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13. VEDIC MATHEMATICS TEACHER'S MANUAL 1 ELEMENTARY LEVEL
This book is designed for teachers of children aged from
about 7 to 11 years. 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. 167 + xi pages. Size: 24cm by 16cm. Paperback. 2005; Author:
Kenneth Williams; Please note that these Manuals do not form a sequence: there is some overlap between the three books.
PREFACE This Manual is the first 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. 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 Cosmic Calculator course there are also many games that help to establish and promote confidence in the ideas used here. 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 By One More than the One Before and Vertically and Crosswise. 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
COMPLETING THE WHOLE Introduction LESSON
2 DOUBLING AND
HALVING Doubling LESSON
3
DIGIT SUMS Adding
Digits LESSON 4
LEFT TO RIGHT Addition: Left to Right LESSON
5 ALL FROM
9 AND THE LAST FROM 10 All From 9 and the Last from 10 LESSON 6
NUMBER SPLITTING Addition LESSON 7
BASE MULTIPLICATION Times Tables LESSON
8 CHECKING AND
DIVISIBILITY Digit Sum Check for Division LESSON 9
BAR NUMBERS Removing Bar Numbers LESSON
10 SPECIAL MULTIPLICATION Multiplication by 11 LESSON
11 GENERAL MULTIPLICATION Revision LESSON
12 SQUARING Squaring
Numbers that end in 5 LESSON
13 EQUATIONS One-step
Equations LESSON
14 FRACTIONS
Vertically
and Crosswise LESSON
15 SPECIAL DIVISION Division
by 9 LESSON
16 THE CROWNING
GEM
Single
Figure on the Flag SUTRAS
AND SUB-SUTRAS
BACK COVER VEDIC
MATHEMATICS MANUAL
ELEMENTARY
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 Elementary Manual is
the first 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 basic Vedic methods.
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