## Cubed Cans

- Lesson

In this lesson, students will use formulas they have explored for the volume of a cylinder and convert them into the same volume for rectangular prisms while trying to minimize the surface area. Various real world cylindrical objects will be measured and converted into a prism to hold the same volume. As an extension, students may design and create a rectangular prism container according to their dimensions to compare and contrast with the cylinder.

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·r^{2}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·r^{2}+ 2·π·r·hSurface 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.

### Reference

Connected Mathematics, Filling & Wrapping, Prob.4.3

- Various cylindrical cans
- Rulers
- Flexible tape measures (optional, if you prefer the circumference measured)
- Calculators
- Cubed Cans Activity Sheet

**Assessment Options**

- Have students measure and create a prism out of paper to match one they came up with during the lesson. Have other students calculate to the volume and surface area to check if the measurements are approximately the same as the cylinder.
- Use the Cubes Cans Activity Sheet as a form of assessment.

**Extensions**

- Have students create a cylinder based on a rectangular prism.
- Give students 36 blocks. Students will need to create a box using the dimensions 1 × 1 × 36. Next, ask students to create other boxes from the same 36 blocks. You will want to clear up any misconceptions that 1 × 36 × 1 is a different box. It is in fact the same as the previous box, but in a different orientation. As students create the boxes, ask them to find the surface area as well. They can do these two ways. They can use the length, width, and height dimension formula or students can count the squares on each side of the box they created. This will allow for a concrete way for students to explore finding surface area and why the formula works. Once students have found all possible boxes, ask students if they see any patterns in the dimensions of the boxes and the surface areas. Lead students in discussing that as the shape of the boxes became closer to a cube, the surface area decreased. The lowest surface area we could create would be the shape that is closest to a cube.

**Questions for Students**

1. How does the volume formula for a cylinder relate to the volume formula of a prism?

[They both require calculating the area of the base first, then multiplying it by the height to find the overall volume.]

2. How can the volume of a cylinder can be determined without filling it with objects?

3. Do cylinders and cubes with the same volume have the same surface area?

**Teacher Reflection**

- How were students manipulating dimensions to create a prism with the same volume as their cylinder?
- How did the students demonstrate understanding of the materials presented?
- Were concepts presented too abstractly? Too concretely? How would you change them?
- Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer?
- What content areas did you integrate within the lesson? Was this integration appropriate and successful?

### Learning Objectives

Students will:

- Use and explore volume formulas for cylinder and prisms.
- Create dimensions for a prism based on a fixed volume.
- Solve problems using the volume formulas.
- Explore surface areas.

### NCTM Standards and Expectations

- Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

- Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

- Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.

### Common Core State Standards – Mathematics

Grade 6, Geometry

- CCSS.Math.Content.6.G.A.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Grade 6, Geometry

- CCSS.Math.Content.6.G.A.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Grade 8, Geometry

- CCSS.Math.Content.8.G.C.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.