Kindle paperback links:
Book 1: $8
Pothi paperback (for India only):
Book 1 Rs.500 Book 2 Rs.500 Book 2 Rs.500 Teacher's Guide Rs.500 Answer Book Rs.300
Description
(PLEASE NOTE: this course was originally titled The Cosmic Computer course.)
Written for 11-14 year old pupils (some of the material in Books 1 and 2 is suitable for children from the age of about eight) this course covers the National Curriculum for England and Wales, Key Stage 3. The full course consists of three Textbooks, a Teacher's Guide and an Answer Book.
THE TEXT BOOKS
Each of the three books has 27 chapters each of which is prefaced by an inspiring quote from a famous mathematician, philosopher etc. Also in each book there are historical notes which relate to the authors of the quotes, a list of Sutras and three other short but interesting sections (e.g. Pascal's Triangle, Fractals).
Book 1 deals mainly with basic arithmetic, proportion, decimals, basic algebra and geometry, polygons, area, volume etc.
Book 2 extends this, covering fractions, probability, sequences, negative numbers, percentages, equations, graphs, charts, transformations, bearings etc.
Book 3 develops this further into recurring decimals, square and cube roots, division, divisibility, the musical scale, formulae, simultaneous equations, quadratic equations, proof, similar triangles, area of a circle, nets, conic sections, loci, motion, vectors, Pythagoras' theorem, triples, coordinate geometry etc.
THE TEACHER'S GUIDE
This contains:
A Summary of the book.
A copy of the Unified Field Chart for that book.
Notes on the content of the chapters- advice, suggestions etc.
Mental Tests (correlated with the books) and answers- which allow earlier work to be regularly revised, give stimulating ideas relevant to the current lesson and which develop themes from earlier tests which may ultimately become the subject of a lesson. Extension Material and answers (about 16 per book)- these consist of a 1 or 2-sided sheet given to children who work fast and get ahead of the rest of the class. Many of these are also very suitable for work with a whole class.
Revision Tests and Answers- There is a revision test every 4 or 5 chapters. This includes a mental test of 10 questions.
Games, Worksheets etc.
THE ANSWER BOOK
This contains answers to all exercises and other numbered questions in the text and should be available for pupils during lessons..
THE COURSE has many unusual and attractive features.
1 It is primarily a system of mental mathematics (though all the methods can also be written down) using simple patterns and methods which are very easy to understand and remember. Each lesson starts with a short mental test.
2 It is extremely coherent and unified and uses sixteen simple word-formulae, called Sutras, like Vertically and Crosswise. These formulae relate to the different ways in which the mind can be used and are therefore a great help to pupils.
3 It makes use of a "Unified Field" chart which shows the whole subject of mathematics at a glance and how the different parts and topics are related.
4 The powerful Vedic methods are delightfully easy and fun. Many problems can be tackled in a variety of ways, from right to left or from left to right, 2 or more figures at a time, etc. The techniques are also interrelated which adds to the beauty and simplicity.
Through this mental approach the course encourages creativity and the use of intuition in mathematics, in contrast to the modern, mainly analytic, approach.
Vedic Mathematics is already being taught with great success in many schools and the response to this course has been extremely encouraging.
Details
Book 1: 214 pages. Book 2: 253 pages. Book 3: 281 pages. Teacher's Guide: 255 pages. Answer Book: 82 pages.
Size: 25cm by 19cm.
Paperback. 2010
Author: Kenneth Williams & Mark Gaskell
ISBN 978-1-902517-29-2.
Reviews
I found the whole course very interesting and really feel I have improved in Maths. I can now even challenge my dad in Maths Sums and beat him! (That is a great improvement). The Maths games were fun and helped a lot . . . - Deborah, aged 13
P.S.: I have your "Cosmic Computer" books on my desk and use this to prepare for my VM class. I enjoy them a lot. Dr S. Sreenath, Professor of electrical engineering and computer science and Vedic Maths tutor in the U.S.A.
I love these books! I have been learning Vedic maths for about two months now. I purchased the Cosmic Calculator course two weeks ago and I can't put the books down! Great stuff! I am also starting to teach my daughter some of the methods in the books. I also purchased the Vedic Mathematics Teacher's Manuals(all three) to assist me in teaching my daughter. Again, I simply love the course so far and have gained a tremendous amount of knowledge from these books. . . . Bill Gaylord, PA, USA
I'm working my way through the Cosmic Calculator, and I'm shocked at how well it's taken me from being a confirmed mathophobe to someone who enjoys math. Thanks for all the work you did to publish this! . . . Richard Shultz, California, USA
Introduction
From The Teacher's Guide
INTRODUCTION TO VEDIC MATHEMATICS
Vedic (pronounced 'Vaydik') Mathematics is an ancient system of mathematics originating in India in Vedic times. It was rediscovered between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960) who studied the ancient Sanskrit texts called the Vedas. The date these texts were written is unknown but the content of the Vedas was passed on by an oral tradition long before writing was invented. The Vedas are said to cover every area and aspect of knowledge, including for example, ethics, grammar, architecture, astronomy. The word 'Veda' literally means 'knowledge'.
Since the publication of his book "Vedic Mathematics" (reference 1) in 1965 interest in this system has been growing and some schools now successfully teach it. Many teachers attending courses and talks on Vedic Mathematics in recent years have expressed a strong interest in teaching the Vedic system in their school, but not without a textbook. This course has been written to meet this need. It is based on careful study and research over the last 27 years and is being tested in various schools.
The Vedic system is attractive to teachers and pupils because it has many striking advantages over the mathematics currently taught. The most significant of these are:
the coherence of the system, the easy and simple Vedic methods,
the emphasis on mental calculation, the use of basic principles or Sutras,
the use of a Unified Field chart, its effectiveness over all ability ranges.
The Vedic system is extremely refined. The methods are simple and complementary, so that for example 'long' division is a simple reversal of the one-line multiplication process; similarly with squaring and square roots.
MENTAL MATHEMATICS
Anyone familiar with the Vedic system will be aware of the remarkable Vedic techniques: 'difficult' problems or huge sums which can be solved immediately by the Vedic method. These striking and beautiful methods are just a part of a complete system of mathematics which is far more systematic than the modern 'system'. Children are inspired by these delightful techniques and often ask "Why was I not shown this before?". Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, direct and easy.
The simplicity of Vedic Mathematics means that calculations can be carried out mentally and this is very much encouraged in the Vedic system. There are many advantages in using a mental system. Starting with a short mental test, the class settle down at the beginning of the lesson and focus on mathematics; they enjoy the challenge of using only their own mind to solve problems; the questions in the test can revise the previous lesson and lessons and introduce ideas to be used in the present lesson. Although the system encourages mental calculation, apart from the initial test at the beginning of the lesson its use in the rest of the lesson is not insisted upon. It is important that the pupil's study is fun and enjoyable and they should not be forced to do what they find too difficult, though the able ones may enjoy doing nearly everything in their head. There are many advantages in a system which emphasises mental mathematics and these are discussed in some detail in the introduction to reference 2.
CREATIVITY
Learning mathematics should be a delightful experience for all children and they should all succeed in it. The Cosmic Computer course offers a complete system of mental mathematics which can be taught in a holistic way. The straightforward and beautifully interrelated Vedic methods mean that mathematics can be done mentally, and this and the many methods of solution which the Vedic system offers, encourages flexibility and innovation. This in turn leads to the development of creativity and intuition. The Vedic system does not insist on a purely analytic approach as many modern teaching methods do. This makes a big difference to the attitude which children have towards mathematics.
Being naturally creative students like to devise their own methods of solution. The Vedic system seeks to cultivate intuition- having a conscious proof or explanation of a method beforehand is not considered essential in the Vedic methodology. Students are shown general methods and also special methods which apply in special cases. This means they do not rigidly have to follow a certain procedure but have a choice and are encouraged to invent their own methods. Every problem is unique and invites its own style of solution.
The Vedic system does not impose unnecessary restrictions- we can for example work from left to right or right to left; with numbers which are partly positive and partly negative; we can work 2 or more figures at a time and can feel comfortable with long numbers.
These benefits of the Vedic system have been observed where it has been used. Pupils become generally more focused, confident and intelligent.
THE SUTRAS OR WORD-FORMULAE
The Vedic system use a collection of sixteen Sutras (or formulae), given in word form, and some sub-Sutras. These are listed in the front of each book and express fundamental principles which run like threads (the word "Sutra" means "thread") through the whole of mathematics, unifying diverse topics. They are extremely useful in education for this reason.
We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic Sutras: they describe the ways in which the mind can work and so they tell the student how to go about solving a problem. It is not necessary for the teacher to be familiar with these. It is best not to stress the Sutras- they become familiar after a while, and seem quite natural. As an illustration:
an
equilateral
triangle appears here
with the three altitudes drawn in
asked how many triangles there are in this figure you can easily get the answer. There are sixteen of these but did you notice how you could mentally make the various shapes stand out- seeing some lines and ignoring others, and then seeing different triangles- at will? The Vedic formula By Alternate Elimination and Retention describes this attribute of our mind and has many more obviously mathematical applications also (in eliminating first x and then y when solving a pair of simultaneous equations, for example).
The universe appears lawful, at every level, and nature is able to respond instantly to every situation that occurs everywhere in the universe. Nature appears to employ a Cosmic Computer which governs all action with the utmost precision.
THE UNIFIED FIELD CHART
The Unified Field chart (invented by Maharishi Mahesh Yogi) is given at the beginning of each Teacher's Guide and shows the whole subject and how it develops sequentially. This means the student can always easily see where they are in relation to the whole. Other subjects- chemistry, computer science for example- all have their own Unified Field chart showing their structure and development, and any part of a Unified Field chart can itself be expanded into a Unified Field chart.
The term "Unified Field" comes from modern Physics and refers to the unified field of all the various forces of nature. This ultimate level of existence corresponds in subjective terms to the self. So the Unified Field is at the basis of creation just as your self is the observer of creation, and therefore any structure, like Mathematics, must unfold from here- from unity into diversity.
As the structure develops from the base the terms get more and more concrete as the three branches of Arithmetic, Algebra and Geometry develop. The first level above the Unified Field: zero, equality, empty space, gives a quality of that branch which also has the infinite, abstract quality of the Unified Field. Next, unit, symbol, point give a more concrete base for the development of the three branches. Each of these, with three further concepts allow the development of numbers and number systems, algebraic expressions and geometrical forms. These develop further into all the various areas of Arithmetic, Algebra and Geometry. In some cases, graphs for example, a topic is a mixture of two or all of the three branches. The large box shows all the various topics studied in the Cosmic Computer books and so the student always knows where they are in relation to the whole. The contents of this box change from book to book, as the topics studied change. The Transcendental Meditation referred to on the chart is a simple technique for bringing the mind to a quiet state where it can consciously experience the Unified Field.
The extent to which this chart is used is left to the discretion of the teacher.
THE COURSE
This course follows the requirements of the National Curriculum for England and Wales and at present three books are available.
The course is intended to cover the first three years of a secondary school course (that is starting with 11 year old pupils), but may be suitable for other classes too. Some of the material is very suitable for children aged from 8 years.
The textbook is intended to supplement the course, rather than being used for individual study and the course is intended to be delivered as whole class teaching. The text is however self-explanatory and so the pupil who misses some lessons in school would still be able to continue. The reason for writing the book in this way is more for the benefit of the teacher- who will probably be unfamiliar with the Vedic system.
STRUCTURE OF THE COURSE
Lessons begin with a mental arithmetic test of 10 questions (included in this Guide). Answers are given at the end of the test and any difficulties are dealt with. The teacher then introduces the lesson for that day.
This Teacher's Guide gives, in the Notes, any points of special interest for each chapter and other suggestions. (Some additional material will also be found there for Book 1, on geometry.) There are plenty of Extension Sheets for more able or fast pupils and many of these are very appropriate for the whole class to work on. Revision Tests are also contained in the Teacher's Guide and these revise the previous 4 or 5 chapters. There are also worksheets, games etc.
There is also an Answer Book which contains answers to all the exercises and other numbered questions that come up in the text. The Answer Book is intended to be made available to the class to check their work.
Reference 3 is a useful addition to the course as it contains many variations and extensions not covered in these books
THE MENTAL TESTS
These tests (also included in this Guide) are carefully structured so that they
a) introduce new ideas,
b) develop ideas encountered earlier,
c) revise work from the previous and earlier lessons (in this way pupils do not forget
what they have learnt and establish it more clearly),
d) introduce or revise ideas needed in the current lesson.
Underlined questions in the tests, and diagrams, are intended to be written up on the board.
The tests are marked immediately at the end- they can mark their own or exchange with a neighbour. They could record their marks.
Any misunderstandings and errors can be sorted out when they are marked- a quick explanation for each answer can also be given.
The tests can also be diagnostic- the books could be taken in to see who got what right or wrong, and it is useful to ask at some stage of the marking 'who got this right?' (which of course tells you who didn't).
The tests should not be too slow: repeat each question but decline to say it a third time unless for a good reason, then go immediately on to the next question. Occasionally a test can be speeded up (with warning). The tests, and answers, should not take long. Hints or reminders can be given if a question draws a blank response.
The tests are also intended to be flexible: questions can be changed if they are too easy or too hard or if extra revision of some topic is needed (but the aim would be for most pupils to get most of the questions right). It should also be easy to improvise extra tests if there are not enough, even if it means giving an earlier test with the numbers altered. In some of the later tests working out could be allowed for question 10.
References:
1. Tirthaji B.K. (1965) Vedic Mathematics, Motilal Banarsidass
2. Williams K.R. (1991) The Natural Calculator, Vedic Mathematics Research Group
3. Williams K.R. (1984) Discover Vedic Mathematics, Vedic Mathematics Research Group
Contents for Book 1
Introduction 1
1) Arithmetic 3
2) Digit Sums and the Nine-Point Circle 5
The Number Nine 6
The 9-Point Circle 6
Your Lucky Number 9
Digit Sum Game 9 GAME
Digit Sum Problems 10
3) Large Numbers 11
Restructuring Numbers 11
Reading and Writing Large Numbers 13
Millions 14
Billions 16
Make a Number Game 16 GAME
4) Digit Sum Check 17
Addition from Right to Left 17
Sums Involving 'Carries' 18
The Digit Sum Check 18
Subtraction 20
Checking Subtraction Sums 22
5) Number Nine 23
By Addition and By Subtraction 24
6) Numbers with Shapes 27
Square Numbers 28
Factor Pairs 29
Factor Rectangles 30
Prime Numbers 31
The Sieve of Eratosthenes 31
Triangular Numbers32
Cube Numbers 32
Summary of Number Sequences 34
REVISION Test 1
7) Geometry 35
The Right Angle 35
Right-Angles Game 36 GAME
Parallel Lines 36
Drawing Geometrical Shapes 37
Right-angled and Equilateral Triangles 40
8) Symmetry 41
Polygons 43 WORKSHEET 1
Rotational Symmetry 45
9) Angles and Triangles 46
Finding Angles 48
Triangles 49
REVISION Test 2
MAGIC SQUARES 52
10) By the Completion 54
Completing the Whole- Fractions 55
Completing the Whole- Shapes 56
11) Doubling and Halving 58
Extending the Multiplication Tables 60
12) Divisibility 62
Divisibility by 2, 5, 10 62
Divisibility by 3 and 9 64
Summary 64
Divisibility by 4 66
Divisibility by 6 67
Divisibility by 15 68
13) Short Multiplication and Division 71
Multiplication 71
Multiplication by 11 72
Division 73
The Digit Sum Check for Division 74
Division by 9 75
14) Powers of Ten and Decimals 77
Adding and Subtracting Decimal Numbers 78
Multiplying and Dividing Decimal Numbers 79
Multiplying & Dividing by Powers of 10 79
Multiplying & Dividing Decimals by 10, 100 etc 81
Metric Units 83
REVISION Test 3
15) Number Splitting 85
Addition 85
Subtraction 87
Multiplication 87
Division 88
Checking Devices 89
Checking Calculations 90
16) Polygons and Coordinates 91
Quadrilaterals 92
Diagonals of Quadrilaterals 93
17) Regular Polygons and Perimeters 95
To Construct an Equilateral Triangle 96
To Construct a Regular Hexagon 96
To Construct a Square 97
To Construct a Regular Octagon 98
Perimeters 98
Perimeter Problems 99
18) All from 9 and the Last From 10 101
All from 9 101
All from 9 and the Last from 10 102
First Extension 103
Second Extension 104
Combining the First and Second Extensions 105
REVISION Test 4
PASCAL'S TRIANGLE 107
19) Bar Numbers 109
Bar Numbers Game 111 GAME
All from 9 and the Last from 10 111
Subtraction 112
Creating Bar Numbers 113
20) On the Flag 115
Addition 115
Multiplication 118
21) Prime and Composite Numbers 120
Factor Trees 121
An Alternative Method 122
Highest Common Factor: HCF 123
By Addition and by Subtraction 124
22) Proportionately 126
Equal Ratios 126 WORKSHEET 2
Simplifying Ratios 127
Finding Equal Ratios 128
Ratio Problems 129
Splitting in a Ratio 130
Extended Ratios 130
REVISION Test 5
23) By One More than the One Before 132
Squaring Numbers that end in 5 132
Multiplying by Numbers whose First Figures are the Same and whose Last Figures Add up to 10, 100 etc 133
Rounding 135 WORKSHEET 3
24) Algebra 138
Using Letters 138
Brackets 140
Factorising 141
Substitution 142
Multiple Substitutions 144
25) Area 145
Rectangles and Squares 145
Irregular Shapes 147
Composite Shapes 149
Parallelograms 150
Triangles 152
Units of Area 154
26) Volume 155
Capacity 157
27) Planets 160
Planet Sizes 160
Orbits of Planets 161
REVISION Test 6
FLEXAGONS 164
HISTORICAL NOTES 166
Contents for Book 2
1) Nikhilam Multiplication 1
Other Bases 3
Numbers Above the Base 4
Proportionately 5
Squaring Numbers Near a Base 6
Multiplying Numbers Near Different Bases 7
A Summary 8
2) Doubling and Halving 9
Multiplying by 5, 50, 25 10
Dividing by 5, 50, 25 11
3) Fractions 13
Top-Heavy Fractions 13
Finding a Fraction of a Number 14
Equivalent Fractions 16
Simplifying Fractions 17
Finding what Fraction One Number is of Another 18
4) Spirals 20
The Isosceles Right-Angled Triangle 21
Spirals from Squares 22
An Infinite Sum 23
5) Decimals and Fractions 24
Converting Decimals to Fractions 24
Changing Fractions to Decimals 25
Comparing Fractions and Decimals 27
Recurring Decimals 27
Block Recurrers 28
Reciprocals 30
Prime Factors 31
REVISION Test 7
6) The Arithmetic of Bar Numbers 33
Addition and Subtraction 33
A Game 34 GAME
Applications in Algebra 36
Multiplication and Division 37
Brackets 38
Nikhilam Multiplication again 39
7) General Multiplication 41
Multiplying 3-Figure Numbers 45
Moving Multiplier 46
Written Calculations 47
8) Algebraic Multiplication 49
Multiplying and Dividing Single Terms 49
Multiplying Binomials 51
Factorising Quadratic Expressions 54
9) Squaring 55
The Duplex 55
Number Splitting 56
Algebraic Squaring 57
REVISION Test 8
THE MOEBIUS STRIP 59
10) Sequences 61
The nth Term 62
Sequences involving Fractions 65
11) Probability 66
Certain, Impossible and Uncertain 66
A Game 67 GAME
Scale of Probabilities 67
Possible Outcomes 68
Theoretical Probabilities 70
12) Equations 71
One Step Equations 71
Two-Step Equations 73
Three-step Equations 76
13) Angles and Triangles 78
Measuring Angles 78
Drawing Angles 81
Angles in a Triangle 82
Constructing Triangles 83
Isosceles Triangles 85
Calculating Angles 87
REVISION Test 9
14) Percentages 88
Converting a Percentage to a Fraction 88
Converting a Fraction to a Percentage 89
Important Percentages 90
Converting between Percentages, Fractions and Decimals 90
Finding a Percentage of a Quantity 91
Forming a Percentage 92
15) Forming Equations 94
16) 2 and 3 Dimensional Shapes 97
Dimensions 97
2-Dimensional Shapes 98
3-Dimensional Shapes 99
17) Straight Line Graphs 103
Sloping Lines 105
Gradient Squares 108
Gradients: By the Completion of the Triangle 109
Gradient and Intercept 109
Alternative Method using Substitution 111
18) Charts 113
Frequency Tables 113
Line Charts and Dot Diagrams 114
Bar Charts 115
Pictograms 116
Averages and Spread 117
REVISION Test 10
FRACTALS 119
19) Divisibility 121
Divisibility by 8 122
Higher Divisors 123
By Addition and By Subtraction 123
Cancelling Zeros 125
Divisibility by 11 127
20) Further Multiplication 128
Multiplying 3-figure Numbers 128
From Right to Left 130
4-Figure Numbers 131
Squaring 132
Special Numbers 133
Proportionately 134
Disguises 135
21) Combining Fractions 137
Addition and Subtraction 137
Comparing Fractions 140
A simplification 140
Multiplication and Division 141
A Simplifying Device 143
22) Arithmetical Operations 144
The Order of Operations 144
Two Puzzles 146
Brackets 146
Cancelling 148
Some Revision of Decimals 149
Multplication of Decimals 149
Decimal Division 151
23) Special Division 152
A Short Cut 153
Divisor Above a Base Number 156
REVISION Test 11
24) Percentage Changes 158
Increasing by 10% 159
Percentages Increases 160
Percentage Reductions 161
25) Transformations 163
Enlargement 163
Reflection 166
Rotation 168
Translation 171
26) Constructions 174
Bisecting a Line 174
Bisecting an Angle 176
Constructing Angles 177
The Golden Rectangle 178
The Pentagram 181
27) Bearings 183
Parallel Lines 183
Scale Drawing 186
Bearings 187
Using Bearings 188
REVISION Test 12
RANGOLI PATTERNS 192
HISTORICAL NOTES 194
Contents for Book 3
1) Recurring Decimals 1
Recurring Decimal Patterns 3 WORKSHEET 4
A Different Denominator 4
A Short Cut 4
Proportionately 6
2) Formulae 7
Rearranging Formulae 10
Applications 11
(a+b)(a-b) 12
3) Squares, Cubes and Roots 14
Square Roots of Perfect Squares 15
Cubing (using Proportionately) 18
Cube Roots of Perfect Cubes 22
4) Straight Division 23
Short Division Digression 24
Longer Numbers 26
Decimalising the Remainder 27
Negative Flag Number 28
5) Equations 31
Some Variations 31
Fractional Answers 32
Two x Terms 33
Forming Equations 35
Quadratic Equations 37
REVISION Test 13
6) Polygons 38
Angle Sum of Polygons 38
Regular Polygons 41
Other Polygon Angles 43
Tessellations 44 WORKSHEET 5
Semi-Regular Tessellations 45
7) Similar Triangles 48
Congruent Figures 56
8) The Musical Scale 57
The Notes of the Octave 58 WORKSHEET 6
The True or Natural Scale 60
Ratios of Notes 62
Another Mode 63
The Ghandava Veda Scale 64
9) Nets and Networks 65
Cuboids and Prisms 66
Pyramids 68
Euler's Formula 69
A Puzzle 71
Networks 71
REVISION Test 14
THE VEDIC SQUARE WORKSHEET 7 73
10) Probability 75
Theoretical Probabilities 75
Relative Frequency 76
Mutually Exclusive Events 78
Expected Number 79
Combined Events 81
11) π 83
To Estimate pi 84
Circumference of a Circle 85
Area of a Circle 86
12) Volumes of Prisms & Pyramids 89
Area of a Trapezium 89
Volume of a Prism 91 WORKSHEET 8
Volume of a Pyramid 93
13) Parabolic Curves 95
A Third Method- By Alternate Elimination and Retention 97
Parabolas 98
14) Sequences 101
Square Numbers 102
Cube Numbers 103
Triangular Numbers 104
A Power Sequence 104
Games and Puzzles 105
The Fibonacci Sequence 108
A Summary 109
REVISION Test 15
15) Loci 110
The Cycloid 112
The Conic Sections 115 WORKSHEET 9
16) Motion 118
Speed 118
Travel Graphs 121
Change of Speed 122
Conversion Graphs 123
17) Auxiliary Fractions 126
Auxiliary Fractions- First Type 126
Denominators Ending in 8, 7, 6 127
Auxiliary Fractions- Second Type 129
Working 2, 3 etc Figures at a TIme 131
18) Surveys 133
Frequency Polygons 133
Pie Charts 134
Grouping Data 136
Designing a Questionnaire 137
Scatter Diagrams 138
Correlation 140
REVISION Test 16
CODES 141
19) Vectors 143
Another Notation 145
Adding Vectors 147
20) Simultaneous Equations 150
Proportionately 153
Solution by Substitution 155
Two Special Types 156
21) Divisibility and Simple Osculators 157
The Ekadhika 157
Osculation 158
Testing Longer Numbers 160
Other Divisors 162
The Negative Osculator 163
22) Square Roots 166
Squaring 166
First Steps 167
Square Root of a Perfect Square 168
General Square Roots 171
REVISION Test 17
23) Quadratic Equations 175
Factorising Quadratic Expressions 176
Solving Quadratic Equations by Factorisation 179
Differential Calculus 180
24) Pythagoras'Theorem 184
An Algebraic Formula 186
Problems 189
The Theorem in Reverse 191
25) Triples 192
Equal Triples 193
Types of Number 194
Perfect Triples 194
The Angle in a Triple 195
Triples for 45°, 30° and 60° 196
Generating Perfect Triples 197
Finding the Code Numbers of a Triple 198
26) Proof 200
Angle Sum of a Triangle 200
Five Proofs using Areas 201
Even and Odd Numbers 203
Representing Numbers Algebaically 205
Nikhilam Multiplication 206
Perfect Triples 207
Quadratic Equations 208
27) Coordinate Geometry 209
Distance Between Two Points 209
Gradient of a Line Joining Two Points 211
Equation of a Line Through Two Given Points 212
Intersection of Two Lines 213
REVISION Test 18
THE PLATONIC SOLIDS 215
HISTORICAL NOTES 217
Contents for Teacher's Guide
BOOK 1
Introduction 1
Unified Field Chart 6
Summary of Book 1 7
Notes on Chapters 8
Mental Tests and Answers 18
Extension Sheets: Summary 32
Extension Sheets 33
Extension Sheets Answers 69
Revision Tests: Mental Tests and Answers 72
Revision Tests and Answers 73
Games, Pattern Cards, Worksheets 1, 2 and 3 85
BOOK 2
Unified Field Chart 98
Summary of Book 2 99
Notes on Chapters 101
Mental Tests and Answers 103
Extension Sheets: Summary 117
Extension Sheets 119
Extension Sheets Answers 151
Revision Tests: Mental Tests and Answers 153
Revision Tests and Answers 155
Games 169
BOOK 3
Unified Field Chart 176
Summary of Book 3 177
Notes on Chapters 178
Mental Tests and Answers 183
Extension Sheets: Summary 197
Extension Sheets 198
Extension Sheets Answers 229
Revision Tests: Mental Tests and Answers 231
Revision Tests and Answers 233
Worksheets 4, 5, 6, 7, 8 and 9 247
