by Ellen Lawsky – Pamoja Math HL Teacher

I had the good fortune to spend spring break with my 5 year old granddaughter. While we had many adventures each day, her favorite activity was playing the board game CandyLand with me. The goal of the game is to be the first to get your figure to the candy castle. After shuffling a deck of cards each player chooses a card. The card indicates how many spaces you are to move. My granddaughter’s method of shuffling the cards was to put them on a table and just mix them around. Amazingly each time we played, she won.

Many people would say she cheated. I choose to say she was strategizing. She would look at the board, compute the number of spaces she would have to move to reach a desired spot, leave the room, stack the shuffled cards in her favor and return, always declaring that she just “happened” to get a “good” card. She understood the cards needed to be shuffled but then felt compelled to rearrange them in her favor.

I subscribe to a math and science journal called Quanta. This past issue had a very interesting article about “smooshing”, a way of randomizing a deck of cards. Persi Diaconis is a professor of statistics at Stanford University in the United States and has won the prestigious McArthur Award.   He and I attended elementary school and religious school together.   So I was excited to read about his current research.

Persi started his professional life as a magician and at some point moved on to the study of mathematics and statistics. He is interested in randomization.

“Smooshing” is the way a 5 year old would randomize a deck of cards – put them out on a table and just mix them around. Persi is exploring questions about smooshing. He proved that if you smoosh long enough, even cards at the edge will get mixed in. Persi is interested in the following questions: Does smooshing mix the cards? (He says (proofs) yes.) How long does it take? Is there a relationship between the size of the deck to the amount of mixing time needed? To answer these questions and others Persi is using ideas from a relatively new field of mathematics call quantitative theory of differential equations. Here is a link to the article.  I would be interested in your response to it.


What is most interesting to me is that looking around you – even at things that 5 year olds do – can lead to interesting and profound mathematical questions.

April 18, 2015
by Tim Knight

Review and Prep for Unit 6- Calculus Integration Test

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As we finish up the weeks 24-27 with a focus on integral calculus we must prepare for the unit 6 test.   In the video below I will walk through 4 questions students have been asking in the last few weeks.   You might already know and feel confident with some of this material other parts might be useful to help you review.

March 24, 2015
by Tim Knight

TOK ideas to explore

Is mathematics discovered or invented? 

 If you have time, listen to the different points of view in these videos to decide what you actually think.

An Ed-Ted Lesson by Jeff Dekofsky ….. would mathematics exist if people didn’t? Did we create mathematical concepts to help us understand the world around us, or is math the native language of the universe itself? Jeff Dekofsky traces some famous arguments in this ancient and hotly debated question.

From PBS – Math is invisible. Unlike physics, chemistry, and biology we can’t see it, smell it, or even directly observe it in the universe. And so that has made a lot of really smart people ask, does it actually even EXIST?!?! Similar to the tree falling in the forest, there are people who believe that if no person existed to count, math wouldn’t be around . .at ALL!!!! But is this true? Do we live in a mathless universe? Or if math is a real entity that exists, are there formulas and mathematical concepts out there in the universe that are undiscovered? Or is it all fiction? Whew!! So many questions, so many theories… watch the episode and let us know what you think!

The GameOverGreggy Show Ep. 44 (Pt. 2) – is the language invented but the properties discovered?

Stephen Wolfram speaking about his views on this issue.

Another video that may interest you

Zeno’s Paradox

You have looked at this before and were asked to post in your blog – see if this makes more sense now.

Judith Shorrocks March 24th 2015

February 28, 2015
by Peter Stevenson

Don’t count your chickens…or don’t your chickens count?

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With spring time coming up, here’s an interesting little story that has been in the news recently. I think that it is a great example of the hidden mathematics of everyday life.


Scientists at the University of Padua in Italy familiarised 3 day old chicks with a target number (dots on a piece of card). Then they presented the chicks with two cards both with a new number on them. If the number was lower than the number they had been familiarised with, the chicks tended to choose the card on the left, if it was higher they tended to go to the card on the right. I have illustrated this below (with possibly the world’s worst ever chicken drawings):

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A very interesting experiment from a biology point of view. Does our human “mental number line” come from culture and education, or is it something that has evolved in our brains?


From a mathematics student’s point of view, it raises some questions

  • “How did the scientists reach their conclusions?”
  • “What proportion of chicks actually chose the ‘correct’ card?”
  • “How many experiments were needed until the scientists were confident with their results?”

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.”

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…

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