You are currently browsing the tag archive for the ‘teaching maths’ tag.

*IB Maths and GCSE Maths Resources for IB Maths explorations and investigations. Real life maths. Theory of Knowledge (ToK). Maths careers. Maths videos. Maths puzzles and Maths lesson resources.*

**IB Maths Resources from Intermathematics**

My name is Andrew Chambers – and I’ve put this site together over several years. There are a huge amount of resources to explore – especially for students doing their IAs and for students looking for revision videos.

**The Mathematics of Cons – Pyramid Selling**

Pyramid schemes are a very old con – but whilst illegal, still exist in various forms. Understanding the maths behind them therefore is a good way to avoid losing your savings!

The most basic version of the fraud starts with an individual making the following proposition, “pay me $1000 to join the club, all you then need to do is recruit 6 more people to the club (paying $1000 each) and you will have made a $5000 profit.”

There are lots of variations – and now that most people are aware of pyramid selling, now normally revolve around multi-level-marketing (MLM). These are often still pyramid schemes, but encourage participants to believe it is a genuine business by actually having a sales product which members have to sell. However the main focus of the business is still the same – taking money off people who then make their money back after having signed up a set number of new recruits.

The following graphic from Consumer Fraud Reporting is a clear mathematical demonstration why these frauds only end up enriching those at the top of the pyramid:

You can see that if the requirement was to recruit 8 new members, that by the 9th level you would need to have 1 billion people already signed up. Even with the need to recruit just 4 new members you still have rapid exponential growth which very quickly means you will run out of new potential members. For pyramid schemes it is only those in the first 3-4 levels (the white cells) that stand any real chance of making money – and these levels are usually filled by those in on the scam.

Ponzi schemes (like that run by Bernie Madoff) use a similar method. A conman takes money from investors promising (say) 10% annual returns. Lots of investors sign up. The conman then is able to use the lump sum investments to pay the 10% annual returns. This scam can last for years, with people thinking that they are getting a good rate of return, only to find out eventually that actually their lump sum investment has gone.

This is a good topic to look at with graphs (plotting exponential growth), interest rates, or exponential sequences – and shows why understanding maths is an important financial skill.

If you like this topic you might also like:

Benford’s Law – Using Maths to Catch Fraudsters – the surprising mathematical law that helps catch criminals.

Amanda Knox and Bad Maths in Courts – when misunderstanding mathematics can have huge consequences .

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

**NASA, Aliens and Binary Codes from the Star**

The Drake Equation was intended by astronomer Frank Drake to spark a dialogue about the odds of intelligent life on other planets. He was one of the founding members of SETI – the Search for Extra Terrestrial Intelligence – which has spent the past 50 years scanning the stars looking for signals that could be messages from other civilisations.

In the following video, Carl Sagan explains about the Drake Equation:

where:

*N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);
R* = the average number of star formation per year in our galaxy
fp = the fraction of those stars that have planets
ne = the average number of planets that can potentially support life per star that has planets
fl = the fraction of planets that could support life that actually develop life at some point
fi = the fraction of planets with life that actually go on to develop intelligent life (civilizations)
fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time for which such civilizations release detectable signals into space*

The desire to encode and decode messages is a very important branch of mathematics – with direct application to all digital communications – from mobile phones to TVs and the internet.

All data content can be encoded using binary strings. A very simple code could be to have 1 signify “black” and 0 to signify “white” – and then this could then be used to send a picture. Data strings can be sent which are the product of 2 primes – so that the recipient can know the dimensions of the rectangle in which to fill in the colours.

If this sounds complicated, an example from the excellent Maths Illuminated handout on codes:

If this mystery message was received from space, how could we interpret it? Well, we would start by noticing that it is 77 digits long – which is the product of 2 prime numbers, 7 and 11. Prime numbers are universal and so we would expect any advanced civilisation to know about their properties. This gives us either a 7×11 or 11×7 rectangular grid to fill in. By trying both possibilities we see that an 11×7 grid gives the message below.

More examples can be downloaded from the Maths Illuminated section on Primes (go to the facilitator pdf).

A puzzle to try:

“If the following message was received from outer space, what would we conjecture that the aliens sending it looked like?”

0011000 0011000 1111111 1011001 0011001 0111100 0100100 0100100 0100100 1100110

Hint: also 77 digits long.

This is an excellent example of the universality of mathematics in communicating across all languages and indeed species. Prime strings and binary represent an excellent means of communicating data that all advanced civilisations would easily understand.

Answer in white text below (highlight to read)

Arrange the code into a rectangular array – ie a 11 rows by 7 columns rectangle. The first 7 numbers represent the 7 boxes in the first row etc. A 0 represents white and 1 represents black. Filling in the boxes and we end up with an alien with 2 arms and 2 legs – though with one arm longer than the other!

If you enjoyed this post you may also like:

Cracking Codes Lesson – a double period lesson on using and breaking codes

Cracking ISBN and Credit Card Codes– the mathematics behind ISBN codes and credit card codes

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

A maths song sung by current flavour of the month One Direction – follow the lyrics to arrive at the total.

Some of the best maths songs are by Learning Upgrade, such as, videos on circle formulae, fractions, exponents, the quadratic formula and the one below, “Mean, Median and Mode”:

**Some other good maths songs:**

Westerville South High School in Ohio have made some great rap-based maths songs such as the trigonmentry song, Getting Triggy With It

The Calculus Rhapsody is a fantastic take on Queen’s famous song – and good for IB SL and HL students.

And James Blunt’s Triangle is good for a KS3 shape and space introduction.

**Does it Pay to be Nice? Game Theory and Evolution**

Game theory is an interesting branch of mathematics with links across a large number of disciplines – from politics to economics to biology and psychology. The most well known example is that of the Prisoner’s Dilemma. (Illustrated below). Two prisoners are taken into custody and held in separate rooms. During interrogation they are told that if they testify to everything (ie betray their partner) then they will go free and their partner will get 10 years. However, if they both testify they will both get 5 years, and if they both remain silent then they will both get 6 months in jail.

So, what is the optimum strategy for prisoner A? In this version he should testify – because whichever strategy his partner chooses this gives prisoner A the best possible outcome. Looking at it in reverse, if prisoner B testifies, then prisoner A would have been best testifying (gets 5 years rather than 10). If prisoner B remains silent, then prisoner A would have been best testifying (goes free rather than 6 months).

This brings in an interesting moral dilemma – ie. even if the prisoner and his partner are innocent they are is placed in a situation where it is in his best interest to testify against their partner – thus increasing the likelihood of an innocent man being sent to jail. This situation represents a form of plea bargaining – which is more common in America than Europe.

Part of the dilemma arises because if both men know that the optimum strategy is to testify, then they both end up with lengthy 5 year jail sentences. If only they can trust each other to be altruistic rather than selfish – and both remain silent, then they get away with only 6 months each. So does mathematics provide an amoral framework? i.e. in this case mathematically optimum strategies are not “nice,” but selfish.

Game theory became quite popular during the Cold War, as the matrix above represented the state of the nuclear stand-off. The threat of Mutually Assured Destruction (MAD) meant that neither the Americans or the Russians had any incentive to strike, because that would inevitably lead to a retaliatory strike – with catastrophic consequences. The above matrix uses negative infinity to represent the worst possible outcome, whilst both sides not striking leads to a positive pay off. Such a game has a very strong Nash Equilibrium – ie. there is no incentive to deviate from the non strike policy. Could the optimal maths strategy here be said to be responsible for saving the world?

Game theory can be extended to evolutionary biology – and is covered in Richard Dawkin’s The Selfish Gene in some detail. Basically whilst it is an optimum strategy to be selfish in a single round of the prisoner’s dilemma, any iterated games (ie repeated a number of times) actually tend towards a co-operative strategy. If someone is nasty to you on round one (ie by testifying) then you can punish them the next time. So with the threat of punishment, a mutually co-operative strategy is superior.

You can actually play the iterated Prisoner Dilemma game as an applet on the website Game Theory. Alternatively pairs within a class can play against each other.

An interesting extension is this applet, also on Game Theory, which models the evolution of 2 populations – residents and invaders. You can set different responses – and then see what happens to the respective populations. This is a good reflection of interactions in real life – where species can choose to live co-cooperatively, or to fight for the same resources.

The first stop for anyone interested in more information about Game Theory should be the Maths Illuminated website – which has an entire teacher unit on the subject – complete with different sections,a video and pdf documents. There’s also a great article on Plus Maths – Does it Pay to be Nice? all about this topic. There are a lot of different games which can be modeled using game theory – and many are listed here . These include the Stag Hunt, Hawk/ Dove and the Peace War game. Some of these have direct applicability to population dynamics, and to the geo-politics of war versus peace.

If you enjoyed this post you might also like:

Simulations -Traffic Jams and Asteroid Impacts

Langton’s Ant – Order out of Chaos

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

**Premier League Finances – Debt and Wages**

This is a great article from the Guardian DataBlog analysing the finances for last season’s Premier League clubs. As the Guardian says, “More than two thirds of the Premier League’s record £2.4bn income in 2011-12 was paid out in wages, according to the most recently published accounts of all 20 clubs. The Guardian’s annual special report of Premier League clubs’ finances shows they spent £1.6bn on wages last season, most of it going to players.”

The first graph (above) shows the net debt levels for different clubs.

The second graph shows the total turnover:

The third graph shows wages as a proportion of turnover:

and the last one is particularly interesting – as it ranks clubs on their wage bills and their league position. This would be an interesting piece of data to test for the strength of correlation:

I’ve used an online scatter plot to calculate both the regression line and the correlation coefficient:

Which clearly shows a strong positive correlation. This would be an interesting exercise for both IGCSE or IB students (especially Maths Studies).

For even more data, a club by club full breakdown is also provided by the Guardian here. I have also made the data above into a word document to be used as a some A4 posters – and you can download that here: Premier League Debt

If you enjoyed this post you might also like:

Which Times Tables do Students Find Difficult? An Investigation.

Why Study Maths? Careers Inspiration

Essential resources for IB students:

2) Exploration Guides and Paper 3 Resources

This is inspired by a fantastic website – we use math – which has a massive amount of information about different careers using mathematics in a really well laid out format.

According to a comprehensive careers survey by Careers Cast – which looked at over 200 different jobs and ranked them for stress, pay, job stability and work environment – 6 of the top 10 jobs require or strongly prefer maths graduates or those with a good mathematical background.

Meanwhile, a 2009 Survey by the National Association of Colleges and Employers – which looked at American students’ job prospects after leaving university found that – “The top 15 highest-earning college degrees all have one thing in common — math skills.”

“Math is at the crux of who gets paid,” said Ed Koc, director of research at NACE. “If you have those skills, you are an extremely valuable asset.”

From the we use math site there is information (with interviews, maths required, job skills etc) on about 50 maths related jobs – as well as some pretty impressive statistics about the usefulness of mathematics:

These are the A-E jobs included on the site:

There is also an accompanying video with interviews from a large range of mathematicians talking about the jobs they have gone into:

Along with the equally brilliant Maths Careers website there should be enough ammunition to never be stumped by the perennial, “When are we ever going to use this in* real life*?”

I’ve used the data above to make a poster for display – it’s downloaded from here: Want to be an astronaut?

Michio Kaku – American Professor of Theoretical Physics and fantastic populariser of mathematics and physics takes us through a 40 minute journey on the importance of physics in explaining the universe:

Great stuff – well worth watching!

**Graham’s Number – literally big enough to collapse your head into a black hole**

Graham’s Number is a number so big that it would *literally* collapse your head into a black hole were you fully able to comprehend it. And that’s not hyperbole – the informational content of Graham’s Number is so astronomically large that it exceeds the maximum amount of entropy that could be stored in a brain sized piece of space – i.e. a black hole would form prior to fully processing all the data content. This is a great introduction to notation for *really* big numbers. Numberphile have produced a fantastic video on the topic:

Graham’s Number makes use of Kuth’s up arrow notation (explanation from wikipedia:)

In the series of hyper-operations we have

1) Multiplication:

For example,

2) Exponentiation:

For example,

3) Tetration:

For example,

- etc.

4) Pentation:

and so on.

Examples:

Which clearly can lead to some absolutely huge numbers very quickly. Graham’s Number – which was arrived at mathematically as an upper bound for a problem relating to vertices on hypercubes is (explanation from Wikipedia)

where the number of *arrows* in each layer, starting at the top layer, is specified by the value of the next layer below it; that is,

and where a superscript on an up-arrow indicates how many arrows are there. In other words, *G* is calculated in 64 steps: the first step is to calculate *g*_{1} with four up-arrows between 3s; the second step is to calculate *g*_{2} with *g*_{1} up-arrows between 3s; the third step is to calculate *g*_{3} with *g*_{2} up-arrows between 3s; and so on, until finally calculating *G* = *g*_{64} with *g*_{63} up-arrows between 3s.

So a number so big it can’t be fully processed by the human brain. This raises some interesting questions about maths and knowledge – Graham’s Number is an example of a number that exists but is beyond full human comprehension – it therefore is an example of a upper bound of human knowledge. Therefore will there always be things in the Universe which are beyond full human understanding? Or can mathematics provide a shortcut to knowledge that would otherwise be inaccessible?

If you enjoyed this post you might also like:

How Are Prime Numbers Distributed? Twin Primes Conjecture – a discussion about the amazing world of prime numbers.

Wau: The Most Amazing Number in the World? – a post which looks at the amazing properties of Wau

Essential resources for IB students:

2) Exploration Guides and Paper 3 Resources

Plus Maths has a large number of great podcasts which look at maths ToK topics:

1) An interview with Max Tegmark (pictured above) about why he thinks that the universe is itself a mathematical structure.

2) An interview with physicists David Berman about how many dimensions exist.

3) A talk with cosmologist John Barrow about infinity.

4) A discussion with Roger Penrose about the puzzle of time.

And many more. Well worth a listen!

5) There’s also a good lecture by Professor Ray Monk on the University of Southampton page (see “useful downloads”) looking at the link between philosophy and mathematics – which takes a fascinating journey through the history of maths and the great ideas of great men.

6) Maths for Primates is a fantastic source of podcasts – 14 and counting, on fractals, Zeno, Hilbert’s hotel and more.