Geoboards

5 X 5, and even 3 X 3 geoboards are great for provoking some careful geometric thinking. We've been using them in Year 5. (Once again there are online environments to use, like this one - http://nrich.maths.org/2883 but we didn't use these this time.)


We used the 5 X 5 ones for investigating possible squares:
How many different-sized squares are there? At first no-one could find any other square than these ones:

Then I gave a clue: I twisted my board round 45°. Immediately, lots of the class saw some others:

and then we went home. It was only when I was checking out a book by Caleb Gattegno, who invented and popularised the geoboard, that I realised we'd all missed a few more. Can you see what they are?

To answer that question, we can label the pegs like I did with Year 5. Then we can name the square by the letters at the four corners:

A	B	C	D	E
F	G	H	I	J
K	L	M	N	O
P	Q	R	S	T
U	V	W	X	Y

The next question was: for all of those possible squares, how many different positions are there?

and, most interesting...
If the smallest square of nails has an area of 1, what is the area of the other squares?

We also asked the same questions of the triangles possible on the 3 X 3 geoboard (there are more).


That bottom right triangle, what is its area? How do you work that out?


Different students had different methods. 

Here are some students talking about their thinking:




One pupil made an observation on method. She said, "If you don't know how to do something, split it into bits." Useful advice in this case.