Today saw the third session of the primary school math circle we’ve been running for kids in Year 2 to Year 6. (I was absent for Session 2, where my wife covered mathematical card tricks.)
- (Medial) graph representations of knot projections, moving backward and forward between them (see http://en.wikipedia.org/wiki/Knots_and_graphs)
- The importance of assigning an (under/over) decision to crossings
- Getting the kids to think of their own ideas for what knot equivalence might mean (shape, size, rotation, deformation – of what type) etc.
The key innovation we used here, which I think really brought the session to life, was to get the kids actually physically making the knots using Wikkistix, wax-coated string which allowed them to make, break, and remake their knots.
The kids really ran with this, and made their own discoveries, in particular:
- One child discovered that some knots corresponding to the graph could be manipulated to produce the unknot, some could not.
- A child discovered that it is possible to produce two interconnected knots, forming a link. Another child came to the same conclusion from the graph representation.
- Consider the graph with vertex set and edge set (is there a name for these graphs?). One child completely independently found that for even, this corresponds to a link of two knots, whereas for odd, it corresponds to a single knot.
I certainly had a lot of fun this afternoon – I hope they did too!