Category: Physics

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My favourite resources for tutoring maths and physics

Here are a few of my preferred resources.

Desmos Graphing Calculator is a web-based calculator. It is very versatile and can plot functions in Cartesian, polar and parametric form. It is very intuitive. You can switch simply between degrees and radians and it is web based so there is no need to download any software.

 

 

 

 

Mr Barton GCSE Maths Takawayย is a GCSE site of course with a lot of resources arranged conveniently by topic. It is a free resource and very popular.

Exam Solutionsย offers video instruction (short video clips) which are tailored to specific syllabuses. For those students who really need to see it done in real time rather than learn from the book, this is excellent. Many of my students have said that they like this website.

I tutor the International Baccalaureate Diploma in maths and physics. An excellent WordPress site isย IB Physics Notes. It is very detailed and offers detailed revision material on all parts of the syllabus broken down by topic.

I also recommendย Hyperphysicsย because it offers a clickable concept map showing all topics within Physics. It’s great for revision too.

Also thoroughly recommended is the inspirational siteย Physics Footnotes. This offers a large range of explanatory video clips that bring to life the principles in Physics.

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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

Continue reading “People I admire”

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Teaching in Kuwait

I taught at a school in Kuwait for a year  from 2003. This was an interesting experience culturally. I had not lived in a Muslim country before. I taught mathematics up to A-level at Kuwait English School. This is one of the better international schools in the country. Some of my classes were difficult and it was hard work. We started the day at 7 am which was the time we had to be on site. Lessons started at 7.30 and finished by the early afternoon. This had its advantages as we could then have the afternoon free. To be honest I was often so tired that I just went home and slept.

I lived in a block about 10 miles from the school which was where the school rented some basic apartments for staff. I hired a car each term, which worked out surprisingly cheap. I think petrol worked out at about 10 US cents per litre. The driving in Kuwait is an experience not to be missed. The standard of driving is amongst the worst I have seen anywhere in the world. The main technique could be called ‘point and push’. It was very scary. There were horrendous accidents every day on the highway which ran south from Kuwait City.

CNV00094 A famous landmark in Kuwait City

The weekends were on Thursdays and Fridays. We called this the virtual weekend. Saturdays and Sundays were normal days. One weekend, we went out for a drive in the desert to see if we could find the so-called tank graveyard which was where the US military had dumped all the Iraqi tanks and trucks that it had attacked during the Iraqi retreat from Kuwait in the first Gulf war. There were hundreds of tanks in row upon row as well as armoured vehicles, unexploded ordnance and so on. It was blisteringly hot in the desert and I only had a rough idea of the location as we were following directions given to us by other teachers. But we found the site in the end.

CNV00101 View from my window – lovely

This huge amount of scrap metal rusting in the desert is worthless. The reason is that depleted uranium, a byproduct of uranium enrichment, is used in the armour piercing shells that the Americans used. The result is a lot of contaminated metal. It will lie there for many years to come.

At the top of the post, you see pictures that we took on that day. Unbeknown to us, there was a military base nearby and we were soon seen by some soldiers who came over and arrested us. We were questioned in broken English and our cameras were confiscated but not before I had removed my camera’s card. We had to report the next day to the army headquarters to collect these items. We were politely questioned by a well-spoken Kuwaiti officer and given tea.