NPW Grade 7- Teacher Edition Sample_clone

Shapes and Designs Generalizing and Using Properties of Geometric Shapes Teacher Edition 4 CONNECTED MATHEMATICS® SAMPLE

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CONNECTED MATHEMATICS 4 ® Teacher Edition Shapes and Designs Generalizing and Using Properties of Geometric Shapes SAMPLE

© Copyright 2025 by Michigan State University. Published by Lab-Aids, Inc., Ronkonkoma, New York, 11779. All rights reserved. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s) or other customer service topics, please contact Lab-Aids, Inc. Connected Mathematics® is used under license from Michigan State University. Acknowledgments: Connected Mathematics® was developed at Michigan State University with financial support from the Michigan State University Office of the Provost and the College of Natural Science. This material is based upon work supported by the National Science Foundation under Grant No. 9150217 and Grant No. ESI 9986372. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Shapes and Designs | Teacher Edition | Connected Mathematics 4 ISBN-13: 979-8-89101-110-6 eISBN: 979-8-89101-162-5 v1 Part ID: CMP7-1-4TE Print number: 01 Print year: 2024 Printed in the United States of America. Photo credits: Page 46, © Bruce Leighty/Alamy Stock Photo; page 83, © American Photo Archive/Alamy Stock Photo; page 151, © Andrey Kuzmin/Alamy Stock Photo Cover image: © iStock.com/balwan (image 607976346) Developed at Published by East Lansing, MI 48824 connectedmath.msu.edu 17 Colt Court Ronkonkoma, NY 11779 lab-aids.com SAMPLE

ABOUT THE AUTHORS Elizabeth Difanis Phillips, a former high school teacher, is a Senior Academic Specialist in the Program in Mathematics Education (PRIME) at Michigan State University. She is interested in the teaching and learning of mathematics, with a special interest in the teaching and learning of algebra across the grades. She is the author of numerous articles and book chapters and a speaker at national and international conferences. Recently, in recognition of her scholarly work, she received an Honorary Doctor of Science degree in 2022 from Michigan State University, the 2023 National Council of Teachers of Mathematics (NCTM) Lifetime Achievement Award, and the 2023 Ross Taylor/Glenn Gilbert National Leadership Award from the National Council of Supervisors of Mathematics (NCSM). Currently, she is the principal investigator for several National Science Foundation grants that are developing a collaborative digital platform for students and teachers of Connected Mathematics® 4. Glenda Lappan is a University Distinguished Professor Emeritus in the Program in Mathematics Education (PRIME) at Michigan State University. Her research and development interests are in the connected areas of students’ learning, mathematics teachers’ professional growth, and change related to the development and enactment of K–12 curriculum materials. She served as president of the National Council of Teachers of Mathematics (NCTM) from 1998 to 2000 and played a major role in NCTM’s 1989 Standards for Curriculum and Evaluation Standards for School Mathematics, and 2000 Principals and Standards for School Mathematics. In addition to authoring numerous articles and book chapters and speaking at national and international conferences, she has won numerous awards, including the 2004 NCTM Lifetime Achievement Award and the 2007 Ross Taylor/Glenn Gilbert National Leadership Award from NCSM. James T. Fey is a Professor Emeritus at the University of Maryland. His consistent professional interests have been development and research focused on curriculum materials that engage middle and high school students in problem-based collaborative investigations of mathematical ideas and their applications. He won the 2005 NCTM Lifetime Achievement Award. Susan N. Friel is a Professor Emeritus of Mathematics Education in the School of Education at the University of North Carolina at Chapel Hill. Her research interests focus on statistics education for middle-grade students and, more broadly, on teachers’ professional development and growth in teaching mathematics K–8. Yvonne Slanger-Grant is an Academic Specialist in the Program in Mathematics Education (PRIME) at Michigan State University. Her professional interests focus on helping teachers develop understanding and agency in the teaching of mathematics. v SAMPLE

She has held various roles in education, including middle school mathematics teacher, elementary teacher, instructional coach, professional development consultant, and mentor to teachers, school leaders, and graduate assistants. Many of these responsibilities have involved the Connected Mathematics Project since its beginning in 1991. Alden J. Edson is a Research Assistant Professor in the Program in Mathematics Education (PRIME) at Michigan State University. His research and development interests center on improving the teaching and learning of mathematics through innovations in curriculum and technology. Specifically, his research studies the enactment of problem-based, inquiry-oriented mathematics curriculum by students and their teacher in a digital world. He also studies the affordances of innovative mathematics curriculum materials as a context for teacher learning. Since joining the Connected Mathematics Project in 2014, AJ has been writing and carrying out research and development grants. He also teaches mathematics education courses, advises doctoral students, and facilitates professional learning with teachers of mathematics. With . . . Kathy Dole and Jacqueline Stewart. Kathy is a recently retired middle school teacher of mathematics at Portland Middle School in Portland, Michigan. Jacqueline is a retired high school teacher of mathematics at Okemos High School, Okemos, Michigan. Both Kathy and Jacqueline have worked on a variety of activities related to the development, professional learning, and implementation of the Connected Mathematics® curriculum since its beginning in 1991. In memory of . . . William Fitzgerald (Deceased) Bill through his making “good trouble” made substantial contributions to conceptualizing and creating Connected Mathematics®. vi About the Authors SAMPLE

ACKNOWLEDGMENTS CONNECTED MATHEMATICS ® 4 DEVELOPMENT TEAM Elizabeth Difanis Phillips, Senior Academic Specialist, Michigan State University Glenda Lappan, University Distinguished Professor Emeritus, Michigan State University James T. Fey, Professor Emeritus, University of Maryland Susan N. Friel, Professor Emeritus, University of North Carolina at Chapel Hill Yvonne Slanger-Grant, Academic Outreach Specialist, Michigan State University Alden J. Edson, Research Assistant Professor, Michigan State University With . . . Kathy Dole, Middle School Mathematics Teacher (Retired), Portland Middle School, Portland, MI Jacqueline Stewart, High School Mathematics Teacher (Retired), Okemos Public Schools, Okemos, MI In Memory of . . . William M. Fitzgerald, Professor (Deceased), Michigan State University CONNECTED MATHEMATICS PROJECT STAFF Elizabeth Difanis Phillips, Senior Academic Specialist, Michigan State University, East Lansing, MI Alden J. Edson, Research Assistant Professor, Michigan State University, East Lansing, MI Taren Going, Postdoctoral Research Associate, Michigan State University, East Lansing, MI Elizabeth Lozen, Consortium Coordinator, Michigan State University, East Lansing, MI Sunyoung Park, Postdoctoral Research Associate, Michigan State University, East Lansing, MI Yvonne Slanger-Grant, Academic Outreach Specialist, Michigan State University, East Lansing, MI Chris Waston, Academic Outreach Specialist, Michigan State University, East Lansing, MI vii SAMPLE

ASSESSMENT TEAM Mary Bouck, Mathematics Education Consultant, Michigan State University, East Lansing, MI Valerie Mills, Mathematics Education Consultant, Ypsilanti, MI WEBSITE AND TECHNOLOGY CONSULTANTS Tyler Knowles, Technology Lead, San Antonio, TX Amie Lucas, Information Technologist, Lansing, MI Emma Craig, Graphic Art/Editing, Detroit, MI CURRICULUM DEVELOPMENT CONSULTANTS Melanie Del Grosso, Mathematics Teacher Consultant, Phoenix, AZ Teri Keusch, Mathematics Teacher Consultant, Lansing, MI Jennifer Kruger, Teacher Guide Consultant, Rochester, NY PROGRAM IN MATHEMATICS EDUCATION GRADUATE STUDENTS (2019–PRESENT) Kate Appenzelle Knowles David Bowers Eli Claffey Ashley Fabry Chuck Fessler Nic Gilbertson Funda Gonulates Ahmad Kohar Merve Kursav Kevin Lawrence Rileigh Luczak Jen Nimtz Michael Quail Molade Osibodu Amy Ray Sasha Rudow Visala Rani Satyam Amit Sharma Brady Tyburski Samantha Wald viii Acknowledgments SAMPLE

CONNECTED MATHEMATICS ® 4 UNDERGRADUATE ASSISTANTS Tyler Boyd Jacob Disbro Shayna Evans Autumn Eyre Cora Haddad Emma Herrera Maya Herrera Sarah Ingemunson Shannon McHugh Maggie Ozias Matthew Phillips Josh Pullen PUBLISHING TEAM Director of Mathematics Publications for Lab-Aids: Denise A. Botelho Project Coordination, Production, Cover and Interior Designs, and Composition: Six Red Marbles Illustrations, Graphics, and Art: Six Red Marbles CONNECTED MATHEMATICS 4 REVIEWERS Illinois Jenesis Byrne, Jennifer Leimberer, and Farah Mahimwalla, UIC/MCMI, Chicago Carolyn Droll, Community Consolidated School District 21, Wheeling Robert Reynolds, Peterson Elementary School, Chicago Public Schools, Chicago Maine Sally Bennett, Chris Driscoll, Joyce Hebert, Sara Jones, and Shawn Towle Falmouth Middle School Michigan Michelle Bortnick, Hillel Day School, Dearborn Gerri Devine, Oakland County Schools, Oakland County Anne Marie Nicoll-Turner, Ann Arbor Public Schools, Ann Arbor Meredith Pelchat, Clague Middle School, Ann Arbor Brian Powell, Ionia Public Schools, Ionia Mary Beth Schmitt, Traverse City West Junior High, Traverse City Dr. Jamie Wernet, Lansing Christian School, Lansing Acknowledgments ix SAMPLE

New York Michaela Marino, East Lower School, Rochester Chi-Man Ng, I.S. 289: Hudson River Middle School, New York City Jennifer Perillo, Brighton Central School District, Rochester Ohio Jim Mamer, Springfield FIELD TESTERS Alabama Rashad Bell, McIntosh High School, McIntosh Arizona Melanie Del Grosso, St. John Bosco School, Phoenix California Estasia Barrientosi and Esther Centers, Santa Cruz Waldorf School, Santa Cruz Anthony Bayro and Traci Jackson, Poway Unified Schools, San Diego Illinois Bryan Becker, Melissa Denton, Erika Inka, Meagan Stass MacDonald, and Josephine Mazzola, Barrington Community Unit 220 School District, Barrington Nancy Kay Berkas, Patrick Black, Laura Bubel, Alexander Laube, Kristy Lutton, Breanna McCann, and Amy Rendino, The Cove School, Northbrook Michael Bryant, Carolyn Droll, Maureen Gannon, Kristen Hale, Debbie Rein, Summer Riordan, and Christopher Schieffer, Community Consolidated School District 21, Wheeling Jenesis Byrne, Jennifer Mundt Leimberer, Farah Mahimwalla, Kathleen Pitvorec, and Margie Pligge, University of Illinois at Chicago, Chicago Mary McKenna Corrigan, Brian Lacey, Fannie Lawson-Rondo, Logan Hammerberg, Héctor Orlando Martinez M., Michael R. Martini, Carla Sever, Gail Smith, Melissa Talaber, and Candice M. Usauskas, Archdiocese of Chicago, Chicago Christine Czarnecki, Samia Haan, and Kristy Regan, Alsip District 126, Alsip Catherine Ditto, Jorge Prieto Math and Science Academy, Chicago Public Schools Aaron Mesh, Chicago Waldorf School, City of Chicago School District 299 Morgan Miller and Margaret Nugent, Oak Lawn-Hometown Middle School, Oak Lawn x Acknowledgments SAMPLE

Robert Reynolds, Mary Gage Peterson Elementary School, Chicago Public Schools Courtney Southward, Percy Julian Middle School, Oak Park Indiana Andrea Leahy, St. Thomas More School, Munster Maine Sally Bennett, Christopher Driscoll, Joyce Hebert, Sara Jones, Craig Shain, and Shawn Towle, Falmouth Middle School, Falmouth Michael H. Hagerty, Jay Harrington, Sr., and Kellie McMahon, Frank H. Harrison Middle School, Yarmouth Massachusetts Andrea Hurley, Hanover Middle School, Hanover Michigan Anna Assaf, Sue Chipman, Jennifer Johnson, and Brian Powell, Saranac Community Schools, Saranac Michelle Bortnick and Abbe Luther, Hillel Day School, Farmington Hills Tom Brighton, Jordan Brown, Amber Proctor, Stacey Schrauben, Kim Schumacher, and Lisa Wandell, Ionia Middle School, Ionia Jenny Douglas and Lisa Roderique, Forest Hills Public Schools, Grand Rapids Sheri Gunns, Marian Murembya, and Heidi Nussdorfer, Okemos Public Schools, Okemos Jesica Eby, Katherine Oberdorf, and Jill Wilson, Monroe Public Schools, Monroe Courtney Henige and Melissa Jacobs, New Lothrop Area Public Schools, New Lothrop Amy Hurley and Jennifer LaCross, Walled Lake Consolidated School District, Walled Lake Mary Lovejoy and Sara Melnik, Holt Junior High, Holt Anne Marie Nicoll-Turner and Meredith Pelchat, Ann Arbor Public Schools, Ann Arbor Mary Beth Schmitt, Traverse City West Middle School, Traverse City Rachel Stelman, Oakview Middle School, Lake Orion Dr. Jamie Wernet, Lansing Christian School, Lansing New Jersey Danielle Dorn, Upper Saddle River School District, Upper Saddle River Acknowledgments xi SAMPLE

New York Karri Ankrom, Scott Dobbs, and Danielle Levy, Village Community School, New York Catherine Klein, West Hempstead Secondary School, West Hempstead Shannon Johnson, Christopher Longwell, Michaela Marino, Sarah Meade, Beth Merritt Jennifer Rees, Liana Spencer, and Tom Street, Rochester City School District, Rochester Mike McNall, Holy Family School, Diocese of Syracuse John Ottomanelli, P.S./I.S. 30, New York Department of Education, New York City Dawn Schafer, I.S. 276 Battery Park City School, Manhattan Jennifer Perillo, Brighton Central School District, Rochester North Carolina Karen Abraham, Darla Jones, and Jessica Wallace, C.C. Griffin STEM Middle School, Cabarrus County Schools, Concord Teresa Fulk, Nancy Lynn Green, and Kim Mann, Canterbury School, Greensboro Ohio Rachael Brookshire, Jamie Chaney, Lynnette Flannery, Megan Hammond, Rebecca Hoffman, Kimberly Kraft, Gaby Tagliamonte, and Kristin Whitt, Hamilton City School District, Hamilton Colin Dietrich, Alexandria Ferguson, Heather S. Gust, and Jennifer Linn, Sylvania City Schools, Sylvania Blake Garberich, Clark-Shawnee Local Schools, Springfield Karin Lauterbach, Leslie Liebig, Braden Short, and Ilona Webel, Oakwood City Schools, Oakwood Pennsylvania Nykeesha Brown, Chester Community Charter School, Aston Danielle Crossey, Mt. Lebanon School District, Pittsburgh Vermont Gerald Bailey, Bellows Free Academy, Franklin West Supervisory Union, Fairfax Heather Estey, Bristol Elementary School, Mt. Abraham Unified School District, Bristol Lorrene Palermo, Fletcher Elementary School, Franklin West Supervisory Union, Fletcher xii Acknowledgments SAMPLE

Virginia Jessica Khawaja, Sabot School, Richmond Tim Malloy and Katrien Vance, North Branch School, Afton Wisconsin Brenda Carlborg, Leah Enwright, and Bianca Gloria, Kenosha Unified School District, Kenosha INTERNATIONAL FIELD TESTERS Brazil Jodie Greve, Pan American Christian Academy, São Paulo Colombia Sandra Moreno Cárdenas, Colegio Los Nogales The Netherlands Jill Broderick, Nadine Galante, Lea anne Windham, The American School of The Hague in Wassenaar United Kingdom Jill Broderick, Laura Brown, Stephanie McBride-Bergantine, and Katherine Muir, American School in London, England Vietnam Lia Garcia Halpin and Jennifer Zimbrick, Concordia International School Hanoi Acknowledgments xiii SAMPLE

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Quick Start Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QS-1 Unit Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UP-1 Unit Planning Chart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-1 Unit Description � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-3 Summary of Investigations � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-4 Mathematics Overview � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-7 Student Edition Mathematical Goals of the Unit � � � � � � � � � � � � � � � � � � � UP-14 Unit Arc of Learning (AoL) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-15 Contents of the Student Edition � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-17 Unit Alignment: Goals, Arc of Learning, Standards, Now What Do You Know?, and Emerging Mathematical Ideas � � � � � � � � � � � � � � � � UP-18 Family Letter � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � UP-28 Investigation 1. Designing Polygons: The Side Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Investigation 1 Planning Chart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �1 Problem 1.1 Designing Triangles Experiment: The Side Connection � � � � � � � 2 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Learning Aids Learning Aid Template: Polystrips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Learning Aid 1.1: Building Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Problem 1.2 Designing Quadrilaterals Experiment: The Side Connection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 16 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Learning Aid Learning Aid Template: Polystrips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Learning Aid 1.2: Building Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 CONTENTS xv SAMPLE

Problem 1.3 Rigidity Experiment � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 29 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Learning Aids Learning Aid Template: Polystrips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Learning Aid 1.3A: Diagonals and Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Learning Aid 1.3B: Diagonals and Rigidity Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Mathematical Reflection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 40 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 Answers Embedded in Student Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Answers Embedded in Applications—Connections—Extensions (ACE) � � � 43 Assessment: Checkup 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 50 Answers for Assessment: Checkup 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 53 Investigation 2. Designing with Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Investigation 2 Planning Chart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 56 Problem 2.1 Four in a Row Game: Angles and Rotation � � � � � � � � � � � � � � � � 57 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . .64 Teaching Aid Teaching Aid 2.1: Introduction to Angle Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Learning Aids Learning Aid 2.1A: Four in a Row Game Boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Learning Aid 2.1B: Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problem 2.2 The Bee Dance and Amelia Earhart: Measuring Angles and Distance � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 75 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Teaching Aid Teaching Aid 2.2: Using an Angle Ruler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86 xvi Contents SAMPLE

Learning Aids Learning Aid 2.2A: Bees and Angle Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 Learning Aid 2.2B: Situation A, Amelia Earhart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Problem 2.3 Vertical, Supplementary, and Complementary Angles � � � � � 90 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Teaching Aids Teaching Aid 2.3A: Parallel and Nonparallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Teaching Aid 2.3B: Compliment Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Learning Aids Learning Aid 2.3A: Initial Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Learning Aid 2.3B: What If . . . ? Situation A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Learning Aid 2.3C: What If . . . ? Situations B and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Mathematical Reflection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 108 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Answers Embedded in Student Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Answers Embedded in Applications—Connections—Extensions (ACE) � � � 111 Assessment: Partner Quiz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 127 Answers for Assessment: Partner Quiz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �131 Investigation 3. Designing Polygons: The Angle Connection Investigation 3 Planning Chart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 135 Problem 3.1 Back to the Bees: Tiling a Plane Experiment � � � � � � � � � � � � � 137 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . 145 Teaching Aid Teaching Aid 3.1: Honeycombs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Learning Aids Learning Aid Template: Shapes Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Learning Aid 3.1: Initial Challenge Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Contents xvii SAMPLE

Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment � � � � � � � � � � � � � � � � � � � � � � � � � � � 156 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . 166 Teaching Aids Teaching Aid 3.2A: Angle Sum of Any Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Teaching Aid 3.2B: Different-Sized Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Learning Aids Learning Aid 3.2A: Initial Challenge Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Learning Aid 3.2B: Angle Sum Patterns in Regular Polygons . . . . . . . . . . . . . . . . . . . . 177 Learning Aid 3.2C: Trevor’s, Casey’s, and Maria’s Strategies . . . . . . . . . . . . . . . . . . . . 178 Learning Aid 3.2D: Zane’s Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Problem 3.3 The Ins and Outs of Polygons: Using Supplementary Angles � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 180 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . . 187 Learning Aid Learning Aid 3.3: Pentagonal Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Problem 3.4 Designing Polygons � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 192 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Extended Launch—Explore—Summarize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Answers Embedded in Student Edition Problems . . . . . . . . . . . . . . . . . . . . . . . . 200 Learning Aids Learning Aid 3.4A: Message Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205 Learning Aid 3.4B: Possible Polygon Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206 Mathematical Reflection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 207 At a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207 Answers Embedded in Student Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .209 Answers Embedded in Applications—Connections—Extensions (ACE) � � �211 Assessment: Unit Test � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 230 Answers for Assessment: Unit Test � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 234 xviii Contents SAMPLE

Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Problem Correlations with Common Core State Standards of Mathematics and Mathematical Practices � � � � � � � � � � � � � 238 Mathematical Practices and Habits of Mind Description � � � � � � � � � � � � � 239 Mathematical Practices and Habits of Mind Examples � � � � � � � � � � � � � � � 241 Assessment Correlation with Common Core State Standards of Mathematics (CCSSM) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 242 Contents xix SAMPLE

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QS-1 QUICK START GUIDE OVERVIEW Connected Mathematics® 4 is developed by a team of educators and teachers at Michigan State University. For more than thirty years, the team at Michigan State University has been designing, field-testing, and evaluating four revisions of the Connected Mathematics Project’s (CMP) curriculum, Connected Mathematics®. These curriculum activities are done alongside several research projects with each informing the other. After the publication of a new edition, the Connected Mathematics Project continues to interact with students, teachers, administrators, teacher educators, and researchers across the United States and several international countries to gather feedback from the field. Each edition reflects the information gathered from the research and the interactions on what it means to understand an important mathematical idea. This knowledge is then used to create an environment where students and teachers come together to define and solve problems with reasoning, insight, inventiveness, joy, and technical proficiency. Feedback from field test teachers of Connected Mathematics® 4 suggests that the new features are successful, with more students engaged in deep mathematics. Teachers are also reporting better pacing and less need for modifications for students with special needs. Connected Mathematics® 4 is a contextualized, problem-based mathematics curriculum. • Important mathematical ideas are identified and embedded in a sequence of contextual problems. • Understanding of an important mathematical idea evolves as concepts are informally introduced and then gradually grow in sophistication over a unit. • New concepts and skills connect to and build on prior understandings. As a result, students deepen their understandings of prior learnings at the same time they are developing the new understandings. • The challenge of the problem provides students with multiple entry points and differentiation throughout the lesson. • As students explore a series of connected problems, they develop an understanding of the embedded ideas and, with the aid of the teacher, abstract powerful mathematical ideas, problem-solving strategies, and ways of thinking. • Students learn mathematics and learn how to learn mathematics. A problem-based curriculum not only helps students to make sense of the mathematics, but it also helps them to process the mathematics in a retrievable way. SAMPLE

QS-2 Quick Start Guide • The set of published research and evaluation studies on the impact of using Connected Mathematics® materials in middle grades classrooms continues to grow. The studies show students in Connected Mathematics® classrooms have greater growth in conceptual understanding and problem-solving abilities than students using more traditional curriculum. The results show that CMP students do as well or better on procedural skills. Students also develop a willingness and flexibility to try new situations that persists throughout high school. For more information, please see the Connected Mathematics Project website. WHAT’S NEW IN CONNECTED MATHEMATICS ® 4? Connected Mathematics® 4 has some exciting new features. • In the student edition, there is a new CMP STEM problem format, new and enhanced contexts including more examples of student thinking as context for promoting student learning, and more problems with embedded card sorts, models, matching, games, and experiments. • In the teacher edition, there is a new Arc of Learning framework that highlights the development of mathematical understanding for each unit. There is a new Attending to Individual Learning Needs framework with embedded supports in the Launch—Explore—Summarize sections to support diverse learners and language development. There are also more explicit answers to mathematics problems that attend to various student strategies and understandings. • Assessment features include a new formative assessment framework that emphasizes how teachers assess students during planning, teaching, and reflection of student learning. WHAT’S IN CONNECTED MATHEMATICS ® 4? Connected Mathematics® 4 is a complete middle grades mathematics program. It includes the following five interconnected items: • Student Edition • Teacher Edition • A Guide to Connected Mathematics® 4 for all grade levels • Classroom Materials Kit for each grade level • Digital Platform with online resources This Quick Start Guide provides an overview of the components of the student edition and the teacher edition. It describes the development team’s goals in designing each component, offers guidance for how to SAMPLE

Quick Start Guide QS-3 engage with the materials for the first time, and provides supporting resources that will be helpful before, during and after your use of Connected Mathematics® 4. For a complete program description and to learn more about the program’s instructional approaches and support provided, please see A Guide to Connected Mathematics® 4. Student Edition Connected Mathematics® 4 provides seven student units for grade 6 and eight student units each in grades 7 and 8. Each unit is organized around a big mathematical idea or cluster of related ideas, such as variables and patterns, area and perimeter, ratio and proportion, linear relationships, or nonlinear relationships. The format of the student material promotes student engagement with an exploration of important mathematical concepts and related skills and procedures. Students develop strategies and conceptual understanding by solving problems and discussing their solutions in class. Mathematical Goals and Looking Ahead The Mathematical Goals guide the development of the big ideas of mathematics for the unit. Each unit opens with three interesting problem situations to draw students into the unit, pique their curiosity and joy in mathematics, and point to the kinds of ideas they will investigate. This is followed by a set of focusing questions that reflect the mathematical goals of the unit. Students can use these questions to help track their progress through the mathematical goals. Students can revisit these questions as part of their reflections on their learning. Investigations Mathematics learning is focused on big ideas of mathematics developed through carefully sequenced investigations. Each unit is comprised of two to four investigations. Each investigation includes the following key elements: • 2–4 problems • Did You Know? • Mathematical Reflection • Applications—Connections—Extensions (ACE) Each investigation builds toward the mathematical goals of the unit. Problems and the CMP STEM Problem Format The problems in the Connected Mathematics® 4 program resemble the work that STEM professionals do to solve problems, build deep knowledge SAMPLE

QS-4 Quick Start Guide 30 Variables and Patterns PROBLEM 2.1 Renting Bicycles: Independent and Dependent Variables The tour operators decide to rent bicycles for their customers. They get information from two bike shops: Rocky’s and Adrian’s. Rocky’s Cycle Center sends a table of rental costs for bikes. Adrian’s Bike Shop sends a graph of its rental costs. They both have a special price list for group rentals. INITIAL CHALLENGE The Ocean Bike Tours partners need to choose a bike rental shop. Suppose they ask for your advice. Bike Rental at Rocky’s Number of Bikes 5 10 15 20 25 30 35 40 45 50 Rental Cost ($) 400 535 655 770 875 975 1,070 1,140 1,180 1,200 400 800 1,200 1,600 0 10 Number of Bikes Adrian’s Weekly Rental Rates for Bikes Rental Cost ($) 20 30 40 50 y-axis x-axis • Which shop would you recommend? How would you justify your choice? and skills, and meaningfully connect these solutions to inform the needs of society. The STEM problem format promotes student engagement and learning as students collaborate to design solutions, make conjectures, offer critiques, and communicate their mathematical understandings. The STEM problem format provides teachers with flexibility to carry out equitable practices that help address the individual needs of all students. Components of the CMP STEM Problem Format The Initial Challenge poses the mathematical challenge. It provides open access for students. The What If . . . ? unpacks the mathematical understandings. Students probe deeper at the mathematics by considering different situations with different quantities, contexts, or strategies. The Now What Do You Know? connects the embedded understandings with prior and future knowledge. It provides student-facing questions for students to self-assess and consolidate their learning. Investigation 2 Determining Tour Needs: Analyzing Relationships Among Variables 31 WHAT IF . . . ? Situation A. Sarah’s Questions Sarah used the answers to the following questions to help her decide which company to use. 1. What are the rental costs from Rocky’s and Adrian’s if the tour needs 20 bikes? 40 bikes? 32 bikes? 2. About how many bikes can be rented from Rocky’s or Adrian’s if a group has $900 to spend? If a group has $400 to spend? 3. What pattern do you see with the relationship between the independent and dependent variables in the table for Rocky’s? The graph for Adrian’s? How are these patterns the same? Different? 4. What is the cost to rent one bike from Adrian’s? Rocky’s? Situation B. Malcolm and Celia Make Predictions Examine each prediction. Do you agree? Explain. Malcolm I think it is possible to predict rental costs for numbers of bikes. We do not need every entry in the table or all points on the graph. Celia I think it is possible to predict costs for Adrian’s rental rates. All the points lie in a straight line. The pattern makes it easier to predict for Adrian’s than for Rocky’s. NOW WHAT DO YOU KNOW? We can examine the relationship between the dependent and independent variables. What information was easier to get from the table? From the graph? Explain why. 2.1 SAMPLE

Quick Start Guide QS-5 Problems are contextualized. The contexts provide opportunities for students to elicit genuine interest and wonderment in the context itself, access to the mathematics problem, anchor instruction in a context to develop understanding, and highlight applications of mathematics in everyday matters. Problems provide students with opportunities to make sense of the world and empower them to use mathematics to solve problems. The contexts may be connected to the real world, whimsical, abstract, mathematical, or imaginary. Students and teachers often refer to the problems by their contextual names. These problems also provide students with opportunities to affirm, value, and build on their experiences through their families, community, and cultural and linguistic funds of knowledge. The Connected Mathematics® 4 program includes the following types of problems: • Card sorts • Games • Experiments • Student thinking • Models and visuals • Matching • Hands-on • Technology and manipulatives • Data sets • Capstone problems • Reflections Students build understanding by exploring, conjecturing, reflecting, connecting, and communicating. The problems need to encourage students to use these processes as they engage in mathematics. Each problem in the Connected Mathematics® 4 program has some or all of the following characteristics: • Includes important, useful mathematics embedded in it • Promotes conceptual and procedural knowledge • Builds on and connects to other important mathematical ideas • Requires higher-level thinking, reasoning, and problem-solving • Provides multiple access points for all students • Engages all students and promotes classroom discourse • Creates an opportunity for teachers to assess student learning • Creates an opportunity for students to know what they know SAMPLE

QS-6 Quick Start Guide Did You Know? Throughout the investigations, there is a Did You Know? feature within a problem that presents interesting facts related to the context of an investigation. This feature highlights interesting and diverse applications of mathematics in everyday life and promotes genuine interest in the use of mathematics in the world. Mathematical Reflection The Mathematical Reflection component provides an opportunity for students to consolidate their learning of the big ideas at a key stage of the development. Students encounter the same unifying questions after each investigation of the unit. The goal of this activity is to have students assess and reflect on their learning. For many students, self-assessment is a new experience, and they may be unfamiliar with this experience at first. By receiving feedback from teachers and using other students’ work to provoke new ideas, students can learn to reflect on their own progress in making sense of mathematics. Students may also return to these ideas in later units. Mathematical MR Reflection In this unit, we are studying variables and the relationship between two variables. We use words, tables, graphs, and equations to represent these relationships and to solve problems. At the end of this investigation, ask yourself: What are the advantages and disadvantages of using different representations to show the relationship between two variables? Application—Connections—Extension (ACE) The ACE occur at the end of each investigation. They are meant to be used as additional practice to apply, connect, and extend students’ mathematical concepts and skills. In the ACE, students are asked to compare, visualize, model, measure, count, reason, connect, and/or communicate their ideas. To truly own an idea, strategy, or concept, a student must apply it, connect it to what they already know or have experienced, and seek ways to extend or generalize it. Investigation 1 Organizing a Bike Tour: Variables, Tables, and Graphs 13 NOW WHAT DO YOU KNOW? Describe how distance changes over time. How is this pattern of change shown in tables and graphs? Did You Know? Assateague (A suh teeg) Island is home to herds of wild ponies. The island has a harsh environment of ocean beaches, sand dunes, and marshes. To survive, these sturdy ponies eat salt marsh grasses, seaweed, and even poison ivy. To keep the population of ponies under control, an auction is held every summer. During the famous “Pony Swim,” the ponies for sale swim across a quarter mile of water to Chincoteague Island. SAMPLE

Quick Start Guide QS-7 English/Spanish Glossary Although students are encouraged to develop their own definitions and examples for key terms, each student edition includes an English/Spanish glossary. Glossaries can serve as a guide for the student, the teacher, and families, as students develop an understanding of key ideas and strategies. Teacher Edition Teachers are an integral part of the learning process. As they implement the curriculum and interact with students, their mathematical content knowledge also increases. Connected Mathematics® 4 provides teachers with ways to think about and enact problem-centered teaching and learning. The result of 68 Shapes and Designs INVESTIGATION 3 APPLICATIONS 1. Multiple Choice Which of the following combinations will not tile a flat surface? A. regular heptagons B. equilateral triangles C. regular hexagons D. squares 2. Multiple Choice Which of the following combinations will tile a flat surface? A. regular heptagons and equilateral triangles B. regular octagons C. regular pentagons and regular hexagons D. regular hexagons and equilateral triangles 3. Multiple Choice Which of the following combinations will tile a flat surface? A. regular heptagons and equilateral triangles B. squares and regular octagons C. regular pentagons and regular hexagons D. regular hexagons and squares 4. A right triangle has one right angle and two acute angles. Without measuring the angles, what is the sum of the measures of the two acute angles? Explain your reasoning. APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) Investigation 3 Designing Polygons: The Angle Connection 73 20. Which of the following statements are true? Be able to justify your answers. a. All squares are rectangles. b. No squares are rhombuses. c. All rectangles are parallelograms. d. Some rectangles are squares. e. Some rectangles are trapezoids. f. No trapezoids are parallelograms. g. Every quadrilateral is a parallelogram, a trapezoid, a rectangle, a rhombus, or a square. 21. Suppose you want to build a triangle with three angles measuring 60°. a. What do you think must be true of the side lengths? b. What kind of triangle is this? c. Would this be a unique triangle? C ONNECTIONS 22. Arun writes the equation d = 6t to represent the distance in miles, d, that riders could travel in t hours at a speed of 6 miles per hour. Make a table that shows the distance traveled every hour, up to 5 hours, if riders travel at this constant speed. Time and Distance Time (h) Distance (mi) ACE Investigation 3 Designing Polygons: The Angle Connection 79 EXTENSIONS 38. Copy and complete the table. Sort the quadrilaterals from the Shapes Set into groups by name and description. Sides and Angles from the Shapes Set Sides and Angles of Quadrilaterals Name Examples in the Shapes Set All sides are the same length. All sides are the same length, and all angles are right angles. All angles are right angles. Opposite sides are parallel. Only one pair of opposite sides are parallel. 39. Multiple Choice Which equation describes the relationship in the table? n 0 1 2 3 4 5 6 c 10 20 30 40 50 60 70 A. C = 10n B. C = 10 + n C. C = 10 D. C = 10 + 10n ACE Applications practice the ideas and strategies using contexts similar to and different from the investigation Connections offer continued review of concepts and skills across the grades Extensions provide new challenges for students to think beyond what is covered in class SAMPLE

QS-8 Quick Start Guide the extensive field testing of the Connected Mathematics® 4 materials is a program rich with teacher support including successful strategies, classroom dialogue and questions, and examples of student solutions and reasoning. The teacher edition for each unit in Connected Mathematics® 4 includes a discussion of the mathematical understandings developed in the investigations, mathematical and problem-solving goals for each investigation, planning charts, connections to other units, in-depth teaching notes, samples of students’ strategies, learning aids, teaching aids, family letters, and an extensive assessment package. Suggestions are made about how to engage the students in the mathematics task in the launch, how to promote student thinking and reasoning during the exploration of the problem, and how to summarize with the students the important mathematics embedded in the problem. Support for this Launch— Explore—Summarize sequence occurs for each problem in the curriculum. The teacher edition provides more in-depth support to engage teachers in a conversation about what is possible in the classroom for each problem. Unit-Level Planning • Each unit includes a Unit Planning Chart that provides an overview of pacing, key terms, and materials and resources needed. • An overview of the context and the mathematics are found in the Unit Description and Summary of Investigations. • The Mathematical Overview uses examples from the unit to highlight the development of student understandings that grow over time. • In the Mathematical Goals section, one or two big concepts are identified for each unit with an elaboration of the essential understandings for each goal. • A unit-specific Arc of Learning framework is provided that describes how each problem is positioned with the development of the mathematical goals. • The Unit Alignment chart describes how the mathematics is developed across the unit through the mathematical goals, the Now What Do You Know? question for each problem, the emerging mathematical student ideas, and the positioning of the problem in the Arc of Learning framework. • A family letter for each unit includes topics, terms, and examples to help families support their student along the way. Investigation- and Problem-Level Planning Mathematics learning is focused on big ideas of mathematics developed through investigations. Each investigation in CMP4 includes an Investigation Planning Chart that provides information on pacing, key terms, materials and resources needed, and suggestions for grouping students. SAMPLE

Quick Start Guide QS-9 For each problem, teachers will find the following support provided: • Learning Aids and Teaching Aids • Recommendations on attending to individual learning needs • Suggested questions to ask students at all phases of the lesson to help teachers support student learning and ongoing formative assessment • Actual classroom scenarios and examples of student thinking are included in the Launch, Explore, and Summarize sections to help stimulate teachers’ imaginations about what is possible. • Examples of student work are embedded in the answers to assist teachers in anticipating student responses. At a Glance The At a Glance includes teacher moves, key pieces from the Extended Launch—Explore—Summarize, and organizational support. 208 Investigation 4 Variability in Numerical and Categorical Data At a Glance In this problem, students will use scatterplots and compare those to a hypothetical modeling line. Students will see unusual points or outliers as distant from the main point cluster. The Initial Challenge has students revisit the bridge layer experiment from Problem 1.1 and write equations for lines of best fit. To test the accuracy of the models, students will be introduced to residuals in What If . . . ? Situation A. What If . . . ? Situation B will introduce students to scatterplots, and What If . . . ? Situation C will have students deciding if data sets have a positive or negative correlation. Arc of Learning Synthesis Exploration Analysis NOW WHAT DO YOU KNOW? How can you find an equation for a linear function that is a good model for a set of data? Describe how finding the residuals for the line can help measure the accuracy of the model. Key Terms Materials residual strong association positive association negative association For every student • Learning Aid 4.1A: Line of Best Fit and Residuals • Learning Aid 4.1B: Scatterplots Pacing 2 days Groups 2 A 1–4 C 13–15 E 24–25 PROBLEM 4.1 Lines of Best Fit Note: If you have a Grade 8 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more Teacher Moves, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE The work in this problem is connected to students’ previous work on data analysis and writing linear equations. Suggested Questions • When looking at a data set, how can you describe the variability? • What data sets were we looking at in Investigation 1? • Did we draw any conclusions about the data? • What is needed to write an equation of a line? PRESENTING THE CHALLENGE Discuss the bridge strength experiments in Investigation 1. In this problem, students will be analyzing data from other students that collected data on breaking weights of bridges. They will determine if the data is linear. If it can be modeled with a line, they will determine the best fit line to represent the relationship. This problem will take 2 days. The Initial Challenge and What If . . . ? Situation A would be on the first day and What If . . . ? Situations B and C on the second day. On Day 1, have a brief summary after the Initial Challenge before launching into What If . . . ? Situation A. You could divide up the models and have different groups do different models and then compare. Problem 4.1 Lines of Best Fit 209 Facilitating Discourse Teacher Moves EXPLORE PROVIDING FOR INDIVIDUAL NEEDS As students are looking for the equations for the lines, take note of the strategies used to create the equations. • Students might estimate the y-intercept on the graph and use two points to find the rise and run on the graph to get the slope. • Students might find several points on the line, create a table, look for the rate of change to find the slope, and then back the table up to x = 0 to get the y-intercept. • Students might use two points to find the slope and then substitute one of the pairs into the equation y = mx + b to find b. Suggested Questions • How could you find the slope? Y-intercept? • How are you using the graph or the equation to make predictions? As you are circulating, listen for conversations on what the students are finding as they calculate the residuals. • How are you finding the residuals? • What do the positive residuals tell you? • What do the negative residuals tell you? PLANNING FOR THE SUMMARY As you are circulating, select student work based on how they are modeling a data set on a graph then using that line to write an equation. Listen for how students are making sense of using residuals to decide the accuracy of linear models. Are they noticing that if the residuals are large, or there isn’t a pattern, the model will not be accurate? SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have students share the equations they wrote for each linear model and the strategies they used to find them. Compare the strategies used. Have students explain which line they think best fits the data and why. Share the data from the work that students did in finding the residuals for all three linear models. Display the tables for each linear model. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • How do the strategies used to find the equations for the linear models compare? • How do the residuals computed compare to the graphs? • When the total of positive residuals is much larger than the total of negative residuals, what will that mean for the predictions from this model? When it is about the same? • Are any of the data sets in What If . . . ? Situations B or C functions? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Compare Thinking Agency, Identity, Ownership 4.1 • Provides an overview of problem that includes a description of the context of the problem; the Arc of Learning phase(s); and the Now What Do You Know? question. • Provides planning information that includes materials needed to implement the problem; ACE exercises to support learning; and suggestions for student grouping and pacing. • Provides teaching support that includes suggested questions for each phase of the lesson. • Provides teacher moves that includes general pedagogical strategies, implementation notes, and Attending to Individual Learning Needs. SAMPLE

QS-10 Quick Start Guide Extended Launch—Explore—Summarize The teacher edition materials are organized around an instructional model that supports a problem-solving STEM classroom. The three instructional phases of a lesson are Launch, Explore, and Summarize. For each instructional phase, specific suggestions that relate to the Attending to Individual Learning Needs framework are provided to support teachers in addressing the learning needs of all students. Support is provided for teachers in each phase of the lesson: • Launch. The teacher launches the lesson with the whole class. Problem Overview Students conduct another experiment to explore a nonlinear relationship as they test how bridge length is related to strength. They look for patterns in their collected data and use the patterns to make predictions. Launch (Getting Started) Connecting to Prior Knowledge As in Problem 1.1, students are asked to make initial conjectures about the relationship between bridge length and strength (for constant thickness) and to test those conjectures with a simulation based on paper bridges. The overarching goal is to develop student sensitivity to and disposition to recognize and use nonlinear patterns in experimental data, especially as a foundation for more explicit study of inverse variation to come in Investigation 3. Presenting the Challenge Prepare a set of paper strips of width 4 __1 4 inches (in.) with lengths 4 inches, 6 inches, 8 inches, 9 inches, and 11 inches in advance. Consider whether you want to spend class time having groups fold their own strips or whether you want to prepare the strips for them. Suggested Questions • How do you think the length of a bridge is related to its strength? (Answers will vary. The longer a bridge is, the weaker it is.) • Are longer bridges stronger or weaker than shorter bridges? (Answers will vary. Weaker.) You can transition to the proposed activity by describing the experiment to be conducted. • What do you expect will happen in this experiment? (Answers will vary. Less weight will probably be needed to collapse the bridge as length increases.) • You are using materials similar to the materials you used before. What are the variables this time? (Length and breaking weight.) • What do you think the data will look like? What shape do you think the graph will have? (Predictions will vary. The graph will probably show breaking weight decreasing as length increases.) You might have students sketch their predicted graphs and then share some of these with the class. Students will probably guess that longer bridges will not support as much weight and that EXTENDED LAUNCH—EXPLORE—SUMMARIZE 20 Investigation 1 Exploring Data Patterns in Building Bridges the relationship will be linear. Talk about the reasoning behind their conjectures, but let them discover the actual shape from the experiment. (Problem-Solving Environment) As in Problem 1.1, establish what it means for a bridge to collapse, and discuss ways to minimize variability, such as marking the books to indicate where the strips will be placed, marking the strips to indicate where the cup will be placed, and using a consistent method for adding pennies to the cup. In addition, labeling each strip with its length will help students avoid errors in recording. Implementation Note: Introduce the term scatterplot if you created one in Problem 1.1. Create a graph template on chart paper or on an electronic whiteboard for groups to record the results of today’s experiment. Each group will record their data points to make a class scatterplot. (This plot will have several points for each length value.) This will get further developed in Investigation 4. (Time) Explore (Digging In) Providing for Individual Needs Have students do the experiment and discuss the questions with their partners, but ask each student to make a table and a graph and write their own answers. You might have groups put their tables and graphs on large sheets of paper for sharing during the Summarize. Have groups record their results in a class scatterplot as they did in Problem 1.1. Suggested Questions • Is there a trend to your data? (Students should look at the general shape of the graph and try to describe what it looks like.) • Is the trend similar to the one in the experiment in Problem 1.1? (In Problem 1.1, the shape of the graph was linear. In Problem 1.2, the shape is nonlinear.) Planning for the Summary What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? What will you do if you do not have evidence? NOW WHAT DO YOU KNOW? How does the relationship between the number of layers in a bridge and its breaking weight compare to the relationship between bridge length and breaking weight? How does a graph or table model the relationship between the variables? LES Extended Launch—Explore—Summarize 21 Providing for Individual Learner Needs. Suggested questions and pedagogical strategies are provided for teachers as they observe and interact with individual and small groups working on the problem. Planning for the Summary. Suggestions are provided for the teachers on collecting evidence from student work and how to use it during the summary discussion around the Now What Do You Know? question for the problem. • Explore. Students work on the problem while the teacher supports the learning. Connecting to Prior Knowledge. Support for the teacher helps them position the new problem within prior understandings and problems. Presenting the Challenge. Support for the teacher is given to help students understand the problem setting, the mathematical context, and the challenge. SAMPLE

Quick Start Guide QS-11 • Summarize. The teacher facilitates students in a discussion to reach the mathematical goal of the problem. As you are circulating during the Explore, look for how groups are deciding if the table/graph are linear or nonlinear, and use these students to lead discussions in the summary. Look for how students are using the data to make predictions. Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Before beginning the summary, have each student jot down their own thoughts on the relationships they saw in the data. Have groups display their data or use the collective class scatterplot. Suggested Questions • Do you have any What If . . . ? questions? (Answers will vary.) • Looking at the graphs from different groups in the room, how are they the same, and how are they different? (Answers will vary. Students should recognize that they do not have a linear feel or look to them but that there is a general curved look to all of them.) • How does the class scatterplot help us? (Answers will vary. Students tend to talk about the errors they made in keeping all other variables isolated to just test length and by combining everyone’s data they can look at an “average” to make decisions.) Making the Mathematics Explicit Suggested Questions • Are there similarities in the results of the different groups? Are there differences? What might have caused those differences? (Sample answers: All the groups found functions with a nonlinear decrease. Differences may have come from different ways of setting up the bridges or different ways of folding the paper.) • As bridge length increases, what happens to the number of pennies the bridge can support? (It decreases.) • As bridge length decreases, what happens to the number of pennies the bridge can support? (It increases.) Focus the students’ attention on the inverse relationship, but don’t define it as inverse yet. • What shape or pattern do you see in your graph? Are the data linear? (No; the points form a curve.) • How can you tell from your table that the graph will be curved? (The difference between consecutive breaking weights decreases by less and less as the length increases. The change is not constant.) LES 22 Investigation 1 Exploring Data Patterns in Building Bridges Assessment Multiple kinds of assessment are included in the program to help teachers see assessment and evaluation as a way to inform students of their progress, apprise families of students’ progress, and guide the decisions a teacher makes about lesson plans and classroom interactions. A Guide to Connected Mathematics® 4 includes a self-assessment, which reinforces the emphasis on helping students know what they know. Formative assessment suggestions are provided throughout the Launch, Explore, and Summarize phases. Diverse kinds of summative assessments are included that mirror classroom practices as well as highlight important concepts, skills, techniques, and problem-solving strategies. The following summative assessments are provided in the Teacher Edition: • Checkup. Short, individual assessments provide insight into student understanding of the baseline mathematical concepts and skills of the unit.  • Partner Quiz. These are more complex than Checkup assessments and more closely resemble the work the students do during class, which prepares them for the work done by STEM professionals. The Partner Quiz includes extensions of ideas students explored in class and provides insight into how students work together to apply the ideas Solutions and Strategies. Suggested questions and pedagogical strategies for teachers are provided to use to facilitate discourse as students share their conjectures and ideas. Making the Mathematics Explicit. Suggested questions and pedagogical strategies for teachers are provided to use to focus the conversation around the mathematical goal of the problem. SAMPLE

QS-12 Quick Start Guide from the unit to new situations. They are done with pairs of students and generally take more time. • Unit Test. Individual assessments that provide information on a student’s ability to apply, refine, modify, and possibly extend the mathematical knowledge and skills acquired in the unit. Digital Platform with Online Resources The Lab-Aids digital portal allows students and teachers to access and interact from any digital device with the Student Edition, Teacher Edition, and additional online resources. With built-in text-to-speech functionality, students can listen to an audio version of the content in many different languages, which improves accessibility to the program. Teachers can assign ACE that students can complete within the online portal. SAMPLE

UNIT PLANNING Investigations and Assessments Total Pacing: 19 Days Materials for All Investigations: calculators student notebooks colored pens, pencils, or markers Investigations and Assessments Key Terms Materials Resources Investigation 1. Designing Polygons: The Side Connection Pacing 3 days polygons regular polygons For each student • Learning Aid 1.1: Building Triangles • Learning Aid 1.2: Building Quadrilaterals • Learning Aid 1.3A: Diagonals and Rigidity • Learning Aid 1.3B: Diagonals and Rigidity Shapes (optional) For each group of 2–3 students • polystrips and fasteners • Learning Aid Template: Polystrips (optional) Mathematical Reflection Pacing __1 2 day For the class • large poster paper (optional) • word bank created by students and/or teacher (optional) Assessment Checkup 1 Pacing __1 2 day For each student • Checkup 1 Investigation 2. Designing with Angles Pacing 6 days angle benchmark angle degree right angle angle measures parallel vertical angles supplementary For each student • Learning Aid 2.1B: Angles • Learning Aid 2.2A: Bees and Angle Measurements • Learning Aid 2.2B: Amelia Earhart • Learning Aid 2.3A: Initial Challenge • Learning Aid 2.3B: What If . . . ? Situation A • Learning Aid 2.3C: What If . . . ? Situations B and C For each pair of students • Learning Aid 2.1A: Four in a Row Game Boards For the class • Teaching Aid 2.1: Introduction to Angle Measures • Teaching Aid 2.2: Using an Angle Ruler • Teaching Aid 2.3A: Parallel and Non-Parallel Lines UNIT PLANNING CHART UP-1 SAMPLE

Investigations and Assessments Key Terms Materials Resources Investigation 2. Designing with Angles (continued) complementary adjacent angles • Teaching Aid 2.3B: Compliment Angles (optional) Mathematical Reflection Pacing __1 2 day For the class • large poster paper (optional) • word bank created by students and/or teacher (optional) Assessment Partner Quiz Pacing 1 day For each pair of students • Partner Quiz Investigation 3. Designing Polygons: The Angle Connection Pacing 6 days tilings tessellations regular polygon irregular polygon angle sum interior angles exterior angles For each pair of students • Learning Aid Template: Shapes Set or plastic Shapes Set • Learning Aid 3.1: Initial Challenge Table • large paper • markers • Learning Aid 3.4A: Message Cards • Learning Aid 3.4B: Possible Polygon Designs • tools to create shapes, such as: polystrips, angle rulers, dot paper, geoboards, geometry technology For each student • Learning Aid 3.2A: Initial Challenge Shapes • Learning Aid 3.2B: Angle Sum Patterns in Regular Polygons • Learning Aid 3.2C: Trevor’s, Casey’s, and Maria’s Strategies • Learning Aid 3.2D: Zane’s Conjectures • scissors (optional) • Learning Aid 3.3: Path • polystrips (optional for Launch) For the class • Teaching Aid 3.1: Honeycombs • Teaching Aid 3.2A: Angle Sum of Any Polygon • Teaching Aid 3.2B: DifferentSized Regular Polygons Mathematical Reflection Pacing __1 2 day For the class • large poster paper (optional) word bank created by students and/or teacher (optional) Assessment Unit Test Pacing 1 day For each student • Unit Test UNIT PLANNING CHART (continued) UP-2 Unit Planning SAMPLE

UNIT DESCRIPTION Shapes and Designs is the second unit in the Connected Mathematics ® 4 geometry strand. It develops students’ ability to recognize, display, analyze, measure, and reason about the shapes and visual patterns that are important features of our world. It builds on students’ elementary school exposure to simple shapes, as they begin analyzing the properties that make certain shapes unique. The unit focuses on polygons and on the side length and angle measure relationships of regular and irregular polygons (circles and other curves are explored in later units). A central theme is designing shapes under constraints. As students learn important criteria that determine shape, they apply these understandings to design figures with specific criteria and solve real-world problems. In the Student Edition, the introduction to Shapes and Designs develops the broad theme of the unit: of all of the shapes we use as basic components in buildings and art, some simple figures occur again and again because certain properties make them especially attractive and useful. The goal of Shapes and Designs is to have students discover and analyze many of the key properties of polygonal shapes that make them useful and attractive. As students become observant of the multitude of shapes that surround them and aware of the reasons that shapes are used for specific purposes, they will be amazed by the visual pleasure and practical insights their new knowledge provides. The approach to geometry in this unit is somewhat unique. First, the primary focus of the unit is on recognizing and generalizing properties of shapes that have important practical and aesthetic implications, not on simple classification and naming of figures. Each investigation focuses on particular key properties of figures and the importance of those properties in applications. We frequently ask students to find and describe places where they see polygons of particular types and to puzzle over why those particular shapes are used. The overarching goal of the unit is to develop the broad principle that form and function of natural and designed objects are intimately related. To reach that goal, it is necessary to develop student understanding and skills that are useful in describing and reasoning about shapes, especially polygons. The shape and function of a polygon are determined by the combination of sides and angles used to construct it. However, some polygons, like triangles, are rigid and often appear in building structures. Quadrilaterals are not rigid but are useful for other parts of structures and in designing folding chairs. The central question for this unit is what important geometric properties determine a particular shape and how this information is useful in designing shapes given specific conditions. The focus is on generalizing the properties of shapes and how these properties are useful in nature and everyday life. Algebraic Unit Planning UP-3 SAMPLE

reasoning is more prominent as students use expressions and equations to solve problems involving angles and polygons. Bees are used as a theme to introduce and explore angles of rotation and again to look at shapes of polygons and why hexagons evolved as the shape for the cells of a honeycomb. The Looking Ahead poses three important application questions that drive the investigations and problems: What properties of a regular hexagon make it the most efficient shape for the cells of a honeycomb? Why do some shapes occur more often than others in the design of craft objects? Why are braces on towers, roofs, and bridges in the shape of triangles and not rectangles or pentagons? As with all of the Connected Mathematics® 4 units, one Mathematical Reflection guides the development of the understanding of the mathematical ideas in the unit. Mathematical Reflection In this unit, we are investigating some general properties of geometric shapes (figures), including angles and polygons, and using this information to design shapes and solve problems. At the end of this investigation, ask yourself: What do you know about geometric shapes? SUMMARY OF INVESTIGATIONS Investigation 1: Designing Polygons: The Side Connection This investigation involves several STEM design experiments involving polystrips to explore the properties and usefulness of polygons. Students discover the relationship among side lengths of polygons needed to build a polygon and which polygons have rigidity. They also sketch shapes given certain constraints about their side lengths. Problem 1.1 focuses on developing the understanding of the Triangle Inequality Theorem, that in a triangle the sum of any two sides must be greater than the third. Problem 1.2 extends the Triangle Inequality Theorem into a somewhat parallel result for quadrilaterals, that the sum of any three side lengths of a quadrilateral is greater than the fourth side length. Additionally, students see that a quadrilateral’s shape will change in response to pressure on the vertex and hence is not rigid. Problem 1.3 looks at the minimum number of diagonals needed to make a shape rigid. Students add diagonals to various polygons to look at patterns of making a shape rigid by adding a triangular structure. A triangular shape is rigid, which explains its popular use in the structure of bridges, buildings, and so on. Investigation 2: Designing with Angles This investigation uses angles of rotation as the shape under investigation. Students estimate and use tools such as protractors and angle rulers UP-4 Unit Planning SAMPLE

to measure angles and to investigate the relationships among right, supplementary, vertical, and complementary angles. Problem 2.1 develops students’ understanding of angles as rotations or turns as geometric shapes and their awareness of common benchmark angles through the game Four in a Row. Problem 2.2 uses the context of the bee dance and Amelia Earhart’s fateful journey for the purpose of developing accuracy in angle measurement. The students develop understanding of two standard tools for angle measurement—the goniometer, or angle ruler, and the protractor. Problem 2.3 develops students’ understanding and skill in working with vertical, supplementary, and complementary angles. Students will investigate the relationships among these angles and write generalizations about these relationships. Investigation 3: Designing Polygons: The Angle Connection This investigation develops fundamental concepts, terminology, and techniques needed to characterize size and shape of polygons, especially the relationship between angle measure and side lengths in polygons. It develops and applies important properties of polygons in order to explain tessellation, structural rigidity/flexibility, and their occurrence in nature. Students use algebraic reasoning to model quantities in the mathematical problem of finding total interior angle sums and the measure of each interior angle in regular and irregular polygons. Students also apply what they learned about angle measures to the concept of tessellations (tilings). The unit ends with several design-under-constraints experiments, which use student knowledge of shapes that have been developed throughout the unit. Problem 3.1 uses results about angle measurements in polygons to explain the tiling properties of regular hexagons. Students are challenged to answer the question “Why do the cells of honeycombs have the shape of hexagons?” The use of tessellations solidifies ideas that students are forming around characteristics of shapes, in particular angle measurements, and provides insights into why the bees choose the hexagon for the shape of their honeycombs. Problem 3.2 adds another piece of information that will help students to construct geometric shapes by developing a formula that predicts the sum of interior angles for a polygon of n sides, especially the instances of that pattern for triangles (180°) and quadrilaterals (360°). Knowing these angle sums is important for determining missing angles in constructing triangles and quadrilaterals. Students examine this relationship for convex and concave polygons. Problem 3.3 uses the context of cycling around a field in the shape of a pentagon to find the number of degrees the cyclists turn in one full lap. Students develop an understanding of the concept of exterior angles. Using the concept of bicycling around polygonal paths, students understand that the sum of the exterior angles of a polygon is always 360°. Problem 3.4 pulls together students’ understanding of the Unit Planning UP-5 SAMPLE

conditions that determine unique shapes. Given a set of design criteria about a shape, including its side lengths and angle measures, students determine if it is possible to design such a shape. If it is, is it unique? Throughout the unit, students develop an understanding of shape and how it is used in design, art, building, and nature by designing, sketching, and building various shapes. UP-6 Unit Planning SAMPLE

MATHEMATICS OVERVIEW Major Focus: geometry, measurement Key Ideas: meaning of angle, measuring angles, properties of angles (corresponding, supplementary, complementary, vertical, adjacent), angle sum of polygons, properties of triangles, relationship of side lengths of polygons Strategic Curriculum Connections: algebra, solving equations Connected Mathematics® 4 approaches this geometry unit from the point of view that shapes have properties and that these properties both define the shapes and are integral to the uses of the shapes in nature and the real world. The shapes, or geometric figures, studied in this unit are angles and polygons. Within the class of polygons, particular attention is paid to triangles and quadrilaterals. The goal is not to have students memorize names of types of triangles or polygons but to investigate the properties that are key to a specific shape. A central theme is designing shapes under constraints. Students apply their understandings about the properties of specific shapes to design figures with specific criteria. The shape and function of a polygon are determined by the combination of sides and angles used to construct it; certain combinations of side lengths and angle sizes produce a unique triangle (or quadrilateral), and some do not. Students find that form and function of natural and designed objects are intimately related; that is, there are properties of a triangle that make triangles uniquely useful in some construction projects, aspects of a quadrilateral that are key to many practical applications, and properties of a hexagon that explain why it occurs frequently in the natural world. In roughly this order, in Shapes and Designs, students: • investigate whether any set of side lengths will produce a triangle (or quadrilateral) and, if so, whether the triangle (or quadrilateral) is unique; • estimate and measure angles and explore the relationships among angles formed by a transversal cutting two lines; UP-7 This unit links to Stretching and Shrinking in grade 7, when scale drawings, or similar figures, are investigated. It also links to the study of similarity, congruence, and transformations in grade 8. In these units, students investigate more formally which combinations of sides and angles make triangles similar or congruent. Link to the Future SAMPLE

• investigate the relationship between the number of sides of a polygon and the sum of the measures of the interior and exterior angles and explore tiling a plane with polygons; and • design polygons under constraints. Comparing Properties: Side Lengths of Triangles and Quadrilaterals A triangle must have three sides. The unit starts with an investigation of the basic questions: Will any three side lengths make a triangle? Will any four side lengths make a quadrilateral? Later in the unit, this question is refined: What information do we need to draw a unique triangle (or quadrilateral)? Comparing Triangles and Quadrilaterals By experimenting with polystrips of different lengths, students discover there is only one specification about side lengths to make a triangle: the sum of the lengths of the two shorter sides must be long enough to “close” the triangle. a b c b + c < a b + c > a b a c This is known as the Triangle Inequality Theorem. That is, the sum of any two sides of a triangle must be greater than the third side. In addition, students find that given three side lengths that make a triangle, that triangle is rigid; the fixing of the side lengths also fixes the angle sizes so that with those side lengths only one triangle is possible. Just as with triangles, there is only one specification about side lengths that makes a quadrilateral: the sum of the three shortest sides must be greater than the longest side length. a b b + c + d > a b c d a c d In contrast to the rigidity of a triangle, a quadrilateral is flexible. That is, given four side lengths that form a quadrilateral, the quadrilateral can be deformed to form other quadrilaterals that have the same side lengths but different angle sizes. a b d b c d a c Form and Function The example of four given side lengths for a quadrilateral being insufficient to define a unique figure is familiar to amateur builders; the bookcase that tilts and the window frame that has settled are examples where the side lengths did not change but the angle sizes did. Compare this to the construction of a triangle. Once the side lengths are fixed, the triangle is rigid. No pressure on a vertex can alter the angle sizes of the triangle. This makes triangles useful in the construction of houses and bridges, for example. Quadrilaterals UP-8 Unit Planning SAMPLE

do not have this property. Four side lengths will not make the quadrilateral rigid. However, the flexibility of a quadrilateral with fixed side lengths makes it useful in other applications. For example, the trapezoid in the picture of the folding chair changes shape while retaining the same side lengths. Angles: Measuring, Reasoning About Relationships Among Angles Angles can be understood three different ways: a rotation of a ray from an initial position to an ending position (compass directions use this idea), a geometric figure comprising two rays from a common vertex (useful to categorize angles), or a corner or wedge (useful to visualize shapes). All three ideas are useful in this unit. Meaning and Measure of Angles Reasoning About Relationships Angles as Rotation: Measuring The mathematical convention is to measure the rotation of the initial ray of an angle counterclockwise. Throughout Connected Mathematics® 4, the angle intended will be clearly marked with an arc showing the rotation. The counterclockwise aspect of the definition will not be emphasized. B A C B A C Two Parallel Lines Cut by an Intersecting Line In the following diagram, there are two parallel lines, L1 and L2, that are cut by another line, t, forming four angles at each intersection point. These angles are formed by rays coming out of each vertex or intersection point. The line t is called a transversal. a b d c e f g h L1 L2 t Because the lines are all straight lines, we have pairs of angles that add to 180°. a + b = 180°, d + c = 180°, b + c = 180°, a + d = 180°. When the measures of two angles add to 180°, we say the angles are supplementary. There are four pairs of supplementary angles at each vertex. Also, the angles opposite each other at each vertex are equal. Students do not prove this; they conjecture and then use measurement to confirm the conjecture that vertical angles are equal. (“Vertical” = “at a vertex.”) This shows angle CBA. It measures a little less than a _1 8 of a full rotation of 360°, about 40°. The rotation of ray BC to the position of ray BA sweeps over ∠CBA. This shows angle ABC. It measures about 320°. The rotation of ray BA to the position of ray BC sweeps over ∠ABC. Unit Planning UP-9 SAMPLE

Meaning and Measure of Angles Reasoning About Relationships Making the rays longer would not alter the size of the rotation. However, as we can see in the case of Amelia Earhart’s lost flight, the distance between the rays increases as she strays farther off course while her direction, or angle, does not change. PAPUA NEW GUINEA Nauru Solomon Islands Tuvalu AUSTRALIA Fiji Islands Samoa Islands Phoenix Islands Gilbert Islands Vanuatu B Howland Island PACIFIC OCEAN CORAL SEA Nikumaroro Island C E F A D 0 625 mi N S W E Lae Thus a = c and b = d and e = h and f = g. (Students may also confirm other conjectures by measuring. a = e, b = f, d = g, c = h. These are in corresponding positions at each vertex. Likewise, d = f and c = e. These are called alternate interior angles.) Angles that share a ray and do not overlap are adjacent. Angles do not have to be adjacent to each other to be supplementary. It is clear a + b = 180° because, side by side, they make a straight angle. Also, a + g = 180°; they are supplementary but not adjacent. Two Nonparallel Lines Cut by a Transversal Vertical angles are equal, and there are adjacent supplementary angles for nonparallel lines. However, corresponding and alternate interior angles are not equal if the lines are not parallel: j ≠ n, m ≠ k. L3 L4 o p m j i l k n Polygons: Interior and Exterior Angle Sum, Tiling Polygons, Interior and Exterior Angle Sum Tiling Angle Sum of a Triangle When we think about the angle sum of the interior angles of a triangle, we naturally think of corners, not rotations. 1 1 2 3 3 2 Students find the sum of angles of a triangle by cutting off the angles of a triangle and rearranging them around a point. They form a straight angle. This is not a proof, but it is convincing if enough students measure and compare results. This experiment is repeated for other polygons. The result is that the sum of the interior angles of a polygon is 180(n − 2), where n is the number of sides of the polygon. (It is possible to prove that the angle sum of a triangle is 180° by using the results about a transversal cutting two parallel lines, as below.) Tiling a plane means covering the entire flat surface with a shape (or shapes), leaving no gaps. Students find that rectangles, parallelograms, regular hexagons, and all triangles tile a plane. Only certain regular polygons tile a plane: an equilateral triangle, a square, and a regular hexagon. The key is the sum of the angles around each vertex. In the case of the regular hexagon, each angle is 120°, so 3 hexagons will fit exactly around a vertex: 3 • 120 = 360. Likewise, 4 squares fit around a vertex: 4 • 90 = 360. And for an equilateral triangle, 6 triangles fit around a vertex: 6 • 60 = 360. In fact, any triangle will tile a plane. Why is this? UP-10 Unit Planning SAMPLE

Polygons, Interior and Exterior Angle Sum Tiling b c b c a L1 L2 Angle Sum of Any Polygon Students also find the angle sum of polygons by using the angle sum of a triangle and subdividing the polygon into triangles. There are several ways to subdivide any polygon into triangles. The most common are shown here: a e K F d g b c h m j l w t n r p q s u y v x z One strategy is to choose one vertex of the polygon and draw all the diagonals from this point (diagram on the left). This divides the polygon into triangles; there will always be 2 fewer triangles than sides of the polygon. The angles of the triangles are all located in the corners of the polygon. So the sum of all the angles of triangles gives the angle sum of the polygon. This strategy confirms the results found by tearing off the angles of a polygon and rearranging them around a point. If there are n sides, there will be n − 2 triangles, and the angle sum is 180(n − 2). Another strategy is to choose a point in the interior and draw lines to each vertex. There are the same number of triangles as sides, so if there are n sides then the sum of the angles of n triangles would be 180n. However, some of the angles of the triangles form 360° around this central point. So to find the sum of the angles of the polygon, we must subtract this 360° from 180n. The angle sum of the polygon is 180n − 360. Strategic Curriculum Connection: Notice that 180n − 360 and 180(n − 2) are equivalent expressions. There are multiple opportunities to use algebraic reasoning about the angle relationships or the angle sum of a polygon. a c b b a c a a a c a b b b b c c c As seen on this sketch of a tiling, because of the angle sum of a triangle, a + b + c = 180, and around the vertex we have 2a + 2b + 2c = 360. (You can also see why parallelograms tile a plane in this figure.) Strategic Curriculum Connections: Tiling is an opportunity to apply the knowledge about the sizes of angles in specific polygons, but it is also rich in connections to other units. Why do only some regular polygons tile a plane? (See the Number Connections unit.) As previously illustrated, the sum of the angles around each vertex must be 360º. Since all the angle measures are equal for a regular polygon, for a tiling, each measure must be a factor of 360: 1, 2, 3, 4, 5, 6, 8 . . . 45, 60, 72, 90, 120, 180, 360. Could there be a regular polygon with n sides and n angles each measuring 45°? Students may argue that as the number of sides in a regular polygon gets larger, so does the angle size. So if a regular polygon with 3 sides has angles of 60°, then we would need to have fewer than 3 sides to make a regular polygon with angles of 45°—and you can’t make a polygon with fewer than 3 sides. (This can also be solved algebraically, though students are not able to do this yet. A regular polygon with n sides has angle sum 180(n − 2). It also has n angles. If each angle is 45°, then the angle sum is also 45n. Solving 180(n − 2) = 45n gives n = 2 __2 3 , not a whole number of sides!) Stretching and Shrinking Connection: In Shapes and Designs, students find that they can tile a plane with any triangle. In Stretching and Shrinking, students also find that a triangle rep-tile creates scale copies, or similar triangles. Unit Planning UP-11 SAMPLE

Polygons, Interior and Exterior Angle Sum Tiling For example: For an octagon the angle sum is A = 180(8) − 360, or 1,080, and if the octagon is regular then each angle is __ A n = _____ 1,080 8 = 135°. Or given that three angle measure of a quadrilateral are 20º, 100º, and 85º, find the fourth angle. A = (n − 2)180 = (4 − 2)180 = 360. So 20 + 100 + 85 + x = 360. 205 + x = 360. x = 155. In the graphic shown here, the largest triangle is a scale copy of the smallest triangle, with a scale factor of 4. You can also see how any triangle tiles a plane in this figure. Tiling and rep-tiling are related ideas. Curriculum Decisions From all the interesting problems that students might investigate and solve, choices must be made. Choosing problems that have connections to several mathematical ideas, and sequencing them judiciously, results in students learning about connected ideas and remembering them in larger cognitive chunks. The tiling problem may seem like an interesting digression, but it is in fact linked to several other ideas and units. Designing Under Constraints Designing Triangles Under Constraints Designing Quadrilaterals Under Constraints If we want to draw a specific triangle, we need particular information about angles and sides. We do not need to know all three angles and all three sides; some of that information would be redundant. Suppose the goal is to draw a triangle that has specific side lengths and angle measures, without using all measures. Specific triangle 46° 3.6 cm 4 cm 3 cm 60° 74° B C A What is the minimum information we need? It turns out that we need three pieces of information. But not any combination works. Three angle measures only. This produces an infinite number of triangles, all scale copies of each other. In fact, knowing only two angle measures would produce the same result. For example, if two angles are 60° and 46°, the third must be 74°. 60°46° A? A? 46° B C? C? We need a side length to pin down which of all these triangles is the one we want to draw. Since three pieces of information are sometimes sufficient to draw a triangle, it is tempting to assume that four carefully chosen measures will be sufficient to draw a specific quadrilateral. In fact, that is never sufficient (unless you are also given some extra information, such as the quadrilateral being a parallelogram or a square). The following are some examples to illustrate this. All four angles. As with triangles, knowing all four angles, even if the order is known, will not produce a unique quadrilateral. For example, all parallelograms shown here have the same angle measures, 60°, 60°, 120°, 120°, but different side lengths. 120° 120° 60° 60° Three angles and one side. As with triangles, knowing three angles is sufficient to be able to calculate the fourth angle. But knowing the length of one side is not enough to fix the lengths of the other sides. UP-12 Unit Planning SAMPLE

Designing Triangles Under Constraints Designing Quadrilaterals Under Constraints Two angles and ANY side. As above, we can see this is in effect three angles and a side. So in the graphic, if we know the length of ANY side, we can draw exactly the triangle we want. One angle and two sides. This combination is tricky. If the angle is included (between the two sides), then this information produces a unique triangle. 4 60° 3 For example, drawing an angle of 60° and measuring the two sides of 3 and 4 units, as shown, ensures that the third side is fixed, and the other angles are fixed. However, if the angle is not between the two sides, then there MAY be two different possible lengths for the side opposite the given angle. For example, if we know only one angle, ∠C = 46°, and BC = 4 and BA = 3, then there are two positions for the side opposite ∠C. A? A? B C 3 3 4 60° 46° One position for BA would match the target ∠B = 60°, and the other would not. More information is needed to remove the ambiguity. Three side lengths only. This produces a unique triangle. You can use a compass or ruler or technology. 3.6 3 4 3.6 4 3 By using only three side lengths to draw the triangle, all three angle measures match the target also. The triangles are identical, just differently oriented. Link to the Future: The conditions to design a unique triangle are investigated informally in Shapes and Designs. The concept of congruent copies will be dealt with more formally in grade 8. 120° 6 100° 60° 80° For example, the sketch here shows several possibilities for different quadrilaterals, all of which have the same angle measures and one side length of 6 units. All four side lengths. With all four side lengths known, we still cannot draw exactly the required quadrilateral. 4 4 4 4 4 4 4 4 Even in a simple example where all four lengths are equal, we have infinite possibilities for the angle measures. Curriculum Decision Offering contrasting ideas is an effective technique for improving understanding. A thorough investigation of the properties of quadrilaterals is beyond the scope of Shapes and Designs. However, an exploration of quadrilateral design emphasizes the properties of a triangle and clarifies the way that form and function are related for polygons. Unit Planning UP-13 SAMPLE

STUDENT EDITION MATHEMATICAL GOALS OF THE UNIT Generalizing and Using Properties of Polygons. Understand the properties of polygons that affect their shape and how this information is useful in solving problems. • Investigate techniques for estimating, measuring, and sketching angles and recognizing the effects of measurement accuracy • Recognize and use information about supplementary, complementary, vertical, and adjacent angles to solve problems • Reason about the properties of angles formed by intersecting lines and by parallel lines cut by a third line and how this information relates to polygons • Explore the relationship between interior and exterior angles of a polygon • Explore the relationships between angle measures, angle sums, and the number of sides in a polygon • Determine which polygons fit together to cover a flat surface and why • Draw or sketch polygons with given conditions using various tools and techniques, such as freehand, geoboards, use of a ruler and protractor, and use of technology • Determine what conditions will produce a unique polygon, more than one polygon, or no polygon, particularly triangles and quadrilaterals • Recognize the special properties of polygons that make them useful in building, design, and nature • Solve multistep problems that involve properties of shapes Algebraic Expressions and Equations. Understand how expressions and equations can be useful to express geometric relationships and how this information is used to solve problems. • Use algebraic equations or expressions to represent geometric patterns and solve problems • Recognize that equivalent expressions can reveal different information about a situation and how the quantities are related UP-14 Unit Planning SAMPLE

UNIT ARC OF LEARNING (AoL) UP-15 Shapes and Designs: Generalizing and Using Properties of Geometric Shapes (AoL) Generalizing and Using Properties of Polygons  Algebraic Expressions and Equations  Introduction Setting the Scene Exploration Mucking About Analysis Going Deeper Synthesis Looking Across Abstraction Going Beyond Investigation 1. Designing Polygons: The Side Connection Problem 1.1 Designing Triangles: The Side Connection 1.1 1.1 1.1 Problem 1.2 Designing Quadrilaterals: The Side Connection 1.2 1.2 1.2 Problem 1.3 Rigidity Experiment 1.3 1.3 1.3 1.3 Mathematical Reflection MR Investigation 2. Designing with Angles Problem 2.1 Four in a Row Game: Angles and Rotations 2.1 2.1 2.1 Problem 2.2 The Bee Dance and Amelia Earhart: Measuring Angles and Distance 2.2 2.2 Problem 2.3 Vertical, Supplementary, and Complementary Angles 2.3 2.3 2.3 Mathematical Reflection MR MR Investigation 3. Designing Polygons: The Angle Connection Problem 3.1 Back to the Bees: Tiling a Plane Experiment 3.1 3.1 Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 3.2 3.2 3.2 Problem 3.3 The Ins and Outs of Polygons: Using Supplementary Angles 3.3 3.3 3.3 Problem 3.4 Designing Polygons 3.4 3.4 3.4 Mathematical Reflection MR MR MR SAMPLE

The goals of Shapes and Designs is to generalize the properties of geometric shapes, angles and polygons, and to use this information to solve problems in mathematics, design, and nature. Students build on an informal study of polygons in earlier grades, where the focus was mostly categorizing shapes and angles. This unit provides deeper understanding of angles and the relationship between angles and polygons, which is fundamental to success in designing shapes and solving problems. This understanding will continue to deepen in the Stretching and Shrinking unit as students investigate similar figures (scale drawings). Further understanding continues in the Filling and Wrapping unit as students use polygonal shapes to investigate shape and measurement of three-dimensional figures. This unit develops understanding of what it means to be congruent, and this information is assumed and required for students to be successful in Flip, Spin, Slide, and Stretch, which is about creating similar and congruent figures. In grade 8, in the Looking for Pythagoras unit, students use triangles and squares to develop formulas for finding distance on a coordinate plane, including the Pythagorean Theorem. Algebraic reasoning is used to represent important geometric properties and to use expressions and equations to solve problems. UP-16 Unit Planning SAMPLE

Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvi Investigation 1. Designing Polygons: The Side Connection . . . . . . . 1 Problem 1.1 Designing Triangles Experiment: The Side Connection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 3 Problem 1.2 Designing Quadrilaterals Experiment: The Side Connection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 6 Problem 1.3 Rigidity Experiment � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 10 Mathematical Reflection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �12 Investigation 2. Designing with Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Problem 2.1 Four in a Row Game: Angles and Rotations � � � � � � � � � � � � � � � 23 Problem 2.2 The Bee Dance and Amelia Earhart: Measuring Angles and Distance � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 27 Problem 2.3 Vertical, Supplementary, and Complementary Angles � � � � � 32 Mathematical Reflection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 36 Investigation 3. Designing Polygons: The Angle Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Problem 3.1 Back to the Bees: Tiling a Plane Experiment � � � � � � � � � � � � � � 50 Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 54 Problem 3.3 The Ins and Outs of Polygons: Using Supplementary Angles � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 59 Problem 3.4 Designing Polygons � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 64 Mathematical Reflection � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 67 CONTENTS OF THE STUDENT EDITION UP-17 SAMPLE

GOALS, ARC OF LEARNING, STANDARDS, NOW WHAT DO YOU KNOW?, AND EMERGING MATHEMATICAL IDEAS Investigation 1: Designing Polygons: The Side Connection Goals Generalizing and Using Properties of Polygons. Understand the properties of polygons that affect their shape and how this information is useful in solving problems. • Investigate techniques for estimating, measuring, and sketching angles and recognizing the effects of measurement accuracy • Recognize and use information about supplementary, complementary, vertical, and adjacent angles to solve problems • Reason about the properties of angles formed by intersecting lines and by parallel lines cut by a third line and how this information relates to polygons • Explore the relationship between interior and exterior angles of a polygon • Explore the relationships between angle measures, angle sums, and the number of sides in a polygon • Determine which polygons fit together to cover a flat surface and why • Draw or sketch polygons with given conditions using various tools and techniques, such as freehand, geoboards, use of a ruler and protractor, and use of technology • Determine what conditions will produce a unique polygon, more than one polygon, or no polygon, particularly triangles and quadrilaterals • Recognize the special properties of polygons that make them useful in building, design, and nature • Solve multistep problems that involve properties of shapes Algebraic Expressions and Equations. Understand how expressions and equations can be useful to express geometric relationships and how this information is used to solve problems. • Use algebraic equations or expressions to represent geometric patterns and solve problems • Recognize that equivalent expressions can reveal different information about a situation and how the quantities are related UNIT ALIGNMENT UP-18 SAMPLE