This project is deceptively simple, and amazingly rich with mathematical applications. It can be used in a variety of ways. The simplest and perhaps most dynamic use of this 3-D model is as a demonstration of the relationship between the volume of a pyramid and the volume of a cube with the same base and altitude. When a student sees the teacher holding a simple cube, which then opens into 3 pyramids, the formula for the volume of a pyramid is indelibly engraved in the students mind.
This project can be used at many levels. The quickest and easiest is, of course, as a teacher demonstration as described above. The most effective way is to have the students construct the project themselves.
The students can be given copies of the pattern below:
The teacher might want to discuss the mathematics involved in creating this figure, as the students work on it. The longest edge is the diagonal of the cube. The edges are interesting numbers; teachers might want to ask students to calculate the lengths of all the sides, beginning with labeling each side of the square "x". The solution to this task is shown below:
The students should follow the instructions, including cutting out the shapes.
Instructions: Cut the figure out of paper, then fold and tape it into a three-dimensional shape, as in the drawing below:
Explain that the figures that result are "lopsided" pyramids, but still pyramids.
The teacher can then ask groups of 3 students to put their 3 models together to form one solid. Most will put them together in such a way as to form a cube, as shown below. Since it takes 3 of these "lopsided" pyramids to form a cube, the volume of a pyramid is clearly seen as exactly one-third the volume of a prism with the same base area and height.
This demonstration shows that since it takes 3 pyramids to form the cube, then the volume of a pyramid is 1/3 the volume of a prism with the same base area and height. Thus the formula V=1/3Bh. The view above shows to 3 pyramids folded up to form a cube. The view below shows the 3 pyramids unfolded. This model is striped so that you can see the different surfaces more clearly:
If more time is available, the teacher could show the pattern to the students (perhaps on an overhead projector) and ask them to construct it themselves, either using The Geometer's Sketchpad or compass and straightedge. Again, the students would then cut out the patterns and put the models together to form a cube.
Hands-on projects such as this one allow students to have hands-on experience with geometric concepts. Research has show that this kind of visual, kinetic, and concrete learning, helps students to truly understand the material, as well as to remember the information.