A4 Containers
Describe the relationship between the surface area and volume of a container
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram, use a model)
This problem is about playing with shapes to see how large a closed container the students can make. We expect that they will mainly try cubical boxes but try to encourage them to use their imagination to produce other shapes.
They many need to be given (or shown where to find) the formula for the particular volume that they have made.
Much of mathematics is about finding the biggest or the smallest of a set of objects. We have other lessons along this line (see for example Square Milk Crates, Rectangular Milk Crates and Penny’s box). One area that looks at these kinds of problems is calculus. Students start on that in Year 12. But there are other optimisation problems where calculus can’t be used. For instance the Travelling Salesperson Problem. Here we have a salesperson that has a country route to cover. She knows the distances between customers and she wants to travel round, visit each customer once and then get home. Of course she wants to do this in minimum time. At the moment there is no good solution to this in the sense that given any configuration of customers we can't tell the salesperson exactly how she should arrange her travel. There are computer programs that will do a reasonable job but nothing that will find the precise answer every time.
Problem
Johann and his friends are playing around with scissors, tape and sheets of A4 paper. They decide to make closed containers that can be constructed from a single sheet of paper.
Johann made a net for his box where the net was a single peice that could be folded or bent to make the container. (Tabs weren't needed as he used tape to join the sides together.)
Mary made a shape where the net was not one piece but could be taped together to make the container (making a more efficient use of paper).
Petra made another shape again where one face was a square.
Who made the container with the biggest volume?
Teaching sequence
- Begin by giving the students a piece of A4 paper and ask them to make a closed container. At this stage keep the requirements open.
- Display the containers and discuss:
Which is the largest?
What criteria have you used in making your decision?
Which container has the largest surface area? How do yo know?
Which container has the smallest surface area?
Which container has the largest volume?
Do any of the containers have square faces? - Pose the problem to the class.
- Ask for their initial ideas about which container has the greatest volume. Ask them to explain the thinking behind their guess.
- Takes votes for the containers. List results on the board.
- Let the students investigate the problem in pairs
- As the students work ask questions that focus on their understanding of area, volume and perimeter.
What is the area of this faces? How did you work it out?
What is the volume of your container? How did you work it out?
What sort of container do you think will have the largest volume? Why?
How did you start on this problem?
What understandings are you using to solve this problem? - When the students think that they have a solution for the largest container ask them to make the container from a single sheet of A4.
Share and discuss containers.
Do you think we have found the container with the largest volume? How do we know?
Solution
There is no one solutions to this problem. The idea here is to explore shapes and to find ways to measure their volume.
| Attachment | Size |
|---|---|
| A4containers.pdf | 55.31 KB |
| A4containersMaori.pdf | 74.45 KB |
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