IB Maths Resources from Intermathematics
IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides
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 →
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 →
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 →
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 →
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 →
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 →
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 →
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 →
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 →
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 →
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 →
Have you got a Super Brain? Adapting and exploring maths challenge problems is an excellent way of finding ideas for IB maths explorations and extended essays. This problem is taken from the book: The first 25 years of the Superbrain challenges. I'm going to see how many different ways I can solve it. The problem... Continue Reading →
3D Printing with Desmos: Stewie Griffin Using Desmos or Geogebra to design a picture or pattern is quite a nice exploration topic - but here's an idea to make your investigation stand out from the crowd - how about converting your image to a 3D printed design? Step 1 Create an image on Desmos or... Continue Reading →
Projectile Motion Investigation II Another example for investigating projectile motion has been provided by fellow IB teacher Ferenc Beleznay. Here we fix the velocity and then vary the angle, then to plot the maximum points of the parabolas. He has created a Geogebra app to show this (shown above). The locus of these maximum points... Continue Reading →
Ramanujan's Taxi Cabs and the Sum of 2 Cubes The Indian mathematician Ramanujan (picture cite: Wikipedia) is renowned as one of great self-taught mathematical prodigies. His correspondence with the renowned mathematician G. H Hardy led him to being invited to study in England, though whilst there he fell sick. Visiting him in hospital, Hardy remarked that... Continue Reading →
Finding the volume of a rugby ball (prolate spheroid) With the rugby union World Cup currently underway I thought I'd try and work out the volume of a rugby ball using some calculus. This method works similarly for American football and Australian rules football. The approach is to consider the rugby ball as an... Continue Reading →
The Shoelace Algorithm to find areas of polygons This is a nice algorithm, formally known as Gauss's Area formula, which allows you to work out the area of any polygon as long as you know the Cartesian coordinates of the vertices. The case can be shown to work for all triangles, and then can be... Continue Reading →
https://www.youtube.com/watch?v=etMJxB-igrc The Van Eck Sequence This is a nice sequence as discussed in the Numberphile video above. There are only 2 rules: If you have not seen the number in the sequence before, add a 0 to the sequence. If you have seen the number in the sequence before, count how long since you... Continue Reading →