Algebra II: Factoring is FOIL in reverse

My students at McKinley Tech are in the midst of learning quadratics. Most recently, we've studied the parabola and used the Vertex Form and the Standard Form to understand the main features of that geometric shape. One of the problems I assigned on the mid-term exam was to translate a parabola's equation from Vertex Form to Standard Form and then to graph the figure.

A day or so after the mid-term, a couple of my students asked: "Now that we can go from Vertex Form to Standard Form, how can we go back the other way?"

That question was music to my ears. "That's what we're going to learn next," I said, happy to have some of them ready to learn about factoring.

Translating a parabola's equation from Vertex Form to Standard Form involves what I call "unpacking" part of the equation and then multiplying and adding all of these unpacked elements until we arrive at a Standard Form equation. A main feature of this unpacking process is to multiply sets of numbers and variables in parentheses. For example, (x + 3) (x + 3).

There are two ways to do this. Here in the U.S., this multiplication process is known as FOIL, an acronym for First Outer Inner Last. Another, more-cumbersome method is known as the Double Distributive process. Suffice it to say we'll leave the Double Distributive process well enough alone.

Well, until last week, my students didn't realize that doing these FOIL calculations led them from Vertex Form through to another form of quadratic equation, the Intercept Form. I didn't bother to tell them about this, since it would have been a distraction from the final objective I had in mind: arriving at the Standard Form equation.

Now, however, as we embark on factoring quadratic equations in Standard Form, it makes to show my students some of the aspects of the Intercept Form.

I think of this as giving my students some perspective on what they're doing. Without this perspective to their studies, what they're missing is any understanding or appreciation of why they're learning. And with no understanding of "Why?" they're learning, students will become bored or disenchanted. And that's a recipe for disaster in the classroom.

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