# New AS level maths SAMs

Having spent most of my Saturday this weekend working through the new AS level sample assessment materials for each exam board I thought I would briefly summarise my thoughts so far. Being a little rusty on statistics and mechanics I have brushed up mostly on M1 and S1, so all the binomial distribution and hypothesis testing questions I haven’t touched for now.

As I am currently teaching the Edexcel specification and so decided to attempt their materials first. They are the only exam board who have proposed a 2 hour core pure paper and a 1 hour applied (statistics and mechanics) paper. The material in both papers seemed reasonable and at similar level to the current qualification.

I then looked at the AQA material which seems to be ever so slightly ‘easier’ than the Edexcel papers. AQA, like OCR A and OCR MEI, have gone with two 1 hour 30 minute papers split into pure and applied. For some of the new material, particularly applicable to differentiation from first principles, AQA have designed a ‘guided’ question (see question 8 in paper 1). In contrast, a question on this topic when assessed in all other boards is just asked up front, e.g. Differentiate $f(x)=3x^2$ from first principles (Edexcel paper 1 question 9). There are also some multiple choice questions which, following a brief conversation on Twitter with Tom Bennison, I am informed is similar to their GCSE papers. This, I think, could be seen in both positive and negative views. It’s quite nice to be able to get students thinking about some of the misconceptions and really thinking about their answers in the exam, however some students could get frustrated over the quantity of work they might need to do for a single mark (I make this statement with respect to my year 12 class who showed me their AS physics sample paper in which they sometimes needed to do large amounts of work to find the correct choice).

OCR A were the next papers I looked at and I very much enjoyed these ones. Some more interesting and different questions cropping up and the statistics element taking up what felt like half the first paper. I particularly liked the trigonometry and vectors questions in paper 1 (question 5 & 6) and the proof question in paper 2 (question 6). The papers were slightly more difficult that the Edexcel papers in my opinion, though with some similarities.

Finally, I looked at the MEI materials. The first paper opened up with a nasty looking question in comparison to other papers, but most of the content again seems to be on par with the OCR A sample materials. Interestingly, MEI are also the only exam board to mix the pure and applied questions together, rather than separate the sections.

I still need to look over the physical specifications for each board, but at this stage I think my preferred set of sample materials for the AS in mathematics are those provided for the OCR A.

Next, I think I will look over the A level mathematics materials before I look at some of the further maths. But before that I need to revise some mechanics and statistics!

Edit (12/06/16): Note also that OCR A has included binomial expansions as part of their statistics section, whilst Edexcel and AQA included this within pure.

# My favourite shape…

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

Möbius strip (Rob Beckett)

My favourite ‘shape’ is the one sided non-orientable surface called the Möbius strip. This can be created by simply twisting a long strip of paper and gluing the ends together. One of the explanations most regularly associated with the Möbius strip is that of MC Escher, who described an ant crawling along its surface. The ant would be able to do this and return to his starting point having not even crossed an edge (or maybe it keeps crawling on indefinitely hoping to find the end!).

In the Numberphile video Möbius bridges and buildings Carlo H Séquin (UC Berkeley) considers using the idea of a Möbius strip to create aesthetic bridges and buildings.

# 3D display with year 8

As we came to the end of the first half term of 2016, my year 8 classes were about to conclude their learning of the shape topic being covered in the department scheme of work. This meant briefly looking at properties of 3D shapes. In a maths Twitter lesson planning (#mathsTLP) session I asked what I might be able to do.

Mr Mattock suggested his jigsaw on Euler’s rule which, whilst I didn’t use it this time, I’m sure will come of use in the future.

It was suggested that I should get the students to make the shapes. Sharon Derbyshire then recommended her post about work with 3D shapes which used to facilitate this.

Using the idea of sweets and cocktail sticks my students completed a number of ‘which shape am I’ tasks to construct various 3D shapes. These then formed my classroom display as Sharon suggested!!

# #ChristMaths15

On the 21st December 2015 Jo Morgan hosted the first ever ChristMaths party event with CPD, networking and of course alcohol! This post contains some of my reflections after reading what I wrote down during the talks.

Strategies for teaching previously ‘Grade C and beyond’ topics to Foundation students – Mel Mundowney (@Just_Maths)

Mel delivered some interesting points in her talk, particularly addressing how many students are now playing ‘catch up’ with the curriculum changes due to different demands and teaching. One of the key things I noted were that there is a need to ‘keep things fresh’ as, more recently, there has been cyclic and repetitive content being taught in a disjointed curriculum. Another was that it’s ‘all about the questions!’ In particular how these differentiate tasks and how it can help develop pupils.

Flexible maths – The Michaela Community School Maths Team (@BodilUK, @danicquinn & @naveenfrizvi)

The Maths team from Michela school delivered a session discussing their approach so far (being a new school they have only year 7 and 8). They showcased their knowledge booklets which have been used to aid lesson teaching and have a strong focus on developing mathematical vocabulary and knowledge through ‘drilling’ the basics. It was interesting to hear their approaches and the booklets that were shared at the event will probably be a valuable resource!

Developing problem solving skills – Colleen Young (@ColleenYoung)

Colleen Young’s talk had a focus on developing the problem solving classroom. She suggests that the teacher-student relationship is a key aspect of this development. Additionally our use of vocabulary and students understanding of this could develop. When tackling ‘problem solving’ questions students should not fear just trying something; there should a resistance in the urge to rely on the teacher as ‘mathematical guru’. She provided lots of resource suggestions and her whole talk can be found on her blog!

A five year GCSE – Kris Boulton (@Kris_Boulton)

Kris talked about how we can better our own teaching, in particular discussing examples where he had not explained or taught topics well and had then adapted or radically them at the next time of teaching. He also identified that there is plenty of time to teach the new curriculum, but we need to focus on teaching well in the first instance and sequencing the content better. He concluded his talk by saying “Mathematics is mathematics!” This was a suggestion that we should be teaching students maths from (at least) year 7 through to the end of (at least) year 11 and we shouldn’t ‘start teaching GCSE in year 9 or year 10’.

Closing remarks – Jo Morgan (@mathsjem)

Jo was the concluding speaker raising some of her talking points and concerns about the new GCSE. One that stood out to me was the mention of Gove claiming the new GCSE would include fundamental mathematical content – but who decided what is fundamental (e.g. trig ratios in foundation)?

Evening networking

Having recently joined the Chalkdust magazine team Jo kindly allowed me to distribute copies around the room for people to take. It was really well received and I hope everyone from #ChristMaths15 enjoys reading issue 2 as much as I did!

The evening of the event saw maths teachers talking, completing Jo’s quiz and solving puzzles from Emma Bell (@El_Timbre).

An excellent ending to a very well organised and successful event! Well done Jo!

# 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.