Now we tackle factoring trinomials that have minus or negative signs. This time the blue tiles are positive and the pink tiles are negative. Each student was given a set of blue and pink tiles, while I made large tiles to use on the class white board.
Here's what I told my students:
If the last sign in the expression is minus (negative), then the positive and negative x tiles must be kept separate. For instance, the negative Xs are built to the right while the positive Xs are built down (or vice versa).
In the example below, we are factoring x^2 - 2x - 8. We start with a blue x^2 tile, 2 pink x tiles, and 8 pink 1 tiles.
I place the x^2 tile in the upper left like I always do and the 2 pink x tiles to the right of it. As you can see, the 1 tiles are just sitting there not fitting in to the rectangle.
There is where things get interesting, I hold up one pink x tile and one blue x tile and ask the students what is the sum of those two tiles. Eventually, they say 0. Then I ask if I would change the value of the expression if I add zero to it. Most students tell me that's okay.
To keep building, I can place one pink and one blue x tile, then fill in as many of the 1 tiles to complete the rectangle. If there are 1 tiles remaining, repeat the process. Finally you can see that the answer is (x + 2)(x - 4)