shift and swap

Start with three people (A,B,C) in positions 1,2,3 as shown.

Start with the single move SHIFT (SH) where everyone shifts one space along (with the end person going to the other end) as demonstrated here:

I then asked students to answer these questions:

Q1 The symbol * means do one move then the next. Can you describe (in words) the sequence of moves SH*SH
(see picture on the right)

Q2 What about SH*SH*SH?

Q3 What positions (combinations of ABC) are possible using only SH? Which are not?

Then I introduced a new move SWAP (SW) where people in positions 1 and 2 switch places like this:

Then I asked these questions:

Q4 What positions can be generated with pairs of moves combining SH and SW?

Q5 Is SH*SW = SW*SH?

This worked well with students working in groups of 3. When students had arrived at their solutions, we discussed ways of showing our results (eg in tables)

Then we extended the task: what happens with 4 people. Are the moves SH and SW enough to generate all the combinations? This was a really valuable exercise; students had to find ways of systematically listing all the possible combinations before they could think about which ones were generated by sets of moves. Here is an example of one group's work; I really like the way they systematically analysed the moves!

Then I asked the question: Can you think of one more move that will generate all the combinations with just pairs of moves?? I don't know the answer to this myself yet!

Next lesson we followed this up with triangle symmetry.