4. VEDIC MATHEMATICS TEACHER'S MANUAL 2

INTERMEDIATE LEVEL
- details

This book is designed for teachers of children in grades 5 to 10. It shows how Vedic Mathematics can be used in a school course but does not cover all school topics (see contents). The book can be used for teachers who wish to learn the Vedic system or to teach courses on Vedic mathematics for this level. Non-teachers who have a background knowledge of mathematics may also find it appropriate (see contents)

Author: Kenneth Williams, 2002, revised 2009. Paperback, 208 pages.
ISBN 978-1-902517-17-9. Revised edition 2010.
Price $27 (including postage).

Please note that these three Manuals do not form a sequence: there is some overlap between the three books.

Review of the three Manuals

I could purchase three volumes of teacher's manuals only today and these have made me blissfully satisfied with every aspect of it.
Teacher's manuals, have brought Vedic methods within reach for teacher's who wish to teach the Vedic system. The message has been blissfully brought home as that Vedic mathematics is a complete system of mathematics, and that it has many surprising properties.
The teachers, no doubt, are to be amply benefited through these manuals. But, the students wishing to approach mathematics with this system, can accept these manuals as text books for the topics taken up in these manuals.
These manuals also have a guiding light for all those who wish to fulfil their goal of being the textbooks writers of Vedic mathematics. The way, the sequence and steps with which the Sutras deserve to be respected for their inherent features of coherence leading to creativity, may be learnt from the sensitivity of handling of them in these manuals.
The bliss with which the virtue of values of pure mathematical concepts have been made a chain of simplified steps to be applied as skills for handling the complex mathematical situations to be within reach as very simple basic arithmetic operations to be worked digit by digit, is so flowing with intensified fragrance throughout that the Author deserves all praise and best wishes for undertaking the unfinished task of covering all remaining topics which could not be covered in these three beautiful Volumes of teacher's manual.
Once again, I congratulate you from core of my heart for this wonderful service for the cause of Vedic mathematics.
Dr. S. K. Cosmic Kapoor 24-11-2005

PREFACE

This Manual is the second of three (elementary, intermediate and advanced) Manuals which are designed for adults with a basic understanding of mathematics to learn or teach the Vedic system. So teachers could use it to learn Vedic Mathematics, though it is not suitable as a text for children (for that the Cosmic Calculator Course is recommended). Or it could be used to teach a course on Vedic Mathematics.

 The sixteen lessons of this course are based on a series of one week summer courses given at Oxford University by the author to Swedish mathematics teachers between 1990 and 1995. Those courses were quite intensive consisting of eighteen, one and a half hour, lessons.

 The lessons in this book however probably contain more material than could be given in a one and a half hour lesson. The teacher/reader may wish to omit some sections, go through the material in a different sequence to that shown here or break up some sections (e.g. recurring decimals).

 All techniques are fully explained and proofs are given where appropriate, the relevant Sutras are indicated throughout (these are listed at the end of this Manual) and, for convenience, answers are given after each exercise. Cross-references are given showing what alternative topics may be continued with at certain points.

 It should also be noted that the Vedic system encourages mental work so we always encourage students to work mentally as long as it is comfortable. In the Cosmic Calculator Course pupils are given a short mental test at the start of most or all lessons, which makes a good start to the lesson, revises previous work and introduces some of the ideas needed in the current lesson. In the Vedic system pupils are encouraged to be creative and use whatever method they like.

 Some topics will be found to be missing in this text: for example, there is no section on area, only a brief mention. This is because the actual methods are the same as currently taught so that the only difference would be to give the relevant Sutra(s).

 

INTRODUCTION

Vedic Mathematics is an ancient system of mathematics which was rediscovered early last century by Sri Bharati Krsna Tirthaji (henceforth referred to as Bharati Krsna).

 The Sanskrit word “veda” means “knowledge”. The Vedas are ancient writings whose date is disputed but which date from at least several centuries BC. According to Indian tradition the content of the Vedas was known long before writing was invented and was freely available to everyone. It was passed on by word of mouth. The writings called the Vedas consist of a huge number of documents (there are said to be millions of such documents in India, many of which have not yet been translated) and these have recently been shown to be highly structured, both within themselves and in relation to each other (see Reference 2). Subjects covered in the Vedas include Grammar, Astronomy, Architecture, Psychology, Philosophy, Archery etc., etc.

 A hundred years ago Sanskrit scholars were translating the Vedic documents and were surprised at the depth and breadth of knowledge contained in them. But some documents headed “Ganita Sutras”, which means mathematics, could not be interpreted by them in terms of mathematics. One verse, for example, said “in the reign of King Kamse famine, pestilence and unsanitary conditions prevailed”. This is not mathematics they said, but nonsense.

 Bharati Krsna was born in 1884 and died in 1960. He was a brilliant student, obtaining the highest honours in all the subjects he studied, including Sanskrit, Philosophy, English, Mathematics, History and Science. When he heard what the European scholars were saying about the parts of the Vedas which were supposed to contain mathematics he resolved to study the documents and find their meaning. Between 1911 and 1918 he was able to reconstruct the ancient system of mathematics which we now call Vedic Mathematics.

 He wrote sixteen books expounding this system, but unfortunately these have been lost and when the loss was confirmed in 1958 Bharati Krsna wrote a single introductory book entitled “Vedic Mathematics”. This is currently available and is a best-seller (see Reference 1).

 There are many special aspects and features of Vedic Mathematics which are better discussed as we go along rather than now because you will need to see the system in action to appreciate it fully. But the main points for now are:

 1) The system rediscovered by Bharati Krsna is based on sixteen formulae (or Sutras) and some sub-formulae (sub-Sutras). These Sutras are given in word form: for example Vertically and Crosswise and By One More than the One Before. In this text they are indicated by italics. These Sutras can be related to natural mental functions such as completing a whole, noticing analogies, generalisation and so on.

 2) Not only does the system give many striking general and special methods, previously unknown to modern mathematics, but it is far more coherent and integrated as a system.

 3) Vedic Mathematics is a system of mental mathematics (though it can also be written down).

 Many of the Vedic methods are new, simple and striking. They are also beautifully interrelated so that division, for example, can be seen as an easy reversal of the simple multiplication method (similarly with squaring and square roots). This is in complete contrast to the modern system. Because the Vedic methods are so different to the conventional methods, and also to gain familiarity with the Vedic system, it is best to practice the techniques as you go along.

 

CONTENTS

PREFACE

LESSON 1           Basic Devices

Introduction
Digit Sums
Left to Right
Addition  
Multiplication 
Advantages of left to Right Calcn
Writing Left to Right Sums
Checking Devices
Subtraction
Subtraction from Left to Right
Checking Subtraction Sums

LESSON 2           MORE BASIC DEVICES

Number Splitting
Addition 14 / Subtraction  
Multiplication 16 / Division
All from 9 and the Last from 10 
Subtraction from a Base  
Calculations Involving Money  
First Extension  
Second Extension  
Combining the Extensions  
Bar Numbers    
Advantages of Bar Numbers   
General Subtraction

LESSON 3       SPECIAL METHODS

Proportionately 
Doubling and Halving            
Extending the Multiplication Tables         
Multiplying by 5, 50, 25                              
All from 9 and the Last from 10: Multiplication
Numbers just below 100  
Geometrical Proof  
Algebraic Proof  
Other Bases  
Numbers above the Base  
One Number ABOVE and one below the Base  
Proportionately  
Multiplying Numbers near Different Bases   
Squaring Numbers near a Base  
Mental Calculations
Special methods  

LESSON 4        BY ONE MORE THAN THE ONE BEFORE

Special Multiplications
Squaring Numbers that End in 5       
A Variation  
Multiplication Summary  
Recurring Decimals
Denominator Ending in 9  
Proof  
A Short Cut  
Proportionately  
Longer Numerators

LESSON 5           AUXILIARY FRACTIONS

Auxiliary Fractions: First Type
Denominators Ending in 8, 7, 6                                                            
Auxiliary Fractions: Second Type   
Denominators Ending in 1  
Alternative Method  
Denominators Ending in 2, 3, 4  
Working 2, 3 etc. Figures at a Time

LESSON 6           VERTICALLY AND CROSSWISE

Fractions
Adding & Subtracting Fractions
Proof  
A Simplification  
Comparing Fractions  
Multiplication and Division  
General Multiplication  
Revision  
Multiplying 2-Figure Numbers  
Explanation  
Explanation of earlier special method  
The Digit Sum Check  
Multiplying 3-Figure Numbers  
Moving Multiplier  
Algebraic Multiplications  
The Digit Sum Check  
Three-Figure Numbers  
Four-Figure Numbers  
Writing Left to Right Sums  
From Right to Left  
setting the sums out  
Using Bar Numbers

LESSON 7           SQUARES AND SQUARE ROOTS

General Squaring   
Two-Figure Numbers  
Number Splitting  
Algebraic Squaring  
Squaring Longer Numbers  
Written Calculations – Left to Right  
Written Calculations – Right to Left  
Square Roots of Perfect Squares

LESSON 8           SPECIAL MULTIPLICATION METHODS

Special Numbers  
Repeating Numbers  
Proportionately  
Disguises  
Using the Average
PROOF  
Multiplication by Nines
Multiplication by 11
Percentages
Increasing  
Reducing

LESSON 9           TRIPLES

Definitions
Triples for 45°, 30° and 60°
Triple Addition   
Double Angle    
Variations of 3,4,5
Quadrant Angles   
Rotations

LESSON 10          SPECIAL DIVISION

Division by Nine       
Adding Digits  
A Short Cut  
Dividing by 8
Algebraic Division  
Dividing by 11, 12 etc.  
Larger Divisors                                           
Divisor just Below a Base  
A Simplification  
Divisor just Above a Base  
Proportionately

LESSON 11           STRAIGHT DIVISION

Single Figure on the Flag
Short Division Digression
Longer Numbers   
Multiplication Reversed  
Decimalising the Remainder
Negative Flag Digits       
Larger Divisors       
ALGEBRAIC DIVISION

LESSON 12 EQUATIONS

Linear
One x Term  
Two x Terms
Quadratic Equations
Simultaneous Equations  
By Addition and By Subtraction
A Special Type  
General Method  
Another Special Type

LESSON 13            APPLICATIONS OF TRIPLES

Triple Subtraction   
Triple Geometry       
Angle Between Two Lines
Half Angle   
Coordinate Geometry
Gradients  
Circle Problems  
Length of Perpendicular

LESSON 14            SQUARE ROOTS

Squaring
Square Root of a Perfect square   
Preamble  
Two-Figure Answer  
Reversing Squaring 
Three-Figure Answer
Reversing Squaring 
General Square Roots
Changing the Divisor  
Heuristic Proof

LESSON 15            DIVISIBILITY

Elementary Parts  
The Ekadhika
Osculation   
Explanation  
Testing Longer Numbers
Other Divisors  
The Negative Osculator

LESSON 16            COMBINED CALCULATIONS

Algebraic
Arithmetic
Pythagoras Theorem

SUTRAS AND SUB-SUTRAS
9-POINT CIRCLES
REFERENCES
INDEX OF THE VEDIC FORMULAE
INDEX

BACK COVER:

VEDIC MATHEMATICS MANUAL

 INTERMEDIATE LEVEL

¯   Vedic Mathematics was reconstructed from ancient Vedic texts early last century by Sri Bharati Krsna Tirthaji (1884-1960). It is a complete system of mathematics which has many surprising properties and applies at all levels and areas of mathematics, pure and applied.

¯   It has a remarkable coherence and simplicity that make it easy to do and easy to understand. Through its amazingly easy methods  complex problems can often be solved in one line.  

¯   The system is based on sixteen word-formulae (Sutras) that relate to the way in which we use our mind.

¯   The benefits of using Vedic Mathematics include more enjoyment of maths, increased flexibility, creativity and confidence, improved memory, greater mental agility and so on.

This Intermediate Manual is the second of three designed for teachers who wish to teach the Vedic system, either to a class or to other adults/teachers. It is also suitable for anyone who would like to teach themselves the Vedic methods.