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.
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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
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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.
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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).
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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:
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 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.
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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.
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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.
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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.
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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.
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 ∝ 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.
