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.
- Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge...
- Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures....
- Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between...
- Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,...
- Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
- GLE 0606.1.3
- Develop independent reasoning to communicate mathematical ideas and derive algorithms and/or formulas.
- GLE 0606.4.3
- Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes.
- GLE 0606.4.4
- Develop and use formulas for surface area and volume of 3-dimensional figures.
- SPI 0606.4.4
- Calculate with circumferences and areas of circles.
- SPI 0606.4.5
- Determine the surface area and volume of prisms, pyramids and cylinders.
- SPI 0606.4.6
- Given the volume of a cone/pyramid, find the volume of the related cylinder/prism or vice versa.
- 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...
- 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...
- Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the...
- Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles,...
- Know and understand the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.
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- 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.
Questions for Students:
- How does the volume formula for a cylinder relate to the volume formula of a prism?
- 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?
- 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.
- Various cylindrical cans
- Flexible tape measures (optional, if you prefer the circumference measured)
- Cubed Cans Activity Sheet