Learn from the maths master

Cillian Fahy, who got straight As in his Leaving Cert, made headlines when he sold his exam notes on eBay for €3,000. In this continuing series, he shares his study experience with readers, today offering advice to fifth- and sixth-year students on Leaving Cert maths. Plus, how to get the best study value out of the Christmas break

WHAT’S THE most important number? Seven. To ensure you do well in the maths exam, you need to have the ability to answer seven questions. You only need to answer six but going in with the minimum will not serve you well.

In maths, you really need to understand every step to every question.

The only way to do this is to keep doing questions. But it’s not enough to just follow the steps in a book. You need to be able to adapt them to yourself and the way you do things. The real secret to my success was that I wrote out every type of question and the steps I took to do them.

Try to think of maths as a language. It’s hard to learn something that someone else has written for you but it is much easier to learn something if you come up with it yourself.

Difficult questions can look daunting but break them down into steps and you’ll find it much easier.

Here’s my step-by-step guide to Higher Paper 1:

ALGEBRA

(Questions one and two)

EU + IMF = 0 may sound like some economic diagnosis but in fact it’s algebra. The most important section in paper one and a section that you need to do well in.

Here’s how:

Strategy

In this exam, strategy is so important. From day one, you need to start looking at every section of the course and ask yourself: “What’s my strategy here?”

Algebra is worth two questions. In the exam, I recommend at least planning to do one of these, if not both. The reason behind this strategy is that in algebra you develop skills that you may need in other questions, such as manipulating equations, so it can be helpful across the board.

Get to the root of the problem There are numerous words in algebra which come up again and again which you need to understand fully.

Examples

Natural numbers, factors and roots. You should know that a quadratic equation has two roots and a cubic has three. Also, that a root on a graph is a place where it cuts the x – axis (y = 0). Also, if you have a root you can sub it in to your equation and it will work out.

Here’s something to look up for yourself: What’s the difference between a factor and a root?

Simple rules

Much of algebra is all about getting the simple things right. So make sure you do.

Understand how to deal with fractions underneath the line and how to combine two fractions into one.

The same goes for inequalities. If you multiply across by a negative number, the direction of the inequality will have to change. You may have been told all these things but make sure you know them.

Once you get the simple rules out of the way, you will find that your maths will improve everywhere.

MATRICES AND COMPLEX NUMBERS

(Question three)

Looking at the entire paper, this question can seem relatively easy. In many ways it is but it’s important that you don’t let your guard down.

Here’s how to get the most out of this question.

Check, check and triple check

With all this “odd” matrix multiplication and the random letter ‘i’ following numbers, it can be easy to make a simple mistake which could cost you marks.

Plan ahead

You should know which order you would prefer to do the questions in. Be aware of where you get caught and what you need to watch out for. The only way to do this is to practise exam papers again and again until you can understand the best plan of action for you.

Watch out for questions crossing over

Just because this is called the matrices and complex numbers question doesn’t mean that you will not see other types of maths. Watch out for sections from other questions crossing over.

SEQUENCES, SERIES AND INDUCTION

(Questions four and five)

Many people ignore these as options. However, you should at least consider one of them if not both. Here’s what you need to look out for in these questions:

Induction

Induction is a key element of question five. Remember that there are different types of induction proofs so if you have difficulty with one, consider the others. Once you get your head around induction the rest of this question is relatively easy.

Sequences and series

Sequences and series usually make up the bulk of question four. It is important when dealing with these that you look closely at exactly what you are being asked. Is it arithmetic or geometric? It took me a while to get a hang of sequences and series but once I got it, all of paper one became much more manageable.

LOGS, INDICES AND INEQUALITIES

(Question five)

In this question, you will be required to manipulate logs and indices using a few rules but these rules are now printed in the table book.

This can make the question an awful lot easier. The inequalities are similar to those in algebra. Exploit that as much as you can.

DIFFERENTIATION

(Questions six and seven)

If you have a strategy, watch for simple mistakes and stay on your guard. If you do, you should find differentiation a manageable section in the exam. Differentiation is algebra. Like algebra, differentiation is worth two questions on the paper so it’s certainly a section that you should look into.

Take it easy

Everyone knows that the (c) part is the most difficult of each question. However, that doesn’t make it the most important. Getting the (a) parts and (b) parts right is where you should focus on.

First principles

Learning differentiation from first principles is important simply because they can come up. In maths, you are not required to learn off very much so you should be willing to learn off what you need for the exam. If you understand each step you will find them quite easy to learn off and should be worth the work on the day.

How does it actually work?

One of the most obvious and yet important questions in differentiation has to be why is it dy/dx?

Examples

Here’s an example: if y = x + 3, then dy/dx = 1. But why is dy on the top and dx on the bottom? It’s because the one on the bottom is always what the equation is in terms of, whereas the piece on top is what it is equal to.

Another example: u = 3x + 9 so du/dx = 3 and z = 5t therefore dz/dt = 5.

Look closely at this because understanding it is important for the exam and can help you do well.

INTEGRATION

(Question eight)

It’s easy to lose marks here but if you stay alert, it should be worth your while.

Compare

The advantage of doing both differentiation and integration is that you can compare the rules of each and see the differences.

Simply put, integration is the opposite of differentiation. However, they appear to be different in some areas and it is also good to compare them there so you can see clearly the differences. For example in trigonometry.

Constant

This question is all about “cs”. Remember to keep putting in your constants constantly. You may have gotten the answer correct but if you don’t put in a “+c” at the end, you will pay for it in marks.

More importantly, learn where you shouldn’t put in “Cs”. This may sound like an obvious and simple rule but it is one you should look at closely to ensure you get those marks.

Checklist

There are so many little bits and pieces in integration that I recommend preparing something of a checklist.

What are the bits that you sometimes forget?

Do you remember to differentiate to find du when using substitution?

Having a brief checklist of your common pitfalls can be a real advantage to you in the exam.

There are so many little bits and pieces in integration that I recommend preparing something of a checklist

Memo to Leaving Cert students: How to get the most out of your Christmas

The break at Christmas is an extremely important opportunity for you to catch up, review and plan ahead for the months to come. It’s crucial that you use this chance to benefit your study.

Relax

The first thing you need to do is relax a little. For a few days you should work on recuperating and enjoying the festivities. If you take some time off, your study afterwards will be much more focused and productive. Of course don’t go to the other extreme and relax all through the holidays or you will regret the time lost afterwards.

Think positively

Time seems to go awful quickly in the Leaving Cert, so you may have expected to have covered more work than you actually managed to get done in the last few months.

That’s ok because it’s the same for everyone.

Don’t get down about the lack of progress you have made. Have confidence in yourself. Of course if you really feel you haven’t done enough work, then Christmas is the best time to take that wake-up call and get your work back on track.

Plan

The real key to success in the Leaving Cert is routine.

The trouble with Christmas is that it’s a break in routine which can really damage you if you have already settled into a nice pace of study. It’s vital that you counteract that by drawing up a new study timetable for the Christmas break.

This will give your study structure and keep you in routine.

Look back

Take a look at all the work that you have covered both in school and in your own study.

Are there any areas which you haven’t fully grasped yet? I think Christmas is the best time to take a look at these and try to crack them. That way you can be happy when you return to school in January that you have that work covered.

Oral

For any language subject, it’s important that you start to look at the oral over Christmas.

If you haven’t started it already, this is a good opportunity to get the basics written up. If you have already done that, then now is a good time to consolidate that learning.

The oral may be four months away but think how fast these last four months have gone. You’ll find the stress of the oral will be greatly reduced if you give it some focus over Christmas.

Subject review

Reviewing the subjects is vital at this stage. You need to sit down and break them up section by section.

The sections which I have shown you in previous articles might be a good template. Check where the problems are and where things have improved since September. Are there any urgent sections which need to be addressed? It is important that you are honest with yourself.

A good idea is to rank the sections of each subject from best to worst. If you did this in September, see if the order has changed.

Grinds

Grind classes are something which you shouldn’t rush into. But if you feel you are falling behind with your school work and are not capable of resolving it on your own, then January is the best time to start with these classes.

The reason why January is the perfect time to start receiving tutoring is that it is just far enough away from June to still make a difference.Don’t leave it any later to make this decision. Consider the subjects you may need extra classes in and discuss it with your parents. There are a number of different ways to track down a good teacher. Word-of-mouth is always good or alternatively you can use the internet.

Trinity College Dublin has its own Leaving Cert grinds database which you can find online. Check if your local university has one too.

Look forward

What are your plans for the next few months? What can you get done in that time and what needs to be done?

Christmas is a good time to look ahead and see what strategy to take for the months to come. Check if you need to adapt your schedule and study more. Based on what you have already covered, identify which areas will cause you trouble. Keep them in mind when planning out your future study plan.

And finally

It’s important that you enjoy Christmas. The Leaving Cert can be a stressful time, I know, but you should see this as a chance to rest the engines before you push it up a gear for the New Year.

Merry Christmas!

Next month: more from Cillian Fahy on maths, plus, how to prepare for the mock exams