Three Way Mix-Up

Some students come to us having little experience of working on unknown problems. When building up confidence I like to start with a few 'content light' problems.

Three Way Mix Up is one of my favourites. I gave it to some trainee teachers who were going to teach secondary school maths. I was fascinated by their approaches.

The Problem: Jack has three blue tiles, three green tiles and three red tiles. He put them together like this to make a square. 

He made a rule that you could not put two tiles of the same colour beside each other. Can you find another way to do it? Can you find ALL the ways to do it?

We discussed whether we would count rotations and reflections as distinct. We agreed we would count them as different arrangements in this case. This is arbitrary though and may have distracted some groups who spent more time considering the rotations than entirely necessary.

I gave them sets of tiles to work with and gave them plenty of time to explore.

I captured some of the recording that occurred. I was intrigued  the varied approaches. I was also able to go round and provoke the groups to refocus by saying...'are you sure there are only 24? That other table has more. How can you be sure you haven't missed any?' And then to the other group, 'how can you be sure that you haven't double counted? The other table are convinced there are only 24'.

I was particularly intrigued by this last contribution. They decided to find all the arrangements and then subtract the number where the tiles touched. I'm not sure that it was the most efficient method!

One thing I particularly like about this task is that you can get as far, or maybe further, by playing with the tiles than by using more advanced techniques such as combinatorics!