Mathematics in Action

On Wednesday 25th November 2015 I took 13 sixth form students from my school to the Mathematics in Action event organised by The Training Partnership. It was an excellent day out and I summarised some of my notes from the day’s lectures in a post for my students which is replicated below.

Host (Tom Evans)

The host of the day was Tom Evans. He opened up the session with two puzzling questions which the audience were asked to attempt during the breaks between speakers. The problems are below: Happy birthday Fermat’s last theorem (Simon Singh)

The opener was Simon Singh talking about Andrew Wiles and Fermat’s last theorem. He opened up the talk with the start of the BBC Horizon episode (see here) he directed on the same topic. In the video Andrew Wiles expresses an analogy for mathematics whereby it can, for a time, feel like you are walking around a dark house stumbling into things all the time. Then, one day, the light switches on and suddenly you see. Wiles becomes very emotional when talking about Fermat’s last theorem, which as Simon explained in his talk is due to the sheer length of time and determination spent solving a problem which had become a childhood dream.

Simon gave us the problem and spent a large part of his talk discussing the back story and some of the people who tried to provide a proof for Fermat’s last theorem across some 300 years. He recommended purchasing his book for further reading (which I did!). If you are wondering the theorem states the following:

No three positive integers $x, y$ and $z$ can satisfy the equation $x^n+y^n=z^n$ for any integer value of $n$ greater than two.

There were also recommendations at the end, for further enriching mathematics, to take a look at Numberphile, Vi Hart and Martin Gardner. Do take a look at all if you haven’t yet!

How big is infinity? (Chris Good)

Chris focused a large part of his talk on infinity around the work of Georg Cantor. He opened the talk by asking a classic question; ‘Which is bigger 1 or 0.99999999999…?’ He used this to clarify that every number between 0 and 1 can be represented as a unique non-terminating decimal (due to one-to-one correspondence) which was introduced by Cantor first in 1874. As a result of much of Cantor’s work there was an implication that there are an infinite number of infinities (i.e. I can always choose a bigger infinity than the one you just chose!). Chris concluded his talk by discussing transcendental numbers.

Exam technique (optional session with Colin Beveridge) I really liked the Venn diagram Colin used at the beginning of his short talk about exam technique to describe the ‘ideal’ experience maths A level students should have (i.e. extended subject knowledge and support with exam technique). One of the things I really liked, and have already used, is the ‘error log’ idea. This is where pupils are asked to write down their mistakes, take note of where they went wrong or got stuck and then include the steps they need/needed to take to rectify the problem. Colin had some really valuable things to say so I am glad I stayed to listen to this (even if my students didn’t!).

7 things you need to know about prime numbers (Vicky Neale)

Fact 1: 1 is not a prime number.

Fact 2: 2 is a prime number (and the only even prime!).

Fact 3: (Theorem) There are infinitely many prime numbers.

This was proved by both Euclid (c.300BC) and Euler (18th century). This theorem of the week blog post gives some detail on both.

Fact 4: (Fundamental theorem of arithmetic) Every integer greater than 1 can be expressed as a product of primes in an essentially unique way.

Fact 5: Every prime is one more or one less than a multiple of six if $p>3$. Therefore, could you prove the following statement (imagine delivering your explanation to a year 9 student)?

If $p$ is a prime number greater than 3, then $p$ is of the form $6n \pm 1$, where $n$ is a natural number.

Fact 6: Let $\pi(x)$ denote the number of primes less than or equal to $x$. For example, $\pi(10)=4$ or $\pi(100)=25$.

(Prime number theorem) Then $\pi(x) \sim \frac{x}{\log{x}}$ where here $\log{x}$ is the number theory notation for $\ln{x}$.

Fact 7: (Twin prime conjecture) There are infinitely many primes $p$ such that $p+2$ is also prime.

This conjecture is yet to be proved, but there have been many recent developments!

Exit question:

If 3 and 5 are twin primes then let us call 3, 5 and 7 a prime triple. Are there any more prime triples?

Geometry and the art of optimisation (Richard Elwes)

Richard opened his talk by discussing how much of his work on optimisation is applicable in everyday life (e.g. traffic lights and train timetabling). He discussed an example of a toy factory where we needed to consider how to maximise the profit from the production line. The calculations made led to the consideration of the feasible set of values which we then needed to consider maximising. Richard then introduced us to the simplex algorithm found by George Dantzig in 1947. He concluded with the Hirsch conjecture to which a counter example was found in 2010.

Cryptography (Keith Martin)

Keith was a very engaging and entertaining speaker, and whilst his talk didn’t contain much ‘maths’ he was able to discuss the importance of cryptography. Some elements of security are ‘lost’ in the cyber world and we need to ensure our information holds its confidentiality, integrity and authentication. Cryptography is a tool used and built by mathematicians to help with encryption and security in cyberspace. Keith discussed ciphers, our data integrity (using ISBN codes as an example) and authentication:

On the internet no one knows if you are a dog.

Keith recommended taking a look at cryptool.org if you are interested in looking further into cryptography. His alternative suggestions were Piper and Murphy’s Introduction to Cryptography book or Simon Singh’s The Code Book.

My favourite function…

The content of this post originally appears on Chalkdust’s ‘What’s your favourite function? Part II’ post.

Rob Beckett’s favourite is a classic: $e^x$

My favourite function is the exponential function $f(x)=e^x$. This is primarily because $\frac{d}{dx}(f(x))=e^x$, in other words the function is growing at a rate which is equal to its current size. This is a really interesting property which comes from the fact that $e=\lim\limits_{n\rightarrow \infty}{(1+\frac{1}{n})^n}$. Just like pi is the ratio between the circumference and diameter of circles, the number $e$ is the base rate of growth and it crops up whenever things grow or decay continuously and exponentially. Our exponential function appears when considering bacterial growth rates, populations, radioactive decay and even occurs in the Black-Scholes formula which is used in the financial market.

Quick post: Plickers cards

This week I brought Plickers cards back into the classroom. Admittedly only for a test run with my two year 7 classes, though I am excited at the prospect of including these in my lesson planning again.

A quality (free) resource for completing multiple choice quizzes, the Plickers cards provide anonymity to the pupils answers which allows you to gauge whether pupils really have ‘got’ something if you are presenting them with hinge point questions. I used these a lot during the second placement on my PGCE as I learnt more about hinge point questioning in the FutureLearn course ‘Assessment for Learning in STEM teaching‘.

I am looking forward to using them more effectively, allowing the pupil responses to help me determine how I proceed with the remainder of the lesson.

‘Evaluating probability statements’ Standards Unit task (S2) with year 8

Having moved on to the topic of probability with my year 8 classes this term I was reminded of my final university tutor observation of a year 8 probability lesson last year, which happened to be one of my better lessons during my PGCE. I had decided to ‘be daring’ and conducted the ‘Evaluating probability statements’ Standards Unit task (link courtesy of @mrbartonmaths) with the class. The feedback was good and I was able to get really fruitful discussions from the class.

The structure I re-used with my two year 8 classes last week, with some tweaks, which resulted in both classes being more engaged with mathematics and ‘real’ mathematical discussion than before!

The task encouraged students to, in pairs, discuss the validity of a number of probability statements. They were asked to ensure that following their discussions they wrote down their thinking in order to refer to it in a later whole class discussion. Some students found some statements difficult to interpret and these were left until the end where pairs were able to join together into a small group to discuss the leftover statements.

After the discussions I wanted to have some class feedback. Using iDoceo’s random student picker I was able to select a student to chose a statement and explain their reasoning as to why they had decided that it was true or false. This then allowed the opportunity to open up a discussion with the class as in many cases there was conflicting opinions or reasons behind pupil answers. It really helped to meet the task’s criteria of clarifying the misconceptions in the statements, and for almost every statement I was able to take a ‘back seat’ listening to my students argue their points and eventually reach an agreed conclusion through their discussion.

The Standards Unit tasks are probably some of the best classroom activities I have come across. They are challenging, promote mathematical thinking and engage all learners in the task. In addition to this they all contain a preamble with a suggested approach to the task which has been helpful on many occasions. If you haven’t used them yet, you should consider it!

You can find all of the Standards Unit tasks on mrbartonmaths.com.