Learning by understanding

I just read responses on a Linked-In forum about how to teach factoring. The answers were full of steps and technical details. Not one linked factoring to something the students knew, or gave them a reason to learn factoring.This was my response:

Before you even start factoring, make sure students have a reason to use it, that they understand WHY they're factoring. Have them graph a simple polynomial equation, like the square of (x+3), using a T-chart for values, and find the zeroes.

I think it is extremely important for the students to understand factoring in polynomials is the same in factoring, say, 96.

They need to know that polynomials are the result of multiplication, so a good way to start is to have them multiply simple things, like the results of the graph they did and other squares, and then, for example, the sum and difference of 2 terms, to see if they discover the pattern, then give them the same problem, plus some similar ones, to factor. Then move on to things like (x+1)(x+3), saving ones with a coefficient other than one for later.

Using Algebra tiles is another way to visualize what's happening, and using the "box" method, which I like for multiplication of polynomials, because it helps keep them straight, is also a good help to reverse the multiplication, which is similar to the Algebra tiles.

But if they have no clue why they are factoring, it just adds to "when will we ever use this in real life?" which is a very legitimate question. They need to know what those zeroes can be used for, too. I'm not sure all that many Algebra I teachers can carry the discussion that far.

When students understand why they're factoring, what it's used for, how the polynomials are graphed and how they came about, I think they will be much more open to the fun puzzle of untangling them as factors.