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jeopardy

One of my favourite resources is the Jeopardy quizzes. For those not familiar with the game (I think it’s American), it’s a gameshow, where you get to choose questions of different levels of difficulty, from a range of categories. I downloaded the template from TES – it’s a ready-made powerpoint which you can click on to take you to relevant questions, and then another click returns you to the home screen. The class can be split into teams, each team given a whiteboard and (say) 2 minutes to answer a question. Teams with the correct answer get those points for their teams. The challenge round adds a bit more excitement – and you can add any general questions or puzzles – such as dingbats or memory challenges (memorise pi to 10 places etc).

I’ve uploaded 20 of the quizzes onto TES here.  There’s 20 different ones –  KS3 Algebra, shape and space, fractions and general revision for different levels, GCSE and IGCSE topic specific quizzes on algebra, geometry, trigonometry, number, probability, matrices, functions, algebra 2 and linear graphs.  I have also done a small number of IB ones – which I’ll link to when I have a few more to share.


What is the sum of the infinite sequence 1, -1, 1, -1, 1…..?

This is a really interesting puzzle to study – which fits very well when studying geometric series, proof and the history of maths.

The two most intuitive answers are either that it has no sum or that it sums to zero.  If you group the pattern into pairs, then each pair (1, -1) = 0.  However if you group the pattern by first leaving the 1, then grouping pairs of (-1,1) you would end up with a sum of 1.

Firstly it’s worth seeing why we shouldn’t just use our formula for a geometric series:

with r as the multiplicative constant of -1.  This formula requires that the absolute value of r is less than 1 – otherwise the series will not converge.

The series 1,-1,1,-1…. is called Grandi’s series – after a 17th century Italian mathematician (pictured) – and sparked a few hundred years worth of heated mathematical debate as to what the correct summation was.

cesaro summation

Using the Cesaro method (explanation pasted from here )

If an = (−1)n+1 for n ≥ 1. That is, {an} is the sequence

1, -1, 1, -1, \ldots.\,

Then the sequence of partial sums {sn} is

1, 0, 1, 0, \ldots,\,

so whilst the series not converge, if we calculate the terms of the sequence {(s1 + … + sn)/n} we get:

\frac{1}{1}, \,\frac{1}{2}, \,\frac{2}{3}, \,\frac{2}{4}, \,\frac{3}{5}, \,\frac{3}{6}, \,\frac{4}{7}, \,\frac{4}{8}, \,\ldots,

so that

\lim_{n\to\infty} \frac{s_1 + \cdots + s_n}{n} = 1/2.

So, using different methods we have shown that this series “should” have a summation of 0 (grouping in pairs), or that it “should” have a sum of 1 (grouping in pairs after the first 1), or that it “should” have no sum as it simply oscillates, or that it “should”  have a Cesaro sum of 1/2 – no wonder it caused so much consternation amongst mathematicians!

This approach can be extended to the complex series, 1 + i + i^2 + i^3 + i^4 + i^5 + \cdots which is looked at in the blog  God Plays Dice

This is a really great example of how different proofs can sometimes lead to different (and unexpected) results. What does this say about the nature of proof?

Ib maths puzzles

There are a lot of good general sequence puzzles on the website Fibonicci.

For example find the next term of:

1) 15, 29, 56, 108, 208

2) 13, -21, 34, -55, 89

3) 52, 56, 48, 64, 32

4) 230, 460, 46, 92, 9.2

5) 68, 36, 20, 12, 8

Answers (in white text – highlight to reveal)

1) 400
2) -144
3) 96
4) 18.4
5) 6

supernova Milky Way

There are some great telescope pictures of the universe on the Guardian Science Gallery this month.  This picture shows the distorted remnants of a supernova explosion – where stars are destroyed.  This particular supernova is 26,000 light years from Earth – meaning that this is a picture 26,000 years into the past.  It is thought that supernova W49B has left a black hole rather than a neutron star.  (Neutron stars are so dense that 1 teaspoon of neutron star matter weighs 5,500,000,000,000 kg.)

big bang

This is the most detailed ever picture created of the aftermath of the birth of the universe.  The full size image is 50 million pixels and has been taken by the European Space Agency’s Planck space telescope.  The most recent estimate is that the universe is 13.82 billion years old – and this picture is a mere 380,000 after the Big Bang – in effect this picture is looking back in time by over 13.8 billion years.

galaxies collide

The ALMA telescope took this spectacular image of galaxies colliding.   ALMA stands for Atacama Large Millimeter Array.  It is the world’s biggest ground based astronomy project – comprised of 66 antenna in a circular pattern with 5 mile radius in Chile.  It has yet to be fully finished, but when it is scientists hope that it will bring ever more clarity and detail to the investigation of the birth of the universe.

So, what has this got to do with maths?  Well, astronomy is applied mathematics – using maths to understand some of the most fundamental questions of all – how did the universe begin and what is our place in it?

puzzles

This is another interesting maths sequence puzzle:

When x = 1, y = 1, when x= 2, y = -1, when x = 3, y = 1,

a) if when x = 4, y = -1, what formula gives the nth term? 

b) if when x = 4, y = 3, what formula gives the nth term?

Answer below in white text (highlight to see)

a) This is a nice puzzle when studying periodic graphs. Hopefully it should be clear that this is a periodic function – and so can be modelled with either sine or cosine graphs.

One possibility would be cos((n-1)pi)

b) This fits well when studying the absolute function – and transformations of graphs. Plotting the first 3 points, we can see they fit a transformed absolute value function – stretched by a factor of 2, and translated by (2,-1). So the function 2abs(x-2) -1 fits the points given.

philosophy maths

This is a huge topic – closely related to some of the Theory of knowledge concepts.  It also gets very complicated.  Here are some of the basics (some of this information is simplified from the Stanford Encyclopedia which goes into far more detail).

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. (“Was mathematics Invented or Discovered”)

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.

tes

This is the first post to just link to a TES contributor rather than a specific resource, but SRWhitehouse has provided a massive number of high quality worksheets on IB (and A level) topics – there’s a huge number of worksheets to choose from, everything from trig graphs to proof by induction to logs to binomial expansion, it’s all been done.  A massive help when it comes to generating additional non-textbook resources.

maths pictionary

This is a great resource from Mr Collins – Maths Pictionary. What I like about this is that it can be incorporated into a large number of classroom activities – from Jeopardy games, to starters to topic revision.  It can also be easily adapted to everything from KS3 to IB – and can be a great way of revising key vocabulary.

Click to download the file here (powerpoint)

Langton's Ant

This is another fascinating branch of mathematics – which uses computing to illustrate complexity (and order) in nature.  Langton’s Ant shows how very simple initial rules (ie a deterministic system) can have very unexpected consequences.  Langton’s Ant follows two simple rules:
1) At a white square, turn 90° right, flip the color of the square, move forward one unit
2) At a black square, turn 90° left, flip the color of the square, move forward one unit.

The ant exists on an infinite grid – and is able to travel N,S,E or W. You might expect the pattern generated to either appear completely random, or to replicate a fixed pattern. What actually happens is you have a chaotic pattern for around 10,000 iterations – and then all of a sudden a diagonal “highway” emerges – and then continues forever. In other words there is emergent behavior – order from chaos. What is even more remarkable is that you can populate the initial starting grid with any number of black squares – and you will still end up with the same emergent pattern of an infinitely repeating diagonal highway.

See a JAVA app demonstration (this uses a flat screen where exiting the end of one side allows you to return elsewhere – so this will ultimately lead to disruption of the highway pattern)

game of life

Such cellular automatons are a way of using computational power to try and replicate the natural world – The Game of Life is another well known automaton which starts of with very simple rules – designed to replicate (crudely) bacterial population growth. Small changes to the initial starting conditions result in wildly different outcomes – and once again you see patterns emerging from apparent random behavior. Such automatons can themselves be used as “computers” to calculate the solution to problems. One day could we design a computer program that replicates life itself? Could that then be said to be alive?

fermat's last theorem

Fermat’s Theorem – one of the most famous and long running puzzles in mathematics is a great way to introduce proof, the history of mathematics and also to show how apparent work on an entirely abstract concept can actually drive the development of techniques which have real world applicability.  The (much abridged!) story is that Fermat, a 17th century mathematician scribbled that he had a proof that the statement at the top of page was correct – ie. that whilst the equation holds when n = 2 (for pythagorian triples), for n greater than 2 there is no solution with positive whole numbers.  Mathematicians spent the next 358 years trying to find that same proof (and now believe that Fermat was actually mistaken), before Andrew Wiles finally proved it in 1995 using mathematics from elliptical curves.

A seemingly simple statement had occupied some of the best mathematical minds for over 3 centuries – everyone agreed that it probably was true – but to go from probably true, to 100% certainty – that knowledge gap required a monumental amount of effort.    How many other things can we genuinely say we are 100% certain of outside mathematics?

See the Horizon documentary on Fermat’s Last Theorem with Simon Singh and Andrew Wiles:

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