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Factoring, up or otherwise

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.

{ 4 } Comments

  1. Elissa | May 10, 2010 at 6:54 am | Permalink

    I was just wondering what you used to create the crosses worksheet. I also use the diamond math method but had not thought of just using x’s instead of the diamonds. Did you just use Microsoft Word?

  2. Mr. K | May 10, 2010 at 9:07 am | Permalink

    @Elissa – I used iWork Pages. Each X is a 3×3 table without borders, on top of which I drew two crossed lines. Even when grouped together, you can still edit the table entries to change the numbers.

    @RIley – Yes, absolutely, for both. And the fact that the framework applies to so many different places means that it’s reinforcing and deepening earlier learning when you revisit the framework with the later topics.

  3. Erin | May 23, 2010 at 9:05 am | Permalink

    I’ve used the exact same method for teaching factoring quadratics this year (my first year of teaching). I started out calling the X problems “diamonds” but found the kids liked it more if I called the whole process “X-box” problems. “X” for the diamond part, “box” for the actual factoring method.

    I started the diamonds (or Xs) early in the year as post-test worksheets. The first one had some partially filled out and asked the kids to figure out the pattern themselves (add the two middle numbers to get the top, multiply to get the bottom of the X). It was fun seeing them pull it together themselves. Then, I was able to have a stack of these ready for exit tickets or extra work when I needed to kill a few minutes or have them work on their multiplication skills (these are LOW level Algebra kids). By the time we got to the actual factoring part, they were pros at the X part.

    Good stuff. :) Glad to see others are using this method with comfort! I think the grouping method is a little more “algebraic”, but this is easier for them in the long run.

  4. Mimi | July 5, 2010 at 9:53 am | Permalink

    I teach this method (which I call “box factoring”) habitually to my Algebra 1 and Algebra 2 kids, and what is neat about it is that those kids continue to prefer it over all other methods, when they need to simplify polynomials of degree n. (Or, whenever they would get down to a possibly factorable quadratic, they would try to break it down by the box method.) It just makes more sense for them.