IB Maths Resources from Intermathematics
IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides
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 →
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 →
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 →
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 →
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 →
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 →
(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 →
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 →
Volume optimization of a cuboid This is an extension of the Nrich task which is currently live - where students have to find the maximum volume of a cuboid formed by cutting squares of size x from each corner of a 20 x 20 piece of paper. I'm going to use an n x 10 rectangle... Continue Reading →
Projective Geometry Geometry is a discipline which has long been subject to mathematical fashions of the ages. In classical Greece, Euclid’s elements (Euclid pictured above) with their logical axiomatic base established the subject as the pinnacle on the “great mountain of Truth” that all other disciplines could but hope to scale. However the status of... Continue Reading →
Modeling hours of daylight Desmos has a nice student activity (on teacher.desmos.com) modeling the number of hours of daylight in Florida versus Alaska - which both produce a nice sine curve when plotted on a graph. So let's see if this relationship also holds between Phuket and Manchester. First we can find the daylight hours... Continue Reading →
Cartoon from here The Gini Coefficient - Measuring Inequality The Gini coefficient is a value ranging from 0 to 1 which measures inequality. 0 represents perfect equality - i.e everyone in a population has exactly the same wealth. 1 represents complete inequality - i.e 1 person has all the wealth and everyone else has nothing.... Continue Reading →
Give your university applications a headstart on other students with Coursera. Applying for university as an international student is incredibly competitive - for the top universities you'll be competing with the best students from around the world, and so giving yourself a competitive advantage to make your university application stand out is really important. ... Continue Reading →
This post is inspired by the Quora thread on interesting functions to plot. The butterfly This is a slightly simpler version of the butterfly curve which is plotted using polar coordinates on Desmos as: Polar coordinates are an alternative way of plotting functions - and are explored a little in HL Maths when looking at... Continue Reading →
Log Graphs to Plot Planetary Patterns This post is inspired by the excellent Professor Stewart's latest book, Calculating the Cosmos. In it he looks at some of the mathematics behind our astronomical knowledge. Astronomical investigations In the late 1760s and early 1770s, 2 astronomers Titius and Bode both noticed something quite strange - there seemed... Continue Reading →
This is a quick example of how using Tracker software can generate a nice physics-related exploration. I took a spring, and attached it to a stand with a weight hanging from the end. I then took a video of the movement of the spring, and then uploaded this to Tracker. Height against time The first... Continue Reading →
Finger Ratio Predicts Maths Ability? Some of the studies on the 2D: 4D finger ratios (as measured in the picture above) are interesting when considering what factors possibly affect mathematical ability. A 2007 study by Mark Brosnan from the University of Bath found that: "Boys with the longest ring fingers relative to their index fingers... Continue Reading →
Amanda Knox and Bad Maths in Courts This post is inspired by the recent BBC News article, "Amanda Knox and Bad Maths in Courts." The article highlights the importance of good mathematical understanding when handling probabilities - and how mistakes by judges and juries can sometimes lead to miscarriages of justice. A scenario to give to... Continue Reading →
Does it Pay to be Nice? Game Theory and Evolution Game theory is an interesting branch of mathematics with links across a large number of disciplines - from politics to economics to biology and psychology. The most well known example is that of the Prisoner's Dilemma. (Illustrated below). Two prisoners are taken into custody and... Continue Reading →
Graham's Number - literally big enough to collapse your head into a black hole Graham's Number is a number so big that it would literally collapse your head into a black hole were you fully able to comprehend it. And that's not hyperbole - the informational content of Graham's Number is so astronomically large that... Continue Reading →
http://www.youtube.com/watch?v=PCu_BNNI5x4 What is the sum of the infinite sequence 1, -1, 1, -1, 1.....? This is a really interesting puzzle to study - which fits very well when studying geometric series, proof and the history of maths. The two most intuitive answers are either that it has no sum or that it sums to zero. ... Continue Reading →
https://www.youtube.com/watch?v=lCjspXB5F4A The Mathematics of Crime and Terrorism The ever excellent Numberphile have just released a really interesting video looking at what mathematical models are used to predict terrorist attacks and crime. Whereas a Poisson distribution assumes that events that happen are completely independent, it is actually the case that one (say) burglary in a neighbourhood... Continue Reading →
This was the last question on the May 2016 Calculus option paper for IB HL. It's worth nearly a quarter of the entire marks - and is well off the syllabus in its difficulty. You could make a case for this being the most difficult IB HL question ever. As such it was a terrible exam... Continue Reading →
IB HL Calculus P3 May 2016: The Hardest IB Paper Ever? IB HL Paper 3 Calculus May 2016 was a very poor paper. It was unduly difficult and missed off huge chunks of the syllabus. You can see question 5 posted above. (I work through the solution to this in the next post). This is so... Continue Reading →
Alan Turing Cryptography Competition Manchester University are running their 5th Alan Turing Cryptography Competition this January. It's aimed at secondary and post 16 students. If you are in the UK and in year 11 or below you can register for the official prizes, for everyone else you can still register and see if you make... Continue Reading →
Crack the Beale Papers and find a $65 Million buried treasure? The story of a priceless buried treasure of gold, silver and jewels (worth around $65 million in today's money) began in January 1822. A stranger by the name of Thomas Beale walked into the Washington Hotel Virginia with a locked iron box, which he gave... Continue Reading →
IB HL Calculus Option Videos For those students studying the IB Maths Higher Level Calculus option, I've just finished putting together video playlists to cover the whole option syllabus. These include both videos teaching the course content and also worked past paper solutions. Hopefully this should make what is a demanding unit a little bit... Continue Reading →
International School Code Breaking Challenge We're now running a huge school code breaking competition. There are a total of 7 competitions to enter - each one with a number of codes to crack. Each time a code is cracked this gives the password to access the next clue. Students who break all codes will be... Continue Reading →
If you are a teacher then please also visit my new site: intermathematics.com. My new site has been designed specifically for teachers of mathematics at international schools. The content now includes over 2500 pages of pdf content for the entire SL/HL Analysis and SL/HL Applications syllabus Some of the content includes: Original pdf worksheets (with... Continue Reading →
Non Euclidean Geometry - Spherical Geometry This article follow on from Non Euclidean Geometry - An Introduction - read that first!Most geometers up until the 19th century had focused on trying to prove that Euclid's 5th (parallel) postulate was true. The underlying assumption was that Euclidean geometry was true and therefore the 5th postulate must... Continue Reading →
The Chinese Remainder Theorem is a method to solve the following puzzle, posed by Sun Zi around the 4th Century AD. What number has a remainder of 2 when divided by 3, a remainder of 3 when divided by 5 and a remainder of 2 when divided by 7? There are a couple of methods... Continue Reading →
Code Crackers, can You Find the Hidden Message? The picture above looks like a normal picture of Albert Einstein - one of the world's greatest ever mathematicians. However, it's concealing a rather surprising secret. Within the picture is a hidden message. This technique of hiding messages in plain sight is called Steganography. This is a... Continue Reading →
Game Theory and Tic Tac Toe The game of Noughts and Crosses or Tic Tac Toe is well known throughout the world and variants are thought to have been played over 2000 years ago in Rome. It's a very simple game - the first person to get 3 in a row wins. In fact it's... Continue Reading →
http://www.youtube.com/watch?v=CegSgiWNgAI Giving students an insight into mathematical savants and other mathematical geniuses is a good way of invoking a sense of wonder about the subject. Calendar savants are able to correctly name the day of the week from any given date in history - almost instantly. Whilst it is still not clear how they do... Continue Reading →