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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Grid Lockout

Age 14 to 16 Challenge Level:

Why do this problem :

In this problem it seems something of a surprise that the square areas which can be constructed should apparently be constrained in this way (you cannot make an area of $4n + 3$ square units). It feels as if there ought to be more freedom. This problem encourages analysis and the forming of conjectures, and especially justifying or accounting for pattern.

Possible approach :

Ask the group to draw a variety of squares using paper with dots, to calculate the area of each and to keep a record.

Ask the group to organise their record in a way they think useful, and invite conjecture about the situation.

Ask the group to look at the way they decided to organise their results and to decide what additional results might usefully be acquired next.

Ask members of the group to share their thoughts.

Once the proposed additional work has been done again invite conjecture about the situation: any comment, or things to try.

Suggest to the students that they look at the remainder when the area values are divided by 4, leave some thinking time before again inviting conjecture.

Key questions :

  • What are the freedoms the problem gives you and what are the constraints?
  • What does this problem invite you to explore?
  • How can you organise your exploration so that analysis can follow most easily?

Possible extension :

Just Opposite

Possible support :

Tilted Squares