When Polina Sabinin moved into her office on the third floor, she brought a collection of vases in different shapes. If you graph the height of water in the vase as a function of volume, how is the shape of the graph related to the shape of the vase?

This question can be posed to students from fifth grade through graduate school. In MATH114, students engaged in this activity remark on changes in the slope of the graph and conjecture that the graph has the same shape as the vase, if you tilt the vase at the correct angle. How would you write this equation? When is it true?

This brief talk will include a shape matching puzzle and a little bit of differential geometry.