How to answer STEP and MAT questions when you're stuck

January 20, 2016

STEP and MAT exams need to be taken for entry to some of the top undergraduate mathematics courses in the country, including Cambridge, Oxford, Imperial College London, Warwick, and UCL. If you're reading this, congratulations on being able to aim so high :-)

However, the style of questions are very different to A-Level questions, and it's easy to get scared when looking at them for the first time, or when you don't even know how to start. Here are a few tips that I continually tell students, as they work for most questions. I'll refer to Q4 of the MAT Specimen Test 1(find it at the link here) to give you an example each time.

I strongly recommend trying the question out yourself before reading the rest of this, because I don't go through the question in order. Do it now! :-)

1) Don't get scared

This is obviously easier said than done, but remember that even I don't know how to do these questions immediately when I look at them (and I have been to Cambridge, and have a PhD in maths!). That is normal - the questions are made up specifically so they hope that you haven't seen a question like it yet. The reason for this is that they want to see whether you can calmly think about how to work something out, without having seen it before. And you can't do that if you immediately tell yourself "Oh my god I have no idea! I'll skip to the next one" (and then repeat the process). It takes time, but hopefully with practice and the next few tips, you'll get used to feeling ok that you don't immediately know what to do.

2) Write down what you know, and use it!

Remember that you already know everything you need to answer the question (make sure you know all the identities and theorems that are on the syllabi beforehand). Either you need something stored in your memory, or you need something that you have already shown earlier on in the question. You will probably have heard your teachers telling you this already, so here are a few examples from Q4 of the MAT Specimen Test 1.

- For part (i) you are asked to show that the area of the triangle ABC is equal to (1/2) bc sin(alpha).

You should recognise this formula from the standard identities you have memorised. Since you have to prove this identity, you can't use it. Now you may be thinking "HOW?? I can't remember how they proved it in the textbook!". That's ok, that's why they put it there! Don't try to remember how it was done, you have to try to work it out yourself. 

So, other things do you know about the area of triangles? Well, the only other thing you know is from GCSE:

Area of triangle = (1/2) x base x height

You have no choice, you'll have to use this. As is most often the case, this is exactly what they want you to do (using AC as the base, and drawing a line so that you can use SOH CAH TOA to find the height). Check the solutions online if you're still stuck on this part.

- For part (ii) you are again asked to find the area of a triangle, PQR. You might have figured out by looking at the answer that you need to work out the area of triangles ARQ, BPR, CPQ (see tip 3). 

So, to work out the area of ARQ: You can try to use Area of triangle = (1/2) x base x height again, but you don't know what the base or the height are. At this point, you know you need another identity from your memorybank, and/or you need to use something from a previous part of the question. In this case, it's both - the only other identity you know for finding areas of triangles is the one that you proved in part (i). Again you have no real choice - you have to use this identity. Smile, because you know you must be on the right track, even if you still have no idea how exactly it will lead to the answer.

To work from here, you have to use more SOH CAH TOA's to get the side lengths of the triangles and the angles (there's a hint in the answer - you need cos(alpha) to turn up!)

3) Don't be afraid to work backwards so that you can see how to work forwards

For part (ii), you have to prove some weird identity that you have never seen before. Don't worry, I haven't either (see tip 1!). Before launching in, look at what you're going to try to prove. Why would the identity look like that?

- Hopefully, you can guess that the identity looks like that because 

Area(PQR) = Area(ABC) - Area(ARQ) - Area(BPR) - Area(CPQ)

Then, you know that for example you need to work out that 

Area(ARQ) = cos^2(alpha) x Area(ABC) (and similar for Area(BPR) and Area(CPQ))

- Now you can try to work backwards again. Why does Area(ARQ) = cos^2(alpha) x Area(ABC) ?

You stare at the triangles, you can't see immediately why cos^2(alpha) comes into play (you might have thought about a double angle formula, but there are no double angles or half angles to use here). Try something else: you know that 

Area(ABC) = (1/2) bc sin(alpha)

So now you just have to show that 

Area(ARQ) = cos^2(alpha) x (1/2) bc sin(alpha)

From here (and combined with the fact that you might have guessed that you need to use that area-of-triangle identity to work out Area(ARQ) (see tip 2), you might be on your way. If not, see tip 4 :-)

4) Make educated guesses

This is not cheating, it's not a less admirable way of doing maths, it's what all mathematicians do. You are trying to prove something, but you can't work forwards. Rearranging the identity and guessing how it comes together, for example by noticing that parts of it look like another identity you already know, is a valid way of working things out and has often led to great leaps of understanding. For example:

- You have to show that 

Area(ARQ) = cos^2(alpha) x (1/2) bc sin(alpha)

Why does the cos^2(alpha) turn up?? We already established it's not part of a double angle formula. Try splitting it up:

Area(ARQ) = (1/2) x b cos(alpha) x c cos(alpha) x sin(alpha)

Notice anything?

Area(ARQ) = (1/2) x [ b cos(alpha) ] x [ c cos(alpha) ] x sin(alpha)

How about now? It's in the form Area of triangle = (1/2) B C sin a 

(I changed the letters to not get confused with Area(ABC) = (1/2) bc sin(alpha) ). To see how the rest goes, look in the solutions. However, even if you're this far and really can't figure out any more, if you blag it no one would notice ;-) Working forwards, you'd write 

Area(ARQ) = (1/2) AR x AQ x sin(alpha)

= (1/2) x [ b cos(alpha) ] x [ c cos(alpha) ] x sin(alpha)

= cos^2(alpha) x (1/2) bc sin(alpha)

= cos^2(alpha) x Area(ABC)

Hey presto! Even if you have no idea why you can go from the 2nd to the 3rd line, it's all correct and the examiner doesn't know that you don't know. Full marks :-)

5) What are the only possibilities?

This is sort of a combination of tips 2 and 4. Sometimes, there are only a few things that the answer could reasonably be, if you had to guess. Try to reason that way. For example:

- For part (iii), they ask "For what triangles ABC does *blah* hold?"

Well, you can't give any answers in terms of specific angles, or lengths. The answer has to be one of the triangles with a name that you know, i.e. equilateral, isosceles, or right-angled. Woop! Now you only have to figure out which one (or two, maybe), and then see why. 

Hopefully, you can see that nothing much would change in the diagrams they give you if ABC were equilateral or isosceles.So why is a right-angled triangle the correct answer? Check what would happen in your earlier answers if ABC had a right angle, and you'll see :-)



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