In this activity, students investigate how the height of water in a graduated cylinder is a function of the volume of water in the graduated cylinder. Students make predictions about how the graph will look and then test their prediction by filling the graduated cylinder with different amounts of water, gathering and graphing the data (MP4).

In the digital version of the activity, students use an applet to do the experiment. The applet allows students to fill a digital cylinder that displays the volume of water added and the height of the water. The applet also lets students change the radius of the cylinder. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.

The goal of this discussion is for students to understand how the experiment can be modeled with a linear function.

Invite 2–3 groups to share their graphs, and display these for all to see. Consider asking students the following questions:

• “What do you notice about the shape of your graph?”
• “What is the independent variable of your graph? Dependent variable?” (Volume in cubic centimeters is the independent variable, height of the water in centimeters is the dependent variable.)
• “How does this graph differ from what you predicted the shape would be?” (I thought the graph would curve because of the round shape of the cylinder, but it's straight.)
• “For the last question, what point did you choose, and what does that point mean in the context of this activity?”

Conclude the discussion by asking students to predict how the graph would change if their cylinder had double the diameter. Alternatively, if it was possible to work with graduated cylinders with different diameters, display those graphs now. After a few responses, display this graph for all to see:

Explain that each line represents the graph of a cylinder with a different radius. One cylinder has a radius of 1 cm, another has a radius of 2 cm, and another has a radius of 3 cm. Ask students to consider which line must represent which cylinder. Ask, “How did the slope of each graph change as the radius increased?” (The larger the radius, the less steep the slope. This is because for a cylinder with a larger base, the same volume of water will not fill as high up the side of the cylinder.)

Previously, students were given a container and asked to draw the graph of the height as a function of the volume. In this activity, students are given the graph and asked to draw a sketch of the container that could have generated that height function (MP2). The goal of this activity is for students to practice connecting words to graphs of functions.

Select students to share the different containers they drew. Display their drawings and the graph for all to see. Ask students to explain how they came up with their drawing and refer to parts in the graph that determined the shape of their container.