Rolling Around
How about asking pupils to imagine this "in their mind's eye" - no paper, no body language and lots of discussion?
Alternatively this can be tackled through a practical activity involving a shape and a coin.
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Is There a Theorem?
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
The Old Goats
A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?
Rolling Triangle
The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.