problem solving strategies (problem books in mathematics) pdf

Problem-Solving Strategies

• © 1998
• Arthur Engel 0

Institut für Didaktik der Mathematik, Johann Wolfgang Goethe-Universität Frankfurt am Main, Frankfurt am Main 11, Germany

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• The most complete training book on the market
• A must-have for instructors wishing to enrich their teaching with interesting non-routine problems
• Provides individuals with a wide array of interesting, challenging problems
• Each chapter starts with typical examples illustrating the central concepts, followed by carefully selected problems and their solutions
• Very few problems have no solutions

Part of the book series: Problem Books in Mathematics (PBM)

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About this book

Problem-Solving Strategies is a unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. The discussion of problem solving strategies is extensive. It is written for trainers and participants of contests of all levels up to the highest level: IMO, Tournament of the Towns, and the noncalculus parts of the Putnam Competition. It will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a "problem of the week", "problem of the month", and "research problem of the year" to their students, thus bringing a creative atmosphere into their classrooms with continuous discussions of mathematical problems. This volume is a must-have for instructors wishing to enrich their teaching with some interesting non -routine problems and for individuals who are just interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. Very few problems have no solutions. Readers interested in increasing the effectiveness of the book can do so by working on the examples in addition to the problems thereby increasing the number of problems to over 1300. In addition to being a valuable resource of mathematical problems and solution strategies, this volume is the most complete training book on the market.

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Table of contents (14 chapters)

Front matter, the invariance principle, coloring proofs, the extremal principle, the box principle, enumerative combinatorics, number theory, inequalities, the induction principle, polynomials, functional equations, further strategies, back matter, authors and affiliations.

Arthur Engel

About the author

Bibliographic information.

Book Title : Problem-Solving Strategies

Authors : Arthur Engel

Series Title : Problem Books in Mathematics

DOI : https://doi.org/10.1007/b97682

Publisher : Springer New York, NY

eBook Packages : Springer Book Archive

Copyright Information : Springer-Verlag New York 1998

Softcover ISBN : 978-0-387-98219-9 Published: 12 December 1997

eBook ISBN : 978-0-387-22641-5 Published: 19 January 2008

Series ISSN : 0941-3502

Series E-ISSN : 2197-8506

Edition Number : 1

Number of Pages : X, 403

Topics : Mathematics, general

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Problem-Solving Strategies (Problem Books in Mathematics) Corrected Edition

Problem-Solving Strategies is a unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. The discussion of problem solving strategies is extensive. It is written for trainers and participants of contests of all levels up to the highest level: IMO, Tournament of the Towns, and the noncalculus parts of the Putnam Competition. It will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a "problem of the week", "problem of the month", and "research problem of the year" to their students, thus bringing a creative atmosphere into their classrooms with continuous discussions of mathematical problems. This volume is a must-have for instructors wishing to enrich their teaching with some interesting non -routine problems and for individuals who are just interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. Very few problems have no solutions. Readers interested in increasing the effectiveness of the book can do so by working on the examples in addition to the problems thereby increasing the number of problems to over 1300. In addition to being a valuable resource of mathematical problems and solution strategies, this volume is the most complete training book on the market.

• ISBN-10 0387982191
• ISBN-13 978-0387982199
• Edition Corrected
• Publisher Springer
• Publication date December 12, 1997
• Part of series Problem Books in Mathematics
• Language English
• Dimensions 6.1 x 0.94 x 9.17 inches
• Print length 413 pages
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• Publisher ‏ : ‎ Springer; Corrected edition (December 12, 1997)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 413 pages
• ISBN-10 ‏ : ‎ 0387982191
• ISBN-13 ‏ : ‎ 978-0387982199
• Item Weight ‏ : ‎ 1.3 pounds
• Dimensions ‏ : ‎ 6.1 x 0.94 x 9.17 inches
• #562 in Mathematics History
• #590 in Mathematics Study & Teaching (Books)
• #6,793 in Core

About the author

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Problem-solving strategies in mathematics : from common approaches to exemplary strategies

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Problem Solving in Mathematics and Beyond: Volume 1

Problem-solving strategies in mathematics.

• By (author): 
• Alfred S Posamentier ( Mercy College New York, USA )  and 
• Stephen Krulik ( Temple University, USA )
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"This is a wonderful book, filled with problems and witty solutions, addressed to anyone interested in problem-solving. Elementary mathematics is used so that the reader can focus on the strategy. The reader will enjoy reading this book and trying to find their own strategies to arrive at the solution of the problems proposed."

This book introduces ten problem-solving strategies by first presenting the strategy and then applying it to problems in elementary mathematics. In doing so, first the common approach is shown, and then a more elegant strategy is provided. Elementary mathematics is used so that the reader can focus on the strategy and not be distracted by some more sophisticated mathematics.

Errata(s) Errata

Sample Chapter(s) Introduction (71 KB) Chapter 1: Logical Reasoning (79 KB)

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• Logical Reasoning
• Pattern Recognition
• Working Backwards
• Adopting a Different Point of View
• Considering Extreme Cases
• Solving a Simpler Analgous Problem
• Organizing Data
• Making a Drawing or Visual Representation
• Accounting for All Possibilities
• Intelligent Guessing and Testing

FRONT MATTER

• Pages: i–xxiii

https://doi.org/10.1142/9789814651646_fmatter

• About the Authors
• Introduction

Chapter 1: Logical Reasoning

• Pages: 1–12

https://doi.org/10.1142/9789814651646_0001

It must seem rather redundant to have a chapter entirely devoted to a strategy referred to as logical reasoning. Regardless of the strategy being used to solve a problem, it would seem that some logical thinking must permeate the use of all the strategies. After all, to many people, problem solving is almost synonymous with logical reasoning, or logical thought. So, why have this chapter and why isolate this strategy…

Chapter 2: Pattern Recognition

• Pages: 13–28

https://doi.org/10.1142/9789814651646_0002

Some of the most inherent beauties of mathematics are the many patterns that occur in problems as we try to resolve them. The famous mathematician, W.W. Sawyer once said that mathematics could be thought of as a search for patterns. One of the major uses of mathematics is to predict things that occur in a regular manner. For example, how many scones will I need for 3 people? For 4 people? For 10 people? For n people…

Chapter 3: Working Backwards

• Pages: 29–43

https://doi.org/10.1142/9789814651646_0003

The very name of this strategy sounds confusing to most people. It is a very unnatural way of doing things. When most of us went to school, we were taught to solve mathematical problems in a direct, straightforward manner. And yet, working backwards is the way many real-life problems are often resolved. As a simple example, suppose you had to pick up your child from football practice at exactly 5:00 p.m. At what time should you leave? Well, let us say it take 30 minutes to get to the ballpark. We would better leave a 5-minute safety valve. Okay, then we need to leave 35 minutes earlier, or not later than 4:25 p.m. Without even thinking about it, we were working backwards! Of course, this is a very simple example of this strategy…

Chapter 4: Adopting a Different Point of View

• Pages: 45–63

https://doi.org/10.1142/9789814651646_0004

Of the many strategies that exist and that are available to us to solve mathematical problems, the one that allows us to avoid “running into a brick wall” — namely avoiding frustration — is that of approaching the problem from a different point of view. Perhaps one of the classic examples — because of its simplicity and dramatic difference in solution method — is the following. This is an example where the common method leads to a correct answer, but is cumbersome and can often provide an opportunity for some arithmetic mistakes. As an example, consider the following problem…

Chapter 5: Considering Extreme Cases

• Pages: 65–82

https://doi.org/10.1142/9789814651646_0005

In order to obtain some help with solving a particular problem, we may consider some variables at their extremes, while holding others constant. If no specificity for the variable has been prescribed, the extreme case might just provide us with some helpful insight. Most of us have used this strategy in real life, without consciously realizing that we were even doing so. We say to ourselves, “What is the worst that can happen?” — for a particular, situation up for a decision. This “worst-case scenario” is an example of using an extreme case that can sometimes help resolve a problem situation in a very clever way. Similarly, suppose you were asked to test a new product, say, laundry soap. You would have to test it in very cold water as well as very hot water — clearly a consideration of two extreme cases that makes the testing worthwhile. If it does well in these two extreme temperatures, it should do well in the temperatures in between…

Chapter 6: Solving a Simpler Analgous Problem

• Pages: 83–95

https://doi.org/10.1142/9789814651646_0006

There are problems that appear to be extremely complex when we first approach them. The numbers in the problem might be extremely large, and thus confusing or possibly distracting. Perhaps there is an excessive amount of data given, some of which may not even be necessary to solve the given problem. Even the way the problem has been presented can sometimes confuse the reader. Regardless of the reason, an excellent approach is to convert the problem into a simpler form, but as an equivalent version of the original. Change the numbers, modify the original drawing, or try another approach to simply alter the form of the problem. By solving this simplified version of the problem, the problem solver can possibly gain some insight into how to approach the original problem…

Chapter 7: Organizing Data

• Pages: 97–113

https://doi.org/10.1142/9789814651646_0007

The strategy of organizing data is one of the most important strategies, even though at first glance we might assume it to be axiomatic. That is, everyone should automatically organize the data that appears in a problem without giving it a moment's thought. It is what we often do in real life without consciously thinking about it…

Chapter 8: Making a Drawing or Visual Representation

• Pages: 115–130

https://doi.org/10.1142/9789814651646_0008

When a problem asks questions about a particular given geometric figure or drawing, it goes without saying that the drawing or visual representation is an integral part of the solution method. It is necessary, and helps to solve the problem. In that sense it is hard to imagine that some mathematicians in ancient times often derived geometric concepts without drawings — or at least they presented their geometric findings without the appropriate sketch. However, there are many problems where a drawing is not implicit in the statement of the problem; yet, actually seeing what is being considered can be of great help. Many people are visual learners; they need a picture rather than merely words to understand what is taking place. And no, it is not a case of daydreaming. Contrary to what some people believe, visualization has nothing to do with daydreaming. While daydreaming is often a waste of time, visualization is a very powerful method that can help you become more familiar with a given situation…

Chapter 9: Accounting for All Possibilities

• Pages: 131–143

https://doi.org/10.1142/9789814651646_0009

We know that organizing the data in a problem can sometimes be very revealing. When we look for a pattern, for example, carefully organized data in a list or table can aid in the search and help us discover the pattern. However, one particular type of list that is extremely important is the exhaustive list. In this kind of organized list, all the possibilities are listed in some systematic way. Somewhere included on this list will be what we are looking for. Constructing an exhaustive list enables us to examine all the possibilities in a carefully-organized manner…

Chapter 10: Intelligent Guessing and Testing

• Pages: 145–159

https://doi.org/10.1142/9789814651646_0010

Somehow, the very idea of guessing as a problem-solving strategy raises some eyebrows. Indeed, many of us can recall a teacher saying to someone who had volunteered an unusual answer, “Do you know , or are you just guessing? ” In some books, this method of guessing and testing is sometimes referred to as the method of “trial and error,” which is sometimes considered a more negative expression. The added adjective, intelligent guessing and testing should put your mind at ease, and reassure you that this is, indeed, a viable, and even an often useful strategy…

BACK MATTER

• Pages: 161–163

https://doi.org/10.1142/9789814651646_bmatter

"I recommend this book to anyone interested in furthering his or her understanding of problem-solving strategies. I would encourage those individuals to read the book, become familiar with the ten strategies, and then spend some time working through the given problems before reviewing the solutions provided because it will most likely offer some new insight and thinking into the problem."

"The problems presented here could be used by teachers to help their students develop problem-solving skills, by students interested in developing their own skills, or by anybody intrigued by mathematics."

"Many of the problems or problem types will be familiar to Gazette readers and some are quite similar to one another. They are attractively presented in a clear, unfussy manner."

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