Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
I started by counting in ones and I got a $12$-gon (that is a $12$-sided polygon - if you like long words you can call it a dodecagon).
Then I ruled lines counting round in $2$s. And I got .....?
Perhaps you do not need to put the numbers round the circles.
I tried $5$s (wow!) and $6$s (well!).
Each time I go on drawing lines until I get to the point where I first started.
Then I tried $7$s, $8$s, $9$s, $10$s, and $11$s.
Something interesting was happening.
Why don't you try it? What patterns do you notice emerging?
And what about counting round in $12$s?
Which shapes are the same? Can you think of a reason why?
Can you see a connection between the number in which you are counting around the circle and the number of sides in the shape you are making?