Introduction
There is a big step up from IGCSE to AS-Level: questions are longer; there is a non-calculator paper; you are given less guidance. Additionally, you will discover that there is a change in the way you are expected to work: your teachers will expect you to work independently; you will be given fewer drill exercises and more challenging problems. You will need greater tenacity and a real desire to persevere.
Those students who work well – and by which we mean who work hard – find AS Mathematics a rewarding and enjoyable course. Others, however, fall by the wayside. This is often because their algebraic skills are weak at the start of the course and they then do not make the effort required to strengthen their skills.
This booklet contains a revision of all the IGCSE material that you need to have mastered before starting the course. There is an emphasis here on being able to complete basic algebra accurately. If you can’t do this confidently already, you need to practice the material in this booklet until you understand all the techniques it contains – and can apply them without error.
You will be tested on this material early in Lower Sixth. If you cannot complete it all accurately by then, you are probably not cut out for the course. We suggest you use this book to help you prepare. We will make answers available on the Maths Department Frog site if you want to check your answers.
This book contains explanations for everything you need, but they are explanations designed to help you revise, not to start from scratch. If you are stuck, you may need a little extra help. We suggest the following sources:
• www.khanacademy.org has a tremendous selection of videos and exercises to help you practice and is completely free.
• www.mymaths.co.uk is a service to which the school has a subscription. You can log in using the school id “sherborne” and our current password “hexagon”. It contains online lessons and exercises which will help you practice.
• CGP revision guides. Many of you will already have some sort of GCSE/IGCSE revision guide. You can refer back to this if you are having difficulty.
We will be offering workshops in the first couple of weeks in September to help you get on top of this material if you are stuck, but the onus is upon you to seek help. Use this book over the summer to work out what you can and cannot do, and then use the workshops to fill in the gaps.
Mastering this material early will enable you to make much better progress through the sixth form. Use this opportunity wisely to get yourself ready.
Good luck Dr Bradshaw
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Sixth Form Kick Start Sherborne School Mathematics Department
Contents
Introduction ......................................................................................................................................................................... 2 Contents ............................................................................................................................................................................... 3 Pre-requisite knowledge for sixth form study ............................................................................................................. 4 Number................................................................................................................................................................................. 6 Fractions ............................................................................................................................................................................... 8 The Laws of Indices.......................................................................................................................................................... 10 Surds .................................................................................................................................................................................... 12 Solving linear equations ................................................................................................................................................... 14 Solving quadratic equations ............................................................................................................................................ 16 Simple trigonometric equations .................................................................................................................................... 18 Linear simultaneous equations ...................................................................................................................................... 20 Non-linear simultaneous equations.............................................................................................................................. 22 Solving linear inequalities ................................................................................................................................................ 24 Polynomials......................................................................................................................................................................... 26 Algebraic Fractions ........................................................................................................................................................... 28 Point geometry .................................................................................................................................................................. 30 Straight-line graphs ........................................................................................................................................................... 32 Sketching quadratic and factorised cubic functions .................................................................................................. 34 Trigonometry..................................................................................................................................................................... 36 Lengths, areas and volumes............................................................................................................................................ 38
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Sherborne School Mathematics Department Sixth Form Kick Start
Pre-requisite knowledge for sixth form study 1. Number
1.1 T ypes of 1.1 (i)
1.1 (ii) 1.1 (iii)
1.2 Fractions 1.2 (i)
1. 2 (ii) 1. 2 (iii) 1. 2 (iv) 1.2 (v) 1. 2 (vi)
2. Algebra
2.1 Key terms 2.1 (i)
2. 1 (ii) 2. 1 (iii) 2. 1 (iv)
2.2 Laws of indices
2.2 (i) 2.2 (ii) 2.2 (iii)
Sum and product formulae.
Negative indices without a calculator. Fractional indices without a calculator.
number
Identification of Natural Numbers, Integers, Rational Numbers, Irrational Numbers. Conversion of fractions to decimals without a calculator
Conversion of decimals to fractions without a calculator
Conversion of top-heavy fractions to mixed-numbers without a calculator Conversion of mixed numbers to top-heavy fractions without a calculator Addition of fractions without a calculator
Subtraction of fractions without a calculator
Multiplication of fractions without a calculator Division of fractions without a calculator
Variables and constants Expressions
Equations
Inequalities
2.3 Use and manipulation of surds.
2.3 (i) Converting surds to index form.
2.3 (ii) Rationalising the denominator where it is of the form n.
2.4 Algebraic manipulation of polynomials
2.4 (i) 2. 4 (ii) 2. 4 (iii) 2. 4 (iv)
Expanding brackets Collecting like terms Factorisation
The difference of two squares
2.5 The solution of linear equations.
2.6 The solution of quadratic equations
2.6 (i) by factorising;
2.6 (ii) by using the quadratic formula.
2.7 The solution of simple trigonometric equations.
2.8 Linear simultaneous equations
2.8 (i) Analytical solution by substitution. 2.8 (ii) Analytical solution by elimination.
2.9 Non-linear simultaneous equations
2.9 (i) Analytical solution by substitution.
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Sixth Form Kick Start Sherborne School Mathematics Department
2.10 Solution of linear inequalities
2.10 (i) Solving inequalities by rearranging.
2.10 (ii) Synthesising the solution of two or more linear inequalities.
2.11 Algebraic fractions
2.11 (i) Simplifying expressions
2.11 (ii) Solving equations involving algebraic fractions
3. Coordinate geometry in the (x, y) plane 3.1 Point geometry
3.1 (i) 3.1 (ii) 3.1 (iii)
3.2 Straight-line 3.2 (i)
3.2 (ii)
3.2 (iii)
Finding the midpoint of two given points. Finding the distance between two given points. Finding the gradient of a line segment.
graphs.
Sketching equations of the form y = mx c. +
Find the equation of a line given a point and the gradient. Perpendicular gradients have product −1.
3.3 Sketching quadratic and factorised cubic functions.
3.3 (i) Use of the word root to describe intersection with the x-axis and y-intercept to
describe intersection with the y-axis.
3.3 (ii) Knowledge that the minimum point on a quadratic lies half-way between the roots.
4. Geometry
4.1 T rigonometry
4.1 (i) 4. 1 (ii) 4. 1 (iii) 4. 1 (iv)
Right-angled triangle trigonometry to find sides and angles The sine rule, including the ambiguous case
The cosine rule
The sine formula for the area of a triangle
4.2 Lengths, areas and volumes
4.2 (i) Formulating expressions for lengths, areas and volumes of compound shapes. 4.2 (ii) Solving equations based on these.
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Sherborne School Mathematics Department
Sixth Form Kick Start
Number
You need to be able to
• Identify the Natural Numbers, Integers, Rational Numbers, Irrational Numbers.
• Convert fractions to decimals
• Convert decimals to fractions
Types of number
The Natural Numbers are the counting numbers {1, 2, 3, 4, 5...}. You may also know them as positive integers. They do not include 0.
The Integers are all the whole numbers, including the negative numbers, e.g. {..., −3, − 2, −1, 0, 1, 2, 3, ...}. They include 0.
The Rational Numbers are all numbers which can be written as fractions. Anything which has a terminating decimal, or a repeating decimal, can be written as a fraction and so is rational. Zero is rational, because 0 = 01 , so it can be written as a fraction.
The Irrational Numbers are all the numbers on the number line which cannot be written as a fraction.
There are very many different types of irrational number, but some examples include 2 , π , 5 +1 . 2
The Real Numbers are all the numbers, rational or irrational, which you can find on a number line. All the numbers you have ever used are Real Numbers. We give them a name because there are some numbers used by mathematicians which do not fit anywhere on the number line. You will not meet these unless you study Further Mathematics.
Converting fractions to decimals
The horizontal line in a fraction represents division, so to express 61 as a decimal we can complete the
division: 1 ÷ 6. Y ou might lay it out as follows:
61. 10 40 40 40 ...
And give your answer as 0.16
If you are not sure how to do this, watch this video:
https://www.khanacademy.org/math/arithmetic/decimals/decimal_to_fraction/v/converting-fractions-to-decimals
Converting decimals to fractions – Terminating decimals
If a decimal terminates then we can convert it easily. Remember that the digits after the decimal point
0. 1 6 6 6 ...
represent tenths, hundredths and thousandths, so 0.842 = 842 . We can simplify this to give 421,
which would be our final answer.
Converting recurring decimals to fractions.
1000 500
If a decimal repeats, e.g. 0.2727272727... = 0.27, you can write it as a fraction by applying the following method:
Let x = 0.27272727... Therefore 100x = 27.27272727... Subtracting the first equation from the second
gives us 99x = 27, so x = 27 3 =. Therefore 0.27 = 3 . If the repeating section had been only one digit 99 11 11
long, we would have multiplied by 10; if it had been three digits long, we would have multiplied by 1000. The repeating section must always line up with itself when you have multiplied by the appropriate power of 10.
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Sixth Form Kick Start Sherborne School Mathematics Department
Exercises
1. For each of the following numbers, decide whether the given number is Rational or Irrational.
(a) 0.7 (b) π
(e) 0 (f) 23 (g) 0.307307...
2. Write each of the following fractions as decimals without using a calculator.
(b) 3 11
(d) 0.4 (h) 0. 125
(g) 14 (h) 5
(c) 2 +1
(a) 3 4
(e) 2 9
(i) 1 8
(m) 6 5
(d) 1
(c) 1
2 3
(f) 3
7 15 37
(j) 13 (k) 3 (l) 3 9 5 16
(n) 33 (o) 8 (p) 4 10 15 3
3. Write each of the following decimals as fractions in their simplest form without using a calculator.
(a) 0.3
(e) 0. 3333... (i) 0. 1625
(b) 0.456
(f) 0. 104104... (j) 0. 101010...
(c) 0.212121... (g) 0. 5555... (k) 0. 363363...
(d) 1.4
(h) 0. 125 (l) 0. 00325
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Sherborne School Mathematics Department
Sixth Form Kick Start
Fractions
You need to be able to
• Convert top-heavy fractions to mixed-numbers without a calculator
• Convert mixed numbers to top-heavy fractions without a calculator
• Add fractions without a calculator
• Subtract fractions without a calculator
• Multiply fractions without a calculator
• Divide fractions without a calculator
Top heavy fractions
Mixed numbers are numbers which look like 112 or 3 73 . They have an integer part and a fractional part.
Top-heavy fractions are just written with a fractional part, although the numerator (the number on
top) may be bigger than the denominator (number on the bottom). It is easier to work with top-heavy
fractions. To convert mixed numbers to top-heavy fractions, write the integer part as if it were a fraction
and then add.
F o r e x a m p l e , 1 = 1+ = + = . S i m i l a r l y , 3 = 3 + = + = .
1 1 2 1 3 3 3 21 3 24 2 2222 7 7777
To go the other way, find the largest multiple of the denominator which can be taken out of the
numerator, and turn this into a whole number part.
Forexample, 23 =20 +3 =4+3 =43. 55555
Adding and subtracting
To add and subtract fractions, they must first be put over a common denominator.
Forexample1+1=2+3=5,and6−1=54 −7 =47. 3 2 6 6 6 7 9 63 63 63
Multiplying and dividing
To multiply fractions, multiply the top lines and the bottom lines together. Don’t forget to cancel
common factors if you can.
14 3 14×3 14×1 2×1 2
For example × 33
= = = .= You may well decide to write this using fewer steps. 33×35 11×35 11×5 55
35
To divide fractions, you have to multiply by the reciprocal of the fraction on the bottom. The reciprocal of a fraction is that fraction written upside down.
Forexample 2 ÷4 =2×3 =1. 93946
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Sixth Form Kick Start Sherborne School Mathematics Department
Exercises
4. Convert the following mixed numbers to top-heavy fractions without using a calculator.
(a) 32 (b) 71 (c) 61 544
(d) −23 (e) 85 (f) 617 7 6 24
5. Convert the following top-heavy fractions to mixed numbers without using a calculator.
(a) 11 (b) 224 (c) 132 5 15 23
(d) 47 (e) 64 (f) 1047 3 11 107
6. Evaluate the following without using a calculator, giving your answer as top-heavy fractions in their lowest terms.
(a) 3+12 (b) 3+9 55 48
(d) 15−5 (e) 13+1−2 4 6 12 8 5
(c) 1−1 730
(f) 1+2+3 234
7. Evaluate the following without using a calculator, giving your answer as top-heavy fractions in their lowest terms.
(a) 3×12 55
(d) 15÷5 46
(b) 3÷9 48
(e) 13×1÷2 1285
(c) 2×11 730
(f) 1÷2÷3 234

