In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?

Find a quadratic formula which generalises Pick's Theorem.

This article for students and teachers tries to think about how long would it take someone to create every possible shuffle of a pack of cards, with surprising results.