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A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

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Eight Dominoes

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

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Holly

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Trominoes

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem :

This problem can be a good way to demonstrate mathematics emerging out of play.

Possible approach :

Using wood blocks (trominoes from three squares, like dominoes from two), arranged as a tower, a game of coordination and experimentation begins as players try not to topple the stack.

Pick up one of the pieces of wood and begin posing some questions about it.

  • Can these blocks form a square ? - other than the three side by side used for each layer in the stacking game.

Let the questions flow and get pursued. Try to encourage adjustment rather than abandonment when a line of questioning appears to run out. For example a five by five square can't be made because each block contributed three squares of area and there are 25 to be covered, so there must be a hole. OK, given there's a hole, can you make a square ?

Key questions :

  • What questions can you pose ? (offer an example : what rectangles can and cannot be made ?)
  • What others ?
  • Which seem like good questions to pursue ?
  • (later) What were our questions ? Where has each question taken us in what we now see or understand ? What came out of this that we didn't know already or didn't expect ?

Possible extension :

Equal Equilateral Triangles is a good next step.

Possible support :

Create a competition to produce all rectangles up to side length 10.

Perhaps use a digital camera or the camera in a mobile phone to catch arrangements quickly. A drawing record may sap energy that might be used more effectively pursuing the task.