Introduction: Activities - Area
Many students will already know how to calculate the area of squares and rectangles. Some will also know how to compute the area of parallelograms and triangles. The geoboard will help these students strengthen their understanding of area even more. But rather than give them formulas to calculate area, we will try to give them conceptual tools that tend to persist in their minds long after the formulas are gone.
For example, if a student were to take a pair of scissors and snip a parallelogram along its height, she would quickly realize (by fitting together the two pieces) that the area of a parallelogram is nothing more than the area of the corresponding rectangle. Similarly, a right triangle may be thought of as half a rectangle. These kinds of associations help students understand area much better, and the geoboard is an excellent device on which to illustrate these concepts.
Still, the teacher must be able to calculate the area of certain polygons on sight, since students will surely ask questions for which you must know the answer (although it is sometimes wise not to give it). We therefore list the common area formulas in Figure 5:
Note that each formula follows readily from geometric considerations: the parallelogram and right triangle, for example, were mentioned above. The formulas for arbitrary triangles are best understood by cloning the given triangle and forming a parallelogram with the resulting pair of congruent triangles. Finally, a trapezoid can be divided into two triangles whose areas can be calculated separately and added. Sample calculations are given in Figure 6:
But what about an arbitrary polygon? For example, how do we find the area of the eight-sided polygon depicted in Figure 7? After spending a lot of time, perhaps days, calculating the area of squares, rectangles and triangles, most students will break up a complicated polygon like this one into more manageable pieces. In fact, take a moment to do just that: break the eight-sided polygon in Figure 7 into squares and triangles, and compute its area. Please, do it now - it's important for what follows.
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