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I’ve just put together a playlist to help students studying for their IGCSE Cambridge 0580 or Cambridge 0607 Maths exams.
Topics included are:
2. Circle Theorems and angles
6. Solving equations using graphs
8. Linear graphs and inequalities
10. Probability and Venn Diagrams
For each topic I have chosen a few past paper questions to talk through. Hopefully this should be useful for students sitting their exams in the coming weeks.
The Gambler’s Fallacy
The above video is an excellent introduction to the gambler’s fallacy. This is the misconception that prior outcomes will have an effect on subsequent independent events. The classic example for this is the gambler who watches a run of 9 blacks on a roulette wheel with only red and black, and rushes to place all his money on red. He is sure that red must come up – after all the probability of a run of 10 blacks in a row is 1/1024. However, because the prior outcomes have no influence on the next spin actually the probability remains at 1/2.
Maths is integral to all forms of gambling – the bookmakers and casino owners work out the Expected Value (EV) for every bet that a gambler makes. In a purely fair game where both outcome was equally likely (like tossing a coin) the EV would be 0. If you were betting on the toss of a coin, the over the long run you would expect to win nothing and lose nothing. On a game like roulette with 18 red, 18 black and 2 green, we can work out the EV as follows:
$1 x 18/38 represents our expected winnings
-$1 x 20/38 represents our expected losses.
Therefore the strategy of always betting $1 on red has an EV of -2/38. This means that on average we would expect to lose about 5% of our money every stake.
Expected value can be used by gamblers to work out which games are most balanced in their favour – and in games of skill like poker, top players will have positive EV from every hand. Blackjack players can achieve positive EV by counting cards (not allowed in casinos) – and so casino bosses will actually monitor the long term fortunes of players to see who may be using this technique.
Understanding expected value also helps maximise winnings. Say 2 people both enter the lottery – one chooses 1,2,3,4,5,6 and the other a randomly chosen combination. Both tickets have exactly the same probability of winning (about 1 in 14 million in the UK) – but both have very different EV. The randomly chosen combination will likely be the only such combination chosen – whereas a staggering 10,000 people choose 1,2,3,4,5,6 each week. So whilst both tickets are equally likely to win, the random combination still has an EV 10,000 times higher than the consecutive numbers.
Incidentally it’s worth watching Derren Brown (above). Filmed under controlled conditions with no camera trickery he is still able to toss a coin 10 times and get heads each time. The question is, how is this possible? The answer – that this short clip was taken from 9 hours of solid filming is quite illuminating about our susceptibility to be manipulated with probability and statistics. This particular technique is called data mining (where multiple trials are conducted and then only a small portion of those trials are honed in on to show patterns) and is an easy statistical manipulation of scientific and medical investigations.
If you liked this post you might also like:
Does it Pay to be Nice? Game Theory and Evolution. How understanding mathematics helps us understand human behaviour
Premier League Finances – Debt and Wages. An investigation into the finances of Premier League clubs.
The video above is a great example of “mathemagic” – magic through maths. Arthur Benjamin’s show at TED (using a mixture of mathematical tricks and savant like numerical ability) shows how numerical calculations can still produce a sense of awe and wonder.
Probably the best resource for “mathemagic” is the TES Word ebook from Stephen Froggatt. This contains over 25 different maths magic tricks with full explanations about how to use them in a classroom setting. As he says,
“Mathematics can be presented as a dry collection of rules and exercises (surely not!) or as a window through which can be seen explanations to many of the world’s mysteries. A magic trick provides the interest, and its explanation the demonstration of the power of mathematics to provide answers. Suddenly all that previous work on simplifying algebraic expressions comes into action when explaining why the Number You Thought Of had to be seven.”
Magic Square Magic
As an example of one of this tricks, he describes how to ask a student his house number – upon answering, “46”, he immediately draws the following grid:
It’s then left to the students to find the connection between this grid and 46. Each row adds to 46, as does each column, and both diagonals, and the 4 corners, and the 2×2 corner squares! The impressive nature of this trick is the speed it can be calculated – and how it can be done with any given numbers. The template needed is:
You simply need to substitute the the given value for N. If you chose to reveal the secret it would be interesting to see if students could work out how to create their own grids with different template numbers.
Lightening Fast Multiplication
Another example from the book include how to multiply by 11 with lightening speed:
Write a large number on the board (eg. 3143221609) and race to see who can multiply this by 11 first. The answer is 34575437699 and can be done in seconds. Simply start with the end digit (in this case 9) and write that down, then working from right to left the next digit is 0+9 = 9. The next digit is 6+0 = 6, the next digit is 1+6 = 7 etc. Each time you just add the consecutive terms of the original number. You finish by writing down the first term (in this case 3).
There are loads of other tricks in the free ebook to utilise. The, “think of a number tricks,” are great for algebra topics, the magic cards make use of binary arithmetic and there is mobius magic for shape and space discussions.
Jan Honnens (also on TES here) has formalised some of this content into an investigation format with some great leading questions for students to follow.
Mind Reading Magic
Another example of a very powerful maths trick – which is very easy to do is given here:
And if all that isn’t enough, there’s a fantastic 96 page ebook pdf also free – available for download from here– which contains a large number of card and number tricks which make use of numerical and algebraic rules.
The Gorilla in the Room and Other Great Maths Investigations
These topics are a great way to add interest to statistics and probability lessons at KS3 and KS4 level, and also a good example of investigations that IB students can conduct. They also have a nice link to ToK – how can we believe what we see or what we hear? To what extent should we trust our senses? And it shows the power of statistics and empirical testing in trying to understand what is externally real and what is our own version of reality.
For each one, have the students make a hypothesis (if possible without giving the endings away!), then collect some data as to how the students react. Then look at how the data could be collected in a larger scale experiment (or how the experiment could be modified).
The first one at the top of the page is the “Fa, Ba” test. This is a really curious experiment that shows that what we “hear” is actually often influenced by what we see.
The second one is the amazing colour changing card trick by Richard Wiseman. This is also a great way of showing how we often fail to see what is really in front of us:
The third video is even more impressive – though it doesn’t work on all students. You have to set this one up so that all students are really intently concentrating on the screen – perhaps a prize for the student who gets the answer correct? Also no talking! Students have to count basketball passes:
The last one is a good test of whether students are “right brain” or “left brain” dominant. They have to stare at a rotating woman – some students will see this going clockwise, others anticlockwise. Some will be able to switch between the 2 views. If they can’t (I initially could only see this going in an anti clockwise direction) near the end of the video it shows the woman rotating in a clockwise direction to help. Then rewinding the video to the start – and as if by magic she had changed direction.
If you liked this post you might also like:
Even Pigeons Can Do Maths A discussion about the ability of both chimps and pigeons to count
Finger Ratio Predicts Maths Ability? A post which discusses the correlation between the two.
Are You Living in a Computer Simulation?
This idea might be familiar to fans of The Matrix – and at first glance may seem somewhat unbelievable. However, Oxford University Professor Nick Bostrom makes an interesting case using both conditional probability and logic as to why it’s more likely than you might think.
The summary of Bostrom’s Computer Simulation argument is the following:
At least one of the following propositions is true: (1) the human species is very likely to go extinct before reaching a “posthuman” stage; (2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof); (3) we are almost certainly living in a computer simulation. It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor-simulations is false, unless we are currently living in a simulation.
The full paper where he makes his argument is available as a pdf here – and is well worth a read. Alternatively Bostrom makes this case in a detailed interview:
Taking the argument step by step, firstly when Bostrom talks about a “posthuman” stage he is referring to an advanced civilisation with significantly greater technological capabilities than we have at present. Such a civilisation would have the ability to run a computer simulation so accurate that it would be indistinguishable from “real life”.
This is a twist on the traditional “Brain in a Vat” thought experiment much loved by philosophers when trying to argue whether we be sure that anything exists outside our own subjective experience:
Based on the same logic, we have no way of genuinely knowing whether we are really “here” or whether we are nothing but a computer model designed to give the impression that we really exist. Interestingly, the possibility that our individual life, the world around us and indeed everything we know about the universe may be false means that we can never truly claim to have knowledge of anything.
I think that most optimists would think that civilisation has the potential to develop into a “posthuman” phase of advanced technology. Indeed, you only need to look at the phenomenal growth in computer power (see Moore’s Law) to have confidence that should we stick around long enough, we will have the computational power possible to run such simulations.
So if we optimistically accept that humans will reach a “posthuman” stage, then it’s even easier to accept the second proposition – that if an advanced civilisation is able to run such civilisations they will do. After all human nature is such that we tend to do things just because we can – and in any case running such ancestor simulations would potentially be very beneficial for real world modelling.
If we do accept both these premises, then this therefore leads to the argument that we are almost certainly living in a computer simulation. Why? Well, an advanced civilisation with the computational power to run ancestor simulations would likely run a large number of them – and if there is only one real world, then our experience of a world is likely to be one of these simulations.
As a ToK topic this is a fantastic introduction to epistemological questions about the limits of knowledge and questions of existence, and is a really good example of the power of logic and mathematics to reveal possibilities about the world outside our usual bounds of thinking.
If you enjoyed this post you might also like:
Imagining the 4th Dimension – How mathematics can help us explore the notion that there may be more than 3 spatial dimensions.
Is Maths Invented or Discovered? – A discussion about some of the basic philosophical questions that arise in mathematics.
A really interesting branch of mathematics is involved in making future predictions about how civilisation will evolve in the future – and indeed looking at how robust our civilisation is to external shocks. This is one area in which mathematical models do not have a good record as it is incredibly difficult to accurately assign probabilities and form policy recommendations for events in the future.
One of the most famous uses of mathematical models in this context was by Thomas Malthus in 1798. He noted that the means of food production were a fundamental limiting factor on population growth – and that if population growth continued beyond the means of food production that there would be (what is now termed) a “Malthusian catastrophe” of a rapid population crash.
As it turns out, agrarian productivity has been able to keep pace with the rapid population growth of the past 200 years.
Looking at the graph we can see that whilst it took approximately 120 years for the population to double from 1 billion to 2 billion, it only took 55 years to double again. It would be a nice exercise to try and see what equation fits this graph – and also look at the rate of change of population (is it now slowing down?) The three lines at the end of the graph are the three different UN predictions – high end, medium and low end estimate. There’s a pretty stark difference between high end and low end estimates by 2100 – between 16 billion and 6 billion! So what does that tell us about the accuracy of such predictions?
More recently academics like Joseph Tainter and Jared Diamond have popularised the notion of civilisations as vulnerable to collapse due to ever increasing complexity. In terms of robustness of civilisation one can look at an agrarian subsistence example. Agrarian subsistence is pretty robust against civilisation collapse – small self sufficient units may themselves be rather vulnerable to famines and droughts on an individual level, but as a society they are able to ride out most catastrophes intact.
The next level up from agrarian subsistence is a more organised collective – around a central authority which is able to (say) provide irrigation technology through a system of waterways. Immediately the complexity of society has increased, but the benefits of irrigation allow much more crops to be grown and thus the society can support a larger population. However, this complexity comes at a cost – society now is reliant on those irrigation channels – and any damage to them could be catastrophic to society as a whole.
To fast forward to today, we have now an incredibly complex society, far far removed from our agrarian past – and whilst that means we have an unimaginably better quality of life, it also means society is more vulnerable to collapse than ever before. To take the example of a Coronal Mass Ejection – in which massive solar discharges hit the Earth. The last large one to hit the Earth was in 1859 but did negligible damage as this was prior to the electrical age. Were the same event to happen today, it would cause huge damage – as we are reliant on electricity for everything from lighting to communication to refrigeration to water supplies. A week without electricity for an urban centre would mean no food, no water, no lighting, no communication and pretty much the entire breakdown of society.
That’s not to say that such an event will happen in our lifetimes – but it does raise an interesting question about intelligent life – if advanced civilisations continue to evolve and in the process grow more and more complex then is this a universal limiting factor on progress? Does ever increasing complexity leave civilisations so vulnerable to catastrophic events that their probabilities of surviving through them grow ever smaller?
Black Swan Events
One of the great challenges for mathematical modelling is therefore trying to assign probabilities for these “Black Swan” events. The term was coined by economist Nassim Taleb – and used to describe rare, low probability events which have very large consequences. If the probability of a very large scale asteroid impact is (say) estimated as 1-100,000 years – but were it to hit it is estimated to cause $35 trillion of damage (half the global GDP) then what is the rational response to such a threat? Dividing the numbers suggests that we should in such a scenario be spending $3.5billion every year on trying to address such an event – and yet which politician would justify such spending on an event that might not happen for another 100,000 years?
I suppose you would have to conclude therefore that our mathematical models are pretty poor at predicting future events, modelling population growth or dictating future and current policy. Which stands in stark contrast to their abilities in modelling the real world (minus the humans). Will this improve in the future, or are we destined to never really be able to predict the complex outcomes of a complex world?
If you enjoyed this post you might also like:
Asteroid Impact Simulation – which allows you to model the consequences of asteroid impacts on Earth.
Chaos Theory – an Unpredictable Universe? – which discusses the difficulties in mathematical modelling when small changes in initial states can have very large consequences.
Sierpinski Triangles and Spirolateral Investigation Lesson Plan
Leaning Objective: Students are introduced to some more complex ideas in mathematics (fractals, infinite perimeter, fractional dimensions), students explore the relationship between maths and art, students conduct an open ended invesigation into patterns and sequences.
Start the lesson with the Mandelbrot Zoom in the background:
Discussion about what this shows (fractal shapes which repeat infinitely). Discuss the coastline paradox – what is the perimeter of the coastline of the UK? Does it have one? What happens when we try and be more accurate with our measurements?
Show video introducing how to create fractals – looking at how to create Sierpinski Triangles and how the Koch snowflake has simultaneously a finite area and an infinite perimeter:
Students create their own Sierpinski Triangle, which can be generated quite easily (instructions here). Students need triangular paper which can also be printed from the link.
Introduction to Spirolaterals – these patterns were first discovered by investigations into fractal designs. Use the computer generator here to show how these shapes are generated. (Teacher notes on this investigation here).
Distribute the Spirolateral worksheet (download here). Students need squared paper and start to investigate different patterns and rules. Which initial starting rules lead to closed patterns? Which ones lead to infinitely repeating pattern? For an extension students could investigate turns of 45 degrees rather than 90 degrees.
If you enjoyed this lesson plan, you might also like:
Lesson Plan on Modelling Asteroid Impacts – a modelling lesson which demonstrates the power of mathematics in making real world predictions.
How Are Prime Numbers Distributed? Twin Primes Conjecture
Thanks to a great post on the Teaching Mathematics blog about getting students to conduct an open ended investigation on consecutive numbers, I tried this with my year 10s – with some really interesting results. My favourites were these conjectures:
1) In a set of any 10 consecutive numbers, there will be no more than 5 primes. (And the only set of 5 primes is 2,3,5,7,11)
2) There is only 1 example of 3 consecutive odd numbers all being primes – 3,5,7
(You can prove both in a relatively straightforward manner by considering that a span of 3 consecutive odd numbers will always contain a multiple of 3)
Twin Prime Conjecture
These are particularly interesting because the study of the distribution of prime numbers is very much a live mathematical topic that mathematicians still work on today. Indeed studying the distribution of primes and trying to prove the twin prime conjecture are important areas of research in number theory.
The twin prime conjecture is one of those nice mathematical problems (like Fermat’s Last Theorem) which is very easy to understand and explain:
It is conjectured that there are infinitely many twin primes – ie. pairs of prime numbers which are 2 away from each other. For example 3 and 5 are twin primes, as are 11 and 13. Whilst it is easy to state the problem it is very difficult to prove.
However, this year there has been a major breakthrough in the quest to answer this problem. Chinese mathematician Yitang Zhang has proved that there are infinitely many prime pairs with gap N for some N less than 70,000,000.
This may at first glance not seem very impressive – after all to prove the conjecture we need to prove there are infinitely many prime pairs with gap N = 2. 70,000,000 is a long way away! Nevertheless this mathematical method gives a building block for other mathematicians to tighten this bound. Already that bound has been reduced to N <60,744 and is being reduced almost daily.
Prime Number Distribution
Associated with research into twin primes is also a desire to understand the distribution of prime numbers. Wolfram have a nice demonstration showing the cumulative distribution of prime numbers (x axis shows total integers x100)
Indeed, if you choose at random an integer from the first N numbers, the probability that it is prime is approximately given by 1/ln(N).
We can see other patterns by looking at prime arrays:
This array is for the first 100 integers – counting from top left to right. Each black square represents a prime number. The array below shows the first 5000 integers. We can see that prime numbers start to “thin out” as the numbers get larger.
The desire to understand the distribution of the prime numbers is intimately tied up with the Riemann Hypothesis – which is one of the million dollar maths problems. Despite being conjectured by Bernhard Riemann over 150 years ago it has still to be proven and so remains one of the most important unanswered questions in pure mathematics.
For more reading on twin primes and Yitang Zhang’s discovery, there is a great (and detailed) article in Wired on this topic.
If you enjoyed this topic, you may also like:
A post on synesthesia about how some people see colours in their numbers.
A discussion about the Million Dollar Maths problems (which includes the Riemann Hypothesis).
Synesthesia – Do Your Numbers Have Colour?
Synesthesia is another topic which provides insights into how people perceive numbers – and how a synesthetic’s perception of the mathematical world is distinctly different to everyone else’s.
Those with synesthesia have a cross-wiring of brain activity between 2 of their senses – so for example they may hear sounds when they see images, sounds may invoke taste sensations, or numbers may be perceived as colours.
Daniel Tammet, an autistic savant with remarkable memory abilities (he can remember pi to 22 thousand places and learn a new language to fluency in one week). He also has number synesthesia which means that he “sees” numbers as each having their own distinct colour. This also allows him to multiply two numbers in his head almost instantaneously by “seeing” the two colours merge into a third one.
Dr Ramachandran (of phantom limb fame) has written a fascinating academic article looking at synesthesia – and estimates that as many as 1 in 200 people may have some form of it. A simple test of grapheme colour synesthesia (where people perceive numbers with colours) is the graphic below:
For people without synesthesia, locating the 2s from graphic on the left is a slow process, but for people with synesthesia, they can immediately see the 2s as standing out – like the graphic on the right. This test is easily able to distinguish that this type of synesthesia is real.
Those with grapheme synesthesia also report that the image below changes colour – depending on whether they look at the whole image (ie. a five) or concentrate on how it is made of smaller constituent parts (of threes):
What is truly remarkable about synesthesia is what it reveals about our brain’s innate capacity for mathematical calculations far beyond what average people can achieve. Francois Galton, the 19th Century polymath who first documented the condition (which he himself had) described how synesthetics often also experienced a tangible number line in their mind – that was not straight but curved and bent and in which some numbers were closer that others (an example is at the top of the page). This allowed him, and others like Temmet, to perform lightening fast mental calculations of unimaginable complexity. In the above video Daniel is able to divide 13 by 97 in a matter of seconds to over 30 decimal places.
Numberphile have also made a short video in which they interview a lady with synesthesia:
Could one day we all unlock this potential? And what does this condition tell us about whether numbers exist in any tangible sense? Do they exist in a more real sense for a grapheme synesthic than someone else?
If you enjoyed this topic you may also like:
Even Pigeons Can Do Maths – a discussion about the ability of both chimps and pigeons to count
Does finger ratio predict maths ability? – a post which discusses the correlation between the two.
Imagining the 4th Dimension
Imagining extra dimensions is a fantastic ToK topic – it is something which seems counter-intuitively false, something which we have no empirical evidence to support, and yet it is something which seems to fit the latest mathematical models on string theory (which requires 11 dimensions). Mathematical models have consistently been shown to be accurate in describing reality, but when they predict a reality that is outside our realm of experience then what should we believe? Our senses? Our intuition? Or the mathematical models?
Carl Sagan produced a great introduction to the idea of extra dimensions based on the Flatland novel. This imagines reality as experienced by two dimensional beings.
Mobius strips are a good gateway into the weird world of topology – as they are 2D shapes with only 1 side. There are some nice activities to do with Mobius strips – first take a pen and demonstrate that you can cover all of the strip without lifting the pen. Next, cut along the middle of the strip and see the resulting shape. Next start again with a new strip, but this time start cutting from nearer the edge (around 1/3 in). In both cases have students predict what they think will happen.
Next we can move onto the Hypercube (or Tesseract). We can see an Autograph demonstration of what the fourth dimensional cube looks like here.
The page allows you to model 1, then 2, then 3 dimensional traces – each time representing a higher dimensional cube.
It’s also possible to create a 3 dimensional representation of a Tesseract using cocktail sticks – you simply need to make 2 cubes, and then connect one vertex in each cube to the other as in the diagram below:
For a more involved discussion (it gets quite involved!) on imagining extra dimensions, this 10 minute cartoon takes us through how to imagine 10 dimensions.
It might also be worth touching on why mathematicians believe there might be 11 dimensions. Michio Kaku has a short video (with transcript) here and Brian Greene also has a number of good videos on the subject.
All of which brings us onto empirical testing – if a mathematical theory can not be empirically tested then does it differ from a belief? Well, interestingly this theory can be tested – by looking for potential violations to the gravitational inverse square law.
The current theory expects that the extra dimensions are themselves incredibly small – and as such we would only notice their effects on an incredibly small scale. The inverse square law which governs gravitational attraction between 2 objects would be violated on the microscopic level if there were extra dimensions – as the gravitational force would “leak out” into these other dimensions. Currently physicists are carrying out these tests – and as yet no violation of the inverse square law has been found, but such a discovery would be one of the greatest scientific discoveries in history.
Other topics with counter-intuitive arguments about reality based on mathematical models are Nick Bostrom’s Computer Simulation Hypothesis, the Hologram Universe Hypothesis and Everett’s Many Worlds quantum mechanics interpretation. I will blog more on these soon!
If you enjoyed this topic you may also like:
Wolf Goat Cabbage Space – a problem solved by 3d geometry.
Graham’s Number – a number literally big enough to collapse your head into a black hole.