Most activities we have done in class have hit every factor on the list of what makes a good math circle problem. For example, engaging, hands-on, minimal lecture, low-level entry, , abstract, etc. I particularly think that the most important thing is that the activities are hands-on and include something that teachers can take back to their classrooms. Two activities that we have done in class come to mind.
The first activity is the activity we did the first day of class where knots were displayed on the front table, we then had to draw them and describe them in various ways on the board, which sparked a great discussion. This activity I think would be a great starter activity to get teachers moving and talking about knots.
The second activity I think we can use in a variety of ways is the tangle dance. Every time we break out the ropes and actually take the time to manipulate things, it really brings everything together and allows us to have great discussions.
I have not yet thought of any specific "problems" per say, but I think it is important to introduce knots on an entry-level basis, with small hands-on activities that will keep teachers engaged throughout the PD event.
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