February 9, 2015
by Tim Knight

Playing and Exploring with Math

During the second year of Math HL you will be required to work on an Exploration – to pick an area and/or a problem in mathematics that is of interest to you, learn about it in depth, play with it and explore its meanings and perhaps it consequences.

I was excited to see a profile in the February 2 issue of the New Yorker Magazine.  The profile is of Yitang Zhang, a Chinese mathematician who received his PhD in algebraic geometry from Purdue University in 1991.  This past September he received a MacArthur Award for solving a theorem on the Distribution of Primes that had be unsolved for more than a hundred and fifty years.

The article describes the problem but it is primarily a profile of the man.  After publishing his result Zhang spent six months at the Institute for Advanced Study in Princeton.  During that time a documentary about Zhang was made for the Mathematical Sciences Research Institute MSRI in Berkeley, California.  Here you can see a trailer of the film and listen to some of what Zhang has to say.

Some friends of mine who also read the article said they think Zhang is dysfunctional but I disagree.  From what I understand he  is a genius who sees the world differently from most of us but functions very well in the world of mathematics.  The article reports that he works “by walking and thinking”, that he is intense, brave, independent and persistent.

While we are not all geniuses I think these qualities are worth considering and nurturing  especially as you begin to think about a topic you might want to study for your Exploration.


December 9, 2014
by Tim Knight

Would you turn down $1,000,000?

Not many of us will get the chance to answer this question, but in 2010 Russian mathematician Grigoriy Perelman got his chance.  Considered to be one of the world’s cleverest people, he was awarded the Clay Institutes $1,000,000 ‘Millennium Prize’ for his solution of the Poincaré Conjecture.  But declined to accept it, statingI’m not interested in money or fame. I’m not a hero of mathematics. I’m not even that successful.”

A million dollars for a mathematics problem?  Actually, 7 million dollars for seven problems.

The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.”  http://www.claymath.org

The prizes were announced at a meeting in Paris on 24th May 2000, and represent seven classic questions that have remained unsolved for many years.  A correct solution to any one of them comes with a $1,000,000 prize and a place in history for the successful mathematician.  If you want to try for yourself! The problems are;

Yang–Mills and Mass Gap   Status: Unsolved

Riemann Hypothesis             Status: Unsolved

 P vs NP Problem                      Status: Unsolved

 Navier–Stokes Equation      Status: Unsolved

 Hodge Conjecture                   Status: Unsolved

 Poincaré Conjecture              Status: Solved

 Birch and Swinnerton-Dyer Conjecture    Status: Unsolved

This is not the only instance of cash prizes being offered for the solution of mathematics problems.  In 1994 Andrew Wiles claimed the $50000 Wolfskehl prize for his solution of Fermats Last Theorem, and still to be claimed is the $1,000,000 prize for a proof of the Beal Conjecture.  One thing all mathematicians agree on though – any cash prize is a very distant second to having your name etched into mathematical history forever!

Want to try something for yourself and get your name in the history books?  You can take part in The Great Internet Mersenne Prime Number Search (GIMPS).  All you need is a reasonably modern computer, some patience and an internet connection, and you could discover the next Mersenne Prime, and claim up to $50,000 for your work!

Neil Bradley

Have Fun With Mathematics

December 5, 2014 by Tim Knight | 0 comments

Now that the holiday will soon be here, and the exams behind you, you can relax and enjoy some fascinating talks and articles about mathematics.

Ted Talks

Adam Spencer: talks about “why I fell in love with monster primes.  They’re millions of digits long, and it takes an army of mathematicians and machines to hunt them down — what’s not to love about monster primes?” Adam Spencer, comedian and lifelong math geek, shares his passion for these odd numbers, and for the mysterious magic of math.

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Arthur Benjamin: “math is logical, functional and just … awesome”. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)

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Peter Donnelly: how juries are fooled by statistics.  Oxford mathematician Peter Donnelly reveals the common mistakes humans make in interpreting statistics — and the devastating impact these errors can have on the outcome of criminal trials.

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Dan Meyer: today’s math curriculum is teaching students to expect — and excel at — paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. In his talk, Dan Meyer shows classroom-tested math exercises that prompt students to stop and think. (Filmed at TEDxNYED.)

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Maths Plus      

Plus is an internet magazine which aims to introduce readers to the beauty and the practical applications of mathematics. A lot of people don’t have a very clear idea what “real” maths consists of, and often they don’t realise how many things they take for granted only work because of a generous helping of it. Apparently, some people even have the idea that it’s boring! Weird. Anyway, we hope that even if you’re such a person now, you won’t be after looking through one or two issues of Plus, and that you’ll come back and read future issues as they come out.  You can subscribe to the newsletter here.  Try some of the puzzles on their website.

Wolfram Alpha

Wolfram Alpha introduces a fundamentally new way to get knowledge and answers – not by searching the web, but by doing dynamic computations based on a vast collection of built-in data, algorithms, and methods.

You can input any equation into the box below and it will compute the answer

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You can choose an unlimited number of random practise problems on any of these topics – click on the image.

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Mathematics Video Links

In the list under WIKIS on the right hand side, I have listed some video websites that may help you when you are studying Mathematics next term although you may feel that there is enough in the Pamoja Content area not to need extra help.

Judith Shorrocks

November 19, 2014
by Tim Knight

Self-inverse rational functions

What makes a function self-inverse?  How can a self-inverse function be spotted from its equation or from its graph?  One way of looking at this is that a self-inverse function generates sequences of length two.  Apply the function to an initial number to get a new number; apply the function again to go back to where you started.

This blog post extends the idea to functions that generate cycles of length 3, so that you have to apply the function three times before you return to your starting number.  The world of bicycle and tricycle functions is well worth investigating.  Will we find a quadricycle function?  Find out the answer in this article…Gerald Newth’s self-inverse blog

Gerald Newth   (November 2014)

November 15, 2014
by Peter Stevenson

Trigonometric Modeling

Mathematicians always have an eye out for mathematics in the world around them. On my early morning stroll today I took a photo of this lovely sine curve on a bridge near my house. Trig curves are all around us!


In this post we will look at a ride on a ferris wheel that ends up with a trig curve, plus a neat trick to remember the exact values of sine, cosine for the special angles.





…and we will finish with a “handy” aide-memoire for the remembering the trig ratios for special angles





November 3, 2014
by Tim Knight

Highlights from Weeks 6-8: Study Help for Unit 2 Test

As we prepare for our second unit test, here are some general highlights from the relevant chapters.

First a little bit similar to the last post about graphing rational functions and their reciprocals; however, you might find it helpful to see these concepts once more:

The second video is from Week 7 content on Logarithmic and Exponential Equations and how much fun they can be to solve:

The final video from Week 8 looks at the graphs of “e” and “ln” and then we get a chance to bang them up a bit with some transformations:

If those videos didn’t help you might want to check out some other nifty math tips below…

October 17, 2014
by Peter Stevenson

Plotting rational and reciprocal functions


Plotting rationals without technology is a satisfying challenge that really increases our understanding of how they work. In this post we will work through plotting a rational and then its reciprocal

Once we have plotted this let’s think about plotting the reciprocal 1/f(x) for which there are different strategies.


…and finally here’s what happens if you ever attempt to divide by zero




October 6, 2014
by Tim Knight

Functions – composite and inverse

Throughout our Math HL course we will be studying a variety of functions.  If we consider functions to be mathematical objects we can look at a variety of ways of combing them and working with them to form new functions. The first video considers Composition of Functions.

Now let us take a look at the graphical representation.

Can you find two functions f(x) and g(x) where the composition is commutative?  The next video answers this question.

Here is a challenge for you:  f(x) =√(x+2).  What is the domain of this function?  Find the inverse of  f(x) and show that f(x) composed with its inverse is equal to the inverse of f composed with f(x).

Roots of Quadratics

September 18, 2014 by Tim Knight | 7 Comments

Quadratic equations have a long history. The Babylonians were studying quadratics 4000 years ago and were creating formulas to solve. I am sure you know it! Here are three videos that look at quadratic equations, specifically relaying the roots of the equations to the coefficients a,b and c in the equation a{x^2} + bx + c = 0.

The first video looks at the determinant \Delta  = {b^2} - 4ac and what it tells us about the number of roots of a quadratic equation.

This video look at an example relevant to the determinant

The final video in this post consider Sum and Product of roots. This is worth looking at carefully since we will later in the Math HL course look at problems relevant to this topic but with higher order polynomials.



December 17, 2013
by Tim Knight


Flatland – A Post by Erik Shofer (Year 2 Math HL Student)In Flatland, a book that is now a movie about a square in a 2D world, the 2 dimensional shapes like the square see each other in 2D and are unable to process the existence of a 3D sphere that comes into their world. Similarly, some line segments that exist on a 1 dimensional line in the 2D square’s world see each other in 1 dimension only and are unable to believe that the world is 2D. I think that, while the idea of different beings existing in different dimensions is an interesting thought (and a good political satire), but I think that the way the shapes see in the book and movie is incorrect.

In Flatland, the shapes on the 2 dimensional plane see each other in 2 dimensions. This is shown by how they know that they are squares, circles, hexagons and so on. However, I believe that if two 2 dimensional shapes were to look at each other, they would only see a line. As they exist in a 2D space, they can only see along their 2D plane. The reason this would cause a line can be explained like this: If you have a plane (or paper) with shapes on it and you looked at it from a parallel plane, (or look at the paper while it is facing you) you would see shapes. However, if you then turned the plane (or paper) so that you were viewing it from a perpendicular plane, (or viewing the paper while it is flat at eye-level) you would merely see a line as from your plane (or view), you would only see the second plane along the line where the two planes (one being your sight and the other being the paper) intersected. This is how the creatures that exist in the 2D plane would see as they can only see along their plane, not from the side, and therefore would only see lines in front of them.

Furthering on that idea, the creatures that existed in 1 dimension would not see each other as lines, but rather as points. An example would be to take a line (or piece of string) that is divided by colors into segments and view it from a perpendicular line (or hold the string straight away from you at eye-level). It would look like a point (or just the end of the string) because from your line (or view), you would only see the other line at the point where it intersects. This is also how a creature existing in 1 dimension would see, a single point in front of it hiding the rest of the line from view. The 1 dimensional creature would also only ever see one point because, no matter where it moves (it can only move along its own line), the lack of a second dimension means that it can never go around the line segment in front of it.

These two earlier statements would make it seem that a 3 dimensional creature would see in 2 dimensions. This would be true, except for one thing. A 3 dimensional space allows for multiple viewpoints facing the same area or general direction, but at different angles (i.e. 2 eyes). These multiple viewpoints allow the view to be transformed into 3 dimensions inside a brain by using the difference between the two views to add depth. A person missing one eye however, would only be capable of seeing in 2 dimensions as they would have no second viewpoint to compare their one view to.

A 2 dimensional shape would still view a 3 dimensional shape as a line because, the 2 dimensional shapes can only see a small part of the 3D shape (the part intersecting with the plane) at a time. A 1 dimensional shape would similarly see everything as a point, even if it was a 2D or 3D shape as it could only view part of it at a time.

This raises the question, what does a 4 dimensional being see?

My answer is that, based on the pattern above, a 4 dimensional shape sees in 3 dimensions. While this seems odd as we 3D creatures see in 3D, it makes sense because we don’t see in 3D. We simply see in 2D twice (each eye) and then compare the two to make a 3D image inside of our brains. A 4D creature would see everything in true 3D. Unlike the movies where true 3D means adding depth, a creature seeing in true 3D would see objects from every possible 3D angle at once. It would be like viewing a cube from below, above, the left side, the right side, behind, in front and every other possible viewpoint all at the same time. Unfortunately, this is something that the human brain is incapable of conceiving or imagining as the furthest dimension it can think in is 3D. It is likely impossible that a human will ever be able to understand, let alone view objects in or from a 4th dimension.

And if a 4 dimensional creature sees in true 3D, what would a 5 dimensional creature’s view be? That is, if it there are even 5 dimensions.