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population graphBlack Swans and Civilisation Collapse

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.

Malthusian Catastrophe

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.

malthus

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?

Complex Civilisations

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

black swan

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.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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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.

prime array 4

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.

prime

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)

prime array 3

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:

prime array

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.

primae array 2The 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).

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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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.

syathesia3

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:

synathesia2

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):

syanthesia4

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.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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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.

manifold

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.

tesseract2

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:

tesseract

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.

inversesquare

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.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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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.

leonard

e’s are good – He’s Leonard Euler.

Having recently starting a topic on the exponential function, I was really struggling to find some good resources online – which is pretty surprising given that e is one of the most important and useful numbers in mathematics. So, here are some possible approaches.

1) e memorisation challenge.

This is always surprisingly popular – and a great starter which reinforces both that e is infinite and also that it’s just a number – so shouldn’t be treated like other letters when it comes to calculus.

5 minutes: How many digits of e can students remember?

Recital at the front. You can make this easier by showing them that  2. 7 1828 1828 45 90 45  they only need to remember 2.7 and then that 1828 repeated, followed by the angles in a triangle – 45, 90, 45.  Good students can get 20 places plus – and for real memory champions here are the first 1000 digits .

2) Introduction to Leonard Euler

Euler is not especially well known outside of mathematics, yet is undoubtedly one of the true great mathematicians. As well as e being named after him (Euler’s number), he published over 800 mathematical papers on everything from calculus to number theory to algebra and geometry.

30-40 minutes – The Seven Bridges of Königsberg
This is one of Euler’s famous problems – which he invented a whole new branch of mathematics (graph theory) to try and solve. Here is the problem:

The city of Königsberg used to have seven bridges across the river, linking the banks with two islands. The people living in Königsberg had a game where they would try to walk across each bridge once and only once. You can chose where to start – but you must cross each bridge only once:

euler

The above graphic is taken from the Maths is Fun resource on Euler’s bridge problem.  It’s a fantastically designed page – which takes students through their own exploration of how to solve similar problems (or as in the case of the 7 bridges problem, understanding why it has no solution).

3) Learning about e

30 minutes – why e ?

This is a good activity for students learning about differentiation for the first time.

First discuss exponential growth (example the chessboard and rice problem ) to demonstrate how rapidly numbers grow with exponential growth – ie. if I have one grain of rice on the first square, two on the second, how many will I have on the 64th square?

Next, students are given graph paper and need to sketch  y = 2^x y = e^x y = 3^x for between x = 0 and 3.  Students can see that y = e^x is between y=2^x and y = 3^x on the graph, so why is e so much more useful than these numbers?   By graphical methods they should find the gradient when the graphs cross the y axis.    Look at how the derivative of e^x is still e^x – which makes it really useful in calculus.   This is a nice short video which explains graphically why e was chosen to be 2.718…

4) The beauty of e.

10-30 minutes (depending on ability), discussion of some of the beautiful equations associated with e and Euler:

a) Euler Identity – frequently voted the most beautiful equation of all time by mathematicians, it links 5 of the most important constants in mathematics together into a single equation.

euler4

b) e as represented as a continued infinite fraction (can students spot the pattern? – the LHS is given by 2 then 1,2,1  1,4,1  1,6,1 etc.

e

c) e as the infinite sum of factorials:

e=\sum_{n=0}^\infty \frac{1}{n!}

d) e as the limit:

\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,

So, hopefully that should give some ideas for looking at this amazing number. (The post title will be lost on anyone not a teenager in England in the 1990s -to find out what you’re missing out on, here’s the song).

If you enjoyed this topic you may also like:

Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2? – a post which looks at the maths behind this particularly troublesome series.

A Maths Snooker Puzzle – a great little puzzle which tests logic skills.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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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.

snookerA Maths Snooker Puzzle

This was suggested by Paul our Physics teacher – and is a nice little maths puzzle.

The maximum break score in snooker is 147 which is achieved by:
15 reds (1 point each) , 15 blacks (7 points each), then yellow ( 2 points), then green (3 points), brown (4 points), blue (5 points), pink (6 points) and finally black (7 points).

Now, if you wanted the maximum break in snooker to be 180, but wanted all the balls to still have a distinct value and for black to remain the highest ball, how could you change the values of the balls to get a 180 top score? And can you prove there is only one answer?

Answer below in white text (highlight to reveal)

1) Firstly we can show that if red has to be 1.  For red-black combinations (2,8) (2,9) we can show that no solution is possible.  For (2,10) and above, and for (3,9) and above we reach 180 without the other colours.
2) The only options are (1,8) (1,9) or (1,10).  (1,8) leaves us needing 37 from 5 balls valued between 2-7 – which we can’t do.  (1,10) leaves us needing 5 from 5 balls – which we can’t do.  (1,9) is the only possible solution – and this requires 21 from 5 balls valued (2,3,4,5,6,7,8).  This can only be achieved with 2,3,4,5,7.

Like puzzles?  Then you might also enjoy some other brain teaser posts here.

Maths Invented or Discovered? 

The PBS Ideas Channel has just released a new video which discusses whether maths is invented by humans, or whether it is discovered (ie whether it can be said to really exist).  It’s an excellent 10 minute introduction to a pretty complicated topic – and certainly accessible for students:

For those interested in more detail – here are some of the basics (some of this information is simplified from the Stanford Encyclopedia which goes into far more detail).  You can also see more discussion of the topic on this site here 

philosophy maths

1) Platonism

The basic philosophical question in maths is ontological – ie concerned with existence.  The Platonic school (named after Greek philosopher Plato) hold that mathematical objects can themselves be said to exist.  Is there a “perfect circle” – in the realm of “ideas” upon which all circles on Earth are simply imitations?  Is this circle independent of human thought?  Does pi exist outside of human experience – and indeed space and time?  The hard Platonists argue that mathematical structures themselves are physically real – and indeed that our universe may be a mathematical structure.   Some other schools of mathematical philosphy include:

2) Logicism

Logicism seeks to reduce all of mathematics to logical thought – if all mathematics is reducible to logic does that mean that mathematics is purely an intellectual exercise?  20th Century efforts by Bertrand Russell and others to reduce mathematics to logical statements have not enjoyed much success.

3) Intuitionism:

“According to intuitionism, mathematics is essentially an activity of construction. The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction… Mathematical constructions are produced by the ideal mathematician, i.e., abstraction is made from contingent, physical limitations of the real life mathematician. But even the ideal mathematician remains a finite being. She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it.”  (Paragraph from Stanford).

Mathematics therefore does not really exist in any physical sense – it is merely a construction of the mind.

4) Fictionalism:

“Fictionalism holds that mathematical theories are like fiction stories such as fairy tales and novels. Mathematical theories describe fictional entities, in the same way that literary fiction describes fictional characters. This position was first articulated in the introductory chapter of (Field 1989), and has in recent years been gaining in popularity.”  (Paragraph from Stanford).

This line of thought tries to explain the amazing effectiveness of mathematics in describing the real world in a novel way – by denying that it does!  The reality that we think is being described by mathematics is nothing more than fiction – there is an underlying reality which we know nothing about.  Think about Nick Bostrom’s Computer Simulation argument – if we were within a computer simulation, then our mathematical laws may very well explain the computer code – but the real reality would be that which existed outside the computer.

Like this topic?  Then you might also enjoy:

Is God a Mathematician? – A Michio Kaku video which looks at how mathematics can be used to model the universe.

Simulations -Traffic Jams and Asteroid Impacts – An example of the power of mathematics in modelling the real world

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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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.

Which Times Tables do Students Find Difficult?

There’s an excellent article on today’s Guardian Datablog looking at a computer based study (with 232 primary school students) on which times tables students find easiest and difficult.  Edited highlights (Guardian quotes in italics):

Which multiplication did students get wrong most often?

times tables1

The hardest multiplication was six times eight, which students got wrong 63% of the time (about two times out of three). This was closely followed by 8×6, then 11×12, 12×8 and 8×12.

The graphic shows the questions that were answered correctly the greatest percentage of times as dark blue (eg 1×12 was answered 95% correctly).  The colours then change through lighter shades of blue, then from lighter reds to darker reds.  It’s interesting to see that the difficult multiplications cluster in the middle – perhaps due to how students anchor from either 5 or 10 – so numbers away from both these anchors are more difficult.

Which times table multiplication did students take the longest time to answer?

times tables2

Maybe unsurprisingly, 1×1 got answered the quickest (but perhaps illustrating the hazards of speed, pupils got it wrong about 10% of the time), at 2.4 seconds on average – while it was 12×9 which made them think for longest, at an average of 7.9 seconds apiece.

It’s quite interesting to see that this data is somewhat different to the previous graph.  You might have expected the most difficult multiplications to also take the longest time – however it looks as though some questions, whilst not intuitive can be worked out through mental methods (eg doing 12×9 by doing 12×10 then subtracting 12.)

How did boys and girls differ?

On average, boys got 32% of answers wrong, and took 4.2 seconds to answer each question.  Girls, by contrast, got substantially fewer wrong, at 22%, but took 4.6 seconds on average to answer.

Another interesting statistic – boys were more reckless and less considered with their answers!  The element of competition (ie. having to answer against a clock) may well have encouraged this attitude.  It would be interesting to see the gender breakdown to see whether boys and girls have any differences in which multiplication they find difficult.

Which times table was the hardest?

times tables 3

As you might expect, overall the 12 times table was found most difficult – closely followed by 8.   The numbers furthest away from 5 and 10 (7,8,12) are also the most difficult.  Is this down to how students are taught to calculate their tables – or because of the sequence patterns are less memorable?

This would be a really excellent investigation topic for IGCSE, IB Studies or IB SL.  It is something that would be relatively easy to collect data on in a school setting and then can provide a wealth of data to analyse.  The full data spreadsheet is also available to download on the Guardian page.

If you enjoyed this post you may also like:

Finger Ratio Predicts Maths Ability?– a maths investigation about finger ratio and mathematical skill.

Premier League Finances – Debt and Wages – an investigation into the finances of Premier League clubs.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

Screen Shot 2021-05-19 at 6.32.13 PM

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.

Wau: The Most Amazing Number in the World?

This is a fantastic video from Vi Hart of Khan Academy. Watch it first and marvel at the properties of this amazing number:

Once you have watched it, watch it again – this time thinking about what number Wau might be – and why you have never heard of Wau before.

This is a great video to show to students – especially IB students, who once they have figured it out can go through the video and see why the infinite sequences, the imaginary numbers, the repeated powers etc all work. There’s one mistake in the video – which is flagged by an annotation. Turn of the annotation and see if anyone in the class spots it.

As a final exercise how about thinking about what other fantastic properties Wau has?

This links really well with the ToK question about where do we get our knowledge from – does the authority of mathematics or science allow people to be mislead more easily? A good example (in a similar vein to Wau is the “health scare” about Dihydrogen Monoxide (DHMO).

dhmo

From the website DHMO:

Dihydrogen Monoxide (DHMO) is a colorless and odorless chemical compound. The atomic components of DHMO are found in a number of caustic, explosive and poisonous compounds such as Sulfuric Acid, Nitroglycerine and Ethyl Alcohol.

Each year, Dihydrogen Monoxide is a known causative component in many thousands of deaths and is a major contributor to millions upon millions of dollars in damage to property and the environment. Some of the known perils of Dihydrogen Monoxide are:
1) Death due to accidental inhalation of DHMO, even in small quantities.
2) Prolonged exposure to solid DHMO causes severe tissue damage.
3) Excessive ingestion produces a number of unpleasant though not typically life-threatening side-effects.
4) DHMO is a major component of acid rain.
5) Gaseous DHMO can cause severe burns.

Sounds pretty scary – and something that should be regulated. And indeed the website has been the cause of numerous petitions to MPs around the world demanding that it be banned. It is however an internet hoax. All the information is correct – it’s just that it refers to……water. People often have a deference to the authority of scientific or mathematical arguments – which can make them a very powerful tool in persuading people what to believe.

For anyone who wants to know what Wau is, the answer is below in white text (highlight to reveal!)

Wau is 1. Now watch the video again!

If you enjoyed this post you might also like:

Graham’s Number – literally big enough to collapse your head into a black hole – a post about an unimaginably big number.

e’s are good – He’s Leonard Euler. – A discussion about the amazing number e.

isbn

Cracking ISBN and Credit Card Codes

ISBN codes are used on all books published worldwide. It’s a very powerful and useful code, because it has been designed so that if you enter the wrong ISBN code the computer will immediately know – so that you don’t end up with the wrong book. There is lots of information stored in this number. The first numbers tell you which country published it, the next the identity of the publisher, then the book reference.

Here is how it works:

Look at the 10 digit ISBN number. The first digit is 1 so do 1×1. The second digit is 9 so do 2×9. The third digit is 3 so do 3×3. We do this all the way until 10×3. We then add all the totals together. If we have a proper ISBN number then we can divide this final number by 11. If we have made a mistake we can’t. This is a very important branch of coding called error detection and error correction. We can use it to still interpret codes even if there have been errors made.
If we do this for the barcode above we should get 286. 286/11 = 26 so we have a genuine barcode.

Check whether the following are ISBNs

1) 0-13165332-6
2) 0-1392-4191-4
3) 07-028761-4

Challenge (harder!) :The following ISBN code has a number missing, what is it?
1) 0-13-1?9139-9

Answers in white text at the bottom, highlight to reveal!

Credit cards use a different algorithm – but one based on the same principle – that if someone enters a digit incorrectly the computer can immediately know that this credit card does not exist.  This is obviously very important to prevent bank errors.  The method is a little more complicated than for the ISBN code and is given below from computing site Hacktrix:

creditcard2credit card 4creditcard3credit card 6credit card 5credit card 8credit card 7

credi card 9 You can download a worksheet for this method here. Try and use this algorithm to validate which of the following 3 numbers are genuine credit cards:

1) 5184 8204 5526 6425

2) 5184 8204 5526 6427

3) 5184 8204 5526 6424

Answers in white text at the bottom, highlight to reveal!


ISBN:
1) Yes
2) Yes
3) No
1) 3 – using x as the missing number we end up with 5x + 7 = 0 mod 11. So 5x = 4 mod 11. When x = 3 this is solved.
Credit Card: The second one is genuine

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NASA, Aliens and Binary Codes from the Stars – a discussion about how pictures can be transmitted across millions of miles using binary strings.

Cracking Codes Lesson – an example of 2 double period lessons on code breaking.

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

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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.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

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P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.

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A Super Exploration Guide with 168 pages of essential advice from a current IB examiner to ensure you get great marks on your coursework.

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