Round and round the circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Problem

Round and Round the Circle printable sheet

I started with a clock without hands or lines showing the minutes (except for those where there is a number). The $12$ was replaced by a $0$ and the numbers placed outside the face.

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A standard clockface with numbers 0 to 11 around the outside.



I drew straight lines to join up the numbers.

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).

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A clockface with lines joining each number around the outside to the two numbers directly next to it.

Then I drew straight lines counting round in $2$s. And I got ...?

 

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A clockface with a line joining 0 and 2, a line joining 2 and 4, and part of a line starting at the number 4.

Perhaps you could try without putting the numbers round the circles.

  

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A clockface without numbers around the outside with the same three lines as the previous clockface.

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?