Abstract
Many students enter ninth grade already familiar with the quadratic formula. Many others learn it in ninth grade. Some can even sing it! Unfortunately, the formula has little meaning for most students. For many, the traditional derivation of the formula by completing the square (fig. 1), if it is shown to them, is more baffling than illuminating. As a teacher, I value student understanding, and early on in my career as an algebra teacher, I found this state of affairs disturbing. My first response was to have students complete the square repeatedly, using numbers at first and then the parameters, in the hope that this process would lead to understanding. Alas, over time I realized that for many if not most of my students, additional symbol manipulation did not throw additional light on the subject. I needed to come at this lack of understanding some other way.
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