools such as ratio tables, tape diagrams, double number lines, and coordinate grids into your sixth-grade team discussion.

By engaging as a team with the same tools students use, you can build a shared understanding of the possible pathways for representing equivalent ratios in a variety of ways. You can record the teaching tools you choose in the Tools and Technology part of the Mathematics Unit Planner (see figure 1.2, page 11). Your team can discuss questions that support student thinking as team members use these tools or write a task to capture evidence of student learning around various models.

When thinking through the levels of proficiency, your team members can discuss how they expect students to demonstrate their thinking, orally and in writing.

Daily Learning Targets

Students will be able to: • Make tables of equivalent ratios involving whole-number measurements.

• Find missing values in tables of equivalent ratios.

• Plot ordered pairs from a ratio table on the coordinate plane.

• Compare ratios using ratio tables.

Students will be able to: • Solve unit rate problems.

• Solve unit rate problems involving unit pricing.

• Solve unit rate problems involving constant speed.

• I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

• I can solve real-world unit rate problems.

6.3. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.4. Solve unit rate problems including those involving unit pricing and constant speed.

© 2021 by Solution Tree Press. All rights reserved.

Figure 3.4: Grade 6 ratios and rates unit standards, essential learning standards, and daily learning targets.

Students will be able to: • Understand the concept of a unit rate.

• Use rate language in the context of a ratio relationship.

• Find the unit rate for a given ratio.

• I can find the unit rate for a given ratio and explain the meaning of the unit rate.

6.2. Understand the concept of a unit rate associated with a ratio, and use rate language in the context of a ratio relationship.

Students will be able to: • Understand a ratio is a relationship between two quantities.

• Describe the relationship between two quantities using ratios.

• Write a ratio using the three different forms: (1) A:B, (2) A to A B, and (3) B .

What students must know and be able to do for each lesson (unwrapped standards) (Storyboard Progression)

• I can write a ratio in different forms and explain the ratio relationship between two quantities.

(Student-Friendly Language)

Essential Learning Standards for Assessment and Reflection

6.1. Understand the concept of a ratio and use ratio language to describe the relationship between two quantities.

(State Standard Language)

Formal Unit Standards

Grade 6 Unit: Ratios and Rates 45

46

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Tuesday

Wednesday

10/28 Explore ratios for understanding: What is a ratio?

10/29 Understand the relationship between quantities; write ratios in different forms.

10/30 Describe relationships between quantities using ratios.

10/31 Common Mid-Unit Assessment: • I can write a ratio in different forms and explain the ratio relationship between two quantities.

11/1 Understand and create equivalent ratios.

11/4 Generate equivalent ratios using ratio tables.

11/5 Generate equivalent ratios using ratio tables by finding missing values in the tables.

11/6 Graph pairs of values from ratio tables.

11/7 Compare ratios using ratio tables.

11/8 Common Mid-Unit Assessment: • I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

11/11 Veterans Day No school

11/12 Flex day based on student needs

11/13 Understand a unit rate using ratio reasoning: What is a unit rate?

11/14 Find unit rates for given ratios or in realworld problems.

11/15 Find unit rates for given ratios or in realworld problems.

11/18 Common Mid-Unit Assessment: • I can find the unit rate for a given ratio and explain the meaning of the unit rate.

• I can solve realworld unit rate problems.

11/19 Solve unit rate problems involving unit pricing and constant speed.

11/20 Review day

11/21 End-of-Unit Assessment: • I can write a ratio in different forms and explain the ratio relationship between two quantities.

• I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

• I can find the unit rate for a given ratio and explain the meaning of the unit rate.

• I can solve realworld unit rate problems.

Figure 3.5: Grade 6 ratios and rates unit calendar.

Thursday

Friday

© 2021 by Solution Tree Press. All rights reserved.

Monday

Grade 6 Unit: Ratios and Rates

If your sixth-grade team decides dates need to change during a unit, it is important to communicate those possible changes with all members. Consult your proficiency map or yearlong pacing guide to ensure you still have time for the remaining units. Record any changes in the Reflection and Notes portion of the Mathematics Unit Planner (see figure 1.2, page 11) to remember them for the next year.

Prior Knowledge Why is the ratios and rates unit at this time of the year? What units have your sixth graders engaged in this year that may contribute to their understanding of this ratios and rates unit (such as multiplication and division)? Which fifth-grade standards connect to the sixth-grade ratio and rate standards? As your team plans for this part of the unit, reflecting on questions such as these can help frame the work ahead.

Once your team identifies prior-knowledge state standards for each essential learning standard in the unit, consider summarizing each one so it is friendly to read in the Mathematics Unit Planner (see figure 1.2, page 11). The far-right column in figure 3.6 (page 48) matches the prior knowledge in the grade 6 Mathematics Unit Planner (see figure 3.11, pages 56–57). (See figure 2.7, page 25, for the template with directions.)

As each teacher on your sixth-grade team designs lessons, your team can connect prior knowledge to the day’s learning target through your use of an intentional task or activity as Kanold, Kanold-McIntyre et al. (2018) discuss in Mathematics Instruction and Tasks in a PLC at Work. (Visit go.SolutionTree.com/ MathematicsatWork for free reproducibles.) Using a prior-knowledge task or activity allows students to make connections from earlier learning to the gradelevel essential standard addressed in each lesson and allows you to formatively assess whether or not additional teaching of prior knowledge is necessary. Your

team can also create a preassessment so you have information about students’ readiness for ratios and proportional relationships.

Your team has now articulated the essential standards and identified the prior knowledge students will use to make connections to new learning. Next, your team determines the mathematics language you expect students to understand and use when communicating their learning and thinking.

Vocabulary and Notations Teachers expect students to interpret and use the language of mathematics as they demonstrate mastery of essential standards. Considering how and when to introduce vocabulary will contribute to a plan that supports student discourse throughout the unit. Are there sentence frames, vocabulary word walls, online resources, or books available to students that support building student understanding of ratios?

As your team discusses the vocabulary and notations element of unit planning, members should determine how students might define and model each term. Are the definitions attached to each term student friendly and still mathematically accurate? Do the drawings and visuals the team chose support student thinking around ratio and rate relationships? How might students describe a unit rate or equivalent ratio? What is the language they will use to describe notations?

Considering questions like these may happen when your team is just starting to plan the unit, and is likely to continue throughout the unit as teachers plan instruction. Adding to and refining this list is an ongoing part of team reflection. In Mathematics Instruction and Tasks in a PLC at Work (Kanold, Kanold-McIntyre et al., 2018), there are several ideas for how to infuse the learning of academic vocabulary and notations into daily lessons as a component of quality lesson design.

Figure 3.7 (page 49) shows a team protocol for determining the meaning of each academic vocabulary word and notation students need to build a conceptual understanding of ratios and rates. (See figure 2.8, page 27, for the template with directions.) List the words and notations in the ratios and rates Mathematics Unit Planner (see figure 3.11, pages 56–57).

© 2021 by Solution Tree Press. All rights reserved.

Once the team has an agreed-on calendar and storyline for the unit, the next step is to discuss the prior knowledge members expect students to be proficient with before the unit of instruction begins. What did students learn earlier this year (or last year) for them to better make sense of ratios and rates in this unit?

47

• I can find the unit rate for a given ratio and explain the meaning of the unit rate.

• I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

• I can solve real-world unit rate problems.

6.2. Understand the concept of a unit rate associated with a ratio, and use rate language in the context of a ratio relationship.

6.3. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.4. Solve unit rate problems including those involving unit pricing and constant speed.

See standard 6.2.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Grade 5: Form ordered pairs consisting of corresponding terms from two patterns, and graph the ordered pairs on a coordinate plane.

Solve real-world problems involving multiplication of fractions and mixed numbers (for example, by using visual fraction models or equations to represent the problem).

Grade 5: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, or the relationship between multiplication and division.

Grade 5: Convert measurements in the same system.

Prior-Knowledge Standards From Prior Grade Level, Course, or Unit

1 2

feet

centimeters

6 24

3 12

36

9

48

12

See standard 6.2.

Complete the two patterns represented in the table.

Explain how you would write ordered pairs from the table.

Plot the ordered pairs on a coordinate plane.

What do you notice about the graph?

Pattern B

Pattern A

What is the relationship between corresponding terms in the following two patterns?

• Use a table to complete a pattern.

• Plot points on a coordinate plane.

A recipe calls for 1 12 cups sugar. If you triple the recipe, how much sugar will you need?

Find 5 ÷ 2. Show how you found your answer.

Divide: 496 ÷ 8.

• Divide whole numbers with up to four-digit dividends and two-digit divisors that have whole number quotients using multiple strategies.

• Solve real-world problems involving multiplication of fractions.

yards = 1

4 meters =

Complete each conversion:

• Convert measurements in the same system.

Prior-Knowledge Summary With Exemplars

© 2021 by Solution Tree Press. All rights reserved.

Figure 3.6: Prior knowledge for grade 6 ratios and rates unit.

• I can write a ratio in different forms and explain the ratio relationship between two quantities.

(Student-Friendly Language)

Essential Learning Standards for Assessment and Reflection

6.1. Understand the concept of a ratio and use ratio language to describe the relationship between two quantities.

(State Standard Language)

Formal Unit Standards

48 MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Grade 6 Unit: Ratios and Rates

49

Unit: Grade 6—Ratios and Rates Vocabulary Ratio

Definition, Explanation, Drawing, or Example A comparison of two or more quantities Some ratios compare quantities with the same units.

• 3 parts of blue paint to 2 parts of red paint • 2 cups of sugar to 4 cups of flour Some ratios compare quantities with different units.

• 5 meters per 12 seconds

Quantity

A measurement of an attribute (for example, lengths, volumes, weights, cups)

Unit rate

A rate showing the relationship between two different quantities, the second of which has a value of 1; words that are commonly used for a unit rate include for each 1, for every 1, and per 1 unit.

Examples: • Timmy sells 3 pounds of fruit for $1.00.

3 pounds $1.00

= 3 pounds per dollar

• Kara makes $24.00 for 3 hours of babysitting. $24.00 3 hours

= $8.00 per hour

• Laura ran 5 miles in 2 hours.

5 miles 2 hours

= 2.5 miles in 1 hour

Ratio and rate language

Language that includes words like for every, for each, per

Equivalent ratios

Two or more ratios expressing the same relationship between two quantities Examples: 4 meters

8 meters because both ratios show the same speed, 2 meters per second.

4 seconds $22.50 to 6 chocolates because both ratios show the same cost, $3.75 per chocolate.

• 2 seconds is equivalent to $7.50

• 2 chocolates is equivalent Ratio table

Notation 4:3

4 3

4 to 3

A structured list of equivalent ratios

Cups of Green Paint

2

4

6

8

Total Cups of Paint

5

10

15

20

Words to Show How to Read the Notation

Cups of Green Paint

Total Cups of Paint

2

5

4

10

6

15

8

20

Example Showing How to Use the Notation in a Mathematics Expression or Equation

4 to 3; 4 for every 3; 4 parts to 3 parts; 4 meters per 3 seconds

The ratio of girls to boys on the soccer team is 4:3.

4 to 3; 4 for every 3; 4 parts to 3 parts; 4 meters per 3 seconds

The ratio of girls to boys on the soccer team is

4 to 3

4 3

.

The ratio of girls to boys on the soccer team is 4 to 3.

Figure 3.7: Vocabulary and notations protocol for grade 6 ratios and rates unit.

© 2021 by Solution Tree Press. All rights reserved.

• $500 per month

50

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Your sixth-grade team may also want to participate in a vertical dialogue with seventh- and eighth-grade teams about vocabulary that connects curriculum year to year. Together, your teams can determine common visuals, definitions, and displays that support proportional reasoning.

Resources and Activities

As your sixth-grade team selects a balance of higherand lower-level-cognitive-demand tasks that match the rigor proficient students demonstrate, you are contributing to a shared bank of resources. It is here—in the selection of resources—that the most significant teacher learning occurs, as the team archives what to use for instruction from year to year. You should digitally and intentionally keep, adjust, or discard your resources based on the evidence of student learning you and your team collect.

Understanding ratios and developing the meaning of equivalent ratios are foundational to a student’s ability to work with ratios and unit rates in future grades.

What are the best types of cognitive-demand tasks, both higher- and lower-level, to develop conceptual understanding and procedural fluency as well as applications? As a sixth-grade collaborative team, or as a middle school mathematics department, you can work together to identify any common resources or activities that will effectively help every student learn the gradelevel essential learning standards.

One of the remaining elements of the Mathematics Unit Planner (see figure 1.2, page 11) is tools and technology. Your team, however, has likely already been discussing tools and technology through your conversations related to unit planning thus far. The Tools and Technology part of the Mathematics Unit Planner provides a location for your team to document your decisions, whether at this point in planning or throughout the unit planning process.

Once your team has clarity on the essential learning standards and examples of mathematical tasks to use for instruction and assessment, it is time to consider your resources. Your team might ask such questions as:

When students are first learning how to name, model, and talk about ratios, what tools and technology most effectively move their learning forward?

Tools are instruments students use to employ a variety of strategies. Some examples include coordinate planes or double number lines, which can help students reason visually about equivalent ratios, or unit cubes, which can help students develop their understanding of the concept of a ratio and the part-to-part and part-towhole relationships.

• Which parts of our school-adopted textbooks correlate with the standards in the unit?

• Which are the best lessons or tasks to use in our books?

• Are there lessons or tasks to omit in a chapter because they don’t match the standards in our unit?

• Is there a need for additional explorations, activities, or lessons because our books do not fully teach each essential learning standard? If so, what other resources should we use or develop?

Figure 3.8 (page 52) provides possible answers to the preceding questions and examples of higher- and lower-level-cognitive-demand tasks for the essential learning standards in the ratios and rates unit. (See figure 2.9, page 28, for the template with directions.)

Tools and Technology

Third- through fifth-grade students develop an understanding of fraction equivalence. For example, 2 4 means the whole remained the same size and 3 6 more equal parts were created. One way to “see” this meaning would be to understand that in the first fraction, the whole is in three equal parts with two parts shaded, and in the second fraction (equivalent to the first), the same-sized whole is in six equal parts with four parts shaded.

However, in sixth grade, when developing an understanding of equivalent ratios, if 2:3 (or 23 ) represents the ratio of red cubes to yellow cubes, then the equivalent

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The next component of the unit planner outlines the instructional resources and activities your team might use in the classroom. Determining key resources and activities as a team is central for delivering a guaranteed and viable curriculum to all learners.

Some parts of the chart are blank because, for example, your sixth-grade team may decide no supplemental or online resources are necessary or available.

Grade 6 Unit: Ratios and Rates

Figure 3.9 (page 53) shows what teams could consider when selecting tools and technology so all members understand each one’s purpose, instructional implications, and assessment use. (See figure 2.10, page 30, for the template with directions.)

Share as a team the management of these materials within the classroom as well. How will students access these tools? Is there a routine in place for ensuring students use these tools appropriately? Are there enough materials for the grade level? How will your team develop student self-efficacy in the choice and use of these tools?

Middle school students need many experiences with concrete materials to make sense of ratios. Sixth graders benefit from exposure to tools such as tape diagrams and double number lines as they use ratio tables or plot points on a graph to solve and reason about ratio and rate problems.

Your team has now thoughtfully considered each element of unit planning, from unwrapping essential standards to selecting resources and tools to use with your students. This systematic process of unit planning would not be complete without embedding reflection into your team practices.

Reflection and Notes Team reflection occurs when planning the unit and during the unit, as you remind each other of strategies to include and consider during the day-to-day details of lesson planning. These are the notes your team takes as it plans together.

Equally important is the intentional ref lection after the unit is complete. At the end of every unit, your sixth-grade team discusses student progress and

considers adjusting the unit in the future based on the evidence of student learning resulting from your team practices throughout the unit.

Figure 3.10 (page 54) illustrates an example of team reflection and notes during planning and after the unit of instruction. (See figure 2.11, page 31, for the template with directions.) Key considerations include listing the essential learning standards, making planning notes, and determining next steps.

The protocol in figure 3.10 (page 54) serves as an archive of your team successes, a living plan of next steps, and an honest dialogue about what is currently working or not working for students. As you build trust among your team, consider the following guiding questions for supporting transparent and honest dialogue.

• Where in the data do you see a trend to celebrate?

• After looking at your classroom data, what questions would you like to ask your teammates?

• What would you like to know more about to support one another’s mathematical teaching practices?

• Where in the data do you see evidence of a need to adjust your team’s approach to teaching the unit standards?

Questions such as these encourage your team to openly reflect on the team’s data and prepare for moving forward instructionally. Through reflection and routine sharing, your team will discover small, yet impactful, practices that help reach more students.

Your sixth-grade team has now navigated through each element of a mathematics unit design. You started by identifying the unit’s essential learning standards and pacing the lessons and assessments using a unit calendar. Together, your team agreed on and clarified prior knowledge and vocabulary and notations. Your team also selected resources, tasks, tools, and technology to support lesson planning. You have included a process for recording notes when planning and for reflecting on the outcomes of the unit. Your team has now successfully generated the information needed to complete the Mathematics Unit Planner for your ratios and rates unit.

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ratio 4:6 (or 46 ) would mean 4 red cubes to 6 yellow cubes, meaning the size of the whole increased from a total of 5 cubes to 10, but the parts stayed the same size. While equivalent fractions and equivalent ratios may appear similar, the meaning differs in terms of parts and wholes. Tools can help with this conceptual understanding of fraction and ratio comparison. When determining tools and technology to use in the ratios and rates unit, your team can clarify how certain tools develop the concepts of ratios and equivalent ratios so students can make connections to the procedures they learn throughout the unit.

51

6

2

6

2

12

8

12

4

8

18

14

18

6

10

6

24

20

24

8

12

8

A golf shop sells 36 golf balls for $12.00. How many golf balls can you buy for $1.00?

Flour

Cups of

Butter

Cups of

Flour

Cups of

Butter

6

4

Martin finished a 12-mile race in 2 hours.

At this rate, how long will it take Martin to finish an 18-mile race?

A 12-pack of pop costs $4.89. A 24-pack of pop costs $8.99. Which case is the better buy? Explain your thinking.

a. Create a ratio table to help you solve the problem. Show your work. b. Graph the values from your table. c. Explain how the table relates to the graph.

A 5-ounce serving of a protein shake has 15 grams of protein. How many grams of protein are in an 8-ounce serving?

Carter wants to buy 3 pairs of jeans for $48.00. His mom says she will pay for 2 pairs of jeans and he will have to pay for one pair. Each pair of jeans costs the same amount of money. How much will Carter have to pay?

What are ratios we could create to describe the school or our class? Create three ratio examples.

Higher-Level Task Example

Lesson 3.4

Lesson 3.3

Lesson 3.2

Lesson 3.1

School-Adopted Textbook Lessons

An activity that has students find real-world unit rates (For example, at a grocery store, students need unit rates to find the best buy when shopping.)

Team-developed exploration activity connecting a ratio table with a graphical representation of plotted ordered pairs

Class brainstorm of ratio examples (Collect examples on poster paper to keep around the room—for example, ratios for paint mixing or baking.)

Unit cube ratio exploration activity

Explorations, Projects, or Activities

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Figure 3.8: Resources and activities protocol for grade 6 ratios and rates unit.

I can solve realworld unit rate problems.

C.

B.

Cups of

Flour

Cups of

Butter

2

A recipe calls for 2 cups of butter for every 6 cups of flour. Which table shows equivalent ratios for cups of butter to cups of flour in the recipe?

I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

A.

Find the unit rate for walking 7 miles in 2 hours.

I can find the unit rate for a given ratio and explain the meaning of the unit rate.

Cups of

There are 3 cups of flour for every 2 cups of sugar. Write the ratio for cups of sugar to cups of flour.

Lower-Level Task Example

I can write a ratio in different forms and explain the ratio relationship between two quantities.

Essential Learning Standard

Extra practice from online practice book: Lesson 3.3

Extra practice from online practice book: Lesson 3.2

Supplemental or Online Resources

52 MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Grade 6 Unit: Ratios and Rates

Tool

Purpose

Instruction

Assessment

Make sense of part-to-whole and part-to-part ratio relationships

Model ratios of part-to-whole and part-to-part.

Not used on common assessments.

Double number line

Conceptual understanding

Model ratios.

May ask students to model a double number line for situations of ratios with different units or to make sense of a given double number line.

Dollars 2

8

?

Exploration

Determine equivalent ratios.

Justification

Explore using benchmark ratios to compare and find missing values.

Tape diagram (bar model) The ratio of blue paint to red paint is 2 to 3.

Conceptual understanding

Model ratios.

Exploration

Create equivalent ratios.

Blue Paint

Justification

Compare ratios.

Conceptual understanding

Model ratios as a preview to linear relationships through a visual model only.

0

3

12

24

Number of Lemons

May ask students to model a question using a tape diagram or make sense of student work using tape diagrams.

Red Paint

Positive coordinate plane

100,000

Students plot points from a ratio table onto a coordinate plane.

10,000 1,000 100 10 1

2

3

4

5

Technology

Purpose

Instruction

Team-developed, technologyenhanced items using a SMART Board

Mirror the technology tools and features of the end-of-year assessment

Engage in guided and independent practice that applies skills to problems with test-like language.

Figure 3.9: Tools and technology for grade 6 ratios and rates unit.

Assessment Common formative exit tickets

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Unit cubes

0

53

2 4 6 8

3 6 9 12

Students loved finding unit rates for real-world situations.

Overall Unit Reflections: Things to Remember or Change for Next Year

Connect real-world unit rate problems to the real-world brainstorming activity for standard 6.1 (see figure 3.8, page 52). Specifically, connect realworld examples to unit pricing and constant speed.

Students can compare ratios using a tape diagram (bar model). This is in addition to using a double number line and ratio table.

Help students understand how to determine the rate of change (later called slope) of the line that is graphed by the equivalent ratios and identify the unit rate from the graph.

Connect the equivalent ratios from the table to the visual of plotted points on a coordinate grid.

y (Height in Meters)

Some students struggled with the number flexibility needed to multiply or divide the two numbers in the ratio to complete the ratio table.

For example: x (Time in Seconds)

Students struggled to see how the ratio table and its graph were related. We need to be more explicit in our instruction at the beginning of these lessons to make the connection.

Students who struggle with division struggled with unit rate problems that have decimal or fraction answers.

Students did a great job rewriting ratios in the different forms.

We definitely had to work through misconceptions about ratios (specifically problems where we gave them a ratio of two quantities and a total amount and we asked them to find the amount of each quantity). But as a team, we monitored this pretty well throughout the unit so students understood by the end-of-unit assessment.

Team Reflections After the Unit

Have students spend time working on creating equivalent ratios using a ratio table to organize their work and look for patterns. Then, learn how to compare ratios and find equivalent ratios using the table.

Include a division number talk at the beginning of this essential learning standard’s lesson.

Focus on the language: for each 1, for every 1, per 1 unit.

Start the lesson by having students identify where unit rates help them or how they have used unit rates in real life.

Include a few multiplication number talks at the beginning of the unit.

Explain the concept of a ratio in the context of problems.

Ratios can have quantities with the same unit or different units.

Notes When Planning: What to Emphasize or Remember for Lessons and Assessments in the Unit

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Figure 3.10: Reflection and notes for grade 6 ratios and rates unit.

• We may need more time for this unit next year.

• Overall, students showed an understanding of both ratios and unit rates, but it would be good to spend a little more time on the models to help build their fluency with the concepts.

I can solve real-world unit rate problems.

I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

I can find the unit rate for a given ratio and explain the meaning of the unit rate.

I can write a ratio in different forms and explain the ratio relationship between two quantities.

Essential Learning Standard

54 MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Grade 6 Unit: Ratios and Rates

challenge as a team is to continue planning for each unit to strengthen both student and teacher learning.

When your team plans a unit (like the ratios and rates unit) before the unit begins, you strengthen your daily lesson design and team common assessments.

Figure 3.11 (pages 56 –57) shows the completed Mathematics Unit Planner for the sixth-grade ratios and rates unit, capturing your team discussions and shared decisions. (See figure 1.2, page 11, for the template with directions.)

Conclusion

The mathematics unit plan in figure 3.11 (pages 56–57), with its accompanying calendar showing daily learning targets and common assessments (see figure 3.5, page 46), scores high on the Mathematics Unit Planning Rubric shared in chapter 2 (see figure 2.2, page 17). How will your team continue planning units for the year and document its discussions so members can learn from one unit to the next and one year to the next? How does each of your team mathematics unit plans score on the rubric in figure 2.2 (page 17)? Your

Although this chapter discusses only one exemplar of a sixth-grade mathematics unit, each unit your team builds creates a foundational building block for members to effectively ensure student learning across your grade-level team. Together, your team’s clarity will maximize individual teachers’ instructional minutes and strengthen practices. Every member of your team and, more important, every student will benefit as members strive to achieve the self-efficacy promise that each and every student can learn the mathematics standards for sixth grade.

The next chapter shows the grade 7 proportional reasoning unit plan examples. Whether or not the standards directly match your state standards, consider the key ideas teachers expect students to learn next year.

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Grade 6 Ratios and Rates Unit Planner

55

56

MAT HEMAT IC S UNIT PL ANNING IN A PLC AT WORK ® , GR A D ES 6 – 8

Unit: Ratios and Rates Start Date: October 28 End Date: November 21 Total Number of Days: 18 • I can write a ratio in different forms and explain the ratio relationship between two quantities.

• I can find the unit rate for a given ratio and explain the meaning of the unit rate.

• I can create a ratio table, plot ordered pairs from a ratio table on a coordinate plane, and compare ratios using ratio tables.

• I can solve real-world unit rate problems.

Prior Knowledge

Grade 5: • Convert measurements in the same system.

• Divide whole numbers with up to four-digit dividends and two-digit divisors that have whole number quotients using multiple strategies.

• Solve real-world problems involving multiplication of fractions.

• Use a table to complete a pattern.

• Plot points on a coordinate plane.

Vocabulary and Notations

Ratio Quantity Ratio language (for every, for each, per) Unit rate Rate language (for each 1, for every 1, per 1 unit) Equivalent ratios Ratio table A Ratio notations: A:B, A to B, B

Resources and Activities

• Textbook: Lessons 3.1–3.4 • Unit cube ratio exploration activity • Class brainstorm of ratio examples (Collect examples on poster paper to keep around the room.)

• Team-developed exploration activity connecting a ratio table with a graphical representation of plotted ordered pairs • An activity that has students find real-world unit rates (For example, at a grocery store, students need unit rates to find the best buy when shopping.)

• Extra practice from online practice book: Lessons 3.2 and 3.3

Tools and Technology

• Unit cubes • Double number line • Tape diagram (bar model) • Positive coordinate plane • Team-developed, technology-enhanced items using a SMART Board

© 2021 by Solution Tree Press. All rights reserved.

Essential Learning Standards

Grade 6 Unit: Ratios and Rates

Reflection and Notes

For example: x (Time in Seconds)

y (Height in Meters)

3

2

6

4

9

6

12

8

• Connect the equivalent ratios from the table to the visual of the plotted points on a coordinate grid.

• Help students understand how to determine the rate of change (later called slope) of the line that is graphed by the equivalent ratios and identify the unit rate from the graph.

• Help students compare ratios using a tape diagram (bar model). This is in addition to using a double number line and ratio table.

• Connect real-world unit rate problems to the real-world brainstorming activity for standard 6.1 (see figure 3.8, page 52). Specifically, connect real-world examples to unit pricing and constant speed.

After the Unit: • Students did a great job rewriting ratios in the different forms. We definitely had to work through misconceptions about ratios (specifically problems where we gave them a ratio of two quantities and a total amount and we asked them to find the amount of each quantity). But as a team, we monitored this pretty well throughout the unit so students understood by the end-of-unit assessment.

• Students who struggle with division struggled with unit rate problems that have decimal or fraction answers.

• Students struggled to see how the ratio table and its graph were related. We need to be more explicit in our instruction at the beginning of these lessons to make the connection.

• Some students struggled with the number flexibility needed to multiply or divide the two numbers in the ratio to complete the ratio table.

• Students loved finding unit rates for real-world situations.

Changes for Next Year: • We may need more time for this unit next year.

• Overall, students showed an understanding of both ratios and unit rates, but it would be good to spend a little more time on the models to help build their fluency with the concepts.

Figure 3.11: Grade 6 ratios and rates unit planner.

© 2021 by Solution Tree Press. All rights reserved.

When Planning: • Remember ratios can have quantities with the same unit or different units.

• Explain the concept of a ratio in the context of problems.

• Include a few multiplication number talks at the beginning of the unit.

• Start the unit rate lesson by having students identify where unit rates help them or how they have used unit rates in real life.

• Focus on the language: for each 1, for every 1, per 1 unit.

• Include a division number talk at the beginning of the unit rate essential learning standard’s lesson.

• Have students spend time working on creating equivalent ratios using a ratio table to organize their work and look for patterns. Then, learn how to compare ratios and find equivalent ratios using the table.

57

“Mathematics Unit Planning in a PLC at Work, Grades 6–8 answers the question, ‘What must we do during our team planning to make a difference for our students?’ From planning essential learning standards to recording reflections and notes, teams to develop necessary structures for the critical work of planning and implementing common units.”

—John W. Staley, NCSM: Leadership in Mathematics Education President 2015–2017, United States National Commission

G R A D E S

for Mathematics Instruction Chair 2018–2020

6–8

through a thorough unit-planning process that ensures deep “This book, with its emphasis on mathematics and useful examples of tools and templates for PLC teacher teams, is an excellent contribution to the field.”

process is critical for teacher teams who collectively believe that every student can learn challenging mathematics content and collectively commit to addressing the specific learning needs of individual students.”

Education, Western Kentucky University

—Becky Walker, Assistant Superintendent of Academics and Innovation, Howard-Suamico School District, Green Bay, Wisconsin

in a PLC at Work ®

—Janet Tassell, Associate Professor, School of Teacher

grade-level mathematics learning for every student. This

Unit Planning

“This book intentionally guides collaborative teams of teachers

M AT H E M AT I C S

this comprehensive text provides guidance for collaborative

Mathematics Unit Planning in a PLC at Work®, Grades 6–8 provides sixth- to eighth-grade mathematics teachers with a seven-step framework for collectively planning units of study. Authors Sarah Schuhl, Timothy D. Kanold, need to know by the 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 collaborative tasks, such as unwrapping standards, generating unit calendars, determining academic vocabulary

6–8

students more equitable learning experiences. The authors share tools and protocols for effectively performing

G R A D E S

Jessica Kanold-McIntyre, Suyi Chuang, Matthew R. Larson, and Mignon Smith help teams identify what students

and rigorous lessons, utilizing and sharing self-reflections, and designing robust proportional reasoning units. 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 proportional reasoning, 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.

ISBN 978-1-951075-27-9 90000

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Schuhl • Kanold Kanold-McIntyre • Chuang • Larson • Smith

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

G R ADES

6–8