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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?

IB Revision

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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:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe 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!

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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.

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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.

Available to download here.

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.

Available to download here.

Modelling Guide


IB Exploration Modelling Guide 

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Modelling Guide includes:

Linear regression and log linearization, quadratic regression and cubic regression, exponential and trigonometric regression, comprehensive technology guide for using Desmos and Tracker.

Available to download here.

Statistics Guide

IB Exploration Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Statistics Guide includes: Pearson’s Product investigation, Chi Squared investigation, Binomial distribution investigation, t-test investigation, sampling techniques, normal distribution investigation and how to effectively use Desmos to represent data.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 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 and Applications students!

Available to download here.

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