Why do this problem?
requires students to draw graphs to represent a real-world situation. Once students have considered how the shapes of vessels affect the way they fill with liquid, they can use an experimental approach to plot the graphs for real.
Display the six pictures on the board (or hand out this worksheet
"Imagine that each container is filled up, 1 cubic centimetre of water at a time, and the height of the water is measured. Can you sketch
what the graph of height against volume would look like for each container?"
Give students time to sketch the graphs, working on their own at first. Then after they've had a chance to sketch all six graphs, ask them to work in pairs:
"Compare your graphs with your partner's. Did you sketch the same shape for each graph?"
Give students time with their partners to discuss similarities and differences between their sketches, and to resolve any differences. Ask them to come up with explanations for the key features of their sketch graphs and to prepare to justify their graphs.
Choose some pairs to come up to the board and sketch their graph for each of the six pictures. Invite them to talk through their thinking, and ask the rest of the class to offer critical feedback on their explanations.Two pairs could draw their graph simultaneously, one at each end of the board, and comparisons could be made.
Finally, suitable vessels could be gathered and data could be collected by adding water in fixed volumes and measuring the height of the water level. The resulting graphs could then be compared to the students' sketches.
When do we get straight lines on the graph?
What different types of curves can we get?
What does the shape of the curve tell us about the shape of the container being filled?
Fill Me up Too
invites students to consider the graphs arising from filling some unusually shaped vessels.
is a challenging extension looking at the functions arising from filling conical vessels.
Students may find it easier to start with the experiment and then explain why the graphs have the shapes they do.