OK, OK, stop screaming! I KNOW C1 is only a few days away, I know you’re underprepared, but panicking isn’t going to help anything.

It’s a pretty common - and heartbreaking - thing for tutors and teachers to see: a student who’s waltzed through GCSE getting to the end of Year 12 and finding that A-level wasn’t quite such a breeze as they’d assumed, and the ideas simply hadn’t sunk in.

It’s not too late to do something about it, though - I can’t promise to get you from a U to an A in a few days, but there are marks you can pick up.

### 1. Spend ten minutes figuring out how surds work

Get your calculator out. Pick a random number under 200. Square root it. Look for patterns. Once you see how they work, it’ll feel completely obvious - but it works best if you discover it yourself.

Bonus: see if you can figure out what the calculator does with things like $\frac{3 + \sqrt 2}{5 - \sqrt{2}}$

### 2. Remember your power laws

This is A* stuff from GCSE: you need to be able to work with fractional and negative powers. Quick reminder: a negative power means you have to use the reciprocal of the thing you’re powering; a fractional power means there will be some rooting involved.

Bonus: look out the rules for products in brackets like $(16x^4)^{-\frac12}$. Oh, and learn your squares up to $20^2$, cubes up to $10^2$ and powers of $2$ up to, let’s say $2^{10}$. You’ll thank me when they come up.

### 3. Learn to integrate and differentiate

A little mnemonic for you: Drink My Drink; I’m Incredibly Drunk.

DMD means: to Differentiate (with respect to $x$), you Multiply by the power of $x$, and then Decrease the power by one: $4x^3$ becomes $12x^2$. Something like $4x$ is the same as $4x^1$, which differentiates to $4x^0 = 4$. But then, you’ve been doing that since year 8. Constants like $3$ are the same as $3x^0$, which become $0$ and vanish.

IID means, to Integrate (with respect to $x$), you Increase the power of $x$, then Divide by the (new) power. $4x^3$ integrates to $x^4$. Something like $4x$ is the same as $4x^1$, which becomes $2x^2$. Constants grow an $x$. And - importantly - add a constant ($+c$) at the end.

Bonus: figure out when to integrate and when to differentiate. That’ll save you a headache.

### 4. Figure out subscript and $\sum$ notation

It’s not that tricky: $u_7$ is just the seventh term, or ‘thing’ in the series called $u$. More to the point, $u_{n+1}$ is just ‘the thing in the series after $u_n$.’ If you’re given a recipe with $u_{n+1}$ in it, it just means “if you know $u_n$, you can apply this recipe to find the next value in the series.

Sum notation isn’t too bad, either - the pointy E is a Greek letter S (sigma), which stands for “Sum” - or, add a load of stuff up. Something like $\sum_{i=7}^{12} u_n$ means ‘add up all of the $u$s from 7 to 12’ - that is, $u_7 + u_8 + u_9 + u_{10} + u_{11} + u_{12}$. The number underneath is where you start; the number up top is where you end.

Bonus: look up the proof for the sum of an arithmetic series and learn it. It doesn’t come up all that often, but a) it’ll help you remember what’s going on and b) it’ll save you from the bloodbath if it does come up.

### 5. Learn to factorise quadratics

It’s taken as read in C1 that you can factorise quadratics, even the pesky ones with a number in front of the $x^2$. There are resources here. Have at it.

Bonus: Learn to sketch them, too. Get into the habit of sketching them even when you don’t have to. You’ve no idea how much stress it’ll save you when you actually do have to draw them.

Don’t say I’m not good to you. Bonus: Learn to spell ‘asymptote’. It’s not going to win you any marks, but it saves you looking like an idiot.

### 7. Figure out gradients and distances

The distance between two points is the length of the hypotenuse of a triangle. Its opposite side is the difference in the $x$ co-ordinates; the adjacent side is the difference between the $y$s.

The same triangle gives you the gradient - only it’s the opposite over the adjacent.

Bonus: drill it into your mind that when you want the gradient of a curve, you need to differentiate to get it.

### 8. Learn to use the equation of a line

This is the biggie. It always comes up. If you never bothered with lines at GCSE because, what the hell, it’s only a few marks, you’re going to come unstuck in C1. You can use $y= mx+c$ if you really want to (although don’t come running to me when the Mathematical Ninja chops off your legs!) - the grown-ups use $(y - y_1) = m(x-x_1)$, where $x_1$ and $y_1$ are the co-ordinates of a point you know, and $m$ is the gradient. Once you rearrange it, it drops out nicely.

Bonus: remember that a line with gradient $m$ is perpendicular to one with the gradient $-\frac{1}{m}$.

### 9. Learn to solve equations

What’s that? You’ve been doing that since Year 7? Jolly good. Then you’ll have no problem finding where a curve crosses the $x$- and $y$-axes? Or another curve?

To find the axis crossing points, you’ll substitute in $y=0$ or $x=0$, as appropriate. To find where curves cross, you solve their equations simultaneously.

Bonus: If your simultaneous equation involves a tangent, be on the lookout for double roots.