iii Shapes and Designs Generalizing and Using Properties of Geometric Shapes 4 CONNECTED MATHEMATICS® Student Edition SAMPLE
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Student Edition CONNECTED MATHEMATICS 4® 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 | Student Edition | Connected Mathematics 4 ISBN-13: 979-8-89101-109-0 eISBN: 979-8-89101-161-8 v1 Part ID: CMP7-1-4SE Print number: 01 Print year: 2024 Printed in the United States of America. Photo credits: Page xvi and 49 (honeybee on honeycomb), © Vetre Antanaviciute-Meskauskiene/Alamy Stock Photo; page xvi and 15 (subway train), © Bruce Leighty/Alamy Stock Photo; page xvii, © Christopher Kane/Alamy Stock Photo; page 1, (left) © Henryk Sadura/Alamy Stock Photo and (right) © Kirk Fisher/Alamy Stock Photo; page 2, © NATUREWORLD/ Alamy Stock Photo; page 22, © Science History Images/Alamy Stock Photo; page 26, ©. Delta Images/Alamy Stock Photo; page 28, © Wavebreakmedia Ltd PH24/Alamy Stock Photo; page 32 (left) Nikolay Malshakov/Alamy Stock Photo and (right) © CrackerClips Stock Media/Alamy Stock Photo; page 35, © Zoonar/Maksim Lashcheuski/Alamy Stock Photo; page 50, © Andrey Kuzmin/Alamy Stock Photo; page 58 (top image): © Zee/Alamy Stock Photo; (middle left image): Ed Pegg Jr., “The 15 Known Tiling Pentagons,” August 3, 2015. Accessed online. (middle right image): Illustrated by Casey Mann. (bottom image): Alamy Stock Photos. 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
v 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 SAMPLE
vi About the Authors helping teachers develop understanding and agency in the teaching of mathematics. 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®. SAMPLE
vii 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 ASSESSMENT TEAM Mary Bouck, Mathematics Education Consultant, Michigan State University, East Lansing, MI Valerie Mills, Mathematics Education Consultant, Ypsilanti, MI SAMPLE
viii Acknowledgments 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 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 SAMPLE
Acknowledgments ix 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 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 SAMPLE
x Acknowledgments 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 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 SAMPLE
Acknowledgments xi 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 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 SAMPLE
xii Acknowledgments 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 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 SAMPLE
xiii 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 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 13 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 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 37 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 CONTENTS SAMPLE
xiv Contents Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . .68 English/Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 SAMPLE
xv 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 recognize 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, such as freehand, geoboards, rulers, a protractor, and 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 used 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 MATHEMATICAL GOALS SAMPLE
LOOKING AHEAD In 1937, the famous aviator Amelia Earhart tried to become the first woman to fly around the world. She began her journey on June 1, when she took off from Miami, Florida. She reached Lae, New Guinea, and then headed east toward Howland Island in the Pacific Ocean. She never arrived at Howland Island. In 2012, investigators found evidence of the crash on the deserted island Honeybees live in colonies. They build nests called hives. A typical hive might be home for as many as 60,000 bees. Bees are small insects, but packing a hive with that many bees and the honey they make is tricky. Why do the bees form their honey storage tubes in the shape of hexagonal prisms? Why not some other shape? From the Pyramids of Giza to Chichen Itza in Mexico, to the Eiffel Tower in Paris and to the World Trade Building in New York City, shapes are everywhere. Why are some braces on towers, roofs, and bridges in the shapes of triangles and not rectangles or pentagons? PAPUA NEW GUINEA Nauru Solomon Islands Tuvalu AUSTRALIA Fiji Islands Samoa Islands Phoenix Islands Gilbert Islands Vanuatu B Howland Island Lae PACIFIC OCEAN CORAL SEA Nikumaroro Island C E F A D 0 625 mi N S W E of Nikumaroro, far off her intended course. An error may have been made in plotting Earhart’s course. How many degrees off course was she? xvi SAMPLE
The world is filled with shapes and designs. Natural and human-made objects come in an endless variety of geometric shapes. But some shapes have properties that make them especially important in science, engineering, construction, crafts, and arts. Kuggen Building by Wingårdh Arkitektkontor In this unit, we will discover the special importance of shapes called polygons. Polygons are two-dimensional shapes formed by linking points called vertices with line segments called sides. Which of these are polygons? Looking Ahead xvii SAMPLE
Polygons are named by the number of sides they have. Regular polygons have side lengths and angle measures that are equal. If the side lengths are not equal, then the polygon is an irregular polygon. Equilateral Trangia Square Regular Pentagon Regular Hexagon Regular Heptagon Regular Octagon Regular Nonagon Regular Decagon Some simple shapes like triangles and rectangles are used again and again. In this unit, many of the problems focus on triangles and quadrilaterals. We will build models and use rulers, protractors, or angle rulers to draw shapes that meet given conditions. These constructions and drawing experiments will show why triangles are frequently used in engineering and construction. We will look for patterns in the measures of interior and exterior angles in regular and irregular polygons. We will develop formulas to find the angle measures without measuring. While working with interior and exterior angles of polygons, we will use facts about supplementary, adjacent, and vertical angles to find angle measures. As we explore the problems in this unit, ask yourself the following questions: What do these polygons have in common? How do they differ from each other? When should I use estimation, freehand drawing, or special tools to measure and construct angles and polygons? How do the side lengths and angles of polygons determine their shape? Why do certain polygons appear so often in buildings, artistic designs, and natural objects? How can I construct polygons that meet the conditions of a given problem? xviii Looking Ahead SAMPLE
1 If we look around us, triangles are everywhere. What role do triangles play in the following buildings? What are some other examples of the use of triangles? Buildings and bridges need strength and rigidity to make sure they do not collapse as shown in the following structures. Designing Polygons: The Side Connection Leonard P. Zakim Bunker Memorial Bridge in Boston, Massachusetts Eiffel Tower in Paris, France But are there other shapes that can provide rigidity and strength? INVESTIGATION 1 SAMPLE
2 Shapes and Designs In this investigation, we will discover properties of polygons that make them useful in buildings, mechanical devices, and crafts. To explore the connections between shapes and their uses in construction, we will build some polygons using polystrips and fasteners. These tools will allow us to build and study polygons with various combinations of side lengths and angles. Later we will explore which shapes are rigid under pressure. Newport Pier in California near popular surfing spot Polystrips and Fasteners SAMPLE
Investigation 1 Designing Polygons: The Side Connection 3 1.1 Designing Triangles Experiment: PROBLEM The Side Connection INITIAL CHALLENGE The best way to discover what is so special about triangles in construction is to build several models and explore the relationship among the side lengths. Use polystrips or other tools to make and study several triangles. Make a Prediction • Will any three side lengths make a triangle? Conduct the Experiment Equipment › polystrips › fasteners (also called brads) Directions › Step 1. Pick three numbers between 2 and 20 for side lengths. Use them as side lengths to build a triangle. › Step 2. Try to make a triangle with the chosen side lengths. If you can build a triangle, try to build a different triangle with the same side lengths. › Repeat Steps 1 and 2 to make and study several other triangles. Record your data in a table with headings similar to the one here. Designing Triangles Experiment Side Lengths Triangle Possible? Sketch Different Triangle Possible? SAMPLE
4 Shapes and Designs Analyze the Data • What pattern do you see that explains why some sets of numbers make a triangle and some do not? Does it agree with your prediction? • If a set of three numbers makes a triangle, is it possible to make another triangle with a different shape? Explain why. • Will your pattern work on the following triangles? Explain why. An equilateral triangle An isosceles triangle A right triangle 1.1 WHAT IF . . . ? Situation A. Building a Tent Geraldo is building a tent. He has • four 3-foot poles; • two 5-foot poles; • one 6-foot pole; and • one 7-foot pole. He wants to make a triangularshaped doorframe. 1. How many ways can he make a triangular doorframe? 2. Which triangle would you use to make the doorframe? Explain your reasoning. 5 ft 3 ft 6 ft 7 ft NOW WHAT DO YOU KNOW? What combinations of three side lengths can be used to make a triangle? How many different triangles are possible for each combination? SAMPLE
Investigation 1 Designing Polygons: The Side Connection 5 When we watch a hockey game at full speed, it is difficult to follow the movement of the puck and explain what is happening in the game. Sport fans often see instant replays on TV where sport commentators add graphics to highlight the player movements and positioning of an important play. As shown in the illustrations, sport coaches often refer to triangle support as a strategy for scoring a goal. Triangle support involves three players often near the hockey goal. Ideally, triangles provide at least two options for hockey players to pass. If the players were spread out in a line, their options would be minimal. While every play is different in hockey, creating passing options is important offensively. Close support in the triangle is often key. If the lengths of the triangle were relatively long, there would be more opportunity for the puck to be intercepted by the other team. If the angles on the triangle were smaller, then a single player on the other team can defend both sides of the triangle. Notice how the hockey players created triangle supports based on the positioning of the other team. Did You Know? 1.1 SAMPLE
6 Shapes and Designs Quadrilaterals, especially rectangles, appear throughout buildings in which we live, work, and go to school. We see rectangles as the mortar around bricks, the frames of windows, and the outlines of large buildings. Rectangles have very different physical properties from triangles. For example, in Problem 1.1 we learned that the sum of any two side lengths of a triangle is greater than the third side length. Do quadrilaterals have the same relationship among their side lengths as triangles? What properties do quadrilaterals have that make them useful? Designing Quadrilaterals Experiment: The Side Connection PROBLEM 1.2 Equipment › polystrips › fasteners (also called brads) Directions › Choose any four numbers between 2 and 20 for side lengths. Use them as side lengths to build a quadrilateral. › If you can build a quadrilateral, try to build a different quadrilateral with the same side lengths. INITIAL CHALLENGE The best way to discover what is so special about quadrilaterals in construction is to build several models and explore the relationship among the side lengths. Use polystrips to make and study several quadrilaterals. Make a Prediction • Will any four side lengths make a quadrilateral? Conduct the Experiment SAMPLE
Investigation 1 Designing Polygons: The Side Connection 7 Analyze the Data • What pattern do you see that explains why some sets of numbers make a quadrilateral and some do not? Does this agree with your prediction? • What combination of side lengths is needed to build squares? Rectangles? Parallelograms? • Test your conjectures on the following side lengths. Which ones form a quadrilateral? Are any of these squares? Rectangles? Parallelograms? Explain why. Shape 1 6, 10, 15, 15 Shape 2 3, 5, 10, 20 Shape 3 8, 8, 10, 10 Shape 4 12, 20, 6, 9 › Record the data in a table similar to the one shown here. › Make several other quadrilaterals and record the data in the table. Designing Quadrilaterals Experiment Side Lengths Quadrilateral Possible? Sketch Different Quadrilateral Possible? WHAT IF . . . ? Situation A. Testing More Polygons Michal wondered if there was a side-length relationship for other polygons. She used polystrips to build pentagons and hexagons. 1. What are some side lengths that make pentagons? Hexagons? 2. What relationship among side lengths could she observe? 3. What is the largest polygon your group can make with your set of polystrips? How many sides does it have? 4. Look back at the angle sizes of the polygons you have made. What do you notice about the measure of the angles as the number of sides of a polygon increases? 1.2 SAMPLE
8 Shapes and Designs 1. What do you think happened when Beatrice pushed down on the vertex of a triangle? On a side or vertex of a quadrilateral? 2. What would happen if she repeated this test on a pentagon or hexagon? Beatrice’s Experiment I wanted to know how the rigidity of triangles and quadrilaterals reacts to stress. I thought of the force applied to a roof with heavy snow or the force applied to a bridge by a car or train. So, I designed an experiment. I chose three different lengths that I knew would create a triangle. And I chose four different lengths that I knew would create a quadrilateral. I wanted to know what would happen when I apply force to a side or vertex. Push Down Push Down Situation B. Beatrice Worries About Rigidity Mechanical engineers use the fact that quadrilaterals are not rigid figures to design linkages. Here is an example of a quadrilateral linkage made from polystrips. Did You Know? Crank Crank A B C D Coupler Frame One of the sides is fixed. It is the frame. The two sides attached to the frame are the cranks. One of the cranks is the driver and the other the follower. The fourth side is called the coupler. In 1883, the German mathematician Franz Grashof suggested an interesting principle for quadrilateral linkages. Find the sum of the lengths of the shortest and longest NOW WHAT DO YOU KNOW? What do we know about side-length relationships among the sides of a polygon? 1.2 SAMPLE
Investigation 1 Designing Polygons: The Side Connection 9 sides. If that sum is less than or equal to the sum of the other two sides, then the shortest side can rotate 360°. Quadrilateral linkages are used for windshield wipers, automobile jacks, and reclining lawn chairs. hip knee ankle bike gear 1.2 SAMPLE
10 Shapes and Designs PROBLEM 1.3 Rigidity Experiment INITIAL CHALLENGE When Beatrice was building polygons in Problem 1.2, she noticed that if she added one diagonal to a quadrilateral, the shape was rigid. Recall that a diagonal is a line segment that joins two nonadjacent vertices of a polygon. She wondered if adding diagonals to other polygons that were not rigid would make them rigid. Make a Prediction • If a polygon with n sides is not rigid, what is the minimum number of diagonals needed to make it rigid? Conduct the Experiment Equipment › polystrips › fasteners (also called brads) Directions › Build a triangle, quadrilateral, pentagon, and hexagon with your polystrips. What is the least number of diagonals needed to make it rigid? Record the data in a table. › Repeat by building a different quadrilateral, pentagon, and hexagon. Record the data in a table. Shape The Minimum Number of Diagonals Make a Sketch triangle quadrilateral pentagon hexagon SAMPLE
Investigation 1 Designing Polygons: The Side Connection 11 Analyze the Data • Was your prediction correct? That is, what is the minimum number of diagonals needed to make a polygon rigid? • If the pattern continues, fill in the table with the minimum number of diagonals needed for a heptagon, octagon, nonagon, and n-gon. • Test your answer by building one of these polygons and adding the minimum number of diagonals. Does it work? WHAT IF . . . ? Situation A. Beatrice Again Shape The Minimum Number of Diagonals Make a Sketch heptagon octagon nonagon n-gon Beatrice’s Observation I noticed that when I add the minimum number of diagonals to a shape, I am creating several embedded triangles. I think this has something to do with why the shape is now rigid. Do you agree with Beatrice? Explain why. NOW WHAT DO YOU KNOW? What do you know about the rigidity of shapes, and how does this explain the frequent use of triangles in building structures? 1.3 SAMPLE
12 Shapes and Designs MR 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? SAMPLE
Investigation 1 Designing Polygons: The Side Connection 13 INVESTIGATION 1 APPLICATIONS 1. For each set of side lengths, follow these directions. • If possible, build a triangle with the side lengths. Sketch your triangle. • Tell whether your triangle is the only one that is possible. Explain. • If a triangle is not possible, explain why. a. 5 cm, 5 cm, 3 cm b. 8 cm, 8 cm, 8 cm c. 7 cm, 8 cm, 15 cm d. 5 cm, 6 cm, 10 cm e. 3 cm, 4 cm, 5 cm f. 2 cm, 4 cm, 6 cm 2. From Exercise 1, which sets of side lengths can make each of the following shapes? a. an equilateral triangle (all three sides equal length) b. an isosceles triangle (two sides equal length) c. a scalene triangle (no two sides equal length) d. a right triangle (having one right angle) 3. Multiple Choice Which of the following could not represent the side lengths of a triangle? A. 5 cm, 5 cm, 5 cm B. 3 cm, 7 cm, 2 cm C. 6 cm, 8 cm, 10 cm D. 4 cm, 7 cm, 4 cm APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) SAMPLE
14 Shapes and Designs 4. Multiple Choice Which of the following could represent the side lengths of a triangle? A. 2 cm, 1 cm, 1 cm B. 2 cm, 3 cm, 4 cm C. 7 cm, 12 cm, 3 cm D. 4 cm, 4 cm, 10 cm 5. Name a set of three lengths that would not form a triangle. How do you know that these lengths would not work? 6. Name a set of lengths that could be used to form a triangle. How do you know that these lengths would work? 7. For each set of side lengths, follow these directions. • If possible, build a quadrilateral with the side lengths. Sketch your quadrilateral. • Tell whether your quadrilateral is the only one that is possible. Explain. • If a quadrilateral is not possible, explain why. a. 5 cm, 5 cm, 8 cm, 8 cm b. 5 cm, 5 cm, 6 cm, 14 cm c. 4 cm, 3 cm, 5 cm, 14 cm d. 8 cm, 8 cm, 8 cm, 8 cm e. 5 cm, 5 cm, 5 cm, 20 cm 8. From ACE 7, which sets of side lengths can make each of the following shapes? a. a square b. a quadrilateral with all angles the same size c. a parallelogram d. a quadrilateral that is not a parallelogram 9. A quadrilateral with four equal sides is called a rhombus. Which set(s) of side lengths from ACE 7 can make a rhombus? 10. A quadrilateral with just one pair of parallel sides is called a trapezoid. Which sets of side lengths from ACE 7 can make a trapezoid? ACE SAMPLE
Investigation 1 Designing Polygons: The Side Connection 15 11. Multiple Choice Which of the following could not represent the side lengths of a quadrilateral? A. 5 cm, 5 cm, 5 cm, 5 cm B. 3 cm, 7 cm, 2 cm, 1 cm C. 6 cm, 8 cm, 10 cm, 4 cm D. 4 cm, 7 cm, 4 cm, 7 cm 12. Multiple Choice Which of the following could represent the side lengths of a quadrilateral? A. 3 cm, 1 cm, 1 cm, 1 cm B. 2 cm, 3 cm, 4 cm, 5 cm C. 7 cm, 12 cm, 3 cm, 1 cm D. 2 cm, 2 cm, 10 cm, 2 cm 13. Draw the polygons described to help you answer the questions. a. To build a square, what must be true of the side lengths? b. Suppose you want to build a rectangle that is a square. What must be true of the side lengths? c. Suppose you want to build a rectangle that is not a square. What must be true of the side lengths? 14. Think about your polystrip experiments with triangles and quadrilaterals. What explanations can you now give for the common use of triangular shapes in structures like bridges and towers for transmitting cell phone and television signals? ACE SAMPLE
16 Shapes and Designs CONNECTIONS 15. Copy the fractions. Insert <, >, or = to make a true statement. a. __5 12 __9 12 b. __15 35 __12 20 c. __7 13 __ 20 41 d. __ 45 36 __ 35 28 16. Find two equivalent fractions for each fraction. Find one fraction with a denominator less than the one given. Find another fraction with a denominator greater than the one given. a. __4 12 b. __9 15 c. __15 35 d. __ 20 12 17. Consider the inequality x < 20. Determine whether the following values for x make this inequality true or false. For each value, explain your reasoning. a. 7 b. 20 c. 28 18. Consider the inequality x > 50. Determine whether the following values for x make this inequality true or false. For each value, explain your reasoning. a. 7 b. 50 c. 51 ACE SAMPLE
Investigation 1 Designing Polygons: The Side Connection 17 19. The following diagram shows two 1-acre squares (Sections 18 and 19) of Tupelo Township. Section 18 Section 19 Lapp Bouck Wong Krebs Stewart Fitz Fuentes Gardella Burg Walker Foley Theule a. Each 1-acre section is about a 70-yard by 70-yard square. How many square feet of land does Lapp own? b. What is the average amount of land owned by landowners in Section 18? 20. Find the area and perimeter of each rectangle. 6 cm 2 cm 2 cm 3 cm 2 cm 3 cm 3 cm L P R M N Q 4 cm 1 cm 2 cm 7 cm 8 cm ACE SAMPLE
18 Shapes and Designs EXTENSIONS 21. Compare the three quadrilaterals. How are the quadrilaterals alike and different? Quadrilateral 1 Quadrilateral 2 Quadrilateral 3 22. 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. 23. Build the following figure from polystrips. The vertical sides are all the same length. The distance from B to C equals the distance from E to D. The distance from B to C is twice the distance from A to B. a. Experiment with holding various strips fixed (one at a time) and moving the other strips. In each case, tell which strip you held fixed, and describe the motion of the other strips. b. Fix a strip between points F and B, and then try to move strip CD. What happens? Explain why this occurs. F E D A B C ACE SAMPLE
19 In Investigation 1 we used experiments to gain information about relationships among the side lengths of triangles and quadrilaterals and which of the shapes were rigid. We collected data and represented the data using tables. We used the patterns we found in the data to model and make predictions about the shapes. Experiments often start with a question that is answered by conducting an experiment to collect data, representing the data, looking for patterns, making predictions, and drawing conclusions. In the grade 6 unit Data About Us, we used representations such as bar graphs, line plots, histograms, and box plots to look more closely at the variability among the data. As we move through the year, we will continue to use data and modeling to solve problems. In the Samples and Population unit, we will look at how to draw samples of data to represent a population. In this investigation, we will look at angles. We will use this knowledge to explore how angles affect shape in Investigation 3. Many Olympic events feature athletes performing exciting flips and spins, including gymnastics, figure skating, snowboarding, and aerial skiing. Judges, competitors, and fans describe the challenge of a flip or spin with numbers like 180, 360, 540, 720, 900, and 1,080. Designing with Angles INVESTIGATION 2 SAMPLE
20 Shapes and Designs A Board Rotating in a Spin A Board Rotating in a Spin 90° 60° 45° 30° 15° 0°/360° 330° 315° 270° 285° 240° 220° 200° 180° 165° 135° 110° Measuring flips and spins involves thinking about an angle as a change in direction called a rotation. In earlier grades, we learned to measure an angle or a rotation with a unit called the degree. The symbol for a degree is a small, raised circle °. We can think of angles as rotations or turns. Angles are measured from 0 degrees to 360 degrees (one full turn) or more. 90° 60° 45° 30° 15° 0°/360° 330° 315° 270° 285° 240° 220° 200° 180° 165° 135° 110° 60° angle of rotation We can think of an angle as the sides of a wedge, like the cut sides of a slice of pizza. ray ray vertex We can think of an angle as a point with two sides extending from the point, like branches on a tree. vertex ray ray Angles are usually marked with a curved angle line indicating a counterclockwise rotation. Since there are two angles indicated by two rays, curved lines are also used to indicate which angle is being used. A one-quarter rotation is 90°. A right angle measures 90°. Right angles are commonly marked with a small square. Suppose we draw a ray to divide a right angle into two angles of equal measure. Each angle would be a 45o angle. We SAMPLE
Investigation 2 Designing with Angles 21 can think of a right angle as a 90° angle or a one-quarter turn. A 45° angle would be a one-eighth turn around the center of a circle. 90° 45° A B C We can identify an angle with the letter used to name its vertex. Angle A or ∠A is a right angle. Angle B or ∠B is a right angle, and angle C or ∠C is a 45° angle. Suppose we draw 89 rays to divide a right angle into 90 angles of equal measure. Each angle would have a measure of 1°. 45° 45° 15° 15° 15° 5° 5° 5° 1° 1° 1° 1° 1° 1° A rotation of one-half turn defines a straight angle. It measures 180°. 180° 180° SAMPLE
22 Shapes and Designs Recall that angles whose measures are less than 90° are called acute angles. Angles whose measures are between 90° and 180° are called obtuse angles. Angles whose measures are more than 180° but less than 360° are called reflex angles. Can you jump and turn through angles of 90°, 180°, 270°, or even 360°? The ancient Babylonians measured angles in degrees. They set the measure of an angle that goes all the way around a point to 360°. They may have chosen 360° because their number system was based on the number 60. They may have also considered the fact that the number 360 has many factors. This makes it easy to measure many fractions of a full turn. It is conjectured that they choose 360 because it was approximately the number of days it takes Earth to rotate around the sun. Did You Know? SAMPLE
Investigation 2 Designing with Angles 23 2.1 PROBLEM Four in a Row Game: Angles and Rotations Estimating and measuring rotation angles is easier if we know some benchmark angles. Playing the Four in a Row game will help us build our angle sense. The Four in a Row game is played on the circular grids shown here. Four in a Row 45º Game Board Four in a Row 30º Game Board 45° A 45° Rotation or Turn Circular Grid O 1 2 3 D 0° O 60° E 30° 0° A 30° Rotation or Turn Circular Grid 1 2 3 The grid on the left has lines at 45° intervals. The grid on the right has lines at 30° intervals. The circles are numbered 1, 2, and 3 as you move out from the center at 0. Point D has coordinates (2, 45°). What is the location of point E? Common angles like 30°, 45°, 60º, 90°, 120°, 180°, and 270 are often used to make estimates of angle sizes. They are called benchmark angles. The following game will help provide some insights into angles of rotation. SAMPLE
24 Shapes and Designs • Play Four in a Row several times. Play games with the 30° and the 45° grids. Describe any winning strategies you used. • Use the circular grids to show angles A, B, C, D, and E described here. Will everyone in class have the same points marked to show the angles? Why or why not? Angle A The angle measure for angle A is greater than 120°. Angle B The angle measure for angle B is equal to 0. Angle C The angle measure for angle C is less than 90° and greater than 30°. Angle D Angle D is a rotation of 1.5 turns. Angle E Angle E is a rotation of 2 turns. Rules Teams can be partners or small groups. › Team 1 chooses a point where a circle and grid line meet. Team 1 says the coordinates of the point. › Team 2 makes sure that the coordinates are correct. If they are, Team 1 marks the point with an X. If they are not, Team 1 does not mark the point. › Team 2 chooses a point where a circle and grid line meet. Team 2 says the coordinates of the point. › Team 1 makes sure that the coordinates are correct. If they are, Team B marks the point with an O. If they are not, Team 2 does not mark the point. › Players repeat the steps. › The first team to get four marks in a “row,” either along a grid line or around a circle, wins the game. 2.1 INITIAL CHALLENGE Four in a Row Game Equipment › Four in a Row Game Boards › pencil, pen, or marker SAMPLE
Investigation 2 Designing with Angles 25 WHAT IF . . . ? Situation A. Using Benchmark Angles to Estimate 1. On a circular grid, mark the benchmark angles 30°, 45°, 60º, 90°, 120°, 180°, and 270°. 2. Amit claims he used benchmark angles to estimate the measure of each angle. Is that possible? Explain. Angle W or ∠W Angle X or ∠X W X Angle Y or ∠Y Angle Z or ∠Z Y Z Situation B. Students’ Estimation Claims Examine each claim. Do you agree with them? Explain why. Juan’s Claim In Situation A, angle W is about 30 degrees more than a _1 2 turn. Maria’s Claim In Situation A, angle W is about 45º less than a _1 2 turn. Tabia’s Claim Angle W is about 30 degrees more than a _1 4turn. Amit’s Benchmark Claim I can use benchmark angles to sketch a rotation angle with approximately the following measures. 220° 50° 130° 300° 2.1 SAMPLE
26 Shapes and Designs Situation C. Sketching Angles Kwun challenges his classmates to sketch an angle with the following directions. Sketch each angle, and label its measure in degrees. Angle 1 Angle 2 Angle 3 Angle 4 Angle 5 One-third of a right angle One and a half times a right angle Three times a right angle Three and a half times a right angle Twice a straight angle Situation D. The Algebra Connection Hank was designing angles and remembers the Algebra Connection game he created for some of the grade 6 units. He added a few more cards to his Algebra Connection game. For each of the following equations, N is the angle measure of a rotation angle. Find a value for N that makes each statement true. Card 1 Card 2 Card 3 What is N? N + 120° + 360° N is a rational number. What is N? N is a rational number. 3 2 N = 45° What is N? N is a rational number. 1 3 N = 20° 2.1 NOW WHAT DO YOU KNOW? How are benchmark angles useful in solving problems? The circular grids used to play Four in a Row are examples of polar coordinate systems. Sir Isaac Newton used polar coordinates in his contributions to mathematics and science. Polar coordinates are commonly used to locate ships at sea, planes in the air, or rain and snowstorms. An object appearing on a radar screen is a moving point or region. It has direction (in degrees) and distance from the radar site. Did You Know? SAMPLE
Investigation 2 Designing with Angles 27 2.2 PROBLEM The Bee Dance and Amelia Earhart: Measuring Angles and Distance Research has shown that honeybees give each other directions to flowers by performing a lively dance! Karl von Frisch was awarded a 1973 Nobel Prize for decoding how the dance uses angles to communicate to hive-mates the location of pollen and nectar sources. Honeybees live in colonies. Each colony has a single queen and thousands of worker bees. The worker bees find flowers to gather nectar and pollen. The nectar is used to make honey. Worker bees build the honeycomb and keep the beehive clean. They take care of the queen bee and the young. They also guard the hive against intruders. On returning to the hive, a bee does a dance to communicate the direction to flowers. The dance is called the waggle dance because of the way the bee moves. To the bees, straight up represents the position of the sun. The bee waggles at an angle away from the sun. This shows the direction to the flowers. The bee repeats the angle, making two semicircles as it turns. The length of the straight run is proportional to the distance to the food source. Unlike bees, we need tools to measure angles. Two common tools for measuring angles are an angle ruler and a protractor. 0 1 2 3 4 5 6 0 1 2 3 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 100 80 110 120 130 140 150 160 170 170 160 150 140 130 120 110 100 80 90 70 60 50 40 30 20 10 Angle Ruler Center Line Center Line Rivet Angle Ruler 90 150 120 60 30 B C Protractor A V 180 0 Protractor Honeybee Dance Direction of Flowers Hive Flowers Honeybee Dance SAMPLE
28 Shapes and Designs INITIAL CHALLENGE A worker bee has located flowers with nectar and is preparing to do her dance. The following picture shows the bee’s search for honey on several different days. The dotted lines represent the angle formed by the hive, the sun, and the flowers. Sun Monday Sun Tuesday Sun Wednesday Sun Thursday Flowers Flowers Flowers Flowers Hive Hive Hive Hive • Estimate the measure of each angle, and then use an angle ruler or protractor to check your estimates. • Suppose on Friday, the angle formed by the two rays to the sun and flower is 180º. Make a sketch of the angle using the hive and sun. • Repeat for an angle of 120º on Saturday. The angle ruler’s formal name is goniometer (goh nee AHM uh tur). Goniometer is Greek for “angle measurer.” Doctors and physical therapists use goniometers to measure flexibility (range of motion) in knees, elbows, fingers, and other joints. Did You Know? 2.2 SAMPLE
Investigation 2 Designing with Angles 29 WHAT IF . . . ? Situation A. Looking for Amelia Earhart Bees know a lot about angles, but angles are also important to humans. In 1937, the famous aviator Amelia Earhart tried to become the first woman to fly around the world. She began her journey on June 1, when she took off from Miami, Florida. She reached Lae, New Guinea, and then headed east toward Howland Island in the Pacific Ocean. She never arrived at Howland Island. In 2012, investigators found evidence of the crash on the deserted island of Nikumaroro, far off her intended course. An error may have been made in plotting Earhart’s course. 2.2 1. Estimate the number of degrees off course Earhart’s crash site was from her intended destination. Check your estimate with a measuring tool. 2. Amelia Earhart apparently flew several degrees south of her intended course. Suppose you start at New Guinea and are trying to reach Howland, but you fly 20° south of your intended course. On which island might you land? How can you use an angle ruler or protractor to find the island? Situation B. The Flight of Two Planes Suppose two planes fly the paths formed by the rays of the angle shown on the map. The planes leave at the same time. They fly at the same speed. The map shows their progress every four hours. Amit uses the scale on the map to estimate the distance between the planes. 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 SAMPLE
30 Shapes and Designs 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 About 80 miles About 160 miles About 240 miles 4 hours 8 hours 12 hours Lae 1. Record the data in a table like this one. Time and Distance Apart Time (hours) 4 8 12 t Distance Apart (miles) d 2. Look for patterns in the table. Use words to describe the relationships in the pattern. 3. Write an equation to represent the relationship between the time t in hours and the distance d apart. 4. Use your equation to find how far apart they are after 10 hours. NOW WHAT DO YOU KNOW? What are the advantages and disadvantages of estimating angle measures? What are the advantages and disadvantages of using a protractor or angle ruler to measure an angle? 2.2 SAMPLE
Investigation 2 Designing with Angles 31 The famous explorer Robert Ballard has found many famous and important sunken ships including the Titanic and the carrier USS Yorktown, the patrol boat that was used by President John F. Kennedy in the Solomon Sea. He has also found ancient boats in the Black Sea filled with mariner skeletons. On August 7, 2019, Ballard—in partnership with the National Geographic Partners and National Geographic Society—went to Nikumaroro, an uninhabited Phoenix Island in the middle of the western Pacific Ocean, to solve the mystery of Amelia Earhart’s disappearance. The article reported that they found debris from an old shipwreck, which also included a soda can. What their search did not find was a single piece of the Lockheed Electra airplane flown in 1937 by Amelia Earhart. Ballard claims it was not a failure, as now they know where the plane is not located. In January 2024, Deep Sea Vision, a Charleston, South Carolina-based team, reported that they used sonar imaging to capture an image in the Pacific Ocean that "appears to be Earhart's Lockheed 10-E Electra" aircraft. The company scanned more than 5,200 square miles of the ocean floor to reveal a plane-shaped image about 16,000 feet below sea level. The location was about 100 miles from Howland Island. The next step is to investigate the area more where they captured the image. Meanwhile the mystery continues. Sources: Horton, Alex. “This Explorer Found the Titantic. His New Mission: Solve Amelia Earhart’s Disappearance.” Washington Post, July 24, 2019. Smith, Stephen. “Amelia Earhart Plane Possibly Detected by Sonar, Exploration Team Says,” CBS News, February 2, 2024. Accessed online. Did You Know? 2.2 SAMPLE
32 Shapes and Designs PROBLEM 2.3 Vertical, Supplementary, and Complementary Angles Parallel lines are everywhere. Real-life examples of parallel lines are railroad tracks and rows in solar panels. The lines on a basketball court are also parallel. Recall that any two lines that run next to each other and remain the same distance apart and never meet, even if extended forever in both directions, are parallel. Without parallel lines, very few things would be straight. How do we determine or construct parallel lines? A way to explore this question is to use lines that cross other lines. Then we can examine the angles that are formed. Here are some examples. The angles on opposite sides of a vertex are called vertical angles. Vertical Angle Examples 4 1 3 2 4 1 3 2 ∠4 and ∠2 ∠1 and ∠3 Two angles that have a common side and a common vertex (corner point) and do not overlap are adjacent angles. Adjacent Angle Examples 4 1 3 2 4 1 3 2 ∠1 and ∠2 ∠2 and ∠3 SAMPLE