AS Maths and Mathematical Thinking

I’ve been with my AS Maths group for 10 teaching weeks now and thought it would be a good opportunity to use the last lesson of the week investigating an nRich problem. We’ve just finished the C1 work on Arithmetic Sequences, so “Prime Sequences” seemed ideal. Also, many of the group are also Further Mathematicians who I thought should benefit from such extension work.

With a significant proportion of international students in the class (who usually sit in native pairs), I first asked them to form new pairs such that there was no shared mother tongue. Their first task was a very simple wordsearch of mathematical terms, with one copy per pair. This served several purposes: to force them to interact and work cooperatively; to reduce the tension of working with somebody new; and to create an atmosphere suggesting that this lesson might be about playing and investigating. Of course there were small packets of Haribo for the winning pair!

I then projected the list of prime numbers provided on the nRich page and asked pairs to contribute any APs they could spot. We used the usual notation (a, d) and commented on the length of the sequences they found (mostly 3, one 4). I tried to emphasise the surprise of the result described at the top of the page but they seemed quite nonplussed!

Subsequent pair-work was then to work through the question prompts from nRich, giving me plenty of time to listen, circulate and intervene. This was my opportunity to tease out their thinking and to play devil’s advocate when they were trying to justify their thoughts. This was the most interesting part of the lesson from my point of view. Even trying to justify that a prime AP couldn’t begin with 2 was a novel challenge for them, but one that they struggled with and eventually worked through to a satisfactory explanation. Great stuff.

Within the hour, they reached the point of debating why d=2 would be impossible, except in the case of 3, 5, 7. Many ideas were circulated, some quite descriptive, and others incorporating more mathematical language. As a whole class we worked together to build the most convincing argument.

Another interesting point was that throughout all their work, the longest AP anyone could find was of length 6 (and even that took significant time to discover). I asked if length 100 or even 1 million might be possible and they strongly doubted it. When I then reiterated Green and Tao’s result, suddenly it was much more exciting for them and, moreover, they really began wondering “how on Earth” such a result could be proven.

As a group who generally prefer to work on the ‘need to know’ exam basis, I am glad I gave them this opportunity and that they immersed themselves in it. Let’s see if on Monday they ask for more, or for a ‘normal’ lesson!

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