Begin your lesson by asking your students the following questions:
- Why do soups and pops and other food containers come in cylindrical containers?”
[Students may answer that the containers are easier to open, easier to make, cheaper, you can fit more food in a cylinder, or other various answers]
- When food companies ship these items, do they fill the shipping boxes completely?
[Students may answer no and point out there is gaps when cylinder objects are stacked together]
- When food companies ship these items, what type of package do they use? Is it a cylinder or prism?
[Students may answer or you direct them that shipping is done in boxes because they stack and are packaged better and more securely.]
- If we ship items in prism boxes for packaging, why do we not make containers that fit this package better? What if we shipped items in prism containers?
Give students the Cubed Cans activity sheet and read through the introduction problem together.
“Why do companies choose to put cylindrical objects in containers that are rectangular prisms? Do cylinders hold more or less volume than prisms with the same volume? Food Containers Corporations has hired you to design a new container for various items they currently ship in cylinders. They would like you to keep the volumes the same, but explore various prisms. Your task is to take one of the cans provided and convert it to a rectangular prism. You will need to prove the volumes for the containers is the same and will want to record your notes and calculations as you report back to the company your findings. The company would like to save money and use as little surface area as possible.”
Once you have briefly given an overview of the task, allow students to choose which containers they will be using. Their task will be to measure out the dimensions of their can. Once they have the measurements, they will be calculating the volume. You will want to point out to students that they are going to have to create prisms that involve decimals or fractions to create a volume that is approximately the same as the cylinder.
Allow your class to direct themselves in self discovery by measurements and calculations. If some students are struggling you can discuss with them steps need. Below are a couple of suggestions and tips to help facilitate struggling students.
Ask struggling students, “What pieces of information will we need from this cylinder?”
[We will need to know the radius, height, and use the formula V = π·r2 to calculate the volume.]
Once students have expressed this information, you will want to review the way volume of a prism is calculated. Students should be refreshed that the same three dimensions are multiplied together to create volume. We will need to know that V = l·w·h to calculate volume of a prism.
Some students will have the volume, but find a problem working backwards to create the dimensions of the prism. As they do, you will want to assist them in realizing it takes three dimensions multiplied together to make that volume. Three dimensions that are the same will create a cube, while dimensions that differ will create a rectangular prism.
Ask students, “If we know the volume of the cylinder, how can we create a prism that has the same or similar volume?”
[We can multiply three dimensions to equal that same volume. For example, if the volume is 45, we can make a prism that is 3 × 3 × 5 = 45.]
Once students have changed their volume to prism calculations, have them think-pair-share their results with others in class or their groups. In doing so, they will see other people’s results.
Since we have created prisms that hold approximately the same volume as the initial cylinder, we are going to explore if the two containers have the same surface area. Objects with less surface area require less material and can save companies money in the long run. Have students calculate the surface area of the cylinder and the surface area of the cube.
Remind students:
Surface area of a cylinder = 2·π·r2 + 2·π·r·h
Surface area of a cube = 2lw + 2wh + 2lh
Have student’s think-pair-share and compare their observations before sharing their findings with the class. Depending on the type of prisms created by the students, their surface areas may be in very different ranges. Allow students to discuss fellow classmates results to see if certain prisms were close to the cylindrical surface area, or quite different. One pattern to look for in the students dimensions are as the prisms became closer to a cube, the surface area would continue to decrease. The least surface area for the prism would be a perfect cubic box. Any prism that is longer on one or both sides will increase the surface area.
