Category: mathematics

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Museums near London for kids who like science

For any kids into their science, the National Museum of Computing at Bletchley Park in Bedfordshire is a must. It has a replica of the Bombe built by Alan Turing during the war years to crack the Enigma Code. It also hosts a rebuilt Colossus, the first electronic computer. It also has a vast range of computer memory devices from car readers to one of the first disc drives. You can call and arrange a guided tour – which is recommended.

The National Maritime Museum has some of John Harrison’s prototype clocks used to solve the longitude problem. This is well worth a visit and is free. Harrison H1 blog.jpg

The Observatory Science Centre at Herstmonceux Castle, the former site of the Greenwich Observatory, hosts courses on astronomy, permanent exhibits on science themes and much more.

Dungeness B Visitor Centre is where you can arrange a visit to a nuclear reactor. What’s more, it is free.

Dungeness B Reactor

 

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Biographies of physicists worth reading

Paul Dirac

I recently read the book by Graham Farmelo about the British physicist Paul Dirac. It is called The Strangest Man. It is a well-written and touching look at the life and personality of the man who was at the heart of the development of Quantum Theory. He grew up with a dominant father who set oppressive rules for hm and his young brother to follow. The young Paul found ways to rebel silently. He won a place at a prestigious grammar school in Bristol, a city in the west of England.

” When you ask what are electrons and protons I ought to answer that this question is not a profitable one to ask and does not really have a meaning. The important thing about electrons and protons is not what they are but how they behave, how they move. I can describe the situation by comparing it to the game of chess. In chess, we have various chessmen, kings, knights, pawns and so on. If you ask what chessman is, the answer would be that it is a piece of wood, or a piece of ivory, or perhaps just a sign written on paper, or anything whatever. It does not matter. Each chessman has a characteristic way of moving and this is all that matters about it. The whole game os chess follows from this way of moving the various chessmen.” 

Paul Dirac

In the background of this engaging narrative is the unfolding of the greatest human achievement of the twentieth century. At the same time, it is a very human story of a man overcoming a domineering and bullying father to become perhaps the second greatest physicist of all time.

Richard Feynman

Genius: Richard Feynman and Modern Physics describes a character almost the complete opposite of Paul Dirac. Richard Feynman is an extrovert genius in the true sense of the word. Working in secret for years on the Manhattan Project as a young man must have been an amazing experience. Richard Feynman was relentless in his search for the truth and in debunking misconceptions. He revealed the cause of the Challenger disaster in 1986 in a public press conference using the simplest of demonstrations. He also wrote the famous Feynman Lectures, and was an advocate for physics education.

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Resources for A-level maths revision

Here are some of my preferred A-level maths revision websites.

http://m4ths.com/teach-yourself-a-level-maths.html

This site is run by retired maths teacher Steve Blades and contains links to a great deal of instructional videos that offer a lot of detailed help on all areas of the syllabus.

https://www.savemyexams.co.uk/a-level/

Save My Exams offers help on maths and the three sciences in not just A-level but pre-U, GCSE and IGCSE exams and is syllabus specific.

https://www.physicsandmathstutor.com/maths-revision/

This site offers syllabus specific support and has solution banks and past papers.

https://www.biochemtuition.com/

Despite its name, this site has extensive support as well as copies of text books (copyright alert!)

https://www.examsolutions.net/

This website is highly rated by my students. It has instructional videos on examination past papers for mathematics A-level. It covers Edexcel, AQA, OCR and MEI syllabuses.

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GCSE maths resources

Many students going back to school in year 10 and 11 quickly realise that they are not prepared for the demands of the course – especially in mathematics. It is a demanding subject and requires discipline and a lot of practice, practice and more practice. It is essential to get into good habits early. I have some resources to recommend as well as some tips.

Don’t neglect to work on your arithmetic, both mental and written. An inability to work quickly and carefully with the basic operations of arithmetic trip up many students. They get muddled and trip up before they even start. To help, I recommend doing online practice wit a website such as www.arithmetic.zetamac.com.

Review the fundamentals before starting to try examination questions. There is no point trying examination questions until you understand the fundamentals. You will find two kinds of questions in your text book and on websites. There are ‘consolidation’ or ‘confidence-building’ exercises and actual exam type questions. If you find that your school provided text book might lack exercises in some topics. In which case, you will need to supplement your revision with some good online resources. Three of my favourite resources are below.

GCSE Maths takeaway

GCSE Higher Level Worksheets

Just Maths

Poor understanding of maths vocabulary can hinder learning. I recommend reviewing GCSE maths vocabulary as a part of revision.

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Ti-nspire graphing calculator resources

The Texas Instrument Ti nSpire is an advanced graphical display calculator able to do complex algebra.

I use the Texas Instrument Ti-nspire pictures, which is a computer algebra system graphing calculator. Some great resources for this calculator are listed below.

http://www.johnhanna.us/TI-nspire.htm

https://learnit.hoonuit.com/2410

 

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

Exam Solutions 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 is IB Physics Notes. It is very detailed and offers detailed revision material on all parts of the syllabus broken down by topic.

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

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

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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 ‘prior learning topics’ 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 ‘The Language of the Universe’ 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 don’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 don’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 ‘exploration’ 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 ‘explorations’ 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 mathematician  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 for Fermat’s principle that explains how light travels and Fermat’s Last Theorem in number theory, which he described in a note at the margin of a copy of his book Diophantus‘ Arithmetica.

Fermat’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 – not a theorem but a conjecture – 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 Fermat’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 don’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, you’ve just been handed a brand new graphical display calculator (GDC) for your IB maths course. There’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. Don’t expect your teacher to show you all the features of the GDC

If you don’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 don’t need it, you don’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. Don’t try to do long-winded calculations by hand when there is a quick method using the GDC. You won’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 don’t get the WINDOW right, you won’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, don’t confuse the ‘subtract’ and the ‘minus’ 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 – you will see what I mean.

10. Practise makes perfect

Don’t leave learning the calculator skills to the last minute.