**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