One of the jobs I have been doing this half-term holiday (in England) is going through the OCR sample assessment material and using the questions to create teaching resources for lessons in the new GCSE course. I don't think we will be using the OCR board to examine our pupils, as their questions seem to be a bit over-engineered to test what they need to test, however this over-engineering does make their questions an ideal classroom resource where pupils can work together to discuss approaches to solving them in an environment supported by a teacher where necessary. One of the exam questions in the OCR SAMs centres around this image:
As a typical example of the over-engineering this question centred around the quadratic sequence created by the bottom row of cubes (as well as a part about plans and elevations) in each pattern, which will make a great practical resource that can be used with multi-link cubes or similar to allow pupils to explore quadratic sequences in class. However what also happened is it got me thinking about building towers like these and shifting focus from looking at how many new cubes, to how many cubes in total. What immediately struck me is that for arithmetic progressions (with positive integer terms) this was describing the difference between sequences and series. Now strictly this is not needed for the new GCSE qualification in England, but it might be the sort of thing you show pupils as an A-Level transition idea, and will definitely be taught to pupils doing A-Level Maths. Out of this I created this image (which was the best that my limited skills in paint could do!):
Now admittedly this is a fairly old fashioned sequence leading to the square numbers, but for me that made it all the more powerful as it comes with ideas that pupils will be familiar with already. Now I don't teach A-Level anymore (the interesting challenge of setting up GCSE maths in a school that had previously only had pupils from 11-14 was enough to drag me away from A-Level teaching) but I thought to myself, "What a great transition activity/introductory activity to have pupils build these towers and focus on the two questions 'How many more cubes are in the next tower' and 'How many cubes in total will there be in the next tower' as a way of illuminating the difference between a sequence (what changes) and the series (what total)." I can see an entire transition/introductory lesson with pupils building different towers and examining the sequence and series arising from each other's towers or alternatively I can see re-introducing sequences using these cubes at A-Level (I can see the towers also providing a good link to the recurrence relation as well) and then using the same towers to explore series in a later lesson.
Some other towers I came up with (and I am sure pupils will be a lot more inventive than I!):