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.

November 5, 2013
by Ellen Lawsky

Encouraging Girls in Math

I was excited when I saw the headline “A Day to Remember The First Computer Programmer Was a Woman” (http://bits.blogs.nytimes.com/2013/10/15/a-day-to-remember-the-first-computer-programmer-was-a-woman/?_r=0) a celebration of Ada Lovelace’s accomplishments.  However, I was saddened as I read the piece, which states that women software developers earn less than their male counterparts and that girls tend to shy away from computer science  because they lack role models and encouragement from their parents.  I wonder whether Pamoja and its structure might contribute to reversing this trend.

I studied mathematics as an undergraduate and statistics as a graduate student and have always been sensitive to the dearth of female peers in my classes.  There was only one other woman in my graduate statistics program at the University of California at Davis, for example.  When I started teaching mathematics in secondary schools, I hoped to encourage and support the young women who were in my classes.  But when I watched a video of one of my advanced math classes, I was shocked.  I seemed to be having a conversation with the boys in the class –  the girls were almost completely silent.  Why was this happening when I felt so strongly about encouraging the girls in my class and the girls were, I knew from test scores,  just as able as the boys?  The conclusion I reached is that the boys were simply quicker to respond to my questions.  I needed to wait longer after I asked a question and to be more sensitive to the different learning styles.  I needed to help the girls share more of their thinking, and the boys in the class to learn to listen more carefully to the other students.

In Pamoja online classes, everyone has more equal opportunity to be a part of the discussion, because students can weigh in on discussion forums at any time. They don’t have to wait to be recognised by a teacher or wait until a fellow student has completed his or her thought.  Perhaps this structure can help girls’ voices be heard and contribute to their sticking with math and computer science courses.



October 9, 2013
by Susan Zagar

The Simple Beauty of Mathematics!

The article below, written by Manil Suri, was recently sent to me by a friend who knows of my passion for mathematics.  As I read it, memories of my ‘mathematical moments’ during my travels returned.  One, in particular, involved a joint venture with a visual arts instructor in the city of Florence teaching the marvels of both mathematics and art to a group of IB students.  Nothing could beat the sheer delight observing that moment our students recognized and began applying branches of mathematics to the art of the Renaissance period.

Today I share this eloquently written and fascinating article with each of you.  The splendor of mathematics surrounds us each day even though we may not be fully cognizant of it at the time.   I welcome you to share your own thoughts/experiences that relate to the essence of this article, namely the appreciation of mathematics.  Enjoy the reading!

September 15, 2013

How to Fall in Love With Math


BALTIMORE — EACH time I hear someone say, “Do the math,” I grit my teeth. Invariably a reference to something mundane like addition or multiplication, the phrase reinforces how little awareness there is about the breadth and scope of the subject, how so many people identify mathematics with just one element: arithmetic. Imagine, if you will, using, “Do the lit” as an exhortation to spell correctly.

As a mathematician, I can attest that my field is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall. Perhaps the most intriguing of these is the way infinity is harnessed to deal with the finite, in everything from fractals to calculus. Just reflect on the infinite range of decimal numbers — a wonder product offered by mathematics to satisfy any measurement need, down to an arbitrary number of digits.

Despite what most people suppose, many profound mathematical ideas don’t require advanced skills to appreciate. One can develop a fairly good understanding of the power and elegance of calculus, say, without actually being able to use it to solve scientific or engineering problems.

Think of it this way: you can appreciate art without acquiring the ability to paint, or enjoy a symphony without being able to read music. Math also deserves to be enjoyed for its own sake, without being constantly subjected to the question, “When will I use this?”

Sadly, few avenues exist in our society to expose us to mathematical beauty. In schools, as I’ve heard several teachers lament, the opportunity to immerse students in interesting mathematical ideas is usually jettisoned to make more time for testing and arithmetic drills. The subject rarely appears in the news media or the cultural arena. Often, when math shows up in a novel or a movie, I am reminded of Chekhov’s proverbial gun: make sure the mathematician goes crazy if you put one in. Hanging thickly over everything is the gloom of math anxiety.

And yet, I keep encountering people who want to learn more about mathematics. Not only those who enjoyed it in school and have had no opportunity to pursue it once they began their careers, but also many who performed poorly in school and view it as a lingering challenge. As the Stanford mathematician Keith Devlin argues in his book “The Math Gene,” human beings are wired for mathematics. At some level, perhaps we all crave it.

So what math ideas can be appreciated without calculation or formulas? One candidate that I’ve found intrigues people is the origin of numbers. Think of it as a magic trick: harnessing emptiness to create the number zero, then demonstrating how from any whole number, one can create its successor. One from zero, two from one, three from two — a chain reaction of numbers erupting into existence. I still remember when I first experienced this Big Bang of numbers. The walls of my Bombay classroom seemed to blow away, as nascent cardinals streaked through space. Creatio ex nihilo, as compelling as any offered by physics or religion.

For a more contemplative example, gaze at a sequence of regular polygons: a hexagon, an octagon, a decagon and so on. I can almost imagine a yoga instructor asking a class to meditate on what would happen if the number of sides kept increasing indefinitely. Eventually, the sides shrink so much that the kinks start flattening out and the perimeter begins to appear curved. And then you see it: what will emerge is a circle, while at the same time the polygon can never actually become one. The realization is exhilarating — it lights up pleasure centers in your brain. This underlying concept of a limit is one upon which all of calculus is built.

The more deeply you engage with such ideas, the more rewarding the experience is. For instance, enjoying the eye candy of fractal images — those black, amoebalike splotches surrounded by bands of psychedelic colors — hardly qualifies as making a math connection. But suppose you knew that such an image (for example, the Julia Set) depicts a mathematical rule that plucks every point from its spot in the plane and moves it to another location. Imagine this rule applied over and over again, so that every point hops from location to location. Then the “amoeba” comprises those well-behaved points that remain hopping around within this black region, while the colored points are more adventurous and all lope off toward infinity. Not only does the picture acquire more richness and meaning with this knowledge, it suddenly churns with drama, with activity.

Would you be intrigued enough to find out more — for instance, what the different shades of color signified? Would the Big Bang example make you wonder where negative numbers came from, or fractions or irrationals? Could the thrill of recognizing the circle as a limit of polygons lure you into visualizing the sphere as a stack of its circular cross sections, as Archimedes did over 2,000 years ago to calculate its volume?

If the answer is yes, then math appreciation may provide more than just casual enjoyment: it could also help change negative attitudes toward the subject that are passed on from generation to generation. Students have a better chance of succeeding in a subject perceived as playful and stimulating, rather than one with a disastrous P.R. image.

Fortunately, today’s online world, with its advances in video and animation, offers several underused opportunities for the informal dissemination of mathematical ideas. Perhaps the most essential message to get across is that with math you can reach not just for the sky or the stars or the edges of the universe, but for timeless constellations of ideas that lie beyond.

Manil Suri is a mathematics professor at the University of Maryland, Baltimore County, and the author, most recently, of the novel “The City of Devi.”


October 2, 2013
by Arno

Mathematical thinking and thinking mathematics

“Mathematics is an esoteric art.” One day in my first year IB mathematics class, this is what Mr Davies stated. I thought it pretty cool, mysterious and attractive. Three years later, studying theoretical physics, I was asked to prepare a presentation to my tutorial group. When I read over it, I can see hints of having been an IB student who had been exposed to thinking about knowledge in a TOK kind of way. I would like to share here those thoughts of mine from 1994 on the nature of mathematics, only slightly edited to change it from a talk to the written form.





Some years ago my mathematics teacher told me: Mathematics is an esoteric art. I thought that interesting, so there the saying was, on the first page of my maths folder. It was not until I started to study physics at Imperial College that the phrase came back to me. Since physics is nearly exclusively written in the mathematical language, the notion of mathematics being an art form started to concern me.

Then when I read a few of Georg Cantor’s letters I got worried. At first for this presentation, I was looking at Transfinite Number Theory, of which Cantor is the founding father. Cantor and another great mathematician, Leopold Kronecker, were in a bitter dispute about the handling of infinities [the 2007 documentary Dangerous Knowledge by David Malone looks in some depth at Georg Cantor, I like to show this to my mathematics as well as TOK class and one can find this documentary online]. I found these disagreements about the foundation of the most certain science not only surprising but, to put it mildly, disconcerting. Unlike physics, which evolves, mathematics is entirely cumulative, it is an ever widening network of logical consistency. So the foundations are very important indeed. A dispute about the foundations of mathematics, due to its close interdependence, would equally have an effect in physics.

Let me give an example. Five men, shipwrecked, find themselves on a deserted island with a monkey, and a pile of coconuts as the only edible thing. They agree to split the coconuts into five integer lots, any remainder will go to the monkey [this is a famous problem in mathematics, in fact, Martin Gardner starts with it in his collection of writings in The colossal book of mathematics (http://goo.gl/KJbBr3) ]. In the middle of the night, the first man wakes up hungry. He decides to take his share of the coconuts, divides the pile into five finding one coconut remaining. He eats his lot and gives one to the monkey, then returns to sleep. Later the second man wakes up. Feeling hungry, he, too, divides the coconuts into five lots, giving the monkey the remainder, which again is one coconut. So do the remaining three men. The next morning, too embarrassed, nobody says anything about their nightly escapades. They divide the remaining coconuts into five lots, giving the remaining one coconut to the monkey. Find the initial number of coconuts.

After some fiddling, you find the simultaneous equation [there is a better explanation in Martin Gardner’s book]

x – 6 = (15625 f + 5385) / 1024

where x is the initial coconuts and f is the share everybody gets in the morning division. There are many, in fact infinite, solutions, but the smallest number is 15621. However, soon after this solution was given [this is what I wrote in 1994, but I read in Gardner’s book that the following is not verified], Paul Dirac came up with the solution of -4, which is clearly a solution! This shows, I think, not only some interesting characteristics about mathematics, namely its independence of reality, but also the potential pitfalls a physicist may encounter when purely relying on mathematics.

So lets take a closer look at it.

There are essentially four interpretations of mathematics: Platonism, Conceptualism, Formalism and Intuitionism. Due to the limited time [I believe, judging by my notes, that the presentation had to be no longer than 12 minutes] I will discuss Platonism only, for reasons which hopefully will become clear.

At the birth of science as we know it today, stood undoubtedly Galileo [a bit of a simplification, but I was young]. A remarkable feature of his thinking was his belief that Nature was ‘a book written in mathematical characters’. This, we will see, is perhaps the single most influential and fundamental idea behind Platonism, and heavily relied on in modern science.

The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious. Some, for example Nobel Prize winner Eugene Wigner, suggest that there is no rational explanation for this. The mathematical formulation of the physicist’s often crude experience leads to many cases of amazingly accurate descriptions of a large class of phenomena. Newton established the Law of Gravity, but could only verify it with an accuracy of about 4%. Today it has proved to be accurate to less than 1 part in 10,000.

[skipping some physics here]

The Platonists would explain that mathematics is so successful in capturing the workings of Nature by saying that Nature is mathematical. Plato’s philosophy is often dubbed as “idealism” and this is represented in how it deals with mathematics. Most remarkable is the assertion that pi really is in the sky. That is, mathematical concepts “exist out there”, independently of you and me. Plato’s belief is that of forms, invariant blueprints from which we can only observe shadows. And so mathematics is discovered rather than invented.

Mathematical ideas like the number ‘seven’ are regarded as immaterial and immutable ideas that exist in some abstract realm, whereas our observations are of specific secondary realisation, like seven dwarfs. Thus, we seem to be able to conceive of perfect mathematical entities -like straight lines or right angles- yet all the examples we see of them are imperfect in some respect -I cannot draw perfectly straight lines or exact right angles [any of my students will tell you that!]. The particulars that we witness in our world are each imperfect reflections of a perfect exemplar or form. These forms exist somewhere and, most strikingly, Plato believes that place is not the human mind; hence, we discover, rather than invent mathematics.

Here perhaps we encounter the most obvious hurdle for this interpretation of mathematics. One might wonder how to clarify the relationship between the universals and the particular examples of them. For as far as our minds are concerned, the universal blueprint is just another particular. This was recognised early on by Aristotle. Aristotle argued that if all men are alike because they share something with the ideal and eternal archetype Man, how can we explain the fact that one man and this archetype Man are alike without assuming a second archetype. And would the same argument not require a third, fourth in an endless regression of ideal worlds?

Nevertheless, the Platonic picture of mathematics is popular among physicists, though less so amongst mathematicians, and it is almost universally regarded as irrelevant by consumers of mathematics such as psychologists, sociologists or economists [who says scientists regard themselves so highly?]

One of the most extreme manifestations of the Platonic assumption of the universality of mathematics is in the search for extraterrestrial intelligence in the universe. For years terrestrial radio telescopes have broadcast signals in the direction of nearby star systems; and equally search for incoming signals. These broadcast signals are all mathematical codes, believing that these ETs would have, if anything, in common mathematical knowledge [three years later the movie Contact came out, you can see the trailer here where it, in fact, mentions this underlying idea of the universality of mathematics


As we were a tutorial group under the wings of a cosmologist [Andy Albrecht, now at UC Davis http://goo.gl/IePyoh], it was of interest to consider the following. We can observe different quasars (http://en.wikipedia.org/wiki/Quasar) so widely separated that there has not been time for light to travel beteween since the “beginning”. Thus they are truly causally separate phenomena and cannot have had mutually influence in any known way. Yet we find that the detailed aspects of the spectrum of light that they emit adhere to identical mathematical analysis. This should surely give us confidence in the existence of some universal substructure that is mathematical in character. Hence, cosmology is the most vivid expression of the Platonic doctrine, assuming that mathematics is something larger than the causally linked physical universe.

So to make up the balance, the advantages of the Platonic picture of mathematics are obvious. It is simple. It makes the effectiveness of mathematics a thoroughly predictable fact of life. It removes human mathematicians from the centre of the mathematical universes where otherwise they must cast themselves in the role of creators. And it explains how we can be so successful in arriving at mathematical descriptions of those aspects of the physical world which are furthest removed from direct human experience.

September 26, 2013
by Kalyani Shankar

Tricky proofs in Mathematics


You would have come across this proof many times. Probably you might have also known the error in one of the steps that had lead to the wrong result.



 The error is in the step number 7 which divides both sides by (a – b) and it is obvious that we are dividing by 0 (as a = b )

Ok. We do accept the weird result is due to this error in dividing by a 0.

Now look at this proof:

 And the proof continues further to get many good results (which are out of the scope of this article)


Here, we had divided the equation by z -1 which assumes  z  not equal to 1 and then we substitute z =1!! Very clever isn’t it? This is mathematics. You can actually cheat – as long as your steps are logical. Also, you would have noticed how cleverly the step number (2) had been rewritten in a convenient form to suit the substitution. If this conversion of


 was not done, making z = 1 would have lead to  form which is unacceptable in mathematical proofs. But still we have used the value of z =1 in the same expression by writing it in a different format.


We can see such approaches in various other places. For example, while using First Principle method to find the derivative,


 we simplify the numerators so that the ‘h’ in the denominator first gets cancelled and make it safe to substitute the limit as h = 0. L’Hospital’s rule basically changes the given function to a form which makes it convenient for the application of the specified limits.

There are number of proofs like this in mathematics. You just need to think out of the box and see how you can proceed without losing the logic.

As a very simple example, consider the quadratic formula. We wouldn’t have got the quadratic formula if someone hadn’t thought of adding and subtracting the extra term   to make it a perfect square. Can we do something similar to cubic? Can we make a normal cubic equation to a perfect cube, for example, and invent a formula for finding the roots? It is possible though so far no one could get the knack of it yet! May be someone in future might come up with a special trick like we saw above.


Hence, all you need is to have a good visualisation of the question and what you have in hand that could be applied in the process of solving it. Keep looking for such approaches while you learn new concepts and you will definitely be able to appreciate the way mathematicians think. Soon, you might also be able to think out of the box and could come out with some wonderful proof. 









Statistics and Causal Claims

September 12, 2013 by John Bradfield | 2 Comments

Year two of the HL course begins with statistics. The problem with statistics is that on one level they are often quite simple. It would be nice to think that the result of this would be that they were well used.

The idea I am putting forward here is the over use of a simple term “cause”.
I have responsibility for statistics to report to the board of trustees at my school. Four years ago they introduced a one for one laptop scheme. As the students using laptops come up through our national examinations (in New Zealand) the question from the board was “have the laptops made a difference?”. This is a good question to ask but not an easy question to answer. I looked at our results and we had done well. In fact it was a very good year as the first group with the laptops came through the examinations. I couldn’t answer their question however. I assured them that the laptops appeared to have done no damage as far as I could see. What might a politician have said in my situation?

“Introduction of laptops has lead to best ever performance in national exams”

Or perhaps

“Laptops deliver stellar results”

What I was stuck with to be true to my mathematical roots was the difference between a random statistical experiment and an observational study. Here we simply observed and there appeared to be a positive improvement in the grades as the laptops came in. Was it the laptops that caused it? We can’t say.

If I could roll back time I could have said to the board. Lets split the student body into two randomized groups. The groups would need to have similar attributes and ability. Then give a laptop to each student in one group and keep computers away from the other. Teach both groups the same material and then see how they compared in the national exam. If all other variables are even and we only manipulate the computer access then we have a case for making a causal claim.

I have stuck to an issue here not likely to inflame but if you keep your head up and listen you will see it all over.

Does smoking cause lung cancer?

Does obesity cause diabetes?

Does Tamiflu help with flu symptoms?

Will the paleo diet help you loose weight?

My challenge is this. As we embark on our statistics topic be critical and just a little cautious. You hear a lot of claims in the media, some more solidly founded than others.
Watch this TED Talk that gives a great view of this issue


September 5, 2013
by Tim Knight

IA Math HL Exploration – The Coke Can – A Solution

We have completed our write up of the Exploration – The Coke Can. You will find the written up task here.

It is recommended that you look through all the stages that led to the final product. Remember that we had to work on our task before it became clear that we would only solve the 2D problem. Also note how we broke the problem into two parts – finding the centre of mass of a quadrilateral and then conserving the volume of our quadrilateral as the can was tilted.

Finally, note how the final solution is really done with technology. An analytic solution would have just been too complex.

As Year 2 Pamoja HL students complete week 34 and are familiar with the criteria it is suggested that they look at the coke can exploration. They may wish to suggest a mark for the different criteria in the comments section.