Olson, Melfried Geometric Selections for Middle School Leac

DOCUMENT RESUME ED 210 166 SE 035 844 AUTHOR Michele, Douglas B.: Olson, Melfried TITLE Geometric Selections for Middle School leachers (5-9). The Curriculum Series. INS1ITUTION National Education Association, Washington, D.C. REPCET NO ISBN-0-8106-1720-X POB DATE 81 NOTE 96p.; Not available in paper copy due tc copyright restrictions. AVAILABLE FROM National Education Association, 1201 Sixteenth St., N.V., Washington, DC 20036 (Stock No. 1720-1-00: no price quoted) . Enps PRICE MF01 Plus Postage. PC Not Available from EDES. DESCRIPTORS Elementary Secondary Education: *Geometric Concepts: *Geometry: Instructional Materials: Learning Activities: Mathematical Enrichment; MatheMatics Curriculum: *Mathematics Education: *Mathematics Instruction: *piddle Schools: Problem Solving: *Teaching Guides: T.:aching Methods AiNCT ThiE document is written for middle school teachers of grades five through nine who do not have specialized backgrounds in geometry. It is arranged in three parts. The first part provides a brief overview of tha geometry curriculum of the middle school that includes the present state of affairs, a rationale for inclusion of geometry in the curriculum, the geometry that is suggested for instruction, and suggestions for teaching methods. Part two covers the following selected topics: axiomatic systems and models, distance, congruence, constructions, and transformational geometry. The material in this part is not designed for immediate use, but requires adaptation to particular classroom settings. Most sections include suggested exercises, learning activities, and selected references. The third part is an extensive bibliography of references for both readings and addi'ional activities in geometry. (MP) *********************************************************************** Reproductions supplied by EDRS are the best that can be made * from the origincl document. ***************w******************************************************* V Geometric Selections for Middle School Teachers (5-9) ,l---, ,.., by Douglas B. Aichele and Melfried Olson The Curriculum Series U.S DEPARTMENT OF EDUCATION NATIONAL INSTITUTE OF EDUCATION "PERMISSION TO REPRODUCE THIS EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) MATERIAL IN MICROFICHE ONLY .....,The document has been reproduced as HAS BEEN GRANTED BY received from the person or organization originating d G. Fe I 4DIN Minor changes have been made to improve reproduction quality Points of new or opinions stated in this docu ment do not necessarily represent official ME TO THE EDUCATIONAL RESOURCES position or poky INFORMATION CENTER (ERIC) National Education Association Washington, D.C. 2 Copyright t 1981 National Education Association of the United States Stock No.. 1720 X-00 Note The opinions expressed in this publication should not be construed as representing the policy or position of the National Education Association. Materials published as part of the NEA Curriculum Series are intended to be discussion documents for teachers who are concerned with specialized interests of the profession. Acknowledgment Excerpts from "Geometry, YesBut How?" by Lewis B. Smith, reprinted fromThe Arithmeii' Teacher,February 1967, copyright © 1967 by the National Council of Teachers of Mathematics. Used by permission. Library of Congress Cataloging in Publication Data Aichele, Douglas B Geometric selections for middle school teachers. (Curriculum series) Includes bibliographies. I. GeometryStudy and teaching. I. Olson, Melfried, joint author. II.Title. III. Series. QA461.A34 516'.007'12 8C-29059 ISBN 0-8106-1720X 3 -.... , , TO KATHRYN ADAM CLINT JUDY TANYA -TRAVIS 4 The Authors Douglas B. Aichele is a Professor and Head of the Department of Curriculum and Instruction, Oklahoma State University. Stillwater. Dr. Aichele isthe editor of Mathematics Teacher Education: Critical Issues and Trendspublished by NEA. Me 'fried Olson is with the Science and Mathematics Teaching Center, University of Wyoming. Laramie. The Advisory Panel Judy Adams, Mathematics teacher, Laramie Junior High School, Wyoming John C. Arch, 6th grade teacher, Park Avenue School, Nashville, Tennessee He ler Neely Cheek, Assistant Professor, Oklahoma State University, Stillwater Genevieve M. Ebbert, 4th and 5th grade teacher, Lexington Public Schools, Mas- sachu.ietts Ray Kurtz, Professor of Curriculum and Instruction, Kansas State University, Manhattan Lee E. Vochko, Algebra and Geometry teacher, Capistrano Valley High School, Mission Viejo, California 5 Contents Introduction 8 Part I The Geometry Curriculum of the Middle School An Overview and Commentary 9 Part H Selected Topics Axiomatic systems and Models 15 What Is an Axiomatic System? 15 Two Familiar Models 21 Distance 25 What Is Distance? 25 Spherical Distance 25 Euclidean Distance 28 Taxicab Distance 33 Congruence 37 What Is Congruence? 37 Triangle Congruence Criteria 41 Triangle Incongruehce 53 Constructions 59 What Are the Basic Constructions? 59 Exploring Triangle-Circle Relationships 64 Transformational Geometry 76\ What Is a Slide? 76 What Is a Flip? 78 What Is a Turn? 82 Exploring Successive Motions 84 Part IIIBibliography 91 a. Introduction This publication is written for middle school teachers of grades 5-9 without specialized backgrounds in geometry. We have arranged it in three parts, hoping that children and young adolescents will thus have an opportunity to experience-some of the activities and suggestions which we have presented. Part 1 is a brief overview of the geometry curriculum of the middle school including the present state of affairs, a rationale for inclusion of geometry in the curriculum, the geometry that we believe should be taught, and suggestions for teaching it. Part 11 is a presentation of several selected topics for study: axiomatic systems and models, distance, congruence, constructions, and transformational geometry. This material is specifically not organized for immediate classroom use, but rather requires adaptation to the particular classroom setting. Most of the sections include thought- provoking supportive exercises worthy of discussion, activities that are ready for immediate classroom use, and references for further study. We purposely omitted all "proofs" from the written presentations, though many of the exercises provide oppor- tunities for this activity. Part III is an extensive collection of references for additional activities as well as readings in the area of geometry. We hope this bibliography will assist teachers in enriching their own geometry curriculum and in developing their own activities. , Douglas B. Aichele Meifried Olsen . 7 8 An Overview andCommentary PRESENT STATE OF AFFAIRS The word "geometry" is from the Greek word "geometrein," whichis composed of tne parts *geo" (earth) and "metrein" (to measure),since geometry was originally the science of measuring the land. It has a rich and beautiful historicaldevelopment which grew primarily out of the human quest tounderstand and quantify the environment. In geometry, as in other subjects, the emphasis on thepragmatic too often overrides the aesthetic in our instruction. Although geometry is a very practical subject, thehistoryof the development of geometry is as interesting as virtually any other voyagethrough time. The most complete attempt to assess individuals' geometric knowledge wasundegItken by the Education Commission of the States through the National Assessment ofEduca- tional Progress. The following two excerpts provide information about whatstudents know and can do. Most of the 'geometry items given to 9-year-olds involved identification of terms. Nearly all could name a circle and triangle, but only aboutone-fourth could identify a cone, sphere, or ellipse. Almost half of the childrencould select a pair of parallel lines from a set of several differentconfigurations of line pairs, but only 20% of them successfully drew a line parallel to agiven line. About one-third of the children could select a right anglefrom the alternatives of right, acute, or obtuse angles (4, p. 4). Some geometric figures are recognized by most 17-year-olds, butthey do not know many other geometric facts. Nearly allthe students named a circle, 75% named a cone, and about 50% named a cylinder.Fewer than one--half named either a cube or a sphere. About 75% knew that the diameterof a circle is twice the length of a radius. Fewer than 40% couldbisect an angle using a straightedge and a compass. Slightly more than one-fourth ofthe 17-year-olds found the area of a square with a perimeter of 12 inches (4, p.81). These statements provide us with indications that some geometryis taught to and learned by students, but they also indicate that improvements can be made concerningthe teaching and learning of geometry. The National Council ofTeachers of Mathematics conducted a study in 1975 of second and fifth grade classrooms andfound that "though geometry is mentioned as being part of texts, objectives,and testing, 78% of the teachers report spending fewer than 15 class periods per year ongeometric topics" (1, p. 13). Thus, we find that while geometry is conk idert.d a viable part of thecurriculum, most teachers seem to pay only lip service to that fact and instead concentrate onteaching number concepts and arithmetic skills. We suspect that teachers'lack of knowledge and previous success in geometry account for their position in thisregard. It is any wonder, then, that children's geometric knowledge is weak? In order to get full benefit from Part H of this book, teachers musthave some background

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