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Copyright © 2016 Looking Glass Ventures. All rights reserved. This book or parts thereof may not be reproduced in any form, stored in any retrieval system, or transmitted in any form by any means—electronic, mechanical, photocopy, recording, or otherwise— without prior written permission of the publisher, except as provided by United States of America copyright law. For permissions and additional information, write to the publisher, at
[email protected].
ISBN: 978-1-944931-00-1 2
Contents
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Note To Reader Welcome to this little Zen Master’s guide on problem solving. I have tried to create a small bag of techniques, strategies, and tricks to help every student solve a wide assortment of problems they are likely to encounter on their middle school journey and beyond. This is based on my journey as a student, mathematician, educator, math coach, mathematics author, and speaker across countries, colleges, and schools for the past two decades. This book is far less about making a student the ultimate problem-solver, but more about engendering an interest in solving problems and evoking a ‘Watson, the game is afoot!’ feeling in every young mind. This title is intended to be equally useful and interesting to educators and 51
students. It may be browsed casually on a digital device or pursued in a more regimented fashion through its online, auto-graded digital problem companion at http://edfinity.com. Like all the other titles in the Zen Master’s series, this guide is available only as an eBook coupled with its Edfinity digital companion (effective June 1, 2016). We, of course, can’t promise that this ebook is the key to unlock supreme success for the worldwide universe of math competitions. However, we offer you: v Interesting ideas on how to go about solving problems. v Thoughts on enjoying the mathematical insights that well crafted problems offer. v Advice on how to keep your emotions in check when things
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look particularly strange and unfamiliar. (It sometimes feels like problem authors are trying to be sneaky! One needs to keep one’s cool.) This is an attribute that will actually serve you well in all aspects of life! This guide includes a compilation of practice problems: the more you work on solving problems, the more confident you’ll become. And you’ll also start to notice familiar ideas and themes that you can use to your advantage. Like any sport, in starting out it all feels hard and strange. But the more you play, the easier it becomes and the more natural it starts to feel. You begin to notice recurring ploys and maneuvers and this makes you extra clever at the sport. And then, before
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you know it, you’re pretty good and other beginners are looking to you as a pro! So, in addition to reading this title, practice! Try lots of different problems. Get them right. Get them wrong. Just keep trying them. The more you practice, the better you get. Seriously, it is as simple as that. And after you get through this title, start working your way through the 5 topicbased titles in the Zen Master’s Middle School series (It’s only up and up from here!)
James Tanton January 2016
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Acknowledgements My deepest thanks and appreciation to Michael Pearson, Executive Director of the Mathematical Association of America, for setting me on the path of joyous mathematical problem solving with the MAA Curriculum Inspirations project, and to Shivram Venkat at Edfinity for inviting me to extend that wonderful work to the global community of younger budding mathematicians. I am so very honored to be part of the unique, and truly remarkable, digital format experience Shivram and Edfinity have developed for the world.
James Tanton January, 2016
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Edfinity’s Zen Master’s Series Edfinity’s Zen Master’s series is a collection of 11 digital titles (6 for Middle School and 5 for High School) created for the modern educator and student. The titles are available only in digital form and consist of carefully crafted problem collections designed to help students master problem solving. Each title guides students through the themes of a specific topic (such as Algebra or Probability), presenting concise expository content, select examples illustrating specific problem solving techniques, and between 150200 problems expertly arranged to help the user achieve complete mastery. The volumes are each accompanied with optional access to an Edfinity ‘digital companion’ presenting all the problems in the title as a self-paced,
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online course with auto-grading and performance analysis. Educators may enroll their students to track their progress, or students/parents may enroll individually. Access to the guides provides educators access to rich, supplemental problem collections for classroom use. The Zen Master’s Series is designed to serve broad usage by educators and students alike, offering substantive general enrichment, development of foundational skills in problem solving, and contest preparation. In addition to helping students prepare effectively for local and major international contests, the problems provide robust attention to standards and guidelines of the Common Core State Standards in Mathematics (USA), GCSE (UK), Singapore’s Math curriculum,
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Australian Curriculum, and most other international syllabi.
ZEN MASTER’S MIDDLE SCHOOL SERIES 8 Tips to Solve Any Problem, by James Tanton Counting and Probability, by James Tanton Numbers and the Number System, by James Tanton Structure, Patterns and Logic, by James Tanton Relationships and Equations, by James Tanton Geometry, by James Tanton Solutions Manual for each title by James Tanton
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ZEN MASTER’S HIGH SCHOOL SERIES Algebra, by David Wells Geometry, by David Wells Number Theory, by David Wells Discrete Mathematics, by David Wells Advanced Topics, by David Wells Solutions Manual for each title by David Wells Enroll at http://edfinity.com/ZenSeries/8tips for online practice with scoring and complete solutions.
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1.Three Mind Issues To Think About MIND ISSUE 1: THE PRESSURE OF TIME When doing competition problems in an actual competition, a clock is counting down your time. This time pressure is psychologically very hard! Before we start on the problem-solving strategies, and before you start in the competition world, we need to think about three tricky mind issues. How are you going to handle each of them?
ADVICE: For starters, know that mathematicians don’t ever do mathematics under a time pressure.
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This idea of having a clock running while thinking mathematics is very strange to professionals. Knowing that, in the big picture of things, people really don’t care how fast you or anyone can solve problems eases the emotions and this feeling of pressure. (Mathematicians can spend days, weeks, even years trying to solve a single problem.) But, in the competition world, the clock is running, and that pressure snags you every now and then. The best way to handle this unpleasant feeling is to: Take a Deep Breath !
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MIND ISSUE 2: THE PRESSURE TO BE RIGHT Mathematics in school is often about getting the right answer and one is left with the impression that answers are always more important than any clever ideas or approaches you discover in getting to those answers. In the competition world, the focus is on answers too. There’s a feeling of pressure to always be “right.”
ADVICE: Know that mathematicians are actually much more interested in ideas and approaches. Answers to questions are only “final details.” The process of getting to answers is what
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mathematicians really care about. Knowing this can help ease this sense of pressure. The best way to handle this tension is: KNOW THAT SUCCESS IN LIFE IS NOT ABOUT SCORES, BUT ABOUT THE THINKING AND THE GOOD IDEAS YOU COME UP WITH.
It’s Not About Scores
Reminding yourself of this will ease the emotional discomfort. And, also, taking a relaxed attitude about this can actually help you improve your scores nonetheless! (Crazy!)
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MIND ISSUE 3: YOUR HUMANESS Mathematics is very much a human experience. People think it is all about procedures and formulas and mechanics. But mathematics is invented and discovered by humans for humans. Competition problems are invented by humans too!
ADVICE: Just know that… IT IS OKAY TO HAVE AN EMOTIONAL REACTION TO A QUESTION. You can say, “This looks scary” or “Why should I care?” or “Wow! Weird!” or “This seems fun” or “I don’t get it!” or anything that is a genuine emotional reaction. Acknowledge your emotions
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(even mutter your reaction out loud) and then go back to my first piece of advice: Take a Deep Breath ! Remember, you are human and it is okay to be human while doing competition problems!
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2.Two Steps In Solving A Problem 1. The first step in problem solving is always the same:
BE SURE TO READ THE QUESTION CAREFULLY.
Take a Deep Breath !
Is the question about the rows or the columns of the table? Do we want to find an actual number? The sum of two numbers? Which area exactly do we need to find? Is it the units digit we want, the tens digit, or both? Are the triangles in the picture important or just the squares? Is it the distance we want or the speed? And so on.
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It is easy to read a question too quickly and not notice what exactly you are meant to focus on. Some people like to underline the important words in a problem just to make sure they actually read them. (No one says you can’t write on any booklets you are handed!). 2. The second step in problem solving is always the same too:
TRY SOMETHING. ANYTHING! Just Do Something! This step is surprisingly hard for most people. If a question looks scary don’t forget to take a deep breath.
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Sometimes problems are designed to look frightening – but they are only looks! Doodling something – anything – on a piece of paper or in a book margin will often get you past the scariness.
Is there a picture or a sketch I can draw? What does the picture look like upside down? (You can always turn the competition booklet upside down if you want!) The number 165 is mentioned. Umm … 165 = 5× 33 and 33 = 3×11 . Do I know a formula for the area of a triangle? Do I know anything about triangles with sides of lengths 3, 4 , and 5 ? Instead of finding the fiftieth number, can I see what the fifth one is? The second one? Can I just guess an answer and see if it is right? Should the answer be even or odd? Is one of the choices offered obviously wrong? Let me write a for Andrew’s age and b for Betty’s age. Then b − a = 5 . If I make a table, can I see a pattern?
The sorts of questions listed above represent general strategies you can
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try. We’ll go through those strategies in the next section. But note: Sometimes competition questions are so strange that it is not even clear if there is a general strategy that will work for them. The only thing to do in these cases really is to Just Do Something! Here are some examples to explain what I mean. ✔ Problems 1-5 -solve on Edfinity.
In your attempt to:
Just Do Something!
You might want to try one or more of the following specific ploys.
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3.Strategy 1: List All The Things You Know Listing the things that you know about the ideas mentioned in the problem often helps to get started with the problem.
What Do I Know? This question seems to be about areas of shapes. Do I know a formula for the area of a circle? Of a rhombus? Of a triangle? Do I know when a number is divisible by 3? By 4 ? By 9 ? What do I know about the sides of a right triangle? If 3 − 4 −5 represent the sides of a right triangle, then so do 6 − 8 −10 and 9 −12−15 and so on.
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If p is the probability of what I want to happen, then 1− p is the probability of what I don’t want to happen. Is that easier to think about? A deck of cards has 52 cards, four suits of 13 cards each. There are 5! = 5× 4 × 3× 2×1 = 120 ways to arrange five different objects in a row. So if two objects are the same…? In a linear relationship, if one quantity changes by 1unit, then the other quantity changes by m units. Here m is the slope of the line I see if I graph an equation for the relationship. Repeating decimals represent fractions. The 85 number 0.85858585... is . 99 ✔ Problems 6-10 -solve on Edfinity.
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4.Strategy 2: Make The Problem Smaller To get a feel for the problem try breaking the problem into smaller parts, or try solving a smaller version of the same problem.
Break It Down
By symmetry I can see I need only work out the area of this smaller part and multiply the answer by six. Maybe I can work out the area of everything outside of what I want instead. That looks easier. Instead of finding the fiftieth number can I first just try to find the third? If I make a first move, is the rest of the game the same problem again? If I take one step, then the rest of the question is about five steps, not six.
✔ Problems 11-15 -solve on Edfinity.
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5.Strategy 3: Eliminate Absurd Answers
You might be able to eliminate some choices offered to you. And if you can eliminate all but one choice, that one option that remains must be the correct answer! Woohoo!
Remove Can I say anything about Absurd what type of answer to Answers expect? How large or small a number should I get? How many zeros should it end with? Should the answer be even or odd? Should the graph be straight or curved? Should the numbers be going up or down?
✔ Problems 16-20 -solve on Edfinity.
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6.Strategy 4: Was The Author Being Sneaky? If something in the question stands out as a bit strange, ask why? The author may have chosen that item in the question for a deliberate reason. See if you can figure out that reason.
Why Does
It Say That? Why the number 37 ? It’s prime. Does that mean anything? Why the number 217 ? It equals 7 × 31 and 31 is its only two-digit factor. Why is the bug walking in a semi-circle? Is it important that it stays the same distance from a certain point? Why do we care about 2s and 5s in this problem? Is it because 2 ×5 = 10 and it is easy to tell when a number is divisible by ten?
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The author said that a card was chosen first and then a coin was flipped. Does it have to be in that order? Does it have to be an equilateral triangle? Does the problem still seem to work if the angle in the corner is not 60! ?
✔ Problems 21-25 -solve on Edfinity.
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7.Strategy 5: Draw A Picture They say that a picture speaks a thousand words! Drawing a picture can shed helpful light on what is happening in the question.
Draw a
Picture Okay, I’ll draw ΔABC . Let me just draw a sketch of these two skyscrapers that are meant to be 500 feet apart. Let me draw the corner of the building and the length of the goat’s leash at that corner. I’ll draw dots for people and see how I can color three of them to mean “on the team.” Instead of writing the square number 25 , let me draw a five-by-five array of dots. Let me highlight the diagonal lines in this picture instead of the vertical ones.
✔ Problems 26-30 -solve on Edfinity.
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8.Strategy 6: Push Things To The Limit Sometimes pushing a question to an extreme can give a hint about how a problem should work. Push to the Limit
Could ALL the people in the room have the same birthday? If this process kept going forever, would the numbers just keep getting bigger? What if the train’s velocity was zero? What if my Annie’s age is exactly the same as Bjorn’s? What if the donkey’s weight was a billion pounds? What if the number had only one meaningful factor? (That is, does this question work for prime numbers too?) What if point P was really, really far away? What if, instead, P is so close to point A that it is on top of it? ✔ Problems 30-35 -solve on Edfinity.
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9.Strategy 7: Just Work It All Out
Sometimes just laboring through the problem is really all one can do to solve it.
Work It Out Okay, I’ll make a list, starting with the first and working my way up to the seventh. Let me make a table and see if I can find a pattern.
✔ Problems 36-40 -solve on Edfinity.
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10.Strategy 8: Work Backwards Or Guess And Check Making a guess at an answer and checking if it works can help solve the problem – either you might be lucky and guess the right answer, or by seeing why something doesn’t work you can gain insight as to what does work. Work Backwards
Let me just try option (B) and see if it could work. The number “ 1200 ” is divisible by all the numbers being asked of me. Could it be the right answer? I half remember something like this from geometry class. Does that mean (D) could be right? ✔ Problems 41-45 -solve on Edfinity.
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Have Fun! One final piece of advice: If you find yourself getting stuck on a long, tedious piece of work and it is not fun. STOP! It is not fun to not have fun! No competition author wants you to spend a long time laboring through something tedious. Consider rethinking your strategy and looking for a simpler, more fun, approach This final piece of advice really says it all. Math is delightful and cheery. Do always let it be fun for you!
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About The Author
JAMES TANTON Visit http://www.maa.org/mathcompetitions/teachers/curriculuminspirations/james-tanton-biography.
ABOUT THE AUTHOR: Believing that mathematics really is accessible to all, James Tanton (PhD, Mathematics, Princeton 1994) is
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committed to sharing the delight and beauty of the subject. In 2004 James founded the St. Mark’s Institute of Mathematics, an outreach program promoting joyful and effective mathematics education. He worked as a fulltime high-school teacher at St. Mark’s School in Southborough, MA (2004-2012), and he conducted, and continues to conduct, mathematics courses and workshops for mathematics teachers across the nation and overseas. James is the author of Solve This: Math Activities for Students and Clubs (MAA, 2001), The Encyclopedia of Mathematics (Facts on File, 2005), Mathematics Galore! (MAA, 2012), Geometry: An Interactive Journey to Mastery (The Great Courses, 2014), Without Words: Volumes 1 and 2
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(Tarquin 2015), Trigonometry: A Clever Study Guide (MAA, 2015), and twelve self-published texts. He is the 2005 recipient of the Beckenbach Book Prize, the 2006 recipient of the Kidder Faculty Prize at St. Mark’s School, and a 2010 recipient of a Raytheon Math Hero Award for excellence in school teaching and currently serves as the Mathematician-at-Large for the Mathematical Association of America. James is the author of Edfinity’s Zen Master’s Series For Middle School Students - a unique collection of digital titles for the modern educator and student.
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Edfinity, a division of Looking Glass Ventures, is an educational technology company headquartered in Silicon Valley that offers transformative educational technology solutions and digital content to educators and students worldwide. Edfinity works with the world’s premier academic associations, research organizations, and educational institutions to provide equitable access to exceptional educational content. Palo Alto | Boston http://edfinity.com Edfinity is a registered trademark of Looking Glass Ventures, LLC. All other trademarks are the property of their respective owners. Copyright 2016 Looking Glass Ventures, LLC. All rights reserved 1/16. ISBN: 978-1-944931-00-1
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