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How far have these students walked by the time the teacher's car reaches them after their bus broke down?

Rolling Around

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

N Is a Number

N people visit their friends staying N kilometres along the coast. Some walk along the cliff path at N km an hour, the rest go by car. How long is the road?

How Far Does it Move?

Age 11 to 14 Challenge Level:


Try to approach the problem systematically.

As you go along try to understand why the graph takes the shape that it does:

  • by relating it to the rolling polygon and the journey of the dot
  • by trying to predict what will happen before you set the polygon rolling

Could the dot have been on the centre of a polygon?
Try for each of the polygons.

Could the dot have been on the centre of the base of a polygon?
Try for each of the polygons.

Could the dot have been on the centre of one of the sloping sides of a polygon?
Try for each of the polygons.

Could the dot have been on the centre of a side opposite the base of a polygon?
Try for each of the polygons.

Could the dot have been on a vertex opposite the base of a polygon?
Try for each of the polygons.

Could the dot have been on a vertex on the base of a polygon?
Try for each of the polygons...

Alternatively...

 

  • try all possible positions of the dot in a triangle,
  • and then in a square,
  • and then in a pentagon,
  • and then in a hexagon...