IB Maths Resources from Intermathematics
IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides
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=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 →
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 →
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 →
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 →
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 →
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 →
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 →
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 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 →
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 →
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 →
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 →
Generating e through probability and hypercubes This is a really beautiful solution to an interesting probability problem posed by fellow IB teacher Daniel Hwang, for which I've outlined a method for solving suggested by Ferenc Beleznay. The problem is as follows: On average, how many random real numbers from 0 to 1 (inclusive) are required... Continue Reading →
Paper 3 investigations Introduction Below are a selection of the Paper 3 investigations I've made over the years. Many of these bridge between Paper 3 practice (exposure to novel or new mathematical ideas) and the Exploration coursework. All of these could be easily adapted to make some very interesting coursework submissions. If you are a... Continue Reading →
IB Maths Exploration Guides Below you can download some comprehensive exploration guides that I've written to help students get excellent marks on their IB maths coursework. These guides are suitable for both Analysis and also Applications students. Over the past several years I've written over 200 posts with exploration ideas and marked hundreds of IAs... Continue Reading →
https://www.youtube.com/watch?v=Ry3B9hJbBfk The Martingale system The Martingale system was first used in France in 1700s gambling halls and remains used today in some trading strategies. I'll look at some of the mathematical ideas behind this and why it has remained popular over several centuries despite having a long term expected return of zero. The scenario You... Continue Reading →
Projectiles IV: Time dependent gravity! This carries on our exploration of projectile motion - this time we will explore what happens if gravity is not fixed, but is instead a function of time. (This idea was suggested by and worked through by fellow IB teachers Daniel Hwang and Ferenc Beleznay). In our universe we... Continue Reading →
https://www.youtube.com/watch?v=1EkoaQJyrfk Classical Geometry Puzzle: Finding the Radius This is another look at a puzzle from Mind Your Decisions. The problem is to find the radius of the following circle: We are told that line AD and BC are perpendicular and the lengths of some parts of chords, but not much more! First I'll look at... Continue Reading →
The Mordell Equation [Fermat's proof] Let's have a look at a special case of the Mordell Equation, which looks at the difference between an integer cube and an integer square. In this case we want to find all the integers x,y such that the difference between the cube and the square gives 2. These sorts... Continue Reading →
https://www.youtube.com/watch?v=RI2p-e7dL3E Can you solve Oxford University's Interview Question? The excellent Youtube channel Mind Your Decisions is a gold mine for potential IB maths exploration topics. I'm going to follow through my own approach to problem posed in the video. The problem is to be able to trace the movement of the midpoint of a ladder... Continue Reading →
https://www.youtube.com/watch?v=ZOiF7ZlboXA Rational Approximations to Irrational Numbers This year two mathematicians (James Maynard and Dimitris Koukoulopoulos) managed to prove a long-standing Number Theory problem called the Duffin Schaeffer Conjecture. The problem is concerned with the ability to obtain rational approximations to irrational numbers. For example, a rational approximation to pi is 22/7. This gives 3.142857 and... Continue Reading →
When do 2 squares equal 2 cubes? Following on from the hollow square investigation this time I will investigate what numbers can be written as both the sum of 2 squares, 2 cubes and 2 powers of 4. i.e a2+b2 = c3+d3 = e4+f4. Geometrically we can think of this as trying to find an... Continue Reading →
Hollow Cubes investigation Hollow cubes like the picture above [reference] are an extension of the hollow squares investigation done previously. This time we can imagine a 3 dimensional stack of soldiers, and so try to work out which numbers of soldiers can be arranged into hollow cubes. Therefore what we need to find is what... 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 →
https://www.youtube.com/watch?v=q6L06pyt9CA Stacking cannonballs - solving maths with code Numberphile have recently done a video looking at the maths behind stacking cannonballs - so in this post I'll look at the code needed to solve this problem. Triangular based pyramid. A triangular based pyramid would have: 1 ball on the top layer 1 + 3 balls... 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 →
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 →
This is a nice example of using some maths to solve a puzzle from the mindyourdecisions youtube channel (screencaptures from the video). How to Avoid The Troll: A Puzzle In these situations it's best to look at the extreme case first so you get some idea of the problem. If you are feeling particularly pessimistic... Continue Reading →
The Telephone Numbers - Graph Theory The telephone numbers are the following sequence: 1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496... (where we start from n=0). This pattern describes the total number of ways which a telephone exchange with n telephones can place a connection between pairs of people. To illustrate this... Continue Reading →
Happy Numbers Happy numbers are defined by the rule that you start with any positive integer, square each of the digits then add them together. Now do the same with the new number. Happy numbers will eventually spiral down to a number of 1. Numbers that don't eventually reach 1 are called unhappy numbers. As... Continue Reading →
https://www.youtube.com/watch?v=0hlvhQZIOQw This carries on the previous investigation into Farey sequences, and is again based on the current Nrich task Ford Circles. Below are the Farey sequences for F2, F3 and F4. You can read about Farey sequences in the previous post. This time I'm going to explore the link between Farey sequences and circles. First... Continue Reading →
This is a mini investigation based on the current Nrich task Farey Sequences. As Nrich explains: I'm going to look at Farey sequences (though I won't worry about rearranging them in order of size). Here are some of the first Farey sequences. The missing fractions are all ones which simplify to a fraction already on... Continue Reading →
Circular Motion: Modelling a ferris wheel This is a nice simple example of how the Tracker software can be used to demonstrate the circular motion of a Ferris wheel. This is sometimes asked in IB maths exams - so it's nice to get a visual representation of what is happening. First I took a video... Continue Reading →
Project Euler: Coding to Solve Maths Problems Project Euler, named after one of the greatest mathematicians of all time, has been designed to bring together the twin disciplines of mathematics and coding. Computers are now become ever more integral in the field of mathematics - and now creative coding can be a method of solving... Continue Reading →
https://www.youtube.com/watch?v=KfAs2hztDtI Measuring the Distance to the Stars This is a very nice example of some very simple mathematics achieving something which for centuries appeared impossible - measuring the distance to the stars. Before we start we need a few definitions: 1 Astronomical Unit (AU) is the average distance from the Sun to the Earth. This... 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 →
A geometric proof for the Arithmetic and Geometric Mean There is more than one way to define the mean of a number. The arithmetic mean is the mean we learn at secondary school - for 2 numbers a and b it is: (a + b) /2. The geometric mean on the other hand is defined... 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 an example of how an investigation into area optimisation could progress. The problem is this: A farmer has 40m of fencing. What is the maximum area he can enclose? Case 1: The rectangle: Reflection - the rectangle turns out to be a square, with sides 10m by 10m. Therefore the area enclosed is... Continue Reading →