SamJShah posted about factoring up, and his concerns that it relies on some magic that hides deeper mathematical understanding.
I share his concerns about that method. Fortunately, I’ve found a way to do it that builds on prior knowledge, rather than relying on legerdemain.
It goes back to teaching binomial multiplication – I rely mostly on the area model, though I allow students to use other methods , but make sure they have at least an understanding of it.
For those not familiar with the area model, it works like this:
This is the algebra tile version of (2x+3). After a while, the kids get tired of drawing rectangles, and you introduce the following shorthand:
You let them learn this, and test them on it, and do some projects where they have to make use of it.
Then, when you’re ready to introduce factoring, you give them a trinomial. Something like $$x^2 + 5x + 6$$. If you’ve carefully left an example of one of the above binomial multiplication problems on the board by accident, and draw the blank area model boxes, they should pretty quickly figure out how to fill out half of the box:
The other half is easily filled in using the usual diamond puzzles. Except I don’t use diamonds, I just use Xs. This worksheet is pretty good at developing the skills for the crosses (I’ve got a slide deck that steps through the first two Xs to explain what’s up, after that the kids tend to take off on their own.)
Once the inside boxes are filled up, it’s simple matter to work backwards to find what the two binomials are.
Aside from the trick of the area model, I find that the kids make pretty good connections with the lesson arc.