# 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.

---------------------------------------------

**Fractions:**

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 6^{2}, 6 is the *base* and 2 is the *power/index*.

6^{2} = 6 x 6 = 36

**Fractional Indicies** – 27^{1/3} is the same as ^{3}√27. (hint: 9^{1/2} is the same as √9 = 3. this is a progression of that.)

**M****ultipication** – For numbers with the same base simply add the powers together.

2^{2} x 2^{3} = 2^{(2 +3)} = 2^{5}.

In algebra, if the terms have the same base – e.g, 2a^{2} x 3a^{4} has the same base, a – they follow the same rule.

a^{(x)y} is the same as a^{x} x a^{y} or a^{xy}.

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

a^{x}/a^{y} = ^{ }a^{x-y}

**Numbers to the power of 0** – for any value of a, a^{0} = 1.

**Negative Indices** – a^{-x} = 1/a^{x}

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

a^{x/y} is the same as (a^{1/x})^{y }or (a^{y})^{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.

**C****hanging 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).

-------------------------------------------------------

**Q****uadratic Equations** – A quadratic equation is of the type y = ax^{2} + 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:

factorisation

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 x^{2} + bx + c = 0 as (x + ^{b}/_{2})^{2} + d = 0 and then solving for x.

**Quadratic Formula** - x = (-b ± √(b^{2} - 4ac))/2a

**Note: The Difference of two squares**

If y = a^{2} – b^{2}

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.

------------------------------------------

**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:

AB^{2} = (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 90^{o}.

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 90^{o}.

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

**Proportion**

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 ∝ y^{2} then x ∝ ky^{2}. 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.