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Geometryassessments Geometry Assessments The Charles A. Dana Center at The University of Texas at Austin With funding from the Texas Education Agency and the National Science Foundation 48810 Frontmatter 59 8/21/02, 11:19 AM 48810 Frontmatter 1 8/21/02, 11:19 AM About the Charles A. Dana Center’s Work in Mathematics and Science The Charles A. Dana Center at The University of Texas at Austin works to support education leaders and policymakers in strengthening Texas edu- cation. As a research unit of UT Austin’s College of Natural Sciences, the Dana Center maintains a special emphasis on mathematics and science education. We offer professional development institutes and produce research-based mathematics and science resources for educators to use in helping all students achieve academic success. For more information, visit the Dana Center website at www.utdanacenter.org. The development of this work is supported in part by the Texas Education Agency, the National Science Foundation under cooperative agreement #ESR-9712001, and the Charles A. Dana Center at The University of Texas at Austin. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessar- ily reflect the views of the Texas Education Agency, the National Science Foundation, or The University of Texas at Austin. Permission is given to any person, group, or organization to copy and distribute this publication, Geometry Assessments, for noncommercial educational purposes only, so long as the appropriate credit is given. This permission is granted by the Charles A. Dana Center, a unit of the College of Natural Sciences at The University of Texas at Austin. First printing August 2002 ©2002, The University of Texas at Austin. All rights reserved. ISBN 0-9707948-8-6 H ii 48810 Frontmatter 2 8/21/02, 11:19 AM Authors Basia Hall, Houston ISD David Eschberger, Region IV Education Service Center Diane Reed, Ysleta ISD Ken Rutkowski, St. Stephen’s Episcopal School, Austin Charles A. Dana Center Production Team Diane McGowan, Editor HeeJoon Kim, Assistant Editor Kathy Park, Proofreader Brandy Windham, Proofreader Phil Swann, Senior Designer Rob Eagleton, Graphics Assistant Aaron Rutkowski, Geometer Sketchpad Graphics Assistant H iii 48810 Frontmatter 3 8/21/02, 11:19 AM Geometry Assessments Advisory Team Heidi Anderson, International School of the Americas, Northeast ISD, San Antonio Brian Barns, Sweetwater High School, Sweetwater ISD Sue Borders, Texas Education Agency, Assessment Division Kathi Cook, Charles A. Dana Center James Epperson, University of Texas at Arlington David Eschberger, Region IV Education Service Center Paula Gustafson, Texas Education Agency, Curriculum and Professional Development Julie Guthrie, Texas Education Agency, Assessment Division Kathy Hale, Region XIV Education Service Center Basia Hall, Houston ISD Mary Alice Hatchett, Round Rock ISD Arthur Howard, Aldine ISD Susan Hudson Hull, Charles A. Dana Center Denise Kraft, New Braunfels ISD Garland Linkenhoger, McAllen ISD Kelli Mallory, Region X Education Service Center Diane McGowan, Charles A. Dana Center Barbara Montalto, Texas Education Agency, Curriculum and Professional Development Joseph Montana, San Antonio USI Pat Miller, Fort Worth ISD Marlene Nelson, Region VII Education Service Center Erika Pierce, Charles A. Dana Center Domingo Ponce, Grand Prairie ISD Diane Reed, Ysleta ISD Barbara Rives, Lamar State College, Orange Elsa Ruiz, Laredo ISD Ken Rutkowski, St. Stephen’s Episcopal School, Austin Carol Salzwedel, Lee High School, Northeast ISD Judy Shahan, Eagle Mt-Saginaw ISD Ann Shannon, University of California, Office of the President Dick Stanley, University of California at Berkeley Walter Stewart, Aldine ISD Susan Thomas, Alamo Heights ISD Judy Wright, Lanier High School, San Antonio ISD With Special Thanks to: Ted Beal, Birdville ISD; Jason Deutsch, Birdville ISD; Donna Harris, Region XI Education Service Center; Pamm Hoffman, Arlington ISD; Cindy McColly, Grand Prairie ISD; Brenda Mesa, Fort Worth ISD; Betsy Norris, Fort Worth ISD; Rebecca Russell, Fort Worth ISD; Joyce Schneider, Carroll ISD; Faith Schwope, Grand Prairie ISD; Caesar J. Thibodaux, III, Pasadena ISD; Melissa Williams, Birdville ISD; Belinda Windham, Birdville ISD H iv 48810 Frontmatter 4 8/21/02, 11:19 AM H v 48810 Frontmatter 5 8/21/02, 11:20 AM TEKS and TAKS Resources The mathematics Texas Essential Knowledge and Skills (TEKS) were developed by the state of Texas to clarify what all students should know and be able to do in mathematics in kindergarten through grade 12. Districts are required to provide instruction that is aligned with the mathematics TEKS, which were adopted by the State Board of Education in 1997 and implemented statewide in 1998. The mathematics TEKS also form the objectives and student expectations for the mathematics portion of the Texas Assessment of Knowledge and Skills (TAKS), which will be implemented in spring 2003 for grades 3 through 10 and for the Grade 11 Exit Level assessment. The mathematics TEKS can be downloaded in printable format, free of charge, from the Texas Education Agency website (www.tea.state.tx.us/ teks). Bound versions of the mathematics and science TEKS are available for a fee from the Charles A. Dana Center at The University of Texas at Austin (www.utdanacenter.org). Resources for implementing the mathematics TEKS, including professional development opportunities, are available through the Texas Education Agency and the Charles A. Dana Center, the state-designated Mathematics Center for Educator Development. Online resources can be found in the Mathematics TEKS Toolkit at www.mathtekstoolkit.org. Additional products and services that may be of interest are available from the Dana Center at www.utdanacenter.org. These include the following: • TEKS, TAAS, and TAKS: What’s Tested at Grades 3–8 charts • Mathematics Abridged TEKS charts • Mathematics TEKS “Big Picture” posters • Mathematics Standards in the Classroom: Resources for Grades 3–5 • Mathematics Standards in the Classroom: Resources for Grades 6–8 • Algebra I Assessments and the corresponding professional development • TEXTEAMS professional development mathematics institutes • TEKS for Leaders professional development modules for principals and other administrators H vi CONTENTS H vii 48810 Frontmatter 7 8/21/02, 11:20 AM CONTENTS Introduction x Solution Guide xiii Mathematics TEKS Alignment xiv Mathematics Grade 11 Exit Level TAKS Alignment xix Chapter 1: Coordinate Geometry 1 Cross Country Cable 5 Whitebeard’s Treasure 13 Quadrilateral Quandary 23 Quite a Quilt 31 Wearable Art 47 Sea Quest 53 Chapter 2: Patterns, Conjecture, and Proof 59 Pizza Delivery Service Regions 63 More Pizza Delivery 77 Conjecture as Discovery and Proof as Explanation 87 Extending the Triangle Midsegment Conjecture 101 Why Doesn’t My Conjecture Work? 109 Steiner’s Point 117 Steiner’s Point Revisited 129 Walking the Archimedean Walk 137 Talking the Archimedean Talk 149 Mad as a Hatter or Hat as a Madder 157 Going the Distance in Taxicab Land 167 Chapter 3: Properties and Relationships 187 of Geometric Figures Angle Bisectors and Parallel Lines 191 Circles and Tangents 199 The Clubhouse 205 Diagonals and Polygons 211 The Most Juice 219 viii 48810 Frontmatter 8 8/21/02, 11:20 AM Chapter 4: Area, Perimeter, and Volume 233 Boxing Basketballs 237 Flower 245 Great Pyramid 253 Walter and Juanita’s Water Trough 257 Greenhouse 263 Nesting Hexagons 267 Chapter 5: Solids and Nets 275 Perfume Packaging 279 Playing with Pipes 293 Circular Security 297 Different Views 301 The Slice Is Right! 307 Chapter 6: Congruence 315 Median to the Hypotenuse of a Right Triangle 319 Tell Me Everything You Can About… 331 Tiling with Four Congruent Triangles 335 Shadow’s Doghouse 345 The School Flag 359 The Shortest Cable Line 369 Chapter 7: Similarity 373 Ancient Ruins 377 Sightseeing Walk 387 Will It Fit? 395 Spotlights 399 References 405 ix 48810 Frontmatter 9 8/21/02, 11:20 AM Introduction The Dana Center has developed Geometry Assessments as a resource for teachers to use to provide ongoing assessment integrated with high school geometry instruction. The National Council of Teachers of Mathematics has identified the following six standards to guide classroom assessment1: Standard 1: Assessment should reflect the mathematics that all students need to know and be able to do. Standard 2: Assessment should enhance mathematics learning. Standard 3: Assessment should promote equity. Standard 4: Assessment should be an open process. Standard 5: Assessment should promote valid inferences about mathematics learning. Standard 6: Assessment should be a coherent process. Implementing these assessment standards may require significant changes in how teachers view and use assessment in the mathematics classroom. Teachers should assess frequently to monitor individual performance and guide instruction. 1. National Council of Teachers of Mathematics. 1995. Assessment Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. H x 48810 Frontmatter 10 8/21/02, 11:20 AM What are the Geometry Assessments? The geometry assessments are problems that reflect what all students need to know and be able to do in a high school geometry course that follows a first-year algebra course. These assessments may be formative, summative, or ongoing. The problems focus on students’ understanding as well as their procedural knowledge. The tasks require more than right or wrong answers; they focus on how students are thinking about a situation. What is the purpose of the Geometry Assessments? The purpose of these assessments is to make clear to teachers, students, and parents what is being taught and learned. Teachers should use evidence of student insight, student misconceptions, and student problem-solving strategies to guide their instruction. Teachers may also use the questions included with the assessments to guide learning and to assess student understanding. The use of these assessments should help teachers enhance student learning and provide them with a source of evidence on which they may base their instructional decisions. What is the format of the Geometry Assessments? This book contains 43 problems.
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