Illuminations: Cubed Cans

# Cubed Cans

 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.

### 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

### Materials

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

### Questions for Students

 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.] How can the volume of a cylinder can be determined without filling it with objects? Do cylinders and cubes with the same volume have the same surface area?

### 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.

### 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.

### 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?

### NCTM Standards and Expectations

 Geometry 6-8Create 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. Measurement 6-8Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.

### References

 Connected Mathematics, Filling & Wrapping, Prob.4.3
 This lesson was prepared by Corey Heitschmidt as part of the Illuminations Summer Institute.

1 period

### NCTM Resources

 More and Better Mathematics for All Students
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