Exploring and noticing

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Can you work out what step size to take to ensure you visit all the dots on the circle?

A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

Can you work out how to produce different shades of pink paint?

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its speed at each stage.

Can you describe this route to infinity? Where will the arrows take you next?

Take a look at the video and try to find a sequence of moves that will untangle the ropes.