Filling and Wrapping mathematical and problem solving goals:
Filling and Wrapping was created to help students:
- Conceptualize volume as a measure of filling an object
- Develop the concept of volumes for prisms and cylinders as stacking layers of unit cubes to fill the object
- Conceptualize surface area as a measure of wrapping an object
- Determine that the total number of blocks in a prism is equal to the area of its base multiplied by its height (the volume)
- Discover that strategies for finding the volume and surface area of a rectangular prism will work for any prism
- Explore the relationship of the surface areas of rectangular prisms and cylinders to the total area of a flat pattern needed to wrap the solid
- Discover the relationships among the volumes of cylinders, cones, and spheres
- Apply the strategies for finding the volumes of rectangular prisms and cylinders to designing boxes with given specifications
- Reason about problems involving the surface areas and volumes of rectangular prisms, cylinders, cones, and spheres
- Determine which rectangular prism has the least (greatest) surface area for a fixed volume Investigate the effects of varying dimensions of rectangular prisms and cylinders on volume and surface area and vice versa
- Estimate the volume of an irregular shape by measuring the amount of water displaced by the solid
- Understand the relationship between a cubic centimeter and a milliliter
The overall goal of the Connected Mathematics curriculum is to help students develop sound mathematical habits. Through their work in this and other geometry units, students learn important questions to ask themselves about any situation that can be represented and modeled mathematically, such as: How can the concept of volume as the number of unit cubes be transferred to finding volumes of shapes that are not rectangular? What techniques can be used to relate the surface area of curved surfaces to familiar area concepts? How is the idea of wrapping an object related to the idea of surface area? How is the surface area of an object related to its volume? What techniques can be used to find the volume of an irregular figure? Where do familiar three-dimensional shapes appear in the real world?
The Mathematics in Filling and Wrapping
In Filling and Wrapping students explore the surface areas and volumes of rectangular prisms and cylinders in depth. They look informally at how changing the scale of a box affects its surface area and volume. They also informally investigate other solids—including cones, spheres, and irregular shapes—to develop volume relationships.
Rectangular Prisms
Students begin by exploring the surface area of a rectangular box. The strategy for finding the surface area of a box is to determine the total area needed to wrap the container. Students create flat patterns that can be folded into boxes. The area of the flat pattern becomes the surface area of the box. This provides a visual representation of surface area as a two-dimensional attribute, though it is an attribute of a three-dimensional object.
The strategy for finding the volume of a rectangular box is to count the number of layers of unit cubes it takes to fill the container. The number of unit cubes in a layer is equal to the area of the base—one unit cube sits on each square unit in the base. The volume of a rectangular prism is the area of its base multiplied by its height.
The same layering strategy is used to generalize the method for finding the volume of any prism. The volume of any prism is the area of its base multiplied by its height.
The surface area and volume of a cylinder are developed in a similar way Students cut and fold a flat pattern to form a cylinder. In the process, they discover that the surface area of the cylinder is the area of the rectangle that forms the lateral surface plus the areas of the two circular ends.
The volume of a cylinder is developed as the number of unit cubes in one layer (the area of the circular base) multiplied by the number of layers (the height) needed to fill the cylinder. Because the edges of the circular base cut through the unit cubes, students will have to estimate the number of cubes in the bottom layer.
Cones and Spheres
Students conduct an experiment to demonstrate the relationships among the volumes of a cylinder, a cone, and a sphere. If all three have the same radius and the same height (the height being equal to two radii), then it takes three cones full of sand to fill the cylinder, and one and a half spheres full of sand to fill the cylinder.
These relationships may also be expressed as follows:
Effects of Changing Attributes
Students also investigate the effects of a change in dimension, surface area, or volume on the other attributes of a three-dimensional object. For example, if 24 unit cubes are arranged in a rectangular shape and packaged in a rectangular box, which arrangement of the cubes will require the least (the most) packaging material? By physically arranging the blocks and determining the surface area of each arrangement, students discover that a column of 24 cubes requires the most packaging, and the arrangement that is the most like a cube (2 by 3 by 4) requires the least amount of packaging. This is similar to ideas students have studied about plane figures: the rectangle that is most like a square has the least perimeter of any rectangle with the same area. (A similar relationship holds for a fixed surface area. The rectangular prism that is the most like a cube will have the greatest volume for a fixed surface area. This relationship is not explored in this unit.)
Through the context of designing an indoor compost box, students also explore the effects that changing a box’s dimension has on the volume of the box. Given the dimensions of a compost box known to decompose a half pound of garbage per day, students investigate what size box would decompose one pound of garbage per day They discover that they need to double only one dimension of a rectangular box to double its volume.
Students also look at the effects of doubling all three dimensions of a box. Making scale models of the original box and the new box helps students visualize the effect of the scale factor. Doubling each dimension of a rectangular prism increases the surface area by 2 x 2 = 4 times (a scale factor of 22) and volume by 2 x 2 x 2 = 8 times (a scale factor of 23). The surface areas of the two prisms, if looked at as flat figures, are similar figures with a scale factor of 2 from the small prism to the large prism. This exploration connects back to ideas in the similarity unit, Stretching and Shrinking.
Irregular Objects
Students are introduced to Archimedes’ principle of determining the volume of an object by finding how much liquid it displaces when placed in a container of water. In the process, they also investigate the relationship between milliliters and cubic centimeters.
Measurement
All measurements are approximations. In the work in this unit, this idea will become more apparent than usual. Students’ calculations of surface area and volume will often involve an approximation of the number it, and they will often use a calculated amount as a value in a subsequent calculation. Be aware that although students’ answers will often differ, many different answers may reflect correct reasoning and correct mathematics.