Lucy Dasgupta pointed us in the direction of the Underground Maths resources, which are available here:
There is a wealth of excellent resources designed to make students think and to make connections between different areas of maths, particularly in relation to some pervasive ideas, such as transforming and invariance. We would like to recommend the two activities we tried, Parabella and Powerful Quadratics, both to be found at the Quadratics Station.
Alexandra Hewitt shared an idea for introducing Eigenvectors by transforming the whole plane into a grid of parallelograms and trying to identify the invariant lines by eye. This helps students to take on board what Eigenvectors actually are, as well as realising the need for a neat algebraic method for finding them.
We also looked at using Tangent Fields to introduce Differential Equations. After creating a tangent field of our own from scratch, we found that drawing in possible solutions onto printed tangent fields (we used Autograph, but other packages will do the same) was the mathematicians' version of colouring books for adults! Again, this was designed to help students understand what was going on when they solve a differential equation.