DOCUMENT RESUME ED 283 721 SE 048 256 AUTHOR Kastner, Bernice TITLE Spacemathematics: A Resource for Secondary School Teachers. INSTITUTION National Aeronautics and Space Administration, Washington, D.C. REPORT NO NASA-EP-175 'PUB DATE 85 NOTE 172p. AVAILABLE FROMSuperintendent of Documents, U.S. Government Printing Office, Washington, DC 20402. PUB TYPE Guides - Classroom Use - Guides (For Teachers) (052) EDRS PRICE MP01/PC07 Plus Postage. DESCRIPTORS *Aerospace Technology; Algebra; Calculus; Computation; Geometry; *Mathematical Applications; *Mathematical Enrichment; Mathematics Education; *Mathematics instruction; Measurement; Probability; *Problem Sets; Science Education; Secondary Education; *Secondary School Mathematics; Space Sciences; Statistics; Trigonometry IDENTIFIERS *National Aeronautics and Space Administration ABSTRACT Designed to_serve as a supplement for high_school level mathematics, this publication contains activities which aim to enhance mathematical:knowledge and skills and to assist students in understanding aerospace technology and its achievements. Thie volume updates an earlier supplement that was published in 1972. Problems are grouped:into chapters according to the predominant mathematical topic and within each chapter the problems are presented by similar themes. Major chapter headings include: (1) "Mathematical Aspects of Some Recent NASA Mi;sions"; (2) "Computation and Measurement"; (3) "Algebra"; (4) "Geometry";_(5) "Probability and Statistics"; (6) "Exponential and Logarithmic Functions"; (7) "Trigonometry"; (8) "Matrix Algebra"; (9) "Conic Sections"; and (10) "Calculus." The appendix consists_of an_analysis of gravitational forces and:the conic section trajectories. A glossary is also included. (ML) ***********************************************1 ************* ReprodLctions supplied by EDRS are the best that can be made from the original document. ****************** *********************************************** A Resource for Secondary SPACEMATHEMAT1CS School Teachers U.S. DEPARTMENT OF EDUCATION Cation of Edunalionai Research and Improvement DUCATIONAL eesoueoes INFORMATION CENTER (ERIC) This documam haa !seen rahrodunna as -hived from the moan Or organization iginating iL 0 Minor changes nave been made xo improve reproduction duality Rants of view or opinions staled in this noinu- rnent tha no+ necessarny represent official OERI pottilion or policy, Sun MOOn X Rs NASA National Aeronautics and Space Administration A Resource for Secondary S ACE MATHEMATICS School Teachers by Bernice Kastner, Ph.D. Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia National Aeronautics and Space Administration A curriculum project prepared by the National Council of Teachers of Mathematics James D. Gates, Ph,D. Executive Director 4 TABLE OF CONTENTS Introduction Preface Chapter 1. Mathe -atical AspcLoi Soe Recent NASA Missions 9 2. Computation and Measurement 15 3. Algebra 37 4. Geom ry 53 5. Probability and Statistics 71 6. Exponential and Logarithmic Functions 87 7. Trigonometry 99 8. Matrix Algebra 125 9. Conic Sections 135 10. Calculus 165 Appendix Gravitational Forces and the Conic Section Trajectories 183 Glossary 191 INTRODUCTINv igh School mathematics teachers have long been aware that their students should know not only something about the development ofpure mathematics but also something about its applications. Several years ago NASA,recog- iizing the appeal of aerospace activities, initiated and supported the development of curriculum supplements for several high school courses. Because attainments in aerospace would not be possible without mathematics, it was most appropriate that a supplementary publication dealing with space activities be prepared for teachers of mathematics. The first mathematics curriculum supplement, Space Mathematics, A Resource for Teachers, was published in 1972. One of the most popular and oft-requested of the supplements, the book has been unavailable for several years. This volumeup- dates the earlier work. We hope that a new generation of students will become more interested in mathematics as the result of seeing some of its significant appli- cations in recent and current space projects. Working problems suchas those in this book should enhance both the mathematical knowledge and skills of students and their appreciation and understanding of aerospace technology and achieve- ments. NASA's Technical Monitor for this project was Muriel M. Thorne, Educational Programs Officer, under the general direction of William D. Nixon, Chief of Edu- cation Services, NASA. National Aeronautics and Space Administration Washington, D.C. September 1985 6 PREFACE In 1972, a collection of mathematical problems related to space science entitled Space Mathematics, A Resource for Teachers was published by the Educational Programs Division of the National Aeronautics and Space Administration (NASA). As an early user of that publication, I can say that it has been both a pleasure and a challenge for me to undertake the revision of that volume of en- richment materials, especially in the light of another twelve years oii: activity in space exploration. This interval has been a period of much progress in both the science and the technology associated with the space program, and it has offered a wealth of new material in which to find applications of high school mathematics. The basic format of the original publication has been retained, as well as many of the classical problems and those which complemented the new material. In devel- oping the examples and problems presented here, we have aimed at preserving the authenticity and significance of the original setting while keeping the level of math- ematics within the secondary school curriculum. The problems have been grouped into chapters according to the predominant mathematical topic. Within each chap- ter we have attempted, as far as possible, to group problems involving similar themes. There is a wide range of sophistation required to solve the various prob- lems. Since this is a resource book for teachers, we have assumed that the reader will be interested not only in problems thatcan be brought directly into the class- room, but also in those that, although beyond the current level of their students, will increase the teacher's own awareness of some of the interesting applications of mathematics in the space program. Perhaps the most valuable potential of a collection such as this lies in its ability to convey a sense of how secondary school mathematics is actually used by practicing scientists and engineers. Attitudes and appraches may thereby be fostered, on the part of teachers, that can help students to be more insightful users of the mathematics they learn. The present school mathematics curriculum, for example, gives no hint that many real-world problems do not have analytic solutions in closed form but may nevertheless be satisfactorily "solved" by using carefully cho- sen approximations or the numerical methods made possible by modern computers. In this connection, we stress that in order to use numerical analysis correctly or to make good approximations, it is necessary to know something of the theoretical background of the subject and to understand the concepts of precision and accu- racy nnd the use of significant digits. Also, methods that reveal meaningful aspects of a procedure are preferable to purely algorithmic prescriptions; the perhaps un- familiar -factor unit" method of unit conversion presented in Chapter 2 is actually quite commonly used in science and engineering. It not only removes all uncer- tainty about whether to multiply or divide by a conversion factoi but also is far more likely to contribute to an understanding of the underlying concepts than, for example, the more usual metric system algorithm expressed in terms of "moving" the decimal point. 7 6 Many NASA staff members contributed time and thought to this project, including personnel at the Goddard Space Flight Center, the Marshall Space Flight Center, the Jet Propulsion Laboratory, the Langley Research Center, and NASA Head- quarters in Washington, D.C. These people, too numerous to mention individu- ally, provided enthusiastic support, which is gratefully acknowledged. Project Associate James T. Fey, of the University of Maryland, and reviewers Louise Rout ledge, Father Stanley J. Bezuszka, Gary G. Bitter, and Terry E. Parks, of the National Council of Teachers of Mathematics (NCTM) provided valuable comments and suggestions. On the editorial and support side, I would like to thank the staff at the NCTM Reston office and Muriel M. Thorne, Educational Programs Officer at NASA Headquarters. Bernice Kastner September 1985 7 " MUM" AMUE EMinimmimummw mom _ mm mommom N MM MMMMENEMEEEMME EMEMMEEEN N MEM EMEEMEEMMEMMEEMMMEEMEMMEEMI ,EME ME N MEEME MM ME EME =NNE ENEMEN_ Chapter One Teachers of mathematics, like most adults in today's world,can hardly fail to be aware of the rapid development of space science. We realize that thespec- tacular achievements of the space program have dependedheavily on mathe- maticsmathematics that is generally complex, advanced, and wellbeyond the level of most secondary school curricula. Even though thisperception is valid, there are many significant aspects of space science thatcan be understood using only high school mathematics. The exploration of space naturally uses the tools and techniquesof astronomy. Astronomy in turn is gaining much new informationas a result of sending