Today we worked mainly on this 'guided investigation'. I chose

*one*theme from a number that the children had been working on in previous lessons - how many triangles on an n-pin geo-board. I did this because (a) I felt it would be useful for the children to experience the benefits of

*working systematically*, and (b) I wanted to explore the effects of more constraint, compared to the freedom I had offered the previous couple of lessons.

Before starting the lesson, we talked about the importance of making choices in mathematics, education and life more generally. After a brief bit of modelling, the children worked in pairs on the tasks, using virtual geo-boards.

There were a few critical points in the lesson. I had a number of conversations with pairs who disagreed over their guesses. It was interesting to talk about the reasons for their guesses. Most children also modified their guesses as they gained more information:

After inviting the children to find the number of triangles on the 7-, 8- and 9-pin board, the worksheet presents a systematic way of finding the number of pins on the 9-pin board, and an opportunity for reflection, as can be seen above. This led to a more systematic way of working on the 12- and 18-pin boards. Here is an example of one pairs' work on the 12-pin board:

I was particularly struck by the spontaneous labelling of each triangle, suggesting a way of exploring the patterns. Here are two examples of children's work on the 18-pin triangle, which were typical:

I showed some of these on the board as a way of focussing attention on interesting approaches, and then the children continued with their work for a while longer. At the end, I asked pairs to discuss what they had learned, and to write one sentence. Here are their sentences:

The comment about "pretty patterns" struck me: that working systematically can lead to something more aesthetically pleasing, and the patterns thus produced can lead to a more 'mathematical' way of seeing. I have also reflected that the additional constraint in this lesson seems to have led to the gaining of a new awareness for these children.