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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

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.)

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.

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?

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.

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.

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.

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.

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)

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?

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number 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 ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

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:

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number 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 ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

This is a really interesting take on a very well known puzzle (courtesy of Ian Stewart’s Cabinet of Mathematical Curiosities).

The puzzle itself is pretty famous:

A farmer wants to cross a river and take with him a wolf, a goat, and a cabbage. There is a boat that can fit himself plus either the wolf, the goat, or the cabbage. If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage. How can the farmer bring the wolf, the goat, and the cabbage across the river?

And the standard way of solving it is trial and error with some logic thrown in.  However, as Ian Stewart points out, we can actually utilise 3 dimensional geometry to solve the puzzle.  We start with a 3D wolf-goat-cabbage (w,g,c) space (shown in the diagram).  All 3 start at (0,0,0).  0 represents this side of the bank, and 1 represents the far side of the bank.  The target is to get therefore to (1,1,1).  In (w,g,c) space , the x direction represents the wolf’s movements, the y direction the goat and z the cabbage.  Therefore the 8 possible triplet combinations are represented by the 8 vertices on a cube.

We can now cross out the 4 paths:

(0,0,0) to (1,00) as this leaves the goat with the cabbages

(0,0,0) to (0,0,1) as this leaves the wolf with the goat

(0,1,1) to (1,1,1) as the farmer would leave the goat and cabbage alone

(1,1,0) to (1,1,1) as the farmer would leave the wolf and goat alone.

which reduces the puzzle to a geometric problem – where we travel along the remaining edges – and the 2 solutions are immediately evident.

(eg.  (0,0,0) – (0,1,0) – (1,1,0) – (1,0,0) – (1,0,1)- (1,1,1)   )

What’s really nice about this solution is that it shows how problems seemingly unrelated to mathematics can be “translated” in mathematics – and also it shows how geometrical space can be used for problem solving.

For a more complicated problem (that can’t be solved without going into 8 dimensions geometrically – so don’t use this method!) try this puzzle. It has a nice animation which allows you to do the problem online.  The problem is as follows:
Find a way to move this group of people across the river. Only 2 persons on the raft at a time. The father cannot stay with any of the daughters without their mother’s presence. The mother cannot stay with any of the sons without their father’s presence.  The thief (striped shirt) cannot stay with any family member if the Policeman is not there.  Only the Father, Mother and the Policeman know how to operate the raft.

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number 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 ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

This is a really nice puzzle we looked at at the IB HL workshop:

When x = 1, y = 1, when x = 2, y = 2, when x = 3, y = 3 but when x = 4, y does not equal 4. Find a sequence which describes these points.

There are an infinite number of answers, though not necessarily easy to find!

If you are interested in the solutions, the answer is written below in white text, highlight to reveal!

Two possible ways of tackling the problem –

1) as a polynomial – if y = (x-1)(x-2)(x-3) + x this satisfies the original question – as the brackets all cancel to zero for 1,2,3 but will remain for x = 4 onwards.

2) modelling as a function with absolute value. Notice that -(abs x ) will satisfy the correct shape – ie a linear increase and then divergence from this. By transformations therefore we can get -(abs(x-3) ) +3. This fits the graph for y=x for 1,2,3 before altering for y=4.

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### IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

1. Investigation essentials,
2. Marking criteria guidance,
3. 70 hand picked interesting topics
4. Useful websites for use in the exploration,
5. A student checklist for top marks
6. Avoiding common student mistakes
7. A selection of detailed exploration ideas
8. Advice on using Geogebra, Desmos and Tracker.

### IB Revision Notes

IB Revision Notes

Full revision notes for both SL Analysis (60 pages) and HL Analysis (112 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis students!

### IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

### IB Exploration Modelling and Statistics Guide

IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

1. Pearson’s Product: Height and arm span
2. How to calculate standard deviation by hand
3. Binomial investigation: ESP powers
4. Paired t tests and 2 sample t tests: Reaction times
5. Chi Squared: Efficiency of vaccines
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7. Linear regression and log linearization.
8. Quadratic regression and cubic regression.
9. Exponential and trigonometric regression.