GCSE to A-Level

Moving from GCSE to A-Levels can be daunting. Here is a refresher of what you need to know when starting your A-level Maths after having completed your GCSE Maths. Reading this guide will give you a strong foundation from which to build your knowledge prior to starting your A-level maths course.


Types of Numbers:


Integers – Any positive or negative whole number, including zero.

Rational – any number that can be written as a fraction. (any whole number can be written as a fraction with 1 as the denominator) – Fractions

Irrational – A number that can't be written exactly, they are non-repeating decimals that never end.

Real Numbers – Any rational or irrational number.

Surds – irrational numbers that include a √ sign. Surds

More on Numbers can be found Here.




Adding and subtracting – the denominator must be the same. Look for the lowest common denominator for all numbers.

If you have mixed numbers turn them into top-heavy, or vulgar, fractions.

(multiplying denominators together will provide a common multiple, but it might not be the lowest common multiple.)


Multiplying – Make mixed numbers into top-heavy fractions. Multiply the numerators. Multiply the denominators. Simplify if possible.


Dividing - Make mixed numbers into top-heavy fractions. Turn the second fraction upside down and multiply the fractions as above.


Find out more details Here




Laws of Indices:


For the number 62, 6 is the base and 2 is the power/index.


62 = 6 x 6 = 36


Fractional Indicies – 271/3 is the same as 3√27. (hint: 91/2 is the same as √9 = 3. this is a progression of that.)


Multipication – For numbers with the same base simply add the powers together.

22 x 23 = 2(2 +3) = 25.


In algebra, if the terms have the same base – e.g, 2a2 x 3a4 has the same base, a – they follow the same rule.

a(x)y is the same as ax x ay or axy.


Division – For numbers with the same base the powers are subtracted.

ax/ay = ax-y


Numbers to the power of 0 – for any value of a, a0 = 1.


Negative Indices – a-x = 1/ax


Simplifying - As far as is possible, write the expression with the minimum number of bases possible (one is ideal).


ax/y is the same as (a1/x)y or (ay)1/x.

The simplest method for finding the answer depends on the values of x and y.


More information on Indices can be found Here.




Factorising - Finding common aspects to each term in an expression. It is the opposite of expanding brackets; you are adding brackets. Easier to do if you know your times table. Factorising and Times Table Test


Algebraic Fractions – The rules are the same as for numeric fractions.


Changing the subject of a formula – To make x the subject of the an equation y = mx + c perform operations on both sides of the equation until x is on its on own one side (it does not have to be the same side it started on).


Quadratic Equations – A quadratic equation is of the type y = ax2 + bx + c where a, b, and c are numbers.

Solved by working out the values of x where y = 0.


3 methods to use when working out:



completing the square

using the quadratic formula


Factorising – Easier if the value of a is 1. To find the values to go with the brackets you need 2 values that when added give you b and when multiplied together give you c.

If c is positive and b is negative then the values that go with the brackets will be negative.

If a is not equal to 1 – the first bracket will contain ax. 2 values for the brackets will multiply to give c.


Completing the square - Write x2 + bx + c = 0 as (x + b/2)2 + d = 0 and then solving for x.


Quadratic Formula - x = (-b ± √(b2 - 4ac))/2a


Note: The Difference of two squares

If y = a2 – b2

then y = (a+b) (a-b)


More information on Quadratic Equations can be found Here.


Linear Simultaneous Equations


Simultaneous equations can be solved more efficiently algebraically.


Elimination method: Multiply or divide either or both equations until you end up with a common term on both sides that can then be eliminated.


Substitution method: Rearrange the equation and substitute it into the other.


More on Simultaneous Equations can be found Here.


Finding the Gradient


Gradient = (change in y)/(change in x)

Make sure the x coordinate and y coordinate of a point occupy the same position in their respective parts of the equation.


Gradients and Graphs.


Y-intercept, Horizontal and Vertical lines


Y-intercept: The Y coordinate of a line when x=0, where the line of an equation crosses the y axis on a graph.


Horizontal lines come from equations of the form y = ?.

Vertical lines come from equations of the form x = ?.


Y = mx + c


m is the gradient

c is the y-intercept

More on this form of equation can be found Here.


Finding the distance between two points


Method 1:

Draw a line between the two points. Complete a right angle triangle and use Pythagoras' theorem to work out the length of the line.


Method 2:

Between points A and B:

AB2 = (Bx – Ax)2 + (By – Ay)2


Essentially Method 2 skips the drawing part of Method 1.


Circle Properties


If line AB is the diameter of a circle and C is a point on the circumference, the angle at C to points A and B will be 90o.


A line perpendicular from the centre of a circle to a chord will bisect that chord.


Knowing a tangent is a point that touches a circle at one point only, the angle between a radius and the point of tangent is 90o.

More on Circles can be found on the Definitions Page and the Theorems Page.





Direct Proportion is when one constant is a constant multiple of the other.


It is written x ∝ y.


Proportionality Equations: if x ∝ y, then x = ky. K is the constant of proportionality or constant multiple.


Squared and cubed terms: If x ∝ y2 then x ∝ ky2. The same goes with cubed terms.


Inverse Proportion: The value of x will increase as y decreases. x ∝ k/y.


More on Proportion can be found Here.

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