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