Category: mathematics

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Seven tips on doing well at IB Mathematics

I have been teaching IB mathematics HL and SL now for twelve years. I want to share some advice with students and their parents that will make their two years study of this compulsory subject more trouble-free and rewarding.

Preparation

Before you start the IB course, prepare for it. This applies whatever level you intend to study. It is a fact that most IB students starting off have weaknesses in basic algebra such as rearranging equations, poor mental arithmetic skills and so forth. There is a list of โ€˜prior learning topicsโ€™ in the IB maths guide that can be used as a framework for preparation. You can download the IB Mathematics Guide for the level you will be studying by doing a simple Google search. Most IB textbooks have a section on prior learning that can be used as a guide to self-study. Be prepared to find out things for yourself. Take an interest in the applications of mathematics in the wider world. There are many books and documentaries on the influence of mathematics on culture. The BBC documentary by Marcus du Satoy entitled โ€˜The Language of the Universeโ€™ is worth watching.

Subject Guide

Read the IB subject guide. This contains essential information such as the structure of the internal and external assessment of the course and the syllabus content. Be prepared. Your teacher will notice and you will be ahead of the game. A bit of preparation before the start of the course will make your life a lot easier.

It’s hard

IB is different from A-level. It is harder but donโ€™t be put off by this. The ethos of the IB is to make you an independent learner as well as a host of other aims. You will be at an advantage compared to your friends who have taken A-levels when you go on to further education.

Choose the level of mathematics carefully in consultation with your teachers and parents. Mathematics is a compulsory subject. Broadly speaking, HL mathematics is for the ablest students who intend to study a mathematics based subject at university such as a science or mathematics itself. SL mathematics is still very challenging for most students, shares a lot of the material in common with HL mathematics, and will demonstrate to universities that you are broadly capable in mathematics. Maths Studies SL is for students who either donโ€™t need a high level of mathematics for further study or who find the subject very challenging.

Graphing calculator

Familiarise yourself with the graphing calculator that you will be using. Your teacher will be able to tell you which model to buy. You will save yourself a lot of time in the classroom if you do this. It always surprises me how many students lack basic understanding of their graphing calculator functions. Some teachers will expect you to teach yourself how to use it. In my opinion, the teaching of the calculator functions should be integrated into the lessons but this is not always done. Regardless of which level you are taking, you will not be able to answer many questions effectively without your GDC. The paper 2 at HL and SL is specifically written with the GDC in mind and needs a different approach to the non-calculator paper.

Coursework

Coursework is a major component of the IB Diploma program and accounts for 20% of the maths course. You will conduct an โ€˜explorationโ€™ of an area of mathematics, usually at the end of the first year or beginning of the second year and will be given a lot of room to decide the area of your research. It is important that you have an opportunity to include mathematics appropriate to the level that you are studying. If the area of choice does not give opportunities for this, you will be better off choosing another area of research. There are exemplar โ€˜explorationsโ€™ available here. Look at the assessment criteria, available in the same link. Study the exemplar material to see where the students either met or failed to meet the criteria. Remember most of all that it is meant to be your own work. Your teacher can give you an oral examination to see if this is really the case.

Work hard

Be prepared for hard work. There are no short cuts. Be realistic whilst at the same time strive to do your best. Prepare for the course. Choose the right level, with advice from your teachers. Practise the calculator skills and improve your core maths skills. Study the course outline and objectives. Learn the meaning of the key words used in the examinations. These have very specific meanings. And most of all, the course will open your eyes to the beauty of mathematics and its applications in the modern world.

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Fermat’s Last Theorem and Andrew Wiles

Any maths student will be familiar with the fact that there are pairs of integers whose squares add up to the square of another integer, for example 3^2 + 4^2 = 5^2.

Examples of these so-called Pythagorean Triples have been known for millenia.

Fermat’s LastTheorem posited that there are no integers for which a^n + b^n = c^n where n is an integer greater than or equal to three.

There is a fascinating documentary by Simon Singh about the proof by the English mathematicianย  Andrew Wiles and his very lengthy proof in 1995.

It may be found here.

Pierre de Fermat was a famous mathematician who lived in the 17th Century in southern France. He is best known forย Fermat’s principleย that explains how light travels andย Fermat’s Last Theoremย inย number theory, which he described in a note at the margin of a copy of his bookย Diophantus‘ย Arithmetica.

Fermatโ€™s Last Theorem is possibly the most well-known theorem in mathematics. It was suggested by Fermat, and indeed he said that he had a proof for it but this was never published. A theorem without a proof is a strange thing indeed โ€“ not a theorem but a conjecture โ€“ a mathematical law which has not been proven.

It took over three hundred years and seven years of work for a British mathematician, Andrew Wiles, based at Princeton University in the USA to solve the problem.

The idea of Fermatโ€™s Last Theorem can easily be understood with a few examples and a calculator. Challenge students to find a case where n is greater than two. They may well not believe that such cases donโ€™t exist.

The documentary lasts 50 minutes and first explains what Pythagoras Theorem is. It then extends the idea to any power to a whole number and explains the hint by Fermat that he had found a proof that there are no integer solutions to the equation

x^2 + y^2 = z^2 for n>2.

It then discusses quite clearly how a problem in one field of mathematics can be translated into a different problem in another area of mathematics. So it was that the original problem was translated into a different problem to which a solution needed to be found. Andrew Wiles, through a flash of inspiration, which he describes vividly, came to this solution.

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Ten tips on using the graphical display calculator

So, youโ€™ve just been handed a brand new graphical display calculator (GDC) for your IB maths course. Thereโ€™s a good chance that you have been given a Texas Instruments Ti-84+ or a Casio FX-9860. If you did the Middle Years Programme, then you may have used a GDC before. But if you took the GCSE or IGCSE, then it will be new to you.

Here are some top tips on how to familiarise yourself with the GDC and make it work for you:

1. Donโ€™t expect your teacher to show you all the features of the GDC

If you donโ€™t understand how to do a particular operation, there are some great tutorials on YouTube (see the list of links at the end of this article). For more complex queries, you may need to refer to the manual, which can be downloaded if you are without a hard copy.

2. Take it to class every day

Although there will be times when you donโ€™t need it, you donโ€™t want to be borrowing one from your neighbour. And remember to put your name on it so it doesn’t get lost! Your classmates will almost all have identical calculators!

3. Use it!

Remember that in paper 2 (Standard Level or Higher Level) and both papers (Studies) you are going to need your calculator to tackle many of the questions. Donโ€™t try to do long-winded calculations by hand when there is a quick method using the GDC. You wonโ€™t get extra credit and you increase your chance of making mistakes. You need a different way of thinking when tackling calculator questions. All good IB textbooks identify whether a question is intended for the calculator or not.

4. Grasp the WINDOW

First and foremost, the graphing calculator can solve equations and inequalities graphically for you. But like drawing any graph, you need to tell the calculator the range of values for the x and y axes. This is called the WINDOW. If you donโ€™t get the WINDOW right, you wonโ€™t see any curve on your display.

5. Use the ZOOM function

All GDCs have this function to zoom in and out of regions of interest on the graph you have plotted.

6. Set the mode

For calculations involving angles (sine, cosine, etc.), you need to know if you are working in degrees or radians. Make sure you know how to change the mode of the calculator. In IB, you are usually working in radians. The sine of ten degrees is not the same as the sine of ten radians.

7. Harness the power of the GDC

The statistical functions on the calculator are very powerful. Make sure that you learn how to enter data sets, display a scatter diagram and work out mean, median and other common statistical functions.

8. Understand the various operations

For example, donโ€™t confuse the โ€˜subtractโ€™ and the โ€˜minusโ€™ operations. These are distinct and not accessed by the same key. Minus for entering a negative number. Subtract for taking away.

9. Beware of raising a negative number to a power

Put all negative numbers in brackets first if they are to be raised to a power. Try both ways โ€“ you will see what I mean.

10. Practise makes perfect

Donโ€™t leave learning the calculator skills to the last minute.

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People I admire

Some of these people are known for their academic brilliance and others for their bravery in standing up for what they believe is right. There is no particular order to the list that I present below. It is just as it comes.

Paul Erdล‘s was a brilliant and eccentric Hungarian mathematician who had a long and productive career. He spent most of his adult life living out of a suitcase and worked ceaselessly. He was truly single-minded in his devotion to the subject.

Paul Erdรถs Paul Erdรถs

Albert Einstein revolutionised physics in the twentieth century. He was responsible for shaking up its foundations and introducing relativity. The idea behind special relativity theory is very simple. The laws of physics must be the same as seen by any observer in the universe.  He was also a respected social commentator. He believed in compromise as the best approach to any problem.

Albert Einstein Albert Einstein

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