a ppt is here
teacher's guide here

















then there are always 3D rooftops to explore...

 width of 2
width of 3
 squares













relating the perimeter to the numbers of inside dots


alternatively all three variables can be related to the (unknown) length (L) for the various (fixed) widths







you could also explore isometric shapes where the unit of area is the small equilateral triangle...

how many ways are there to join 6 dots with a continuous line, not taking your pencil off the paper?

• each point must have only two lines (arcs) coming from it
• rotations and reflections are not counted as being different

there are 12
the regions are 2-coloured to better show the patterns:

Kobon Fujimura posed the problem (fairly recently) of finding the maximum numbers of triangles (not overlapping) for a given number of straight lines.

Students could be asked to explore this, for up to 10 lines:

some diagrams for
n = 3 , 5 and 7 are shown, giving totals of
1 , 5 and 11 respectively

for n = 4 , 6 and 8 the totals are 2 , 7 and 16

can students produce diagrams for these?




beyond small numbers of lines the diagrams become complex, with tiny triangles
for solutions see e.g. Wolfram

 (there could be 26 for 10 lines)



some examples:

n = 9 giving 21 triangles
n = 10 (shown symmetrically) giving 25 triangles (it might be possible to produce 26)









n = 18


n = 20

















Saburo Tamura proved that an upper limit to the number of triangles is given by the largest integer less than or equal to
n(n - 2)/3


art work