Explaining, convincing and proving

The area of a square inscribed in a circle with a unit radius is 2. What is the area of these other regular polygons inscribed in a circle with a unit radius?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Charlie has moved between countries and the average income of both has increased. How can this be so?

Various solids are lowered into a beaker of water. How does the water level rise in each case?