Representing Three-Dimensional Figures in Two Dimensions
In the junior grades students are expected to move flexibly between two- and three-dimensional representations of a figure. Two-dimensional representations can include nets, “rectangular” views, and isometric sketches.
A net is a pattern that can be folded into a three-dimensional figure. A net must include all of the two-dimensional faces of the figure. For example, a net of a cube must have six squares, a net for a triangular prism must have two triangles and three rectangles, and a net for a pentagonal pyramid must have one pentagon and five triangles.
In addition to knowing the faces of a figure, students must develop a spatial awareness of how the faces “fit” together. Consider as an example these two patterns:
Although both of these patterns are composed of the correct number of faces to form a cube, only the one on the left can actually be folded to form a cube.
Initial experiences with nets should be concrete. Paper folding and Polydron pieces are examples of manipulatives that serve as an entry point for investigations involving nets of solids. Polydron shapes can be snapped together and taken apart easily, and folded and snapped together to form a solid. They are limited, however, in that the number of shapes is finite. Paper folding takes longer to prepare and is less “user-friendly”, but it has no limits with respect to the type and size of shapes.
Student experiences with nets should be varied, and should include working from net to solid and from solid to net. When students are able to “unfold” a solid, they are more likely to understand the relationships between the faces. These relationships include shared edges and edge length. Similarly, students should be given examples and non-examples of nets to fold into solid figures. Consider these two nets for triangular prisms:
Net A can be folded into a triangular prism, but net B cannot.
For more information on representing three-dimensional figures in two dimensions, please visit the following resources: