Tried something new in College Algebra, working with ellipses in their two formats. One in which the vertex is plainly visible, and one in which it is obscured. As an exercise in reversibility, I wanted students to see how we can manipulate terms to reformat the equations, using skills we already possess. Along the way I want them to discover complete the square as a useful pattern, rather than an arbitrary algorithm they’re handed. And we aren’t going to call it complete the square either.
We start with some binomials, and pose the question, is it possible to rewrite these?
Students remember that (x+3)^2 is the same as (x+3)(x+3), and distribution yields a quadratic with three terms. After a few of these I ask if they’ve observed a pattern. When simplifying, the middle term appears to be double the constant I started with, and the third term is that same constant squared. Eventually we can expand a binomial in one step, knowing that our distribution will created a “double” and a “square.”
Recognizing how a binomial is constructed is going to help us deconstruct it later.