One of the most beautiful results that can be illustrated by blocks is the fact that a cube can be decomposed into three identical pieces meeting along a diagonal of the cube (Figure 8), just as a square is decomposed into two congruent triangles by a diagonal line (Figure 7).
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Figure 7. The diagonal subdivision of a square into two congruent triangles serves as a prelude to a similar decomposition in three dimensions. |
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Figure 8. The diagonal decomposition of a cube into congruent pyramids can be illustrated by blockes built from corresponding templates. |
Decomposition models illustrate deeper ideas than do comparisons of volumes since they not only demonstrate relationships but also show why these relationships hold. Students should eventually come to see that all geometric relationships are based on reasons.
This particular decomposition property of the cube can be a bit misleading because it doesn't quite work for other rectangular solids. Although a diagonal always decomposes a rectangle into two congruent triangles, the diagonal decomposition of a rectangular solid will usually not produce three congruent pyramids (Figure 9). The three pyramidal parts will all have the same volume but not the same shape. This can be seen by pouring sand into plastic pyramid containers, but greater insight comes from a different model-playing cards.
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Figure 9. The diagonal decomposition of a rectangular solid yields three pyramids of different shapes but the same volume. |
Think of a pyramid constructed of thick rectangular cards stacked above the base. If we double the thickness of each card in the stack, then the base stays the same while both the height and the weight of the stack (and therefore its volume) also double. If we keep the width and the thickness of each card the same and double the length, then the volume also doubles. Doubling any single dimension causes the volume to double: in general, multiplying a single dimension by any number will multiply the entire volume by that same number.
This procedure enables us to obtain the volume of any pyramid formed by a diagonal decomposition of a rectangular solid — that is, of any pyramid with a rectangular base whose top vertex is directly over a corner of the base. Further work with pyramid-shaped blocks will quickly show that any pyramid with a rectangular base can be built up from pyramids of this special type, all with the same height. Taken together, these demonstrations show why, in general, the volume of a pyramid with a rectangular base is one-third the volume of the right rectangular prism with the same base and height.
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Figure 10. The same set of rods that forms a rectangle can also form a parallelogram of the same dimensions. Hence the areas of the rectangle and the parallelogram must be the same. |
Experiments with stacks of cards or thin rods can lead easily to a powerful idea known to mathematicians as Cavalieri's principle for shear transformations. First observe how the same set of rods that fills a parallelogram will also fill a rectangle with the same base and height. Hence their areas must be equal (Figure 10). The same principle works in space as well as in the plane. The same pack of cards that fills a straight box can fill a slanted box with the same base and height. Similarly, an off-center pyramid can be approximated with the same collection of square cards that approximate a centered one (Figure 11).
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Figure 11. The same set of cards that forms an off-centered pyramid can be rearranged to form a centered pyramid of the same base, height, and volume. |
Students who explore models of pyramids with sets of blocks and stacks of cards throughout their early school years are certainly more likely to understand and appreciate the formal proofs presented for such theorems in calculus classes; students who have never thought about properties of volumes until they arise in calculus will not get nearly as much out of their experience. We now spend a great deal of effort getting students ready for the algebraic techniques needed for advanced mathematics. We should be just as concerned for their geometric preparation as well.
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area. |