Cutting Corners
I really feel that it is good to have such an open investigation
every now and again. From my experience it will need to be
introduced in some enthusiastic way that will capture the pupils'
interest and imagination.
The pupils' understanding of the properties of these isosceles
triangles will no doubt be enhanced. With some pupils you may get
into the whole idea of pythagoras and square numbers and square
roots. The concept of area is often challenged in this
investigation as our eyes seem to play trick on us and we have to
verify things in a more certain way.
Take for example this, where the half triangle has been placed in
the bottom right hand corner.
Triangles small-drawfile3

It looks to many people that it does not show a half, and what
about putting the smaller triangle in the top corner so that it
fits snugly.
There is also a good opportunity in this investigation to consider
Infinity, and particularly in respect of fractions.
It's not possible to predict what will happen but I have found
these activity to be very rich in what the pupils learn, develop,
image and communicate. There's not much more that we would
want!