Fill Me Up
2 out of 3
Can you sketch graphs to show how the height of water changes in different containers as they are filled?
Problem
Below are some images of containers. Imagine you put them under a steady stream of water.
What would the graphs look like if you plotted the height of the water level against the volume of water as the containers fill up?
Draw a sketch graph for each container. You will need to consider which portions of each graph will be straight and which will be curved. Can you suggest suitable units and scales for the axes?
Beaker Image
| Conical flask Image
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Boiling tube Image
| Round-bottomed flask Image
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Pint glass Image
| Volumetric flask Image
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Once you have sketched the graphs, you might like to plot the graphs by collecting some data.
One way to do this is to collect suitable containers, add water in fixed amounts and measure the height at each stage. Do your experimental graphs match your sketches?
Getting Started
Why not try filling some containers with water, perhaps 15ml (a tablespoon) at a time, and measure the height of the water level at each stage?
Record the heights and plot graphs to show the shape.
Which parts of the graph are straight lines? Which parts are curved? Can you explain why?
Student Solutions
We received these good sketches from Christian and Kodai from the Munich International School, though I don't think the graphs should have ever become horizontal.
It is also worth noting that the graphs didn't always take into account the varying widths of some of the containers - e.g. when the water reaches the narrow top of the volumetric flask, how will the height of the water level be affected by an increase in the volume of water?
We also received good sketches from Priya, Stephanie and Aishwarya, also from the Munich International School.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
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.
Possible approach
Display the six pictures on the board.
"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 explain their reasoning 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.
Key questions
Possible support
Students may find it easier to start with the experiment and then explain why the graphs have the shapes they do.
Possible extension
Fill Me up Too is a challenging extension looking at the functions arising from filling conical vessels.