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 enrichment materials, especially in the light of another twelve years of 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 developing the examples and problems presented here, we have aimed at preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. The problems have been grouped into chapters according to the predominant mathematical topic. Within each chapter we have attempted, as far as possible, to group problems involving similar themes. There is a wide range of sophistication required to solve the various problems. Since this is a resource book for teachers, we have assumed that the reader will be interested not only in problems that can be brought directly into the classroom, 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 approaches 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 chosen 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 accuracy and the use of significant digits. Also, methods that reveal meaningful aspects of a procedure are preferable to purely algorithmic prescriptions; the perhaps unfamiliar "factor unit'' method of unit conversion presented in Chapter 2 is actually quite commonly used in science and engineering. It not only removes all uncertainty about whether to multiply or divide by a conversion factor 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.
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 Headquarters in Washington, D.C. These people, too numerous to mention individually, provided enthusiastic support, which is gratefully acknowledged.
Project Associate James T. Fey, of the University of Maryland, and reviewers Louise Routledge, 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 .