IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides
FOBISIA Code Breaking Competition 2025 Back by popular demand – we’re going to run the FOBISIA Code Breaking Maths Challenge again. Last year we had close to 120 schools taking part with over 22,000 students completing at least one code level. We hope that this year’s event will be equally successful. The dates will be: Between... Continue Reading →
Teachers can also download hundreds of IB maths resources from my new site intermathematics.com. See below for a flavour of the type of content available! 1. Worksheets All worksheets are designed to take 40 mins to 1 hour and can be used very effectively as homework sheets or classroom activities. Worksheets are designed to hit... Continue Reading →
Pi appears again - a probability problem This is a nice maths problem which has a surprising result: If you take 2 randomly generated numbers between 0 and 1 (lets call them x and y), then divide y by x, what is the probability that the resultant number will round to an even number? The... Continue Reading →
Cardioids in Coffee Cups Numberphile have just done a nice video on how a cardioid shape is formed when a light is shone against the side of a mug. You can see this effect above (from the Numberphile video here). So, I decided to recreate this using Geogebra to get to understand some of the... Continue Reading →
Looking for a pre-Ice Age civilisation? An interesting thought experiment is to consider the rise of modern civilisation and to ask whether civilisations could have risen in the long distant past. Let's look at some graphs to see the plausibility of the climatic side of this. Firstly we can see on this graph (source here)... Continue Reading →
Solar Gravitational lens: Seeing alien planets Let’s say in the future we pick up a signal from Proxima Centauri b – an exoplanet which is orbiting the star Proxima Centauri around 4.2 light years away. It would be nice to jump in a spaceship to explore further – but even travelling at 61,500km/h (the speed... Continue Reading →
Plotting asteroids - will 2024 YR4 hit Earth? According to current estimates there is approximately a 1.6% chance of a collision between asteroid 2024 YR4 and Earth. This would not be an extinction level event - but could be enough to flatten a city. So I thought I'd work backwards from the calculated orbital paths... Continue Reading →
Buffon's needle: Calculating pi The following problem, first posed in the 1700s by the Comte de Buffon has a surprising solution which can be used to generate pi. This is a nice example of probability games which can generate mathematical results over repeated trials (the Monte Carlo method). Here is the original problem: "Suppose we... Continue Reading →
Lattice based cryptography With the growing possibility of quantum computing being able to crack RSA encryption (the encryption technique which currently secures most banking and digital communications), the search is underway to find quantum-computing proof encryption. One potential possibility is lattice based cryptography - so I will explore the basics of this below! Creating a... Continue Reading →
Maths and Evolutionary Biology Mathematics is often utilised across many fields - lets look at an example from biology, evolutionary biology and paleontology, in trying to understand the development of homo-sapiens. We can start with a large data set which gives us the data for mammal body mass and brain size in grams (downloaded from... Continue Reading →
https://www.youtube.com/watch?v=vF_-ob9vseM&t=920s A Cat and Mouse Game The Numberphile video above talks through an investigation in which a mouse is swimming in a pond, with a hungry cat prowling around the edge. The cat can't swim, but can run at a speed of 4m/s. The mouse can swim at a speed of 1m/s and can run... Continue Reading →
Aliquot sequence: An unsolved problem At school students get used to the idea that we know all the answers in mathematics - but the aliquot sequence is a simple example of an unsolved problem in mathematics. The code above (if run for long enough on a super-computer!) might be enough to disprove a conjecture about... Continue Reading →
Time dependent gravity exploration In our universe we have a gravitational constant - i.e gravity is not dependent on time. If gravity changed with respect to time then the gravitational force exerted by the Sun on Earth would lessen (or increase) over time with all other factors remaining the same. Interestingly time-dependent gravity was first... 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 →