The teacher says: ”The area of a trapezoid is given by the formula h(b1+b2)/2, where h is the height, b1 and b2 are the bases. Here is a worksheet where you can practice this.”
The worksheet includes 20 examples, each with different numbers for the bases and the height. The students practice in silence. Many students like this, because they know exactly what to do. Other students don’t like it, because they find it boring. All of them know they will soon have to calculate some trapezoid areas on a quiz, and they hope that this practice will help them remember the formula. Some will make an effort to memorize the formula in preparation for the quiz. Most will have forgotten the formula a week, a month, or a year later. This is because they will not use the formula again, unless they take calculus many years hence. In any case, whether they remember it or not, doing the exercises does not help them understand the formula.
(In fact, see how many strategies you can find.)
Students who find one quickly can be encouraged to look for more. Some students may not like the activity, because they are not told exactly what to do. The teacher can offer hints to them, or encourage them to get help from neighbors.
Once some strategies have been found, the teacher can lead a discussion where students demonstrate their approaches. All strategies will reveal that the lengths of the legs do not contribute to the final answer. In fact all strategies will yield the same answer for the area. A general formula can be the final punch line: applying any of the strategies to a generic trapezoid always yields the same formula.
(Scissors are not absolutely necessary. For example, the activity can be carried out on paper, without any cutting. Whether that is preferable will depend on the specifics of a given class.)
Even if the first approach includes a brilliant teacher explanation of the formula, I claim that the second approach is preferable. Many students who cannot remember the formula at some point in the future will be able to use one of the strategies that came up in the course of the exploration, either to find a particular trapezoid’s area, or to reconstruct the formula. This approach also carries the message that formulas can make sense, that there are many ways to solve a given problem, and that not everything needs to be memorized. A perhaps unexpected bonus is that the different solutions to this essentially geometric problem yield different interpretations of the formula, and some apparently different but actually equivalent formulas. Discussing this can help improve symbol sense. Finally, if the teacher has an excellent explanation of the formula that was not found by the students, nothing prevents him or her from sharing it. Starting with the hands-on activity does not prevent the teacher from offering an explanation, but it does mean that more students will understand the explanation.