This lesson is the beginning of a sequence of lessons that interweaves the development of the function concept with the development of formulas for volumes of cylinders and cones. 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 function. 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 later lessons they will learn a formula for that relation.
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. 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?
- Create a graph of a function from collected data, and interpret (in writing) a point on the graph.
- Draw a container for which the height of water as a function of volume would be represented as a piecewise linear function, and explain (orally) the reasoning.
- Interpret (orally and in writing) a graph of heights of certain cylinders as a function of volume, and compare the rates of change of the functions.
Let’s fill containers with water.
Students work in groups of 3–4 for the activity Height and Volume. Each group needs 1 graduated cylinder and water.
- I can collect data about a function and represent it as a graph.
- I can describe the graph of a function in words.
A cylinder is a three-dimensional figure like a prism, but with bases that are circles.