This activity was devised as part of the ‘Researchers in Residence’ regional project in the South West. You can also read about this activity on the Researchers in Residence website. The resource was inspired by Jack, a researcher at Exeter University - here he is talking about his research.
The resource consists of
I also used a file for the Interactive whiteboard so I could move cards and symbols about in front of the students.
You will also need to make something with three coloured strings as in the photos. We used three ropes tied to coffee mugs or three coloured braids from a craft shop attached to a lump of wood with holes drilled in it.
This activity shows students how something very concrete can be modelled by means of abstract symbols conforming to a set of rules. The second half of the activity shows how an algorithm can be used with the symbols to predict what is happening in reality
What I did
I used this activity in two, one hour slots for a lunch time Maths Club. In the first slot I introduced the idea of the 5 symbols to represent moves of the three strings, talking through the worksheet, but with slides on the board rather than giving the students the worksheet. I put the first example up as cards and symbols and asked them to replicate the braid with actual bits of string attached to a lump of wood.
They then showed that the braid undid if you pulled the ends of the strings.
I then showed them a plait in cards and asked them to write down the symbols, and showed them a string of symbols and asked them to arrange the cards for those symbols.
I then asked them to arrange cards for aba and bab and we discussed whether these were in fact the same. Once everyone was happy that they were the same I introduced the names “delta” and “delta inverse” and gave out the statement cards, one copy of each statement, shared between the students. I put a big label saying “TRUE” on one table and one saying “FALSE” on the other, and students worked alone, in pairs or in groups to decide on which table to put their statements.
Once all statements were placed, we gathered round the TRUE table and I got the students to notice similarities. They put the statements into sets and used this to challenge some of the statements, which had been put onto the FALSE table, which they then reclaimed.
This took us to the end of the first hour.
In the second session I gave out the sheet of true statements. I was going to go through the example of simplification on the board, but the computer wasn’t working, so I had to give out the sheet and talk through the three stages of the algorithm. The students worked through the example for themselves on a fresh sheet, line by line, then I put up the two 10 symbol tangled braid examples up on the board and asked them to apply the algorithm to see which was the least tangled.
Those who finished first tried braiding the actual three strings by the 10 symbol instructions, then pulled and looked to see if they could see the reduced moves, which they could.
When everyone had finished I showed them the video of Jack talking about applying these ideas to quantifying energy from the sun and talked a little about groups.
I did this with a voluntary maths club, which is a self-selecting group of first and second year A level and IB students, so they were very bright and quick on the uptake. Many of them are taking Further Mathematics. They took to it like ducks to water although it was all new to them, and came back for the second session the next week. There were few enough of them to make it possible for everyone to see one of the two blocks of wood with three strings and for everyone to get to actually play with one at some point, but some preferred to move to the cut out cards very swiftly.
The second session was a lot more challenging than the first, so the first could easily be used with younger students I think.
I teach groups to the higher IB students, so I will be interested to see whether those who came to this activity tie it in with the syllabus work. I feel it is rather a nice gentle introduction to that kind of abstraction, with a real application to give it a context and a reason for its existence.