given the sides, make a polygon

powerpoint: sides

and another, that goes off the grid

maxagon

finding the maximum number of sides that a (closed) polygon can have on a square dotty grid

generalisations can emerge
and created patterns can be justified by relating them to the diagram(s)

a ppt is here

for these tasks:

• it is simple enough to experiment with and record (dotty paper being better than squared)
• you can appreciate that longer diagonal lines don't seem to help achieve a maximum
• it is helpful to have a regular pattern, with symmetries sometimes
• different 'routine' growing patterns can be compared
• number patterns are created from a diagram so justifications can relate to the pictures
• odd and even 'n' numbers need to be considered - as is often the case 
• there are depth (proof) and breadth (e.g. angle sums) extensions 

the 36 sided maxagon for a 6 by 6 grid: