groups of six
This is a reflection of a lesson where year 7 students were grouped into sixes. It is based on activities from the excellent book People Maths, available from the ATM.
Video 1: Students stand in height order; by swapping with neighbours, what is the minimum number of swaps that will reverse the order? Can you see a pattern for different size groups? What's interesting about this video is that Harrison at the front, who is usually one of the 'less engaged' students in the class, is fully involved and spotted the patterns before anyone else! Video 2: Students stand in a line of 6 and 'sing' a note each in an ascending scale. Move A is to swap all adjacent pairs (ie 3 swaps). Move B is to swap the middle 2 pairs. If you repeatedly alternate between Move A then Move B, making sounds after each move, what happens? This task was fun but the 'singing' was awful! It might be fun to do this with keyboards or other musical instruments. What other sets of moves have this property? Video 3: Students stand in rows of six, facing either left or right. It is up to them which way they face at the start. They can turn 180 degrees if they are facing someone, but can not move if they are not. The objective is for the group to complete as many rounds as possible until no one can move. What is the best starting position? Extend this to larger groups, maybe even the whole class  how many moves are possible for a line of n people? What starting positions are best? 
