For this lesson, I invited the children (upper primary, ages 8-11) to work on one of these tasks, based on their work from previous lessons.

**finding the number of triangles**on various geo-boards. We talked a bit about what it meant to 'work systematically' at the start of the lesson, as well as what made a triangle the same or different.

Harry was one of a few children who were not sure how to work systematically on this, but realised that "finding triangles in random" resulted in duplication :

Sarah and Sophie discovered increasingly sophisticated methods:

**reflective symmetry**. Chloe found shapes with various numbers of lines of symmetry:

Charlotte found it hard to find shapes that had more than 1 line of symmetry on a 9-pin board, apart from the "nineagon":

Milos, the originator of the questions on symmetry, continued his work, conjecturing that shapes with various lines of symmetry could be found on 12-pin boards, but that "you can't make a four lines of sim on a nine pin board":

Jack continued his work on the symmetry of regular shapes. He made these wonderful conjectures:

**'overlapping' patterns**, following on from Stewart's work last week:

Children who worked on this didn't seem to be working as mathematically as the others, and I wondered about my choice to offer this as something to work on. However, a few children developed it in a nice way.

Elsie and Ellie chose a shape from our work on quadrilaterals (e.g. trapezium, arrow-head), and rotated this around the board to create some interesting patterns:

Callum was exploring 'rotating rectangle' patterns, and wrote a detailed description of what he did. I really like his account of experimenting with different types of board:

He seemed to have developed an efficient way of finding the interior angle of an n-gon. I asked him to find the interior angle of a hexagon, this is what he did:

There was a lot of excellent mathematics done here. On the negative side, a few children did not work as mathematically, or productively, as I would like. I wonder if this was a result of the freedom offered, perhaps of their view of my expectations, or the fact that it was Sports Relief day!

I can see four themes developing here: rotating patterns, reflective symmetry, angles in polygons and counting (triangles). There are conjectures that can be developed, such as Connor's angle method, or Milos' conjecture about squares on a n-pin board. Next week I intend for the children to work on virtual boards so that they can explore one of these four themes on boards with any number of pins.