Speed Math for Kids PDF

SPEED MATH for Kids The Fast, Fun Way to Do Basic Calculations Bill Handley John Wiley & Sons, Inc. ffirs.indd i 1/5/07 11:44:29 AM ftoc.indd iv 1/5/07 11:44:54 AM ...

SPEED MATH for Kids The Fast, Fun Way to Do Basic Calculations Bill Handley John Wiley & Sons, Inc. ffirs.indd i 1/5/07 11:44:29 AM ftoc.indd iv 1/5/07 11:44:54 AM  SPEED MATH for Kids The Fast, Fun Way to Do Basic Calculations Bill Handley John Wiley & Sons, Inc. ffirs.indd i 1/5/07 11:44:29 AM  First published in 2005 by Wrightbooks, an imprint of John Wiley & Sons Australia, Ltd. © 2005 by Bill Handley. All rights reserved Published by Jossey-Bass A Wiley Imprint 989 Market Street, San Francisco, CA 94103-1741 www.josseybass.com Wiley Bicentennial Logo: Richard J. 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Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Jossey-Bass books and products are available through most bookstores. To contact Jossey-Bass directly, call our Customer Care Department within the U.S. at 800-956-7739, outside the U.S. at 317-572-3986, or fax 317-572-4002. Jossey-Bass also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Handley, Bill, date. Speed math for kids : the fast, fun way to do basic calculations / Bill Handley.—1st ed. p. cm. Originally published: Australia : Wrightbooks, 2005. Includes index. ISBN 978-0-7879-8863-0 (paper) 1. Mental arithmetic—Study and teaching (Elementary) I. Title. QA135.6.H36 2007 372.7—dc22 2006049171 Printed in the United States of America First Edition 10 9 8 7 6 5 4 3 2 1 ffirs.indd ii 1/5/07 11:44:29 AM  CONTENTS Preface v Introduction 1 1 Multiplication: Getting Started 4 2 Using a Reference Number 13 3 Numbers Above the Reference Number 21 4 Multiplying Above & Below the Reference Number 29 5 Checking Your Answers 34 6 Multiplication Using Any Reference Number 43 7 Multiplying Lower Numbers 59 8 Multiplication by 11 69 9 Multiplying Decimals 77 10 Multiplication Using Two Reference Numbers 87 11 Addition 106 12 Subtraction 116 13 Simple Division 130 14 Long Division by Factors 141 15 Standard Long Division Made Easy 149 16 Direct Long Division 157 17 Checking Answers (Division) 166 iii ftoc.indd iii 1/5/07 11:44:54 AM  iv Contents 18 Fractions Made Easy 173 19 Direct Multiplication 185 20 Putting It All into Practice 195 Afterword 199 Appendix A Using the Methods in the Classroom 203 Appendix B Working Through a Problem 207 Appendix C Learn the 13, 14 and 15 Times Tables 209 Appendix D Tests for Divisibility 211 Appendix E Keeping Count 215 Appendix F Plus and Minus Numbers 217 Appendix G Percentages 219 Appendix H Hints for Learning 223 Appendix I Estimating 225 Appendix J Squaring Numbers Ending in 5 227 Appendix K Practice Sheets 231 Index 239 ftoc.indd iv 1/5/07 11:44:54 AM  PREFACE I could have called this book Fun with Speed Mathematics. It contains some of the same material as my other books and teaching materials. It also includes additional methods and applications based on the strategies taught in Speed Mathematics that, I hope, give more insight into the mathematical principles and encourage creative thought. I have written this book for younger people, but I suspect that people of any age will enjoy it. I have included sections throughout the book for parents and teachers. A common response I hear from people who have read my books or attended a class of mine is, “Why wasn’t I taught this at school?” People feel that with these methods, mathematics would have been so much easier, and they could have achieved better results than they did, or they feel they would have enjoyed mathematics a lot more. I would like to think this book will help on both counts. I have definitely not intended Speed Math for Kids to be a serious textbook but rather a book to be played with and enjoyed. I have written this book in the same way that I speak to young v fpref.indd v 1/5/07 11:44:41 AM  vi Preface students. Some of the language and terms I have used are definitely non-mathematical. I have tried to write the book primarily so readers will understand. A lot of my teaching in the classroom has just been explaining out loud what goes on in my head when I am working with numbers or solving a problem. I have been gratified to learn that many schools around the world are using my methods. I receive e-mails every day from students and teachers who are becoming excited about mathematics. I have produced a handbook for teachers with instructions for teaching these methods in the classroom and with handout sheets for photocopying. Please e-mail me or visit my Web site for details. Bill Handley [email protected] www.speedmathematics.com fpref.indd vi 1/5/07 11:44:41 AM  INTRODUCTION I have heard many people say they hate mathematics. I don’t believe them. They think they hate mathematics. It’s not really math they hate; they hate failure. If you continually fail at mathematics, you will hate it. No one likes to fail. But if you succeed and perform like a genius, you will love mathematics. Often, when I visit a school, students will ask their teacher, can we do math for the rest of the day? The teacher can’t believe it. These are kids who have always said they hate math. If you are good at math, people think you are smart. People will treat you like you are a genius. Your teachers and your friends will treat you differently. You will even think differently about yourself. And there is good reason for it—if you are doing things that only smart people can do, what does that make you? Smart! I have had parents and teachers tell me something very interesting. Some parents have told me their child just won’t try when it comes to mathematics. Sometimes they tell me their child is lazy. Then the 1 cintro.indd 1 1/5/07 11:44:16 AM  2 Speed Math for Kids child has attended one of my classes or read my books. The child not only does much better in math, but also works much harder. Why is this? It is simply because the child sees results for his or her efforts. Often parents and teachers will tell the child, “Just try. You are not trying.” Or they tell the child to try harder. This just causes frustration. The child would like to try harder but doesn’t know how. Usually children just don’t know where to start. Both child and parent become frustrated and angry. I am going to teach you, with this book, not only what to do but how to do it. You can be a mathematical genius. You have the ability to perform lightning calculations in your head that will astonish your friends, your family and your teachers. This book is going to teach you how to perform like a genius—to do things your teacher, or even your principal, can’t do. How would you like to be able to multiply big numbers or do long division in your head? While the other kids are writing the problems down in their books, you are already calling out the answer. The kids (and adults) who are geniuses at mathematics don’t have better brains than you—they have better methods. This book is going to teach you those methods. I haven’t written this book like a schoolbook or textbook. This is a book to play with. You are going to learn easy ways of doing calculations, and then we are going to play and experiment with them. We will even show off to friends and family. When I was in ninth grade I had a mathematics teacher who inspired me. He would tell us stories of Sherlock Holmes or of thriller movies to illustrate his points. He would often say, “I am not supposed to be teaching you this,” or, “You are not supposed to learn this for another year or two.” Often I couldn’t wait to get home from school to try more examples for myself. He didn’t teach mathematics like the other teachers. He told stories and taught us short cuts that would help us beat the other classes. He made math exciting. He inspired my love of mathematics. cintro.indd 2 1/5/07 11:44:17 AM  Introduction 3 When I visit a school I sometimes ask students, “Who do you think is the smartest kid in this school?” I tell them I don’t want to know the person’s name. I just want them to think about who the person is. Then I ask, “Who thinks that the person you are thinking of has been told they are stupid?” No one seems to think so. Everyone has been told at one time that they are stupid—but that doesn’t make it true. We all do stupid things. Even Einstein did stupid things, but he wasn’t a stupid person. But people make the mistake of thinking that this means they are not smart. This is not true; highly intelligent people do stupid things and make stupid mistakes. I am going to prove to you as you read this book that you are very intelligent. I am going to show you how to become a mathematical genius. HOW TO READ THIS BOOK Read each chapter and then play and experiment with what you learn before going to the next chapter. Do the exercises—don’t leave them for later. The problems are not difficult. It is only by solving the exercises that you will see how easy the methods really are. Try to solve each problem in your head. You can write down the answer in a notebook. Find yourself a notebook to write your answers in and to use as a reference. This will save you writing in the book itself. That way you can repeat the exercises several times if necessary. I would also use the notebook to try your own problems. Remember, the emphasis in this book is on playing with mathematics. Enjoy it. Show off what you learn. Use the methods as often as you can. Use the methods for checking answers every time you make a calculation. Make the methods part of the way you think and part of your life. Now, go ahead and read the book and make mathematics your favorite subject. cintro.indd 3 1/5/07 11:44:17 AM  Chapter 1 +-=x0123456789()%+-=x0123456789() MULTIPLICATION: %+-=x0123456789()%+-=x0123456789 ()%+-=x0123456789()%+-=x012345678 9()%+-=x0123456789()%+-=x01234567 GETTING STARTED 89()%+-=x0123456789()%+-=x0123456 789()%+-=x0123456789()%+-=x012345 6789()%+-=x0123456789()%+-=x0123 How well do you know your multiplication tables? Do you know them up to the 15 or 20 times tables? Do you know how to solve problems like 14 × 16, or even 94 × 97, without a calculator? Using the speed mathematics method, you will be able to solve these types of problems in your head. I am going to show you a fun, fast and easy way to master your tables and basic mathematics in minutes. I’m not going to show you how to do your tables the usual way. The other kids can do that. Using the speed mathematics method, it doesn’t matter if you forget one of your tables. Why? Because if you don’t know an answer, you can simply do a lightning calculation to get an instant solution. For example, after showing her the speed mathematics methods, I asked eight-year-old Trudy, “What is 14 times 14?” Immediately she replied, “196.” I asked, ‘“You knew that?” 4 c01.indd 4 1/9/07 8:42:48 AM  Multiplication: Getting Started 5 She said, “No, I worked it out while I was saying it.” Would you like to be able to do this? It may take five or ten minutes of practice before you are fast enough to beat your friends even when they are using a calculator. WHAT IS MULTIPLICATION? How would you add the following numbers? 6+6+6+6+6+6+6+6=? You could keep adding sixes until you get the answer. This takes time and, because there are so many numbers to add, it is easy to make a mistake. The easy method is to count how many sixes there are to add together, and then use multiplication to get the answer. How many sixes are there? Count them. There are eight. You have to find out what eight sixes added together would make. People often memorize the answers or use a chart, but you are going to learn a very easy method to calculate the answer. As multiplication, the problem is written like this: 8×6= This means there are eight sixes to be added. This is easier to write than 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 =. The solution to this problem is: 8 × 6 = 48 c01.indd 5 1/9/07 8:42:48 AM  6 Speed Math for Kids THE SPEED MATHEMATICS METHOD I am now going to show you the speed mathematics way of working this out. The first step is to draw circles under each of the numbers. The problem now looks like this: 8 × 6 = We now look at each number and ask, how many more do we need to make 10? We start with the 8. If we have 8, how many more do we need to make 10? The answer is 2. Eight plus 2 equals 10. We write 2 in the circle below the 8. Our equation now looks like this: 8 × 6 = 2 We now go to the 6. How many more to make 10? The answer is 4. We write 4 in the circle below the 6. This is how the problem looks now: 8 × 6 = 2 4 We now take away, or subtract, crossways or diagonally. We either take 2 from 6 or 4 from 8. It doesn’t matter which way we subtract— the answer will be the same, so choose the calculation that looks easier. Two from 6 is 4, or 4 from 8 is 4. Either way the answer is 4. You only take away one time. Write 4 after the equals sign. 8 × 6 = 4 2 4 c01.indd 6 1/9/07 8:42:48 AM  Multiplication: Getting Started 7 For the last part of the answer, you “times,” or multiply, the numbers in the circles. What is 2 times 4? Two times 4 means two fours added together. Two fours are 8. Write the 8 as the last part of the answer. The answer is 48. 8 × 6 = 48 2 4 Easy, wasn’t it? This is much easier than repeating your multiplication tables every day until you remember them. And this way, it doesn’t matter if you forget the answer, because you can simply work it out again. Do you want to try another one? Let’s try 7 times 8. We write the problem and draw circles below the numbers as before: 7 × 8 = How many more do we need to make 10? With the first number, 7, we need 3, so we write 3 in the circle below the 7. Now go to the 8. How many more to make 10? The answer is 2, so we write 2 in the circle below the 8. Our problem now looks like this: 7 × 8 = 3 2 Now take away crossways. Either take 3 from 8 or 2 from 7. Whichever way we do it, we get the same answer. Seven minus 2 is 5 or 8 minus 3 is 5. Five is our answer either way. Five is the first digit of the answer. You only do this calculation once, so choose the way that looks easier. c01.indd 7 1/9/07 8:42:48 AM  8 Speed Math for Kids The calculation now looks like this: 7 × 8 = 5 3 2 For the final digit of the answer we multiply the numbers in the circles: 3 times 2 (or 2 times 3) is 6. Write the 6 as the second digit of the answer. Here is the finished calculation: 7 × 8 = 56 3 2 Seven eights are 56. How would you solve this problem in your head? Take both numbers from 10 to get 3 and 2 in the circles. Take away crossways. Seven minus 2 is 5. We don’t say five, we say, “Fifty...” Then multiply the numbers in the circles. Three times 2 is 6. We would say, “Fifty... six.” With a little practice you will be able to give an instant answer. And, after calculating 7 times 8 a dozen or so times, you will find you remember the answer, so you are learning your tables as you go. Test yourself Here are some problems to try by yourself. Do all of the problems, even if you know your tables well. This is the basic strategy we will use for almost all of our multiplication. a) 9 × 9 = e) 8 × 9 = b) 8 × 8 = f) 9 × 6 = c) 7 × 7 = g) 5 × 9 = d) 7 × 9 = h) 8 × 7 = c01.indd 8 1/9/07 8:42:48 AM  Multiplication: Getting Started 9 How did you do? The answers are: a) 81 b) 64 c) 49 d) 63 e) 72 f ) 54 g) 45 h) 56 Isn’t this the easiest way to learn your tables? Now, cover your answers and do them again in your head. Let’s look at 9 × 9 as an example. To calculate 9 × 9, you have 1 below 10 each time. Nine minus 1 is 8. You would say, “Eighty...” Then you multiply 1 times 1 to get the second half of the answer, 1. You would say, “Eighty... one.” If you don’t know your tables well, it doesn’t matter. You can calculate the answers until you do know them, and no one will ever know. Multiplying numbers just below 100 Does this method work for multiplying larger numbers? It certainly does. Let’s try it for 96 × 97. 96 × 97 = What do we take these numbers up to? How many more to make what? How many to make 100, so we write 4 below 96 and 3 below 97. 96 × 97 = 4 3 What do we do now? We take away crossways: 96 minus 3 or 97 minus 4 equals 93. Write that down as the first part of the answer. What do we do next? Multiply the numbers in the circles: 4 times 3 equals 12. Write this down for the last part of the answer. The full answer is 9,312. 96 × 97 = 9,312 4 3 c01.indd 9 1/9/07 8:42:49 AM  10 Speed Math for Kids Which method do you think is easier, this method or the one you learned in school? I definitely think this method; don’t you agree? Let’s try another. Let’s do 98 × 95. 98 × 95 = First we draw the circles. 98 × 95 = How many more do we need to make 100? With 98 we need 2 more and with 95 we need 5. Write 2 and 5 in the circles. 98 × 95 = 2 5 Now take away crossways. You can do either 98 minus 5 or 95 minus 2. 98 – 5 = 93 or 95 – 2 = 93 The first part of the answer is 93. We write 93 after the equals sign. 98 × 95 = 93 2 5 Now multiply the numbers in the circles. 2 × 5 = 10 Write 10 after the 93 to get an answer of 9,310. 98 × 95 = 9,310 2 5 c01.indd 10 1/9/07 8:42:49 AM  Multiplication: Getting Started 11 Easy. With a couple of minutes’ practice you should be able to do these in your head. Let’s try one now. 96 × 96 = In your head, draw circles below the numbers. What goes in these imaginary circles? How many to make 100? Four and 4. Picture the equation inside your head. Mentally write 4 and 4 in the circles. Now take away crossways. Either way you are taking 4 from 96. The result is 92. You would say, “Nine thousand, two hundred...” This is the first part of the answer. Now multiply the numbers in the circles: 4 times 4 equals 16. Now you can complete the answer: 9,216. You would say, “Nine thousand, two hundred... and sixteen.” This will become very easy with practice. Try it out on your friends. Offer to race them and let them use a calculator. Even if you aren’t fast enough to beat them, you will still earn a reputation for being a brain. Beating the calculator To beat your friends when they are using a calculator, you only have to start calling the answer before they finish pushing the buttons. For instance, if you were calculating 96 times 96, you would ask yourself how many to make 100, which is 4, and then take 4 from 96 to get 92. You can then start saying, “Nine thousand, two hundred...” While you are saying the first part of the answer you can multiply 4 times 4 in your head, so you can continue without a pause, “... and sixteen.” You have suddenly become a math genius! c01.indd 11 1/9/07 8:42:49 AM  12 Speed Math for Kids Test yourself Here are some more problems for you to do by yourself. a) 96 × 96 = e) 98 × 94 = b) 97 × 95 = f) 97 × 94 = c) 95 × 95 = g) 98 × 92 = d) 98 × 95 = h) 97 × 93 = The answers are: a) 9,216 b) 9,215 c) 9,025 d) 9,310 e) 9,212 f) 9,118 g) 9,016 h) 9,021 Did you get them all right? If you made a mistake, go back and find where you went wrong and try again. Because the method is so different, it is not uncommon to make mistakes at first. Are you impressed? Now, do the last exercise again, but this time, do all of the calculations in your head. You will find it much easier than you imagine. You need to do at least three or four calculations in your head before it really becomes easy. So, try it a few times before you give up and say it is too difficult. I showed this method to a boy in first grade and he went home and showed his dad what he could do. He multiplied 96 times 98 in his head. His dad had to get his calculator out to check if he was right! Keep reading, and in the next chapters you will learn how to use the speed math method to multiply any numbers. c01.indd 12 1/9/07 8:42:49 AM  Chapter 2 +-=x0123456789()%+-=x0123456789() USING A REFERENCE %+-=x0123456789()%+-=x0123456789 ()%+-=x0123456789()%+-=x012345678 9()%+-=x0123456789()%+-=x01234567 NUMBER 89()%+-=x0123456789()%+-=x0123456 789()%+-=x0123456789()%+-=x012345 6789()%+-=x0123456789()%+-=x0123 In this chapter we are going to look at a small change to the method that will make it easy to multiply any numbers. REFERENCE NUMBERS Let’s go back to 7 times 8: 10 7 × 8 = The 10 at the left of the problem is our reference number. It is the number we subtract the numbers we are multiplying from. The reference number is written to the left of the problem. We then ask ourselves, is the number we are multiplying above or below the reference number? In this case, both numbers are below, so we put the circles below the numbers. How many below 10 are they? Three and 2. We write 3 and 2 in the circles. Seven is 10 minus 3, so we put a minus sign in front of the 3. Eight is 10 minus 2, so we put a minus sign in front of the 2. 13 c02.indd 13 1/9/07 8:42:33 AM  14 Speed Math for Kids 10 7 × 8 = –3 –2 We now take away crossways: 7 minus 2 or 8 minus 3 is 5. We write 5 after the equals sign. 10 7 × 8 = 5 –3 –2 Now, here is the part that is different. We multiply the 5 by the reference number, 10. Five times 10 is 50, so write a 0 after the 5. (How do we multiply by 10? Simply put a 0 at the end of the number.) Fifty is our subtotal. Here is how our calculation looks now: 10 7 × 8 = 50 –3 –2 Now multiply the numbers in the circles. Three times 2 is 6. Add this to the subtotal of 50 for the final answer of 56. The full calculation looks like this: 10 7 × 8 = 50 –3 –2 + 6 56 Answer Why use a reference number? Why not use the method we used in Chapter 1? Wasn’t that easier? That method used 10 and 100 as reference numbers as well—we just didn’t write them down. c02.indd 14 1/9/07 8:42:34 AM  Using a Reference Number 15 Using a reference number allows us to calculate problems such as 6 × 7, 6 × 6, 4 × 7 and 4 × 8. Let’s see what happens when we try 6 × 7 using the method from Chapter 1. We draw the circles below the numbers and subtract the numbers we are multiplying from 10. We write 4 and 3 in the circles. Our problem looks like this: 6 × 7 = –4 –3 Now we subtract crossways: 3 from 6 or 4 from 7 is 3. We write 3 after the equals sign. 6 × 7 = 3 –4 –3 Four times 3 is 12, so we write 12 after the 3 for an answer of 312. 6 × 7 = 312 –4 –3 Is this the correct answer? No, obviously it isn’t. Let’s do the calculation again, this time using the reference number. 10 6 × 7 = 30 –4 –3 + 12 42 Answer That’s more like it. c02.indd 15 1/9/07 8:42:34 AM  16 Speed Math for Kids You should set out the calculations as shown above until the method is familiar to you. Then you can simply use the reference number in your head. Test yourself Try these problems using a reference number of 10: a) 6 × 7 = b) 7 × 5 = c) 8 × 5 = d) 8 × 4 = e) 3 × 8 = f) 6 × 5 = The answers are: a) 42 b) 35 c) 40 d) 32 e) 24 f) 30 Using 100 as a reference number What was our reference number for 96 × 97 in Chapter 1? One hundred, because we asked how many more do we need to make 100. The problem worked out in full would look like this: 100 96 × 97 = 9,300 –4 –3 + 12 9,312 Answer The technique I explained for doing the calculations in your head actually makes you use this method. Let’s multiply 98 by 98 and you will see what I mean. c02.indd 16 1/9/07 8:42:34 AM  Using a Reference Number 17 If you take 98 and 98 from 100 you get answers of 2 and 2. Then take 2 from 98, which gives an answer of 96. If you were saying the answer aloud, you would not say, “Ninety-six,” you would say, “Nine thousand, six hundred and...” Nine thousand, six hundred is the answer you get when you multiply 96 by the reference number, 100. Now multiply the numbers in the circles: 2 times 2 is 4. You can now say the full answer: “Nine thousand, six hundred and four.” Without using the reference number we might have just written the 4 after 96. Here is how the calculation looks written in full: 100 98 × 98 = 9,600 –2 –2 + 4 9,604 Answer Test yourself Do these problems in your head: a) 96 × 96 = b) 97 × 97 = c) 99 × 99 = d) 95 × 95 = e) 98 × 97 = Your answers should be: a) 9,216 b) 9,409 c) 9,801 d) 9,025 e) 9,506 c02.indd 17 1/9/07 8:42:34 AM  18 Speed Math for Kids DOUBLE MULTIPLICATION What happens if you don’t know your tables very well? How would you multiply 92 times 94? As we have seen, you would draw the circles below the numbers and write 8 and 6 in the circles. But if you don’t know the answer to 8 times 6 you still have a problem. You can get around this by combining the methods. Let’s try it. We write the problem and draw the circles: 100 92 × 94 = We write 8 and 6 in the circles. 100 92 × 94 = –8 –6 We subtract (take away) crossways: either 92 minus 6 or 94 minus 8. I would choose 94 minus 8 because it is easy to subtract 8. The easy way to take 8 from a number is to take 10 and then add 2. Ninety-four minus 10 is 84, plus 2 is 86. We write 86 after the equals sign. 100 92 × 94 = 86 –8 –6 Now multiply 86 by the reference number, 100, to get 8,600. Then we must multiply the numbers in the circles: 8 times 6. c02.indd 18 1/9/07 8:42:34 AM  Using a Reference Number 19 If we don’t know the answer, we can draw two more circles below 8 and 6 and make another calculation. We subtract the 8 and 6 from 10, giving us 2 and 4. We write 2 in the circle below the 8, and 4 in the circle below the 6. The calculation now looks like this: 100 92 × 94 = 8,600 –8 –6 –2 –4 We now need to calculate 8 times 6, using our usual method of subtracting diagonally. Two from 6 is 4, which becomes the first digit of this part of our answer. We then multiply the numbers in the circles. This is 2 times 4, which is 8, the final digit. This gives us 48. It is easy to add 8,600 and 48. 8,600 + 48 = 8,648 Here is the calculation in full. 100 92 × 94 = 8,600 –8 –6 + 48 –2 –4 8,648 Answer You can also use the numbers in the bottom circles to help your subtraction. The easy way to take 8 from 94 is to take 10 from 94, which is 84, and add the 2 in the circle to get 86. Or you could take 6 from 92. To do this, take 10 from 92, which is 82, and add the 4 in the circle to get 86. c02.indd 19 1/9/07 8:42:34 AM  20 Speed Math for Kids With a little practice, you can do these calculations entirely in your head. Note to parents and teachers People often ask me, “Don’t you believe in teaching multiplication tables to children?” My answer is, “Yes, certainly I do. This method is the easiest way to teach the tables. It is the fastest way, the most painless way and the most pleasant way to learn tables.” And while they are learning their tables, they are also learning basic number facts, practicing addition and subtraction, memorizing combinations of numbers that add to 10, working with positive and negative numbers, and learning a whole approach to basic mathematics. c02.indd 20 1/9/07 8:42:34 AM  Chapter 3 +-=x0123456789()%+-=x0123456789() NUMBERS ABOVE THE %+-=x0123456789()%+-=x0123456789 ()%+-=x0123456789()%+-=x012345678 9()%+-=x0123456789()%+-=x01234567 REFERENCE NUMBER 89()%+-=x0123456789()%+-=x0123456 789()%+-=x0123456789()%+-=x012345 6789()%+-=x0123456789()%+-=x0123 What if you want to multiply numbers above the reference number; above 10 or 100? Does the method still work? Let’s find out. MULTIPLYING NUMBERS IN THE TEENS Here is how we multiply numbers in the teens. We will use 13 × 15 as an example and use 10 as our reference number. 10 13 × 15 = Both 13 and 15 are above the reference number, 10, so we draw the circles above the numbers, instead of below as we have been doing. How much above 10 are they? Three and 5, so we write 3 and 5 in the circles above 13 and 15. Thirteen is 10 plus 3, so we write a plus sign in front of the 3; 15 is 10 plus 5, so we write a plus sign in front of the 5. 21 c03.indd 21 1/9/07 8:42:20 AM  22 Speed Math for Kids +3 +5 10 13 × 15 = As before, we now go crossways. Thirteen plus 5 or 15 plus 3 is 18. We write 18 after the equals sign. +3 +5 10 13 × 15 = 18 We then multiply the 18 by the reference number, 10, and get 180. (To multiply a number by 10 we add a 0 to the end of the number.) One hundred and eighty is our subtotal, so we write 180 after the equals sign. +3 +5 10 13 × 15 = 180 For the last step, we multiply the numbers in the circles. Three times 5 equals 15. Add 15 to 180 and we get our answer of 195. This is how we write the problem in full: +3 +5 10 13 × 15 = 180 + 15 195 Answer If the number we are multiplying is above the reference number, we put the circle above. If the number is below the reference number, we put the circle below. If the circled number is above, we add diagonally. If the circled number is below, we subtract diagonally. c03.indd 22 1/9/07 8:42:20 AM  Numbers Above the Reference Number 23 The numbers in the circles above are plus numbers and the numbers in the circles below are minus numbers. Let’s try another one. How about 12 × 17? The numbers are above 10, so we draw the circles above. How much above 10? Two and 7, so we write 2 and 7 in the circles. +2 +7 10 12 × 17 = What do we do now? Because the circles are above, the numbers are plus numbers, so we add crossways. We can either do 12 plus 7 or 17 plus 2. Let’s do 17 plus 2. 17 + 2 = 19 We now multiply 19 by 10 (our reference number) to get 190 (we just put a 0 after the 19). Our work now looks like this: +2 +7 10 12 × 17 = 190 Now we multiply the numbers in the circles. 2 × 7 = 14 Add 14 to 190 and we have our answer. Fourteen is 10 plus 4. We can add the 10 first (190 + 10 = 200), then the 4, to get 204. Here is the finished problem: +2 +7 10 12 × 17 = 190 + 14 204 Answer c03.indd 23 1/9/07 8:42:20 AM  24 Speed Math for Kids Test yourself Now try these problems by yourself. a) 12 × 15 = f) 12 × 16 = b) 13 × 14 = g) 14 × 14 = c) 12 × 12 = h) 15 × 15 = d) 13 × 13 = i) 12 × 18 = e) 12 × 14 = j) 16 × 14 = The answers are: a) 180 b) 182 c) 144 d) 169 e) 168 f) 192 g) 196 h) 225 i) 216 j) 224 If any of your answers were wrong, read through this section again, find your mistake, then try again. How would you solve 13 × 21? Let’s try it: 10 13 × 21 = We still use a reference number of 10. Both numbers are above 10, so we put the circles above. Thirteen is 3 above 10, 21 is 11 above, so we write 3 and 11 in the circles. Twenty-one plus 3 is 24, times 10 is 240. Three times 11 is 33, added to 240 makes 273. This is how the completed problem looks: +3 +11 10 13 × 21 = 240 + 33 273 Answer c03.indd 24 1/9/07 8:42:20 AM  Numbers Above the Reference Number 25 MULTIPLYING NUMBERS ABOVE 100 We can use our speed math method to multiply numbers above 100 as well. Let’s try 113 times 102. We use 100 as our reference number. +13 +2 100 113 × 102 = Add crossways: 113 + 2 = 115 Multiply by the reference number: 115 × 100 = 11,500 Now multiply the numbers in the circles: 2 × 13 = 26 This is how the completed problem looks: +13 +2 100 113 × 102 = 11,500 + 26 11,526 Answer SOLVING PROBLEMS IN YOUR HEAD When you use these strategies, what you say inside your head is very important, and can help you solve problems more quickly and easily. Let’s try multiplying 16 by 16. c03.indd 25 1/9/07 8:42:20 AM  26 Speed Math for Kids This is how I would solve this problem in my head: 16 plus 6 (from the second 16) equals 22, times 10 equals 220 6 times 6 is 36 220 plus 30 is 250, plus 6 is 256 Try it. See how you do. Inside your head you would say: 16 plus 6... 22... 220... 36... 256 With practice, you can leave out a lot of that. You don’t have to go through it step by step. You would only say to yourself: 220... 256 Practice doing this. Saying the right thing in your head as you do the calculation can better than halve the time it takes. How would you calculate 7 × 8 in your head? You would “see” 3 and 2 below the 7 and 8. You would take 2 from the 7 (or 3 from the 8) and say, “Fifty,” multiplying by 10 in the same step. Three times 2 is 6. All you would say is, “Fifty... six.” What about 6 × 7? You would “see” 4 and 3 below the 6 and 7. Six minus 3 is 3; you say, “Thirty.” Four times 3 is 12, plus 30 is 42. You would just say, “Thirty... forty-two.” It’s not as hard as it sounds, is it? And it will become easier the more you do. DOUBLE MULTIPLICATION Let’s multiply 88 by 84. We use 100 as our reference number. Both numbers are below 100, so we draw the circles below. How many below are they? Twelve and 16. We write 12 and 16 in the circles. c03.indd 26 1/9/07 8:42:21 AM  Numbers Above the Reference Number 27 Now subtract crossways: 84 minus 12 is 72. (Subtract 10, then 2, to subtract 12.) Multiply the answer of 72 by the reference number, 100, to get 7,200. The calculation so far looks like this: 100 88 × 84 = 7,200 –12 –16 We now multiply 12 times 16 to finish the calculation. +2 +6 10 12 × 16 = 180 + 12 192 This calculation can be done mentally. Now add this answer to our subtotal of 7,200. If you were doing the calculation in your head, you would simply add 100 first, then 92, like this: 7,200 plus 100 is 7,300, plus 92 is 7,392. Simple. You should easily do this in your head with just a little practice. Test yourself Try these problems: a) 87 × 86 = c) 88 × 87 = b) 88 × 88 = d) 88 × 85 = c03.indd 27 1/9/07 8:42:21 AM  28 Speed Math for Kids The answers are: a) 7,482 b) 7,744 c) 7,656 d) 7,480 Combining the methods taught in this book creates endless possibilities. Experiment for yourself. Note to parents and teachers This chapter introduces the concept of positive and negative numbers. We will simply refer to them as plus and minus numbers throughout the book. These methods make positive and negative numbers tangible. Children can easily relate to the concept because it is made visual. Calculating numbers in the eighties using double multiplication develops concentration. I find most children can do the calculations much more easily than most adults think they should be able to. Kids love showing off. Give them the opportunity. c03.indd 28 1/9/07 8:42:21 AM  Chapter 4 MULTIPLYING ABOVE +-=x0123456789()%+-=x0123456789() %+-=x0123456789()%+-=x0123456789 ()%+-=x0123456789()%+-=x012345678 & BELOW THE 9()%+-=x0123456789()%+-=x01234567 89()%+-=x0123456789()%+-=x0123456 789()%+-=x0123456789()%+-=x012345 REFERENCE NUMBER 6789()%+-=x0123456789()%+-=x0123 Until now, we have multiplied numbers that were both below the reference number or both above the reference number. How do we multiply numbers when one number is above the reference number and the other is below the reference number? NUMBERS ABOVE AND BELOW We will see how this works by multiplying 97 × 125. We will use 100 as our reference number: 100 97 × 125 = Ninety-seven is below the reference number, 100, so we put the circle below. How much below? Three, so we write 3 in the circle. One hundred and twenty-five is above, so we put the circle above. How much above? Twenty-five, so we write 25 in the circle above. 29 c04.indd 29 1/9/07 8:42:06 AM  30 Speed Math for Kids +25 100 97 × 125 = –3 One hundred and twenty-five is 100 plus 25, so we put a plus sign in front of the 25. Ninety-seven is 100 minus 3, so we put a minus sign in front of the 3. We now calculate crossways, either 97 plus 25 or 125 minus 3. One hundred and twenty-five minus 3 is 122. We write 122 after the equals sign. We now multiply 122 by the reference number, 100. One hundred and twenty-two times 100 is 12,200. (To multiply any number by 100, we simply put two zeros after the number.) This is similar to what we have done in earlier chapters. This is how the problem looks so far: +25 100 97 × 125 = 12,200 –3 Now we multiply the numbers in the circles. Three times 25 is 75, but that is not really the problem. We have to multiply 25 by minus 3. The answer is –75. Now our problem looks like this: +25 100 97 × 125 = 12,200 – 75 –3 c04.indd 30 1/9/07 8:42:06 AM  Multiplying Above & Below the Reference Number 31 A shortcut for subtraction Let’s take a break from this problem for a moment to have a look at a shortcut for the subtractions we are doing. What is the easiest way to subtract 75? Let me ask another question. What is the easiest way to take 9 from 63 in your head? 63 – 9 = I am sure you got the right answer, but how did you get it? Some would take 3 from 63 to get 60, then take another 6 to make up the 9 they have to take away, and get 54. Some would take away 10 from 63 and get 53. Then they would add 1 back because they took away 1 too many. This would also give 54. Some would do the problem the same way they would when using pencil and paper. This way they have to carry and borrow in their heads. This is probably the most difficult way to solve the problem. Remember, the easiest way to solve a problem is also the fastest, with the least chance of making a mistake. Most people find the easiest way to subtract 9 is to take away 10, then add 1 to the answer. The easiest way to subtract 8 is to take away 10, then add 2 to the answer. The easiest way to subtract 7 is to take away 10, then add 3 to the answer. What is the easiest way to take 90 from a number? Take 100 and give back 10. What is the easiest way to take 80 from a number? Take 100 and give back 20. What is the easiest way to take 70 from a number? Take 100 and give back 30. If we go back to the problem we were working on, how do we take 75 from 12,200? We can take away 100 and give back 25. Is this c04.indd 31 1/9/07 8:42:06 AM  32 Speed Math for Kids easy? Let’s try it. Twelve thousand, two hundred minus 100? Twelve thousand, one hundred. Plus 25? Twelve thousand, one hundred and twenty-five. Easy. So back to our example. This is how the completed problem looks: +25 100 97 × 125 = 12,200 – 75 = 12,125 Answer –3 25 With a little practice you should be able to solve these problems entirely in your head. Practice with the problems below. Test yourself Try these: a) 98 × 145 = e) 98 × 146 = b) 98 × 125 = f) 9 × 15 = c) 95 × 120 = g) 8 × 12 = d) 96 × 125 = h) 7 × 12 = How did you do? The answers are: a) 14,210 b) 12,250 c) 11,400 d) 12,000 e) 14,308 f) 135 g) 96 h) 84 Multiplying numbers in the circles The rule for multiplying the numbers in the circles is: When both circles are above the numbers or both circles are below the numbers, we add the answer. When one circle is above and one circle is below, we subtract. c04.indd 32 1/9/07 8:42:06 AM  Multiplying Above & Below the Reference Number 33 Mathematically, we would say: when we multiply two positive (plus) numbers, we get a positive (plus) answer. When we multiply two negative (minus) numbers, we get a positive (plus) answer. When we multiply a positive (plus) by a negative (minus), we get a minus answer. Let’s try another problem. Would our method work for multiplying 8 × 42? Let’s try it. We choose a reference number of 10. Eight is 2 below 10 and 42 is 32 above 10. +32 10 8 × 42 = –2 We either take 2 from 42 or add 32 to 8. Two from 42 is 40, times the reference number, 10, is 400. Minus 2 times 32 is –64. To take 64 from 400 we take 100, which equals 300, then give back 36 for a final answer of 336. (We will look at an easy way to subtract numbers from 100 in the chapter on subtraction.) Our completed problem looks like this: +32 10 8 × 42 = 400 – 64 = 336 Answer –2 36 We haven’t finished with multiplication yet, but we can take a rest here and practice what we have already learned. If some problems don’t seem to work out easily, don’t worry; we still have more to cover. In the next chapter we will have a look at a simple method for checking answers. c04.indd 33 1/9/07 8:42:06 AM  Chapter 5 +-=x0123456789()%+-=x0123456789() CHECKING %+-=x0123456789()%+-=x0123456789 ()%+-=x0123456789()%+-=x012345678 9()%+-=x0123456789()%+-=x01234567 YOUR ANSWERS 89()%+-=x0123456789()%+-=x0123456 789()%+-=x0123456789()%+-=x012345 6789()%+-=x0123456789()%+-=x0123 What would it be like if you always found the right answer to every math problem? Imagine scoring 100% on every math test. How would you like to get a reputation for never making a mistake? If you do make a mistake, I can teach you how to find and correct it before anyone (including your teacher) knows anything about it. When I was young, I often made mistakes in my calculations. I knew how to do the problems, but I still got the wrong answer. I would forget to carry a number, or find the right answer but write down something different, and who knows what other mistakes I would make. I had some simple methods for checking answers I had devised myself, but they weren’t very good. They would confirm maybe the last digit of the answer or they would show me that the answer I got was at least close to the real answer. I wish I had known then the method I am going to show you now. Everyone would have thought I was a genius if I had known this. 34 c05.indd 34 1/9/07 8:41:50 AM  Checking Your Answers 35 Mathematicians have known this method of checking answers for about 1,000 years, although I have made a small change I haven’t seen anywhere else. It is called the digit sum method. I have taught this method of checking answers in my other books, but this time I am going to teach it differently. This method of checking your answers will work for almost any calculation. Because I still make mistakes occasionally, I always check my answers. Here is the method I use. SUBSTITUTE NUMBERS To check the answer to a calculation, we use substitute numbers instead of the original numbers we were working with. A substitute on a football team or a basketball team is somebody who takes another person’s place on the team. If somebody gets injured, or tired, they take that person off and bring on a substitute player. A substitute teacher fills in when your regular teacher is unable to teach you. We can use substitute numbers in place of the original numbers to check our work. The substitute numbers are always low and easy to work with. Let me show you how it works. Let us say we have just calculated 12 × 14 and come to an answer of 168. We want to check this answer. 12 × 14 = 168 The first number in our problem is 12. We add its digits together to get the substitute: 1+2=3 Three is our substitute for 12. I write 3 in pencil either above or below the 12, wherever there is room. The next number we are working with is 14. We add its digits: 1+4=5 c05.indd 35 1/9/07 8:41:50 AM  36 Speed Math for Kids Five is our substitute for 14. We now do the same calculation (multiplication) using the substitute numbers instead of the original numbers: 3 × 5 = 15 Fifteen is a two-digit number, so we add its digits together to get our check answer: 1+5=6 Six is our check answer. We add the digits of the original answer, 168: 1 + 6 + 8 = 15 Fifteen is a two-digit number, so we add its digits together to get a one-digit answer: 1+5=6 Six is our substitute answer. This is the same as our check answer, so our original answer is correct. Had we gotten an answer that added to, say, 2 or 5, we would know we had made a mistake. The substitute answer must be the same as the check answer if the substitute is correct. If our substitute answer is different, we know we have to go back and check our work to find the mistake. I write the substitute numbers in pencil so I can erase them when I have made the check. I write the substitute numbers either above or below the original numbers, wherever I have room. The example we have just done would look like this: c05.indd 36 1/9/07 8:41:50 AM  Checking Your Answers 37 +2 +4 10 12 × 14 = 160 3 5 + 8 168 6 If we have the right answer in our calculation, the digits in the original answer should add up to the same as the digits in our check answer. Let’s try it again, this time using 14 × 14: 14 × 14 = 196 1 + 4 = 5 (substitute for 14) 1 + 4 = 5 (substitute for 14 again) So our substitute numbers are 5 and 5. Our next step is to multiply these: 5 × 5 = 25 Twenty-five is a two-digit number, so we add its digits: 2+5=7 Seven is our check answer. Now, to find out if we have the correct answer, we add the digits in our original answer, 196: 1 + 9 + 6 = 16 To bring 16 to a one-digit number: 1+6=7 Seven is what we got for our check answer, so we can be confident we didn’t make a mistake. c05.indd 37 1/9/07 8:41:50 AM  38 Speed Math for Kids +4 +4 10 14 × 14 = 180 5 5 + 16 196 7 A shortcut There is another shortcut to this procedure. If we find a 9 anywhere in the calculation, we cross it out. This is called casting out nines. You can see with this example how this removes a step from our calculations without affecting the result. With the last answer, 196, instead of adding 1 + 9 + 6, which equals 16, and then adding 1 + 6, which equals 7, we could cross out the 9 and just add 1 and 6, which also equals 7. This makes no difference to the answer, but it saves some time and effort, and I am in favor of anything that saves time and effort. What about the answer to the first problem we solved, 168? Can we use this shortcut? There isn’t a 9 in 168. We added 1 + 6 + 8 to get 15, then added 1 + 5 to get our final check answer of 6. In 168, we have two digits that add up to 9, the 1 and the 8. Cross them out and you just have the 6 left. No more work to do at all, so our shortcut works. Check any size number What makes this method so easy to use is that it changes any size number into a single-digit number. You can check calculations that are too big to go into your calculator by casting out nines. For instance, if we wanted to check 12,345,678 × 89,045 = 1,099,320,897,510, we would have a problem because most c05.indd 38 1/9/07 8:41:50 AM  Checking Your Answers 39 calculators can’t handle the number of digits in the answer, so most would show the first digits of the answer with an error sign. The easy way to check the answer is to cast out the nines. Let’s try it. 12,345,678 0 89,045 = 8 1,099,320,897,510 0 All of the digits in the answer cancel. The nines automatically cancel, then we have 1 + 8, 2 + 7, then 3 + 5 + 1 = 9, which cancels again. And 0 × 8 = 0, so our answer seems to be correct. Let’s try it again. 137 × 456 = 62,472 To find our substitute for 137: 1 + 3 + 7 = 11 1+1=2 There were no shortcuts with the first number. Two is our substitute for 137. To find our substitute for 456: 4+5+6= We immediately see that 4 + 5 = 9, so we cross out the 4 and the 5. That just leaves us with 6, our substitute for 456. Can we find any nines, or digits adding up to 9, in the answer? Yes, 7 + 2 = 9, so we cross out the 7 and the 2. We add the other digits: 6 + 2 + 4 = 12 1+2=3 Three is our substitute answer. c05.indd 39 1/9/07 8:41:50 AM  40 Speed Math for Kids I write the substitute numbers in pencil above or below the actual numbers in the problem. It might look like this: 137 × 456 = 62,472 2 6 3 Is 62,472 the right answer? We multiply the substitute numbers: 2 times 6 equals 12. The digits in 12 add up to 3 (1 + 2 = 3). This is the same as our substitute answer, so we were right again. Let’s try one more example. Let’s check if this answer is correct: 456 × 831 = 368,936 We write in our substitute numbers: 456 × 831 = 368,936 6 3 8 That was easy because we cast out (or crossed out) 4 and 5 from the first number, leaving 6. We cast out 8 and 1 from the second number, leaving 3. And almost every digit was cast out of the answer, 3 plus 6 twice, and a 9, leaving a substitute answer of 8. We now see if the substitutes work out correctly: 6 times 3 is 18, which adds up to 9, which also gets cast out, leaving 0. But our substitute answer is 8, so we have made a mistake somewhere. When we calculate it again, we get 378,936. Did we get it right this time? The 936 cancels out, so we add 3 + 7 + 8, which equals 18, and 1 + 8 adds up to 9, which cancels, leaving 0. This is the same as our check answer, so this time we have it right. Does this method prove we have the right answer? No, but we can be almost certain. c05.indd 40 1/9/07 1:32:37 PM  Checking Your Answers 41 This method won’t find all mistakes. For instance, say we had 3,789,360 for our last answer; by mistake we put a 0 on the end. The final 0 wouldn’t affect our check by casting out nines and we wouldn’t know we had made a mistake. When it showed we had made a mistake, though, the check definitely proved we had the wrong answer. It is a simple, fast check that will find most mistakes, and should get you 100% scores on most of your math tests. Do you get the idea? If you are unsure about using this method to check your answers, we will be using the method throughout the book so you will soon become familiar with it. Try it on your calculations at school and at home. Why does the method work? You will be much more successful using a new method when you not only know that it does work, but you understand why it works as well. First, 10 is 1 times 9 with 1 remainder. Twenty is 2 nines with 2 remainder. Twenty-two would be 2 nines with 2 remainder for the 20 plus 2 more for the units digit. If you have 35¢ in your pocket and you want to buy as many candies as you can for 9¢ each, each 10¢ will buy you one candy with 1¢ change. So, 30¢ will buy you three candies with 3¢ change, plus the extra 5¢ in your pocket gives you 8¢. So, the number of tens plus the units digit gives you the nines remainder. Second, think of a number and multiply it by 9. What is 4 × 9? The answer is 36. Add the digits in the answer together, 3 + 6, and you get 9. Let’s try another number. Three nines are 27. Add the digits of the answer together, 2 + 7, and you get 9 again. c05.indd 41 1/9/07 8:41:51 AM  42 Speed Math for Kids Eleven nines are 99. Nine plus 9 equals 18. Wrong answer? No, not yet. Eighteen is a two-digit number, so we add its digits together: 1 + 8. Again, the answer is 9. If you multiply any number by 9, the sum of the digits in the answer will always add up to 9 if you keep adding the digits until you get a one-digit number. This is an easy way to tell if a number is evenly divisible by 9. If the digits of any number add up to 9, or a multiple of 9, then the number itself is evenly divisible by 9. If the digits of a number add up to any number other than 9, this other number is the remainder you would get after dividing the number by 9. Let’s try 13: 1+3=4 Four is the digit sum of 13. It should be the remainder you would get if you divided by 9. Nine divides into 13 once, with 4 remainder. If you add 3 to the number, you add 3 to the remainder. If you double the number, you double the remainder. If you halve the number, you halve the remainder. Don’t believe me? Half of 13 is 6.5. Six plus 5 equals 11. One plus 1 equals 2. Two is half of 4, the nines remainder for 13. Whatever you do to the number, you do to the remainder, so we can use the remainders as substitutes. Why do we use 9 remainders? Couldn’t we use the remainders after dividing by, say, 17? Certainly, but there is so much work involved in dividing by 17, the check would be harder than the original problem. We choose 9 because of the easy shortcut method for finding the remainder. c05.indd 42 1/9/07 8:41:51 AM  Chapter 6 MULTIPLICATION +-=x0123456789()%+-=x0123456789() %+-=x0123456789()%+-=x0123456789 ()%+-=x0123456789()%+-=x012345678 USING ANY 9()%+-=x0123456789()%+-=x01234567 89()%+-=x0123456789()%+-=x0123456 789()%+-=x0123456789()%+-=x012345 REFERENCE NUMBER 6789()%+-=x0123456789()%+-=x0123 In Chapters 1 to 4 you learned how to multiply numbers using an easy method that makes multiplication fun. It is easy to use when the numbers are near 10 or 100. But what about multiplying numbers that are around 30 or 60? Can we still use this method? We certainly can. We chose reference numbers of 10 and 100 because it is easy to multiply by 10 and 100. The method will work just as well with other reference numbers, but we must choose numbers that are easy to multiply by. MULTIPLICATION BY FACTORS It is easy to multiply by 20, because 20 is 2 times 10. It is easy to multiply by 10 and it is easy to multiply by 2. This is called multiplication by factors, because 10 and 2 are factors of 20 (20 = 10 × 2). So, to multiply any number by 20, you multiply it by 2 and 43 c06.indd 43 1/9/07 8:41:36 AM  44 Speed Math for Kids then multiply the answer by 10, or, you could say, you double the number and add a 0. For instance, to multiply 7 by 20, you would double it (2 × 7 = 14) and then multiply your answer by 10 (14 × 10 = 140). To multiply 32 by 20, you would double 32 (64) and then multiply by 10 (640). Multiplying by 20 is easy because it is easy to multiply by 2 and it is easy to multiply by 10. So, it is easy to use 20 as a reference number. Let us try an example: 23 × 21 = Twenty-three and 21 are just above 20, so we use 20 as our reference number. Both numbers are above 20, so we put the circles above. How much above are they? Three and 1. We write those numbers above in the circles. Because the circles are above, they are plus numbers. +3 +1 20 23 × 21 = We add diagonally: 23 + 1 = 24 We multiply the answer, 24, by the reference number, 20. To do this, we multiply by 2, then by 10: 24 × 2 = 48 48 × 10 = 480 We could now draw a line through the 24 to show we have finished using it. The rest is the same as before. We multiply the numbers in the circles: c06.indd 44 1/9/07 8:41:36 AM  Multiplication Using Any Reference Number 45 3×1=3 480 + 3 = 483 The problem now looks like this: +3 +1 20 23 × 21 = 24 480 + 3 483 Answer Checking answers Let us apply what we learned in the last chapter and check our answer: 23 × 21 = 483 5 3 15 6 The substitute numbers for 23 and 21 are 5 and 3. 5 × 3 = 15 1+5=6 Six is our check answer. The digits in our original answer, 483, add up to 6: 4 + 8 + 3 = 15 1+5=6 This is the same as our check answer, so we were right. Let’s try another: 23 × 31 = c06.indd 45 1/9/07 8:41:36 AM  46 Speed Math for Kids We put 3 and 11 above 23 and 31: +3 +11 20 23 × 31 = They are 3 and 11 above the reference number, 20. Adding diagonally, we get 34: 31 + 3 = 34 or 23 + 11 = 34 We multiply this answer by the reference number, 20. To do this, we multiply 34 by 2, then multiply by 10. 34 × 2 = 68 68 × 10 = 680 This is our subtotal. We now multiply the numbers in the circles (3 and 11): 3 × 11 = 33 Add this to 680: 680 + 33 = 713 The calculation will look like this: +3 +11 20 23 × 31 = 34 680 + 33 713 Answer We check by casting out the nines: c06.indd 46 1/9/07 8:41:36 AM  Multiplication Using Any Reference Number 47 23 × 31 = 713 5 4 11 2 Multiply our substitute numbers and then add the digits in the answer: 5 × 4 = 20 2+0=2 This checks with our substitute answer, so we can accept that as correct. Test yourself Here are some problems to try. When you have finished them, you can check your answers yourself by casting out the nines. a) 21 × 24 = d) 23 × 27 = b) 24 × 24 = e) 21 × 35 = c) 23 × 23 = f) 26 × 24 = You should be able to do all of those problems in your head. It’s not difficult with a little practice. MULTIPLYING NUMBERS BELOW 20 How about multiplying numbers below 20? If the numbers (or one of the numbers to be multiplied) are in the high teens, we can use 20 as a reference number. Let’s try an example: 18 × 17 = c06.indd 47 1/9/07 8:41:36 AM  48 Speed Math for Kids Using 20 as a reference number, we get: 20 18 × 17 = –2 –3 Subtract diagonally: 17 – 2 = 15 Multiply by 20: 2 × 15 = 30 30 × 10 = 300 Three hundred is our subtotal. Now we multiply the numbers in the circles and then add the result to our subtotal: 2×3=6 300 + 6 = 306 Our completed work should look like this: 20 18 × 17 = 15 –2 –3 300 + 6 306 Answer Now let’s try the same example using 10 as a reference number: +8 +7 10 18 × 17 = Add crossways, then multiply by 10 to get a subtotal: c06.indd 48 1/9/07 8:41:36 AM  Multiplication Using Any Reference Number 49 18 + 7 = 25 10 × 25 = 250 Multiply the numbers in the circles and add this to the subtotal: 8 × 7 = 56 250 + 56 = 306 Our completed work should look like this: +8 +7 10 18 × 17 = 250 + 56 306 Answer This confirms our first answer. There isn’t much to choose between using the different reference numbers. It is a matter of personal preference. Simply choose the reference number you find easier to work with. MULTIPLYING NUMBERS ABOVE AND BELOW 20 The third possibility is if one number is above and the other below 20. +14 20 18 × 34 = –2 We can either add 18 to 14 or subtract 2 from 34, and then multiply the result by our reference number: 34 – 2 = 32 32 × 20 = 640 c06.indd 49 1/9/07 8:41:37 AM  50 Speed Math for Kids We now multiply the numbers in the circles: 2 × 14 = 28 It is actually –2 times 14 so our answer is –28. 640 – 28 = 612 To subtract 28, we subtract 30 and add 2. 640 – 30 = 610 610 + 2 = 612 Our problem now looks like this: +14 20 18 × 34 = 32 –2 640 – 28 = 612 Answer 2 Let’s check the answer by casting out the nines: 18 × 34 = 612 9 7 9 0 0 Zero times 7 is 0, so the answer is correct. USING 50 AS A REFERENCE NUMBER That takes care of the numbers up to around 30 times 30. What if the numbers are higher? Then we can use 50 as a reference number. It is easy to multiply by 50 because 50 is half of 100. You could say 50 is 100 divided by 2. So, to multiply by 50, we multiply the number by 100 and then divide the answer by 2. c06.indd 50 1/9/07 8:41:37 AM  Multiplication Using Any Reference Number 51 Let’s try it: 50 47 × 45 = –3 –5 Subtract diagonally: 45 – 3 = 42 Multiply 42 by 100, then divide by 2: 42 × 100 = 4,200 4,200 ÷ 2 = 2,100 Now multiply the numbers in the circles, and add this result to 2,100: 3 × 5 = 15 2,100 + 15 = 2,115 50 47 × 45 = 4,200 –3 –5 2,100 + 15 2,115 Answer Fantastic. That was so easy. Let’s try another: +3 +7 50 53 × 57 = Add diagonally, then multiply the result by the reference number (multiply by 100 and then divide by 2): 57 + 3 = 60 60 × 100 = 6,000 c06.indd 51 1/9/07 8:41:37 AM  52 Speed Math for Kids We divide by 2 to get 3,000. We then multiply the numbers in the circles and add the result to 3,000: 3 × 7 = 21 3,000 + 21 = 3,021 Our problem will end up looking like this: +3 +7 50 53 × 57 = 6,000 3,000 + 21 3,021 Answer Let’s try one more: +1 +14 50 51 × 64 = Add diagonally and multiply the result by the reference number (multiply by 100 and then divide by 2): 64 + 1 = 65 65 × 100 = 6,500 Then we halve the answer. Half of 6,000 is 3,000. Half of 500 is 250. Our subtotal is 3,250. Now multiply the numbers in the circles: 1 × 14 = 14 Add 14 to the subtotal to get 3,264. c06.indd 52 1/9/07 8:41:37 AM  Multiplication Using Any Reference Number 53 Our problem now looks like this: +1 +14 50 51 × 64 = 6,500 3,250 + 14 3,264 Answer We could check that by casting out the nines: 51 × 64 = 3,264 6 1 6 Six and 4 in 64 add up to 10, which added again gives us 1. Six times 1 does give us 6, so the answer is correct. Test yourself Here are some problems for you to do. See how many you can do in your head. a) 45 × 45 = e) 51 × 57 = b) 49 × 49 = f) 54 × 56 = c) 47 × 43 = g) 51 × 68 = d) 44 × 44 = h) 51 × 72 = How did you do with those? You should have had no trouble doing all of them in your head. The answers are: a) 2,025 b) 2,401 c) 2,021 d) 1,936 e) 2,907 f) 3,024 g) 3,468 h) 3,672 c06.indd 53 1/9/07 8:41:37 AM  54 Speed Math for Kids MULTIPLYING HIGHER NUMBERS There is no reason why we can’t use 200, 500 and 1,000 as reference numbers. To multiply by 200, we multiply by 2 and 100. To multiply by 500, we multiply by 1,000, and halve the answer. To multiply by 1,000, we simply write three zeros after the number. Let’s try some examples. 212 × 212 = We use 200 as a reference number. +12 +12 200 212 × 212 = Both numbers are above the reference number, so we draw the circles above the numbers. How much above? Twelve and 12, so we write 12 in each circle. They are plus numbers, so we add crossways. 212 + 12 = 224 We multiply 224 by 2 and by 100. 224 × 2 = 448 448 × 100 = 44,800 Now multiply the numbers in the circles. 12 × 12 = 144 (You must know 12 times 12. If you don’t, you simply calculate the answer using 10 as a reference number.) c06.indd 54 1/9/07 8:41:37 AM  Multiplication Using Any Reference Number 55 44,800 + 144 = 44,944 +12 +12 200 212 × 212 = 224 44,800 + 144 44,944 Answer Let’s try another. How about multiplying 511 by 503? We use 500 as a reference number. +11 +3 500 511 × 503 = Add crossways. 511 + 3 = 514 Multiply by 500. (Multiply by 1,000 and divide by 2.) 514 × 1,000 = 514,000 Half of 514,000 is 257,000. (How did we work this out? Half of 500 is 250 and half of 14 is 7.) Multiply the numbers in the circles, and add the result to our subtotal. 3 × 11 = 33 257,000 + 33 = 257,033 c06.indd 55 1/9/07 8:41:37 AM  56 Speed Math for Kids +11 +3 500 511 × 503 = 514