Visualising and representing

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance up the wall which the ladder can reach?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

What is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

An introduction to bond angle geometry.

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

What is the minimum number of squares a 13 by 13 square can be dissected into?