Most calculus students do not realize that the area of a trapezoid is the average of the area of two rectangles. Areas of trapezoids come up in calculus when approximating the value of a definite integral, where the focus of the lesson is on improving the approximation. As a side note, I sketch a diagram of the trapezoid with the rectangles along with the formula that students recall for the area of a trapezoid (b1+b2)h/2, and add (b1•h+b2•h)/2. “I never thought of it that way!” is a common student reaction.
It has been some time since I taught geometry, and I looked up an NCTM Illuminations resource for finding the area of a trapezoid. The resource suggests different proofs for the area formula, and incorporating the notion of averages can provide another perspective.
The simple diagrams can help geometry students build convincing arguments about the relationship between the areas of the trapezoid and the rectangles:
In calculus, the trapezoids students use in approximating integrals have two right angles as shown below. It might be easier for geometry students to make observations using the following series of diagrams instead.
Whether students model with paper by cutting and rearranging the pieces or use measurement as their tool, I believe it would help them get away from memorization and get closer to seeing formulas from a meaning-making perspective.