When you said you wanted to do maths A-level, I imagine people pursed their lips and said “oo, it’s a big step up.” They’re right, to an extent: every October I get deluged with panicked students who realise that the ‘coast through it all doing just enough’ approach that served them very well at GCSE is suddenly failing them at A-level… and I don’t want you to be one of those people!

With that in mind, I’ve put together an ‘A-level ready’ checklist to make sure you’re up to speed with the GCSE knowledge you’ll need at your fingertips when you start A-level.

Arithmetic

… or, as the more prosaically-minded boards call it, number.

You need to be totally au courant with basic arithmetic – you need to be able to multiply two- and three-digit numbers together (and, in principle if not in practise, any length of numbers). You need to be able to divide numbers – again, in principle, of any length, but two-digit numbers are plausible. And, probably most importantly, you need to know your squares and square roots up to about $20^2=400$ and your cubes up to about $10^3 = 1,000$.

Other powers (especially of 2 – up to about $2^{10} = 1,024$, and 3 – up to about $3^6=729$) are helpful; knowing your power laws, including negative and fractional indices, are essential. You’ll also want to be happy with surds, especially multiplying (things like $(3+\sqrt{7})(2 - \sqrt{7}) = -1 - \sqrt{7}$) and simplifying (e.g., $\sqrt{250} = 5\sqrt{10}$)

You need to be able to work with fractions – the four basic operations are taken as read, and you need to be able to take powers of fractions. Luckily, that’s simple. For instance, $\left(\frac 3 4\right)^3 = \frac{3^3}{4^3} = \frac{27}{64}$.

Oh, and you need to be OK with negative numbers.

Algebra

There’s an awful lot of algebra in A-level – at a very bare minimum, you need to be comfortable with solving linear equations such as $4x + 7 = 2x + 23$ and expanding brackets. If that’s all you can do, though, you’ll struggle; you’ll also want to be able, roughly in order of importance, to:

  • Solve quadratic equations using any method
  • Factorise two-term expressions (like $x^2 - 5x$)
  • Factorise simple quadratics (such as $x^2 - 6x + 8$)
  • Find the gradient of a straight line from its equation
  • Solve linear simultaneous equations
  • Solve non-linear simultaneous equations
  • Solve inequalities (linear for sure, quadratic also nice)
  • Manipulate algebraic fractions

Graphs

If there’s one thing I’d recommend working on hard over the summer, it’s straight lines. In terms of the number of marks available and the number of students I have struggling with a topic, it’s a hands-down winner for Year 12 as the most useful thing to take time to work on.

Make sure you understand the idea of the gradient of a line, finding whether a point is on a line, working out the equation of a line, finding parallel and perpendicular lines… if you master those skills, C1 and C2 will be enormously easier.

In addition, you’ll do well to know and recognise the graph shapes for quadratic and cubic graphs, as well as reciprocal ($y = \frac 1x$) and exponential ($y=a^x$) graphs, and understand a bit about transforming graphs by replacing $f(x)$ with $f(x-3)$ and so on.

Geometry

… or ‘shape’, if you’re into that whole new-fangled Anglo-Saxon nonsense.

Triangles are clearly the best shape in the world, a fact that A-level recognises. As a result, you must be comfortable with Pythagoras and finding the area of a triangle as $\frac 12 bh$.

Especially if you’re doing Mechanics, SOH CAH TOA is extremely handy, as is an idea of how bearings work; for Core 2, the sine and cosine rules are useful, as well as the area formula ($Area = \frac 12 ab \sin(C)$).

In terms of less-interesting shapes, you’ll want to be able to find the area and perimeter of a range of shapes, including the circle, rectangle, trapezium and (of course) triangle.

Circles crop up in many contexts, especially C2, and having an idea about area of a sector and arc length will stand you in good stead, as will circle theorems (especially the facts that a tangent is perpendicular to a radius, a right-angle on the circumference of a circle is opposite a diameter, tangents to the same point are equal, and the bisector of a chord is a diameter).

Statistics

If you’re doing S1, which I would guess is the most popular optional module, it’ll help you if you understand probability and can find means, medians and modes of date presented in different ways (lists, tables, and grouped tables).

You’ll also find that histograms and box plots show up – hopefully you’re familiar with them already!

This isn’t meant to be a comprehensive list ((If you think of anything else that ought to be here, let me know and I’ll update with thanks)) – and you’ll cover almost everything here in class. However, knowing all of this before you show up will seriously ease the notoriously tough transition into A-level maths.

Good luck!

Edited 2015-06-10 to fix formatting.