ebook img

Ants, Bikes, and Clocks: Problem Solving for Undergraduates PDF

175 Pages·2004·10.47 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Ants, Bikes, and Clocks: Problem Solving for Undergraduates

ants. JUKCl& CL* OCK* S This page intentionally left blank ants.m&sc CLOCKS PROBLEM SOLVING FOR UNDERGRADUATES William Briggs University of Colorado at Denver Denver, Colorado slam. Society for Industrial and Applied Mathematics Philadelphia Copyright © 2005 by the Society for Industrial and Applied Mathematics. 109876543 21 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. MATLAB® is a registered trademark of The MathWorks, Inc. For MATLAB product information please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7101, [email protected], www.mathworks.com/. Library of Congress Cataloging-in-Publication Data Briggs, William L. Ants, bikes, and clocks : problem solving for undergraduates / William Briggs. p. cm. Includes bibliographical references and index. ISBN 0-89871-574-1 (pbk.) 1. Problem solving. I. Title. QA63.B72 2005 510-dc22 2004058720 513JTL is a registered trademark. Contents 1 Introduction 1 1.1 Exercises 4 1.2 Hints and Answers 6 2 A Problem-Solving Framework 7 2.1 Polya's Method 7 2.2 Exercises 12 2.3 Hints and Answers 14 3 Problem-Solving Strategies 17 3.1 Exercises 30 3.2 Hints and Answers 32 4 How Do You Do It? 35 4.1 Exercises 38 4.2 Hints and Answers 40 5 Parts of the Whole 43 5.1 Basic Uses of Percentages 43 5.2 Assorted Problems 46 5.3 Exercises 48 5.4 Hints and Answers 51 6 A World of Change 53 6.1 Exercises 57 6.2 Hints and Answers 60 7 At Any Rate 63 7.1 Exercises 68 7.2 Hints and Answers 70 8 Difference Equations 73 8.1 Background 73 8.2 Exercises 82 8.3 Hints and Answers 84 V vi Contents 9 Insight and Computing 87 9.1 P61ya's Method for Computing 87 9.2 Root Finding 92 9.3 Exercises 98 9.4 Hints and Answers 102 10 Take a Chance 105 10.1 Discrete Probability 105 10.2 Geometric Probability 113 10.3 Exercises 118 10.4 Hints and Answers 121 11 Toward Modeling 123 11.1 Exercises 123 11.2 Hints and Answers 125 12 Solutions 127 Bibliography 163 Index 167 Chapter 1 Introduction Skills are to mathematics what scales are to music or spelling is to writing. The objective of learning is to write, to play music, or to solve problems —not just to master skills. — Everybody Counts For students of mathematics, science, and engineering, the name of the game is problem solving. Whether it's proving a theorem, writing a computer program, designing a statistical experiment, or solving for the stresses in a bridge, the essential challenge is problem solving. Mathematical problem solving has been the subject of countless books, maga- zines, anthologies, and, more recently, web sites. It has been promoted and enjoyed by diverse audiences that include students, teachers, and recreational aficionados. Given the immense scope of problem solving, it's clear that one book cannot possibly do the subject justice. It has been said that the To explain why this book may be different than many others, let's attempt a most ancient of all puzzles is the Riddle of the Sphinx: brief taxonomy of mathematical problem solving. Greatly simplifying the matter, What creature walks on we might identify the following categories of problems that mathematics students four feet in the morning, typically encounter (the list has no special order, and the categories certainly overlap): two feet at noon, and three feet in the evening? Many 1. Recreational problems are concise intellectual challenges often associated people failed to answer the with puzzle connoisseurs such as Sam Loyd [22, 23] and Martin Gardner riddle until Oedipus, the fu- ture King of Thebes, gave [10, 8, 9,11]. These problems may or may not be mathematical in nature, but the intended answer: Man they generally require keen critical thinking and ingenious strategies. Books crawls on all fours as a of many flavors and difficulties are devoted to recreational problems, and any baby, walks upright as an adult, and uses a staff in his serious problem solver should be familiar with them. If nothing else, they old age. provide excellent mind calisthenics and occasionally come in handy as party tricks. 2. Contest problems are precisely formulated mathematical problems that often appear in formal exams and competitions such as the American High School Math Exam (AHSME), the USA Mathematical Olympiad (USAMO), the 1 2 Chapter 1. Introduction International Mathematical Olympiad (IMO), and the Putnam Exam. Such contest problems generally require a fair amount of mathematical background Information about the Put- and sophistication. nam exam can be found at http://math.scu.edu/ putnam/. 3. Logic problems are generally qualitative in nature and often take the form of a story. Their solution requires organized thinking and often formal logic. Collections of logic problems abound; in fact, monthly magazines of logic problems can be found in supermarkets. These problems provide excellent thinking exercises and are often used as training for standardized exams such as the GRE, LSAT, and MCAT. 4. Modeling or story problems are quantitative problems that are posed in a realistic context. A key distinction of these problems is that they are not posed explicitly as mathematical problems. For this reason, their solution requires an essential preliminary step that may be the crux of the solution. That step, often called modeling, is to transform the stated problem from words into mathematics. Having formulated the problem in mathematical terms, it must still be solved! The only formal exam that emphasizes such problems is the More information about the Mathematical Contest in Modeling. Mathematical Contest in Modeling can be found at 5. The previous four categories require analytical techniques—traditional http://www.comap.com/ methods carried out with pencil and paper (and brain). Twenty-five years undergraduate/contests/ ago, the list would have ended here. However, we must now acknowledge that mem/. there is another tool in the problem-solving arsenal: the computer. With its numerical, graphical, and symbolic capabilities, the computer is the laboratory of mathematics; it is a tool of exploration and discovery. Without diminishing the role of analytical methods, it is fair to say that fluency with a programming language (for example, C++ or Java) or a mathematical environment (for example, MATLAB, Maple, Derive, or Mathematica) is an essential skill for all mathematics students. This observation leads to another problem category: Computational problems are those problems for which the powers of a computer are used for insight, exploration, or solution. Even if oversimplified, these problem categories certainly demonstrate that problem solving is a diverse and complex enterprise. In choosing the focus of this book, choices had to be made. Because problems posed in a realistic context are common and important, and because solving such problems is a valuable skill, this book highlights modeling or story problems. This is not to say that the other categories are neglected, but the emphasis of the book is decidedly on the following two-step process: • transforming a problem in context (a story problem) into a mathematical problem, and • using both analytical and computational tools to solve the resulting mathe- matical problem. If you have gotten this far, you have read about 1% of this book! Before going any further you should know a few honest facts. Most importantly, this book (or any problem-solving book) will not give you a universal formula for problem solving; 2 Introduction such a formula just doesn't exist. However, this book (and many others) can help you become a better problem solver. For everyone, problem solving power comes with practice. Through practice, you see similar problems and patterns reappear. Through practice, you master tech- niques and variations on those techniques. And, perhaps most important of all, through practice, you gain confidence. Creativity can solve almost It's easy to say that practice leads to mastery, but practice is not always easy. any problem. The creative act, the defeat of habit by A pianist cannot practice enough to improve unless she finds some enjoyment in originality, overcomes ev- practice. Analogously, to improve at problem solving, you really must find some erything. enjoyment in mathematics and problem solving. And just as a competitive cyclist — George Lois cannot seriously train unless he has a desire to win, proficiency in problem solving will be hard-earned without the desire to be a better problem solver. Hopefully, this book can provide enjoyment and instill desire. A word about answers and solutions is in order. A lone answer, which is typically a numerical result such as | or 8.23 miles, is never acceptable. It must always be accompanied by a complete solution, which is a full account of how you arrived at the answer. One goal of the book is to improve mathematical communication, both written and verbal. With mathematical proofs, the standards of exposition are fairly clear. With the more open-ended story problems that appear in this book, the rules may not be so evident. However, here are some guidelines. Solutions must be compelling and convincing to others who have never seen the problem, but who have the mathematical background needed to understand the solution. Graphs, tables, and figures can often make a solution much more digestible. You should write a solution so that if you were to read it five years from now, you could make sense of your own writing. The Lord's Prayer is 66 Throughout the book, you are encouraged to focus as much on the process of words, the Gettysburg Ad- dress is 286 words, and problem solving as on the final answer. It often helps to step back and watch yourself there are 1,322 words in in the act of problem solving. Are certain environments more conducive to successful the Declaration of Inde- problem solving? For example, do you do better on a bus, in the shower, surrounded pendence. Yet, government by silence, or inside a set of vibrating headphones? Are certain times of day more regulations on the sale of cabbage total 26,911 words. effective? Are you more successful after a big pasta dinner or on an empty stomach? — David Mclntosh What strategies worked? Were there breakthrough moments when you experienced a key Aha! or Eureka! moment? Does working in groups help or hinder your problem solving? Try to follow your efforts, both the victories and the frustrations. You will learn a lot about problem solving by watching yourself in the act. Having said that the end result isn't everything, it is always rewarding to know when you devise a correct solution. In this book, you will find hints and answers for most problems at the end of each chapter. Solutions to selected odd problems (marked with a <>) appear at the back of the book. Remember that very little is gained by reading the solution to a problem before seriously attempting to solve it. Finally, mathematical problems are like folklore: The origins of many problems are lost in the shadows of time. Problems fall into obscurity and are rediscovered; story lines change as problems are passed on. As an inveterate collector, I have regrettably forgotten where I first saw or heard many problems, and even if I could remember the source, it might not be the primary source. I have done my best to give credit for problems used in this book by citing my primary source. I apologize (and would like to be informed) if I have failed to recognize the creator of a good problem.

Description:
Mathematics educators agree that problem solving is one of the essential skills their students should possess, yet few mathematics courses or textbooks are devoted entirely to developing this skill. Supported by narrative, examples, and exercises, Ants, Bikes, and Clocks: Problem Solving for Undergr
See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.