IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides
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 →
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 →
https://www.youtube.com/watch?v=4aMtJ-V26Z4 Narcissistic Numbers Narcissistic Numbers are defined as follows: An n digit number is narcissistic if the sum of its digits to the nth power equal the original number. For example with 2 digits, say I choose the number 36: 32 + 62 = 45. Therefore 36 is not a narcissistic number, as my answer... Continue Reading →
https://www.youtube.com/watch?v=fcfQkxwz4Oo Quantum Mechanics - Statistical Universe Quantum mechanics is the name for the mathematics that can describe physical systems on extremely small scales. When we deal with the macroscopic - i.e scales that we experience in our everyday physical world, then Newtonian mechanics works just fine. However on the microscopic level of particles, Newtonian mechanics... 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 →
Is Intergalactic space travel possible? The Andromeda Galaxy is around 2.5 million light years away - a distance so large that even with the speed of light at traveling as 300,000,000m/s it has taken 2.5 million years for that light to arrive. The question is, would it ever be possible for a journey to the... 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 →
http://www.youtube.com/watch?v=skM37PcZmWE Zeno's Paradox - Achilles and the Tortoise This is a very famous paradox from the Greek philosopher Zeno - who argued that a runner (Achilles) who constantly halved the distance between himself and a tortoise would never actually catch the tortoise. The video above explains the concept. There are two slightly different versions to... Continue Reading →
Fourier Transform The Fourier Transform and the associated Fourier series is one of the most important mathematical tools in physics. Physicist Lord Kelvin remarked in 1867: "Fourier's theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly... Continue Reading →
Non Euclidean Geometry - An Introduction It wouldn't be an exaggeration to describe the development of non-Euclidean geometry in the 19th Century as one of the most profound mathematical achievements of the last 2000 years. Ever since Euclid (c. 330-275BC) included in his geometrical proofs an assumption (postulate) about parallel lines, mathematicians had been trying... 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 →
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 →
Statistics to win penalty shoot-outs With the World Cup upon us again we can perhaps look forward to yet another heroic defeat on penalties by England. England are in fact the worst country of any of the major footballing nations at taking penalties, having won only 1 out of 7 shoot-outs at the Euros and... 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 →
Modelling more Chaos This post was inspired by Rachel Thomas' Nrich article on the same topic. I'll carry on the investigation suggested in the article. We're going to explore chaotic behavior - where small changes to initial conditions lead to widely different outcomes. Chaotic behavior is what makes modelling (say) weather patterns so complex. f(x)... 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 →
Modelling Chaos This post was inspired by Rachel Thomas' Nrich article on the same topic. I'll carry on the investigation suggested in the article. We're going to explore chaotic behavior - where small changes to initial conditions lead to widely different outcomes. Chaotic behavior is what makes modelling (say) weather patterns so complex. Let's start... Continue Reading →
Modelling tides: What is the effect of a full moon? Let's have a look at the effect of the moon on the tides in Phuket. The Phuket tide table above shows the height of the tide (meters) on given days in March, with the hours along the top. So if we choose March 1st (full... 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 →
The Folium of Descartes The folium of Descartes is a famous curve named after the French philosopher and mathematician Rene Descartes (pictured top right). As well as significant contributions to philosophy ("I think therefore I am") he was also the father of modern geometry through the development of the x,y coordinate system of plotting algebraic... 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 →
Spotting Asset Bubbles Asset bubbles are formed when a service, product or company becomes massively over-valued only to crash, taking with it most of its investors' money. There are many examples of asset bubbles in history - the Dutch tulip bulb mania and the South Sea bubble are two of the most famous historical examples.... 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 →
The Remarkable Dirac Delta Function This is a brief introduction to the Dirac Delta function - named after the legendary Nobel prize winning physicist Paul Dirac. Dirac was one of the founding fathers of the mathematics of quantum mechanics, and is widely regarded as one of the most influential physicists of the 20th Century. This... Continue Reading →
The Rise of Bitcoin Bitcoin is in the news again as it hits $10,000 a coin - the online crypto-currency has seen huge growth over the past 1 1/2 years, and there are now reports that hedge funds are now investing part of their portfolios in the currency. So let's have a look at... 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 →
Euler's 9 Point Circle This is a nice introduction to some of the beautiful constructions of geometry. This branch of mathematics goes in and out of favour - back in the days of Euclid, constructions using lines and circles were a cornerstone of mathematical proof, interest was later revived in the 1800s through Poncelot's projective... 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 →
Predicting the UK election using linear regression The above data is the latest opinion poll data from the Guardian. The UK will have (another) general election on June 8th. So can we use the current opinion poll data to predict the outcome? Longer term data trends Let's start by looking at the longer term trend... 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 →
Cracking ISBN and Credit Card Codes ISBN codes are used on all books published worldwide. It’s a very powerful and useful code, because it has been designed so that if you enter the wrong ISBN code the computer will immediately know – so that you don’t end up with the wrong book. There is lots... Continue Reading →
Modelling a Nuclear War With the current saber rattling from Donald Trump in the Korean peninsula and the instability of North Korea under Kim Jong Un (incidentally a former IB student!) the threat of nuclear war is once again in the headlines. Post Cold War we've got somewhat used to the peace afforded by the... Continue Reading →
I've just put together a couple of playlists to help students studying for their IGCSE Cambridge 0580 or Cambridge 0607 Maths exams, and also their Additional Mathematics Cambridge 0606 exams. IGCSE Maths Cambridge Playlist: https://www.youtube.com/watch?v=YwTgJrE0UiY&list=PLTzMaA6R8ce0vUIWWwP_-8_tmVK4q0SeR Videos included are: 1. Number 2. Circle Theorems and angles 3. Algebra 4. Volume 5. Statistics 6. Solving equations using... Continue Reading →
NASA, Aliens and Binary Codes from the Star The Drake Equation was intended by astronomer Frank Drake to spark a dialogue about the odds of intelligent life on other planets. He was one of the founding members of SETI - the Search for Extra Terrestrial Intelligence - which has spent the past 50 years scanning... Continue Reading →
Sequence Investigation This is a nice investigation idea from Nrich. The above screen capture is from their Picture Story puzzle. We have successive cubes - a 1x1x1 cube, a 2x2x2 cube etc. The cubes are then rearranged to give the following shape. The puzzle is then to use this information to discover a mathematical relationship.... Continue Reading →
http://www.youtube.com/watch?v=vIsDjbhbADY Benford's Law - Using Maths to Catch Fraudsters Benford's Law is a very powerful and counter-intuitive mathematical rule which determines the distribution of leading digits (ie the first digit in any number). You would probably expect that distribution would be equal - that a number 9 occurs as often as a number 1. But... Continue Reading →
IB Standard Level Revision Videos With the IB SL Maths exams fast approaching here are some video resources to help with revision: Algebra, Functions, Trigonometry past paper questions This is a playlist with past paper questions covering: 1) Algebra:arithmetic and geometric sequences and series, logs, binomial expansion, 2) functions: Inverses, completing the square, sketching quadratics... Continue Reading →
Simulations -Traffic Jams and Asteroid Impacts Why do traffic jams form? How does the speed limit or traffic lights or the number of lorries on the road affect road conditions? You can run a number of different simulations - looking at ring road traffic, lane closures and how robust the system is by applying an... Continue Reading →
Even Pigeons Can Do Maths This is a really interesting study from a couple of years ago, which shows that even pigeons can deal with numbers as abstract quantities - in the study the pigeons counted groups of objects in their head and then classified the groups in terms of size. From the New York... Continue Reading →
Maths of Global Warming - Modeling Climate Change The above graph is from NASA's climate change site, and was compiled from analysis of ice core data. Scientists from the National Oceanic and Atmospheric Administration (NOAA) drilled into thick polar ice and then looked at the carbon content of air trapped in small bubbles in the... Continue Reading →
This is a great puzzle which the Guardian ran last week: Fill in the equations below using any of the basic mathematical operations, +, –, x, ÷, and as many brackets as you like, so that they make arithmetical sense. 10 9 8 7 6 5 4 3 2 1 = 2017 There are many... 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 →
Modelling Radioactive decay We can model radioactive decay of atoms using the following equation: N(t) = N0 e-λt Where: N0: is the initial quantity of the element λ: is the radioactive decay constant t: is time N(t): is the quantity of the element remaining after time t. So, for Carbon-14 which has a half life of... 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 →