Chapter 36: Area of Plane Figures: Situation 30 From the MACMTL–CPTM Situations Project

A teacher in a geometry class introduced formulas for the areas of parallelograms, trapezoids, and rhombi. She removed the formulas from the overhead projector and posed several area problems to her students. One student volunteered the correct answers very quickly. Another student asked, “How did you memorize the formulas so fast?” The first student responded, “I didn't memorize the formulas. I can just see what the area should be.”

The four Foci for this situation reflect relationships among three classes of figures: parallelograms, trapezoids, and rhombi. In Foci 1 and 2, strategies for a class of figures are applied to one of its subclasses. The two Foci differ in terms of whether the strategy is the application of a known formula or the application of a method that develops the known formulas. Foci 3 and 4 involve decomposition of quadrilaterals, with the former emphasizing efficient calculation and the latter targeting the logical development of mathematics from what is known to what is needed.