Alternatively, you could explore the problem using this interactivity.

Draw some polygons by joining the dots on a $3$ by $3$ grid.

What about on a $3$ by $4$ grid, or a $3$ by $5$ grid?

What about on a $3$ by $n$ grid?

Can you explain the pattern by which the 'number of sides' increases?

Explore some polygons on grids that are $4$ dots high.

What is the maximum number of sides a polygon could have on a $4$ by $n$ grid?

Can you explain how you know?

What is the maximum number of sides a polygon could have on a $6$ by $6$ grid?

AndÂ on a $6$ by $n$ grid?