Precious Pentominoes

Here is a task that I made up that I have recently tried which provoked a lot of thinking with students and adults alike.

From a set of 12 pentominoes, choose two tiles to make a closed shape that has symmetry. For example these are NOT allowed:

Once you have a design, then work out the perimeter:

the area:

and the number of sides:

Now calculate a 'cost' for your design as follows:

What is the most 'expensive' design you can make?

The students got stuck into this immediately using the pentominoes to create a variety of designs:

They recorded their thinking as they went:

 

After a while, we recorded the costs that students had found on the board. Now the students were really keen on finding a more expensive design. There were shouts and screams of delight as groups found new, more expensive designs. After a while, all groups had reached a consensus of what the more expensive design was (I won't spoil it for you by revealing the answer!)

As there was still time left, we extended this by allowing three pentominoes and removing the symmetry constraint. 

As we wrapped up, we asked them what strategies they used. They noticed that the area was always the same but that they had to take care when calculating the perimeter and the number of sides to avoid 'double' counting:

They also noticed that it was important to maximise the number of sides as this was the multiplying factor. So long, 'straight' pentominoes were not as good as the pentomino shaped as a cross.

But perhaps the best comment was as I was leaving:
"Please come again soon and please bring some more questions that will give me a mathematical headache!"

We also tried this with our educators at a Capacity Building session today. Again, the level of engagement was high. We tried also bringing some of the ideas from Peter Liljedahl's research about using vertical non-permanent surfaces, giving verbal instructions, creating visible random groups, and only answering 'keep thinking questions'.

This is when one knows that it is a good task: that adults and students alike are captivated by it.