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ED043476.Pdf DOCUMENT RESUME ED 043 476 SE 007 523 'TT LE The Shapes of Tomorrow. INSTITUTION Vermont Univ., Burlington. SPONS AGENCY National Aeronautics and Space Administration, Washington, D.C. Educational Programs Div. PUB DATE 67 NOTE 201p. AVAILABLE FROM Superintendent of Documents, U. S. government Printing Office, Washington, D.C., 20t302 (Cat. No. 0-2116-252, $1.0) ERRS PRICE DRS Price MF$1.00 PC Not. Available from EDPS. DESCRIPTORS *Aerospace Technology, Computers, *Geometric Concepts, *Instructional Materials, Integrated Curriculum, Mathematics, Measurement, Scientific Concepts, Secondary School Mathematics, *Secondary School Science APSTRACT This hook, written by classroom teachers, introduces the application of secondary school mathematics to space exploration, and is intendel to unify science and mathematics. 7n early chapters geometric concepts are used with general concepts of space and rough approximations of space measurements. Later, these concepts are refined to include the use of coordinates e.nd measurements in space exploration. The first three chapters may be used as supplementary material to challenge students with space applications, as well as resource materials for the teachers. Chapters U and 5 provide basic concepts of science and mathematics needed to understand space travel. These concepts may supplement regular science computers in treating meaningful problems. (PP) F. /v As,q MATHEMATICS IN SPACE SCIENCE_ C t ....... A411%*% s,.4:,.,.'' % 6. i 0-nt. ''.:'.... -7 .)... -t? "h. 1 01#:\;1.-:*;:, ..* a .". i"7.1*--V.1.4.4. ii,. T.4 4 Orv.. t -110,,- EV r..I,,-. : "II.0'04 ;.'?* a...,,....1,,........t.!.. A .- ...,1 ...rvt,,,s, ,fr\ CS 1. .; t rmank.a..'. e 0 TOMORROW , tesi A supplement in space oriented geometry for secondary levels v 1 memo 01 KILN IRWIN I 11$1111i Oflia Of IN(1110i Ipnt Ittimlil ifl till IltINMX111 Inn! It ttiova Mx tril t.) 19$0 N ONII/11101 WIWI* IINoll of viw 0110WIS PM N 10111LISSIK! IIINIsuil 011Kin 0110 Of 10.:11p0o 3 HAIN 01 14$1(1 n , Prepartd by the National Aeronautics and Spew Administration in cooperation with the Unitedstates Mc* ot Education reN uJ The SHAPES of TOMORROW A supplement in space oriented geometry for secondary levels Prepared from mattrlola furnished by the National Aeronautics and Space Administration in cooperation with the United State,: Office of Education bya Committee on Space Science Oriented Mathematics 1967 For sale by the Superintendentof Oecumenic, US. Government Printing Mike Washington, D.C., 20402Price $1,60 FOREWORD This book introduces the application of secondary school mathematics to the exploration of space. It provides a unifica- tion of science and mathematics. This is nota textbook, al- though there are exercises to provide the reader withoppor- tunities to test and extend his understanding. An inquiring mind, rather than a specified grade lave' in school, is a pre- requisite for an enjoyable exploration of the ideas in this book. While various treatments of this mateilal may be found else- where and some elementary concepts ate included, these are presented here in an easily accessible format, for supplemen- tary classroom uses. There has been a del berate effort to in- clude enough familiar topics to make a meaningful transition from previous experiences to new concepts. As the reader progresses through this book he encounters a spiral development of ideas. Elementary geometric concepts are introduced in Chapi:.er 1 for readers who have not formally studied geometry. These geometric concepts are esed with gen- eral concepts of space and rough approximations of space meas- urements. Then, in later chapters geometric concepts, space concepts and measurements are gradually refined as the ma- turity of the reader increases. In Chapter 2 the use of coordi- nates In the study of space is explored. In Chapter 3 the uses of mt.surements in our explorations of apace are described. These first three chapters may be used effectively to supplement the usual materials at any one of several secondary-school grade levels, to challenge students with space applications, and as source materials for teachers (and writers) as they strive to capture the imagination of their students. Chapters 4 and 6 provide basic concepts of science and mathe- matics that are needed to understand space travel. These con- cepts are appropriate as supplements for either science or mathematics courses. Chapter 6 illustrates the effective man- ner in which electronic comp tters may he used to treat mean- ingful problems. The authors of this material are experienced classroom teach- ers who visited the NASA/Goddard Space Flight Center, con- sulted personally with many of its scientists and engineers, and studied important aspects of the U.S. space program. This book Is their effort to share these experiences by providing other teachers with enrichment materials for use in motivating their students. BRUCE E. MESERVE University of Vermont Iii CONTENTS Chapter 1 DESCRIBING THE SHAPES OF THINGS 1 Chapter 2 THE UNIVERSE WE LIVE IN 23 Chapter 3 MEASLIREMENT, A WINDOW TO THE UNIVERSE_ 67 Chapter 4 MOTION IN SPACE 91 Chapter 5 SPACE MECHANICS 119 Chapter 6 COMPUTERS ARE NEEDED 169 ANSWERS TO EXERCISES 179 INDEX 203 About This Book It is logical to presume that major achievements in the explo- ration of apace rest with the youth of today and with the education they receive. It is therefore our sincere concern that the space program be conducted in close cooperation with our Nation's educational institutions. Understandably, young peo- ple throughout the land are fascinated by the era of space travel. Teachers and students have long demonstrated their eagerness to relate the subjects they study to the apace program. However, frequently there appears to be a lack of available reference material suitable for classroom use. In view of the continuing demand for such material, espe- cially for elementary and secondary levels, the NASA/Goddard Space Flight Center, in cooperation with the U.S. Office of Edu- cation, has initiated a program of summer workshops to de- velop space-oriented mathematics supplements. The program Is directed by a Committee on Space-Oriented Mathematics consisting of Dr. Patricia Spross, Specialist in Mathematics, U.S. Office of Education, and Mr. Alfred Rosenthal, of the NASA /Goddard Space Flight Center. Mr. Elva Bailey, Goddard Educational Programs Officer, serves: as materials coordinator. This publication, the third in the series, focuses on the use of geometric concepts. It was prepared during a summer work- shop held at the University of Vermont, t3urlington, Vermont, under the direction of Dr. Bruce E. Meserve, Department of Mathematics, College of Technology. Overall guidan a and direction for this project has been pro- vided by the Office of Educational Programs, National Aero- nautics and Space Administration. MICHAEL J. VACCARO Chairman ,Committee on Space-Oriented Mathematics Chapter 1 DESCRIBING THE SHAPES OF THINGS evcip Brayton East Lanai's/7 High School East Lansing, Michigan DESCRIBING THE SHAPES OF THINGS 1-1 Shapes on Earth Think of the part of our Earth over which you may have walked. Except for hills and valleys, mountains and water- ways, it can be considered flat. Walking is a very simple way of getting from one place to another. And as you look toward the horizon, it appears that there is an edge of Earth and perhaps the sky comes down to meet it.It is not surprising, therefore, that before the time of Columbus most people believed that their Earth was flat, and that sooner or later, if they traveled far enough, they would come to the edge and drop off! Their World was the one they could see by walking around upon it. As time passed, inen invented Improved ways of travel by using wheels on and and sailboats on water. Travel with wheels, Figure 1 -1 however, required the constructionof was beyond the sight of land. The stars roads. The roads on water were already had long been guides for travel at night. there: It was natural that men turned to The invention of the compass made it pos- coats for traveling greater distances. So sible to chart a course both day and night. long as they kept land in sight they could The art of navigation reached such a re- start and stop when they pleased, could tell finement that men could travel farther where they were, and felt sure that they from land than they had ever dared to would not "drop off." venture before. Finally when Columbus set Man's curiosity and courage impelled out many people were certain that both him to develop instruments that would en- men and ships would be lost over the edge able him to travel on the water to find what of the World forever. P --r-A 4 ) r if(74, 4 WelA311Eglill; I 10 AtZiodigalgibasSil.EltrilVil or11111)44,ffil.,'1111111011f1t44244 .". IIIiiiinberid;Alai riallOP"Atellin Miffidiail i iii 1 Pipit( I-4 gys Although Columbus did not prove the termine a line. If every point of each line World to be round, nor find a new route to that is determined by points of a surface India, he did find a "New World" and the also lies on the surface, then the surface is maps began to be changed. flat and is called a plane. A table top is an However, man has newel ae;,eared to be example of a plane surface. We consider satisfied in his search for Knowledge and a line to be.of infinite length and a plane explorations have continued to tell us more to extend infinitely far in all directions. A about our Earth end the space in which it plane lika a line is a collection of points. revolves. It does not have thickness. You are just beginning to study about 1-1 Exercises those whose contributions in science and Shapes on Earth mathematics make travel in space possible.
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