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14. VEDIC MATHEMATICS
TEACHER'S MANUAL 2
INTERMEDIATE LEVEL
- details
This book is designed for teachers of children aged from
about 9 to 14 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. Non-teachers who
have a background knowledge of mathematics may also find it appropriate
(see contents)
201 + xi pages. Size: 24cm by 16cm. Paperback. 2005; Author:
Kenneth Williams;
ISBN 81-208-2787-2. Price 12.50 pounds.
Please note that these Manuals do not form a sequence:
there is some overlap between the three books.
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
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.
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