Making big shapes out of little shapes

We have been using the PASA material from Jo Mulligan and ACER and came across something really interesting with the kids. 

One question looked at a conventional tangram puzzle and asked:
 - how many of the small triangle will fit into the middle sized triangle?
 - how many of the small triangle will fit into the big triangle?

I thought this would be pretty obvious to the kids.

It wasn't.

They had lots of confusion and lots of struggle.

Seemed to me that we needed to spend some quality time with the pattern blocks.

So we got the blocks out and started to look at how we could use the small shapes to replicate the bigger shapes.

My favourite shape is the yellow hexagon. I was interested to see how the students might construct a similar hexagon using the other shapes.

Here are all the possible solutions they came up with.

I wonder if that is all of them - or can you do it another way?

Then we had to consider other shapes, of course. We have a very large set (super size) of the pattern blocks. Some students got a few of these and tried to remake them using the smaller blocks.

This is just making the same shape using different sized pieces. The combination of the trapezium and the equilateral triangle is interesting though. 

This one is my favourite - it took some patience.

Then we started to explore other material. We found a square can be made from 4 triangles or from 10 of these rectangles.

Someone had fun making this big triangle.

And finally we had some fun using a random selection of shapes to make some squares.

The yellow one is cheating - it is 5 right angles triangles overlapping each other.

Then we had one student attempt to make a square from equilateral triangles but she got stuck.

"This is impossible!" she said.

"Why?" I asked, innocently.

"Well, the corner on the square is too fat. The green corner is always too little."

And there we have it ladies and gentlemen - you can never make a square if all you have is equilateral triangles.

Nice work, Year 2.