IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides
Are you a current IB student or IB teacher? Do you want to learn the tips and tricks to produce excellent Mathematics coursework? Gain the inside track on what makes a good coursework piece from an IB Maths Examiner as you learn all the skills necessary to produce something outstanding. This course is written for... Continue Reading →
Teachers can also find 3000 pages of pdf content for IB Maths at my new site: http://www.intermathematics.com. This includes hundreds of worksheets with full worked solutions, treasures hunts, tests and more. I've tried to build connections with real life maths, Theory of Knowledge (ToK) and ideas for maths careers. There are also maths videos, puzzles and... Continue Reading →
Cowculus - the farmer and the cow The Numberphile video linked the end of this is an excellent starting point for an investigation - so I thought I'd use this to extend the problem to a more general situation. The simple case is as follows: A farmer is at point F and a cow at... Continue Reading →
Roller Coaster design This post continues from the previous post on Lissajous Curves. Make sure to read that one first! We can design a rollercoaster track by using the following Lissajous Curve: This gives the following graph: Ground level is given by the line y = −50. Distances are in metres and t is measured... Continue Reading →
AI Masters Olympiad Geometry The team behind Google's Deep Mind have just released details of a new AI system: AlphaGeometry This has been specifically trained to solve classical geometry problems - and already is now at the level of a Gold Medalist at the International Olympiad (considering only geometry problems). This is an incredible achievement... Continue Reading →
Lissajous Curves Lissajous Curves were explored by French Physicist Jules Lissajous in the 1850s. The picture above (Wikimedia Commons) shows him investigating Lissajous curves through a telescope. Lissajous curves include those which can be written in the form: This parametric form allows us to represent complicated curves which are difficult to write in terms of... Continue Reading →
Using matrices to make fractals We start with a triangle ABC, with coordinates 𝐴(0,0) , 𝐵(1,0) , 𝐶( 0,1) as shown above. We can this triangle F_0 and we then write this as the following matrix: We then have the following algorithm to generate the next triangle F_1. In effect this means that the triangle... Continue Reading →
Chi Square: Language Recognition II I thought I would build on the last post by making a simple spreadsheet that can then easily show which language is being used. I chose the groupings of letters such that as long as there are at least 1000 letters in the text it will satisfy the Chi square... Continue Reading →
Google Page Rank: Billion dollar maths In the early 1990s search engines used to be text based – and would rank pages based on how many times a key word appeared. But this did not discriminate between useful pages and less useful pages. Larry Page and Segei Brin used some maths to come up with... Continue Reading →
Ladybirds vs Aphids At t=0 we have a ladybird on the edge of a leaf at point A(0,10) in cm, and an aphid at point B(0,10). The ladybird is in pursuit of the aphid. In each time interval of 1 second the ladybird travels 1cm by heading towards the aphid following the shortest straight-line path. ... Continue Reading →
https://www.youtube.com/watch?v=4dyytPboqvE This year's TOK question for Mathematics is the following: "How can we reconcile the opposing demands for specialization and generalization in the production of knowledge? Discuss with reference to mathematics and one other area of knowledge" This is a nice chance to discuss the Langlands program which was recently covered in a really excellent... Continue Reading →
https://www.youtube.com/watch?v=nlm07asSU0c Winning at Snakes and Ladders The fantastic Marcus de Sautoy has just made a video on how to use Markov chains to work out how long it will take to win at Snakes and Ladders. This uses a different method to those I've explored before (Playing Games with Markov Chains) so it's well worth... Continue Reading →
Chi Square: Language Detection + Code Breaking We can use the power of maths to allow computers to accurately recognise which language someone is writing in - even without needing to have understanding of any language at all. How? With the Chi Square goodness of fit test. Every language in the world has its own... Continue Reading →
Roll or bust? A strategy for dice games Let's explore some strategies for getting the best outcome for some dice games. Game 1: 1 dice, bust on 1. We roll 1 dice. However we can roll as many times as we like and add the score each time. We can choose to stop when we... Continue Reading →
Cooling Curves: Dead bodies and fridges All the maths behind this fits for cooling bodies - whether objects placed in fridges or dead bodies cooling over time - and this idea is used in CSI investigations to work out the time of death of bodies. I will do this investigation with a Microbit - which... Continue Reading →
https://www.youtube.com/watch?v=mhlc7peGlGg A brief summary of the Monty Hall problem. There are 3 doors. Behind 2 doors are goats and behind 1 door is a car. You choose a door at random. The host then opens another door to reveal a goat. Should you stick with your original choice or swap to the other unopened door?... Continue Reading →
Toads and snakes: an investigation! We have 2 populations: Toads who live inside a circle (a pond) and snakes which live inside a square (field). If the circle is completely surrounded by the square then no toads can live, and if the square is completely surrounded by the circle, no snakes can live. We want... Continue Reading →
Climate Change: Modelling Global Sea Ice Modelling the change of sea ice over time (global sea ice extent) is an important metric for understanding one of the (many) effects of climate change. This is a good example of how we can use some good quality secondary data, CSV files and Desmos to represent this data.... Continue Reading →
New IB teacher and IB student resources added I've just added a lot of new free content to support both students and teachers in the IB Mathematics course. This includes: Paper 3 Paper 3 resources: 13 full exploration questions with full markschemes. This is a selection of the Paper 3 investigations I’ve made over the... Continue Reading →
(Photograph: Photograph: WWL-TV, from The Guardian) Teenagers prove Pythagoras using Trigonometry The Guardian recently reported that 2 US teenagers discovered a new proof for Pythagoras using trigonometry. Whilst initial reports claimed incorrectly that this was the first time that Pythagoras had been proved by trigonometry, it is nevertheless an impressive achievement. I will go through... Continue Reading →
GPT-4 vs ChatGPT. The beginning of an intelligence revolution? The above graph (image source) is one of the most incredible bar charts you’ll ever see – this is measuring the capabilities of GPT4, Open AI’s new large language model with its previous iteration, ChatGPT. As we can see, GPT4 is now able to score in... Continue Reading →
https://www.youtube.com/watch?v=rHdYv62F5fs The Perfect Rugby Kick This was inspired by the ever excellent Numberphile video which looked at this problem from the perspective of Geogebra. I thought I would look at the algebra behind this. In rugby we have the situation that when a try is scored, there is an additional kick (conversion kick) which can... Continue Reading →
Creating a Neural Network: AI Machine Learning A neural network is a type of machine learning algorithm modeled after the structure and function of the human brain. It is composed of a large number of interconnected "neurons," which are organized into layers. These layers are responsible for processing and transforming the input data and passing... Continue Reading →
Can Artificial Intelligence (Chat GPT) Get a 7 on an SL Maths paper? ChatGPT is a large language model that was trained using machine learning techniques. One of the standout features of ChatGPT is its mathematical abilities. It can perform a variety of calculations and solve equations. This advanced capability is made possible by the... Continue Reading →
(Header image generated from here). ECDSA: Elliptic Curve Signatures This is the second post on this topic - following on from the first post here. Read that first for more of the maths behind this! In this post I'll look at this from a computational angle - and make a simple Python code to create... Continue Reading →
The mathematics behind blockchain, bitcoin and NFTs. If you've ever wondered about the maths underpinning cryptocurrencies and NFTs, then here I'm going to try and work through the basic idea behind the Elliptic Curve Digital Signature Algorithm (ECDSA). Once you understand this idea you can (in theory!) create your own digital currency or NFT -... Continue Reading →
Finding planes with radar PlusMaths recently did a nice post about the link between ellipses and radar (here), which inspired me to do my own mini investigation on this topic. We will work in 2D (with planes on the ground) for ease of calculations! A transmitter will send out signals - and if any of... Continue Reading →
Proving Pythagoras Like Einstein? There are many ways to prove Pythagoras' theorem - Einstein reputedly used the sketch above to prove this using similar triangles. To keep in the spirit of discovery I also just took this diagram as a starting point and tried to prove this myself, (though Einstein's version turns out to be... Continue Reading →
I've just made a big update to both the teacher and student resources sections: Student resources These now have some great free resources for students to help them with the IB maths course - including full course notes, formula books, Paper 3s, an Exploration guides and a great mind-map. Make sure to check these all... Continue Reading →
Finding the average distance in a polygon Over the previous couple of posts I've looked at the average distance in squares, rectangles and equilateral triangles. The logical extension to this is to consider a regular polygon with sides 1. Above is pictured a regular pentagon with sides 1 enclosed in a 2 by 2 square. ... Continue Reading →
Finding the average distance in an equilateral triangle In the previous post I looked at the average distance between 2 points in a rectangle. In this post I will investigate the average distance between 2 randomly chosen points in an equilateral triangle. Drawing a sketch. The first step is to start with an equilateral triangle... Continue Reading →
What is the average distance between 2 points in a rectangle? Say we have a rectangle, and choose any 2 random points within it. We then could calculate the distance between the 2 points. If we do this a large number of times, what would the average distance between the 2 points be? Monte Carlo... Continue Reading →
https://www.youtube.com/watch?v=tkC1HHuuk7c Plotting Pi and Searching for Mona Lisa This is a very nice video from Numberphile - where they use a string of numbers (pi) to write a quick Python Turtle code to create some nice graphical representations of pi. I thought I'd quickly go through the steps required for people to do this by... Continue Reading →
https://www.youtube.com/watch?v=_MscGSN5J6o&t=514s Witness Numbers: Finding Primes The Numberphile video above is an excellent introduction to primality tests - where we conduct a test to determine if a number is prime or not. Finding and understanding about prime numbers is an integral part of number theory. I'm going to go through some examples when we take the... Continue Reading →
Maths Games and Markov Chains This post carries on from the previous one on Markov chains - be sure to read that first if this is a new topic. The image above is of the Russian mathematician Andrey Markov [public domain picture from here] who was the first mathematician to work in this field (in... Continue Reading →
New Paper 3s for Applications! I've just finished making six Paper 3 practice papers for HL students sitting the Applications examination. The Paper 3 pack is 41 pages and includes over 180 marks of questions and full typed up markscheme. I've paid close attention to the IB's provided examples for the course to make sure... Continue Reading →
Life on the Beach with Markov Chains Markov chains are exceptionally useful tools for calculating probabilities - and are used in fields such as economics, biology, gambling, computing (such as Google's search algorithm), marketing and many more. They can be used when we have the probability of a future event dependent on a current event.... Continue Reading →
My colleague has just made a fantastic set of video tutorials for the SL Applications syllabus. So far he's made over 60 videos - with more to come! I really like the on screen use of the GDC and clear examples given. Very useful for Applications students. Playlist 1 (Number and Algebra) https://www.youtube.com/playlist?list=PL_lHi5M90vWqESNSvV0BL5-06gMOXrSLq Playlist 2... Continue Reading →
https://www.youtube.com/watch?v=WHeOrISYWDA Spotting fake data with Benford's Law In the current digital age it's never been easier to fake data - and so it's never been more important to have tools to detect data that has been faked. Benford's Law is an extremely useful way of testing data - because when people fake data they tend... Continue Reading →
Weaving a Spider Web II: Catching mosquitoes First I thought I would have another go at making a spider web pattern - this time using Geogebra. I'm going to use polar coordinates and the idea of complex numbers to help this time. Parametrically I will define my dots on my web by: Here r will... Continue Reading →
Weaving a Spider Web I often see some beautiful spider webs near my house, similar to the one pictured above (picture from here). They clearly have some sort of mathematical structure, so I decided to have a quick go at creating my own. Looking at the picture above there are 2 main parts, an inner... Continue Reading →
Elliptical Curve Cryptography Elliptical curves are a very important new area of mathematics which have been greatly explored over the past few decades. They have shown tremendous potential as a tool for solving complicated number problems and also for use in cryptography. Andrew Wiles, who solved one of the most famous maths problems of the... Continue Reading →
Prime Spirals - Patterns in Primes One of the fundamental goals of pure mathematicians is gaining a deeper understanding of the distribution of prime numbers - hence why the Riemann Hypothesis is one of the great unsolved problems in number theory and has a $1 million prize for anyone who can solve it. Prime numbers... Continue Reading →
Anscombe's Quartet - the importance of graphs! Anscombe's Quartet was devised by the statistician Francis Anscombe to illustrate how important it was to not just rely on statistical measures when analyzing data. To do this he created 4 data sets which would produce nearly identical statistical measures. The scatter graphs above generated by the Python... Continue Reading →
Hailstone Numbers Hailstone numbers are created by the following rules: if n is even: divide by 2 if n is odd: times by 3 and add 1 We can then generate a sequence from any starting number. For example, starting with 10: 10, 5, 16, 8, 4, 2, 1, 4, 2, 1... we can see... Continue Reading →
Chaos and strange Attractors: Henon's map Henon's map was created in the 1970s to explore chaotic systems. The general form is created by the iterative formula: The classic case is when a = 1.4 and b = 0.3 i.e: To see how points are generated, let's choose a point near the origin. If we take... Continue Reading →
The Barnsley Fern: Mathematical Art This pattern of a fern pictured above was generated by a simple iterative program designed by mathematician Michael Barnsely. I downloaded the Python code from the excellent Tutorialspoint and then modified it slightly to run on repl.it. What we are seeing is the result of 40,000 individual points - each plotted... Continue Reading →
Galileo's Inclined Planes This post is based on the maths and ideas of Hahn's Calculus in Context - which is probably the best mathematics book I've read in 20 years of studying and teaching mathematics. Highly recommended for both students and teachers! Hahn talks us though the mathematics, experiments and thought process of Galileo as... Continue Reading →
Finding focus with Archimedes This post is based on the maths and ideas of Hahn's Calculus in Context - which is probably the best mathematics book I've read in 20 years of studying and teaching mathematics. Highly recommended for both students and teachers! Hard as it is to imagine now, for most of the history... Continue Reading →
Finding the average distance between 2 points on a hypercube This is the natural extension from this previous post which looked at the average distance of 2 randomly chosen points in a square - this time let's explore the average distance in n dimensions. I'm going to investigate what dimensional hypercube is required to have an... Continue Reading →
https://www.youtube.com/watch?v=i4VqXRRXi68 Find the average distance between 2 points on a square This is another excellent mathematical puzzle from the MindYourDecisions youtube channel. I like to try these without looking at the answer - and then to see how far I get. This one is pretty difficult (and the actual solution exceptionally difficult!) The problem is... Continue Reading →