Speed Mathematics E-Book

Bill Handley

• Herausgeber: John Wiley & Sons
• Kategorie: Bildung
• Sprache: Englisch

Seitenzahl: 346

Veröffentlichungsjahr: 2012

Beschreibung

This new, revised edition of the bestselling Speed Mathematics features new chapters on memorising numbers and general information, calculating statistics and compound interest, square roots, logarithms and easy trig calculations. Written so anyone can understand, this book teaches simple strategies that will enable readers to make lightning-quick calculations. People who excel at mathematics use better strategies than the rest of us; they are not necessarily more intelligent. With Speed Mathematics you'll discover methods to make maths easy and fun. This book is perfect for students, parents, teachers and anyone who enjoys working with figures and even those who are terrified of numbers!

Leseprobe

Contents

Preface

Introduction

Chapter 1: Multiplication: Part one

Multiplying numbers up to 10

To learn or not to learn tables?

Multiplying numbers greater than 10

Racing a calculator

Chapter 2: Using a reference number

Using 10 as a reference number

Why use a reference number?

When to use a reference number

Using 100 as a reference number

Multiplying numbers in the teens

Multiplying numbers above 100

Solving problems in your head

Combining methods

Chapter 3: Multiplying numbers above and below the reference number

A shortcut for subtraction

Multiplying numbers in the circles

Chapter 4: Checking answers: Part one

Substitute numbers

Casting out nines

Why does the method work?

Chapter 5: Multiplication: Part two

Multiplication by factors

Checking our answers

Multiplying numbers below 20

Numbers above and below 20

Multiplying higher numbers

Doubling and halving numbers

Using 200 and 500 as reference numbers

Multiplying lower numbers

Multiplication by 5

Chapter 6: Multiplying decimals

Chapter 7: Multiplying using two reference numbers

Using factors expressed as a division

Why does this method work?

Chapter 8: Addition

Two-digit mental addition

Adding three-digit numbers

Adding money

Adding larger numbers

Checking addition by casting out nines

Chapter 9: Subtraction

Subtracting one number below a hundreds value from another that is just above the same hundreds number

Written subtraction

Subtraction from a power of 10

Subtracting smaller numbers

Checking subtraction by casting out nines

Chapter 10: Squaring numbers

Squaring numbers ending in 5

Squaring numbers near 50

Squaring numbers near 500

Numbers ending in 1

Numbers ending in 2

Numbers ending in 9

Squaring numbers ending with other digits

Chapter 11: Short division

Using circles

Chapter 12: Long division by factors

Division by numbers ending in 5

Rounding off decimals

Finding a remainder

Chapter 13: Standard long division

Chapter 14: Direct division

Division by two-digit numbers

A reverse technique

Division by three-digit numbers

Chapter 15: Division by addition

Dividing by three-digit numbers

Possible complications

Chapter 16: Checking answers: Part two

Casting out elevens

Chapter 17: Estimating square roots

Square roots of larger numbers

A mental calculation

When the number is just below a square

A shortcut

Greater accuracy

Chapter 18: Calculating square roots

Cross multiplication

Using cross multiplication to extract square roots

Comparing methods

A question from a reader

Chapter 19: Fun shortcuts

Multiplication by 11

Calling out the answers

Multiplying multiples of 11

Multiplying larger numbers

A maths game

Multiplication by 9

Division by 9

Multiplication using factors

Division by factors

Multiplying two numbers that have the same tens digits and whose units digits add to 10

Multiplying numbers when the units digits add to 10 and the tens digits differ by 1

Multiplying numbers near 50

Subtraction from numbers ending in zeros

Chapter 20: Adding and subtracting fractions

Addition

Another shortcut

Subtraction

Chapter 21: Multiplying and dividing fractions

Multiplying fractions

Dividing fractions

Chapter 22: Direct multiplication

Multiplication by single-digit numbers

Multiplying larger numbers

Combining methods

Chapter 23: Estimating answers

Practical examples

Chapter 24: Estimating hypotenuse

Where the lengths of the sides are the same or similar

Where the lengths of the sides are different

Chapter 25: Memorising numbers

Phonetic values

Rules for phonetic values

Phonetic values test

Chapter 26: Phonetic pegs

An impressive demonstration

Six rules for making associations

Memorising square roots

Chapter 27: Memorising long numbers

Memorising pi

Mental calculation

Chapter 28: Logarithms

What are logarithms?

Components of logarithms

Why learn the table of logarithms?

Antilogs

Calculating compound interest

The rule of 72

Two-figure logarithms

Chapter 29: Using what you have learned

Travelling abroad

Temperature conversion

Time and distances

Money exchange

Speeds and distances

Pounds to kilograms

Sporting statistics

Sales tax or GST

Estimating distances

Miscellaneous hints

Apply the strategies

Afterword

Appendix A: Frequently asked questions

Appendix B: Estimating cube roots

Appendix C: Finding any root of a number

Appendix D: Checks for divisibility

Appendix E: Why our methods work

Appendix F: Casting out nines — why it works

Appendix G: Squaring feet and inches

Appendix H: How do you get students to enjoy mathematics?

Appendix I: Solving problems

Glossary

Index

Dedicated to

Benyomin Goldschmiedt

Also by Bill Handley: Teach Your Children Tables, Speed Maths for Kids and Fast Easy Way to Learn a Language

(published by and available from Wrightbooks)

Third edition first published 2008 by Wrightbooks

an imprint of John Wiley & Sons Australia, Ltd

42 McDougall Street, Milton Qld 4064

Office also in Melbourne

First edition 2000

Second edition 2003

Typeset in 11.5/13.2 pt Goudy

© Bill Handley 2008

The moral rights of the author have been asserted

National Library of Australia

Cataloguing-in-Publication data:

Author:Handley, Bill.Title:Speed mathematics / author, Bill Handley.Edition:3rd ed.Publisher:Camberwell, Vic. : John Wiley and Sons Australia, 2008.ISBN:9780731407811 (pbk.)Subjects:Mental arithmetic — Study and teaching.Dewey Number:513.9

All rights reserved. Except as permitted under the Australian Copyright Act 1968 (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address above.

Cover design by Rob Cowpe

Author photograph by Karl Mandl

Preface

I made a number of changes to the book for the second edition of Speed Mathematics; many minor changes where I, or my readers, thought something could be explained more clearly. The more important changes were to the chapters on direct long division (including a new chapter), calculating square roots (where I included replies to frustrated readers) and to the appendices. I included an algebraic explanation for multiplication using two reference numbers and an appendix on how to motivate students to enjoy mathematics.

I decided to produce a third edition after receiving mail from around the world from people who have enjoyed my book and found it helpful. Many teachers have written to say that the methods have inspired their students, but some informed me that they have had trouble keeping track of totals as they make mental calculations.

In this third edition, I have included extra material on keeping numbers in your head, memorising numbers, working with logarithms and working with right-angled triangles. I have expanded the chapters on squaring numbers and tests for divisibility and included an idea from an American reader from Kentucky.

I have produced a teachers’ handbook with explanations of how to teach these methods in the classroom with many handout sheets and problem sheets. Please email me or visit my website for details.

Many people have asked me if my methods are similar to those developed by Jakow Trachtenberg. He inspired millions with his methods and revolutionary approach to mathematics. Trachtenberg’s book inspired me when I was a teenager. After reading it I found to my delight that I was capable of making large mental calculations I would not otherwise have believed possible. From his ideas, I developed a love for working, playing and experimenting with numbers. I owe him a lot.

My methods are not the same, although there are some areas where our methods meet and overlap. We use the same formula for squaring numbers ending in five. Trachtenberg also taught casting out nines to check answers. Whereas he has a different rule for multiplication by each number from 1 to 12, I use a single formula. Whenever anyone links my methods to Trachtenberg’s, I take it as a compliment.

My methods are my own and my approach and style are my own. Any shortcomings in the book are mine.

Some of the information in Speed Mathematics can be found in my first book, Teach Your Children Tables. I have repeated this information for the sake of completeness. Teach Your Children Tables teaches problem-solving strategies that are not covered in this book. The practice examples in my first book use puzzles to make learning the strategies enjoyable. It is a good companion to this book.

Speed Maths For Kids is another companion which takes some of the methods in this book further. It is a fun book for both kids and older readers, giving added insight by playing and experimenting with the ideas.

My sincere wish is that this book will inspire my readers to enjoy mathematics and help them realise that they are capable of greatness.

Bill Handley

Melbourne, Australia, January 2008

<[email protected]>

<http://www.speedmathematics.com>

Introduction

Imagine being able to multiply large numbers in your head — faster than you could tap the numbers into a calculator. Imagine being able to make a ‘lightning’ mental check to see if you have made a mistake. How would your colleagues react if you could calculate square roots — and even cube roots — mentally? Would you gain a reputation for being extremely intelligent? Would your friends and colleagues treat you differently? How about your teachers, lecturers, clients, management?

People equate mathematical ability with intelligence. If you are able to do multiplication, division, squaring and square roots in your head in less time than your friends can retrieve their calculators from their bags, they will believe you have a superior intellect.

I taught a young boy some of the strategies you will learn in Speed Mathematics before he had entered Grade 1 and he was treated like a prodigy throughout primary and secondary school.

Engineers familiar with these kinds of strategies gain a reputation for being geniuses because they can give almost instant answers to square root problems. Mentally finding the length of a hypotenuse is child’s play using the methods taught in this book.

As these people are perceived as being extremely intelligent, they are treated differently by their friends and family, at school and in the workplace. And because they are treated as being more intelligent, they are more inclined to act more intelligently.

Why teach basic number facts and basic arithmetic?

Once I was interviewed on a national radio programme. After my interview, the interviewer spoke with a representative from the mathematics department at a leading Melbourne university. He said that teaching students to calculate is a waste of time. Why does anyone need to square numbers, multiply numbers, find square roots or divide numbers when we have calculators? Many parents telephoned the network to say his attitude could explain the difficulties their children were having in school.

I have also had discussions with educators about the value of teaching basic number facts. Many say children don’t need to know that 5 plus 2 equals 7 or 2 times 3 is 6.

When these comments are made in the classroom I ask the students to take out their calculators. I get them to tap the buttons as I give them a problem. ‘Two plus three times four equals..?’

Some students get 20 as an answer on their calculator. Others get an answer of 14.

Which number is correct? How can calculators give two different answers when you press the same buttons?

This is because there is an order of mathematical functions. You multiply and divide before you add or subtract. Some calculators know this; some don’t.

A calculator can’t think for you. You must understand what you are doing yourself. If you don’t understand mathematics, a calculator is of little help.

Here are some reasons why I believe an understanding of mathematics is not only desirable, but essential for everyone, whether student or otherwise:

People equate mathematical ability with general intelligence. If you are good at maths, you are generally regarded as highly intelligent. High-achieving maths students are treated differently by their teachers and colleagues. Teachers have higher expectations of them and they generally perform better — not only at mathematics but in other subject areas as well.Learning to work with numbers, especially mastering the mental calculations, will give an appreciation for the properties of numbers.Mental calculation improves concentration, develops memory, and enhances the ability to retain several ideas at once. Students learn to work with different concepts simultaneously.Mental calculation will enable you to develop a ‘feel’ for numbers. You will be able to better estimate answers.Understanding mathematics fosters an ability to think laterally. The strategies taught in Speed Mathematics will help you develop an ability to try alternative ways of thinking; you will learn to look for non-traditional methods of problem-solving and calculations.Mathematical knowledge boosts your confidence and self-esteem. These methods will give you confidence in your mental faculties, intelligence and problem-solving abilities.Checking methods gives immediate feedback to the problem-solver. If you make a mistake, you know immediately and you are able to correct it. If you are right, you have the immediate satisfaction of knowing it. Immediate feedback keeps you motivated.Mathematics impacts on our everyday lives. Whether you are at the football or buying groceries, there are many practical uses of mental calculation. We all need to be able to make quick calculations.

Mathematical mind

Is it true that some people are born with a mathematical mind? Do some people have an advantage over others? And, conversely, are some people at a disadvantage when they have to solve mathematical problems?

The difference between high achievers and low achievers is not the brain they were born with but how they learn to use it. High achievers use better strategies than low achievers.

Speed Mathematics will teach you better strategies. These methods are easier than those you have learnt in the past so you will solve problems more quickly and make fewer mistakes.

Imagine there are two students sitting in class and the teacher gives them a maths problem. Student A says, ‘This is hard. The teacher hasn’t taught us how to do this. So how am I supposed to work it out? Dumb teacher, dumb school.’

Student B says, ‘This is hard. The teacher hasn’t taught us how to do this. So how am I supposed to work it out? He knows what we know and what we can do so we must have been taught enough to work this out for ourselves. Where can I start?’

Which student is more likely to solve the problem? Obviously, it is student B.

What happens the next time the class is given a similar problem? Student A says, ‘I can’t do this. This is like the last problem we had. It’s too hard. I am no good at these problems. Why can’t they give us something easy?’

Student B says, ‘This is similar to the last problem. I can solve this. I am good at these kinds of problems. They aren’t easy, but I can do them. How do I begin with this problem?’

Both students have commenced a pattern; one of failure, the other of success. Has it anything to do with their intelligence? Perhaps, but not necessarily. They could be of equal intelligence. It has more to do with attitude, and their attitude could depend on what they have been told in the past, as well as their previous successes or failures. It is not enough to tell people to change their attitude. That makes them annoyed. I prefer to tell them they can do better and I will show them how. Let success change their attitude. People’s faces light up as they exclaim, ‘Hey, I can do that!’

Here is my first rule of mathematics:

The easier the method you use to solve a problem, the faster you will solve it with less chance of making a mistake.

The more complicated the method you use, the longer you take to solve a problem and the greater the chance of making an error. People who use better methods are faster at getting the answer and make fewer mistakes, while those who use poor methods are slower at getting the answer and make more mistakes. It doesn’t have much to do with intelligence or having a ‘mathematical brain’.

How to use this book

I have tried to write a non-technical book that anyone can understand. By the end of this book, you will understand mathematics as never before. Mathematics is an extremely satisfying subject. Solving a difficult problem in mathematics or logic can be rewarding and give a high beyond most people’s imagination. This book will teach you how.

Each section has a number of examples. Try them, rather than just read them through. You will find the examples are not difficult, and it is by trying them that you will really learn the strategies and principles and you will be really motivated. It is only by trying the examples that you will discover how easy the methods really are.

Some readers have written to say they have read chapter 1 and couldn’t get the methods to work for some calculations. The book teaches the methods and principles in order, so you may have to read several chapters until you completely master a strategy.

I encourage you to take your time, and practise the examples both by writing them down and by calculating the answers mentally. Work your way through the book, and you will be amazed that maths can be so easy and so enjoyable.

Chapter 1

Multiplication: Part one

How well do you know your basic multiplication tables?

How would you like to master your tables up to the 10 times tables in less than 10 minutes? And your tables up to the 20 times tables in less than half an hour? You can, using the methods I explain in this book. I only assume you know the 2 times tables reasonably well, and that you can add and subtract simple numbers.

Multiplying numbers up to 10

We will begin by learning how to multiply numbers up to 10 × 10. This is how it works: