Loving Math

Advanced Problems with Solutions, Applications and Comments

This book was written for high school students and teachers who love exploring beyond standard math curricula for a deeper understanding of the principles and applications of mathematics. It is also for anyone who loves the pursuit of a problem solution, including both professional and amateur mathematicians. The vehicle that transports us through this exploration is the study and solution of classical and advanced math problems.

As a high school math student, an engineer, a businessman and, ultimately, a high school math teacher, I collected and created math problems and solutions that can be used for advanced study. Some of the problems may be very familiar to you; some may not. A few may be quite easy to do; others will take more time. Included are classical proofs and their extensions that are often omitted in today's curricula.

Beyond the pure enjoyment of this exploration, we also attempt to find a "deeper understanding" of the math. We address four larger aspects of "understanding," namely: convention, evidence, perspective and connection. A portion of these aspects is addressed in the solutions, themselves. The rest is in comments, which come after the solutions. The comments range widely, including: additional points regarding the math itself, historical factoids, linguistics, suggestions for teachers, some personal experiences regarding the material, etc. Readers who only skim the problems and solutions might still find the applications and comments quite interesting.

It is hoped that this book will assist teachers and students alike in exploring the subject of mathematics in a new way, whether using material that is thousands of years old, or recently developed. Each problem can be used as a single assignment, done in a few minutes, or a term project that could require intuition, technique, research and/or fortitude (to plow through it). The material can be adapted for use in the standard classroom, subject to students' ability and the constrictions of uniform curricula. It is, perhaps, more applicable to classrooms with the freedom to experiment with project learning and with longer assignment periods. School math clubs or math teams might find this text a handy reference to hone skills, learn new techniques and satisfy the quest for more exciting material beyond the routine.

Although the primary focus here is the application of math principles to math problems, these studies are extended to interdisciplinary examples in the sciences, engineering, finance, social studies, etc. The subject material itself is organized into groups. There are twenty-two geometry/trigonometry problems, many of which are "classic proofs." Though some have been forgotten or ignored at large, they are offered here with some new ideas and approaches. There are ten algebra problems, all of which are extensions of a standard curriculum, and offer fresh insights when studied as a group. Statistics, the newest subject to be added to the high school curriculum, has three problems. And calculus, which is not always studied in high schools, has five problems.