# The Definitive Guide to A Level Maths

There is so much content in A Level Maths, it can be overwhelming to know where to start. Hopefully, some of the hints and tips here will help!

What is the difference between maths and Further Maths?

Further Maths is just that – more maths! It covers some similar topics to A Level Maths but in more detail, or with fewer restrictions – it builds on the work done in A Level Maths.

For example, you will have been introduced to vectors at GCSE. In A Level Maths, this is extended to find the direction and length of vectors and their applications in mechanics. In A Level Further Maths, you will consider intersecting 3D vectors and planes (imagine a metal rod meeting a piece of paper).

This is just one example, other topics that you study in maths aren’t revisited in Further Maths, such as binomial expansion. Other topics are brand new in Further Maths, like hyperbolic functions or discrete maths.

Further Maths is a lot more demanding than maths; it’s the perfect preparation for people looking to study maths, physics or engineering later in life.

What are the overarching themes?

You may have come across Assessment Objectives (AO1, AO2 and AO3). The overarching themes are similar; they are key skills that will be tested in every question. The three overarching themes are:

• Mathematical argument, language and proof;
• Mathematical problem solving;
• Mathematical modelling.

Most of these will have come up throughout your A Level – you may not have even noticed it!

Mathematical argument, language and proof.

Example 1:

Given that $f(x) = x^3 - 3x^2 -25x + 75$

a. Show that $(x - 3)$ is a factor of $f(x)$.

b. Fully factorise $f(x)$.

The first part of this question is checking your mathematical argument – you need to go through logical steps to show that $(x - 3)$ is a factor of $f(x)$.

It also checks that you understand function notation, which is mathematical language. In the answer, an examiner would expect to see $f(3)$ to show that you are substituting in 3 and a sentence concluding that $f(3) = 0$ , therefore $(x - 3)$ is a factor of $f(x)$.

Finally, it is testing mathematical proof! You are given the outcome: $(x - 3)$ is a factor, but you need to prove its validity.

The second question is more focused on mathematical argument, although there is mathematical language in every question – in this case, it can be seen in the notation and in the keyword “factorise”.

Mathematical problem solving

Example 2:

From the resource Differentiating Functions of the form x^n.

Joan East is making cylindrical cans of tuna. She wants to minimise the amount of tin she uses while maintaining a volume of 250cm³.

a. Write an expression for the height, h, of the cylinder, in terms of the radius, r.

b. Find the optimum height and radius of the tin.

This question could also be classed as mathematical modelling, but we are going to look at it from a point of view of problem-solving.

You are asked to minimise the amount of tin; from this keyword, you need to recognise this is a differentiation question. You will need to use the surface area and volume of cylinder – skills you will have from GCSE. This is a very challenging question

Mathematical modelling

Example 3:

Over summer, the amount of water in a reservoir decreases by 1% each day. On the 1st June, the reservoir holds 4 billion litres of water

a. By writing an equation representing the amount of water in the reservoir, find the amount of water on the 30th June.

b. Explain why your model may not be suitable to estimate the amount of water in the reservoir in December.

The most common place for these questions to come up is in statistics for mechanics, but there is no reason it can’t be in pure maths too! One of the key parts of mathematical modelling is knowing the limits to the model.

A mathematical model estimates the situation, it is unlikely that the water level decreases by exactly 0.1% – some days it will rain and some days it will be sunny, and these will have different effects on the water level. However, it is reasonable to estimate the average decrease across summer.

It isn’t reasonable to extrapolate the percentage throughout the year – the change in the water level in December will not be very similar to June. At some point, the reservoir will start refilling so this would need a different model.

Solutions

Example 1:

Given that $f(x) = x^3 - 3x^2 - 25x + 75$

a. Show that $(x - 3)$ is a factor of $f(x)$.

$f(3) = (3)^3 - 3(3)^2 - 25(3) + 75 = 27 - 27 - 75 + 75 = 0$

b. Fully factorise $f(x)$.

$f(x)=(x - 3)(x^2 - 25) = (x - 3)(x + 5)(x - 5)$

Example 2:

Joan East is making cylindrical cans of tuna. She wants to minimise the amount of tin she uses while maintaining a volume of 250cm³.

a. Write an expression for the height, h, of the cylinder, in terms of the radius, r.

We know the volume is 250cm³ so we can set:

$250 = r^2 h$

Making h the subject gives:

$h = \frac{250}{\pi r^2}$

b. Find the optimum height and radius of the tin.

$A=2r^2 + 2r$

$A=2r^2 + \frac{500}{r}$

To find the minimum value of A, we differentiate with respect to r:

$\frac{dA}{dr} = 4 \pi r - \frac{500}{r^2}$

$4 \pi r - \frac{500}{r^2} = 0$

$4 \pi r^3 = 500$

So, r = 3.41cm and h = 6.84cm.

Example 3:

Over summer, the amount of water in a reservoir decreases by 1% each day. On the 1st June, the reservoir holds 4 billion litres of water

a. By writing an equation representing the amount of water in the reservoir, find the amount of water on the 30th June.

Using W to represent the amount of water (in “billion litres”) in the reservoir, and d the number of days since 1st June gives us:

$W = 4 \times 0.99d$

On 30th June, 29 days have passed (30 – 1) so d = 29:

$W=4 \times 0.99^{29}=2.99$billion litres

b. Explain why your model may not be suitable to estimate the amount of water in the reservoir in December.

The weather is significantly different in December rather than June, therefore the decrease of 1% may not still be valid.

How do I revise A Level Maths?

The first step is to work out which areas you need to revise. Make sure you know how your exam boards papers are structured. You don’t want to be revising statistics when your paper is only on pure!

Look through the topics and decide which topics you need to spend time one. It might help to organise or colour-code your notes: split them into mechanics, statistics and pure maths, or into how hard you find the topics, or both!

These topics should be prioritised but shouldn’t take up all your time. Do a mixture of topic specific revisions to make sure you have the knowledge and mixed revision to make sure you have exam technique. The ability to determine if a question is coordinate geometry, vectors or calculus is just as important as being able to do the maths.

For each of the prioritised topics, look through your notes for each of the sections and make revision cards. Ask yourself these questions:

• What formulae do I need to know?
• What notation do I need?
• What are key bits of information?
• Are there any mistakes that you often make?

This might bring up some gaps in your knowledge; now is the time to fix them! In fact, as soon as you spot a gap in your knowledge, fix it. Look at the examples you have and try some questions on them.

Now is a great time to try an exam paper! Make sure you are in a quiet environment and get rid of distractions like phones – decide if you are going to limit your time or not. There are benefits to both, so try and do a mixture.

Doing this paper may have brought up some new gaps in your knowledge or may have shown that you hadn’t quite grasped something you thought you had! That’s fine, go back and look specifically at those topics. Don’t be afraid to go back to basics, those questions are there for a reason.

Keep repeating this pattern as often as you can. Once you’ve mastered a topic, don’t just forget about it! You need to keep it fresh in your mind or you might end up back where you started. Doing regular exam papers should help with this.

Here are some quick tips for revision:

• Make an exam timetable and stick to it. You could try our free Revision Timetable Template.
• Take regular breaks, you will be more productive if you do. Here are 5 Top Tips for Effective Revision Breaks.
• Manage your time well and prioritise key topics.
• Don’t just read your notes, understanding maths rarely comes from just reading – you need to be active. Try completing the examples yourself and comparing the answers, or writing a list of instructions.
• Don’t neglect the topics you’re good at – think of them as a chance to guarantee marks. Keep doing exam questions on them.
• Make posters of key terms and formulae and stick them up where you regularly see them – in the kitchen, the bathroom and your bedroom.

Useful Resource – 5 Top Tips for Revision

How am I assessed?

Each exam board has a slightly different set out for exams. It’s really important you know what to revise and when!

A Level Maths

AS Level Maths

A Level Further Maths

AS Level Further Maths

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