The goal of this lesson is to start interweaving the development of the function concept with the development of formulas for volumes. This work will start with cylinders and progress to cones and spheres in later lessons. 

Because students have not yet learned these formulas, the context of filling a cylindrical container with water is useful for developing the abstract concept of functions. It makes physical sense that the height of the water is a function of its volume even if we cannot write down an equation for the function. At the same time, considering how changing the diameter of the cylinder changes the graph of the function helps students develop a geometric understanding of how the volume is related to the height and the diameter. 

In this lesson, students fill a graduated cylinder with different amounts of water and draw the graph of the height as a function of the volume (MP4). They next consider how their data and graph would change if their cylinder had a different diameter. The following activity turns the situation around: When given a graph showing the height of water in a container as a function of the volume of water in the container, can students create a sketch of what the container must look like?