Today I introduced factoring the difference of squares using the same drawing I use for multiplying binomials. This was actually one of the best ways I've ever used to provide visually why it works the way it does.
I started by drawing a representation of 4X^2 -16 so they could see the two parts that existed. I asked what is this drawing missing. The students noticed there was no x's in the drawing. I asked them why would there be nothing showing. After a bit of thinking and guessing they finally came up with the idea that the missing elements were opposites of each other.
So in a different color, I added in the missing x's. One set of four is positive, one set is negative. We drew in the missing parts to this square. It was great because this is the first time students could connect the drawing with the equation and add in the missing parts.
The final step in the process of drawing was to add the lengths of the side. The 4x^2 was easy because it is 2x due to 2 x's. The numbers were 4 and -4
It was so easy for them to see all. I just realized, I should have cut the x's in half so I had 8 positive and 8 negative values. I'll change that the next time I do it.
In Algebra I, I introduced multiplying binomials whose result is the difference of squares. I drew the boxes, filled in the values and showed how the x values cancel each other out by erasing those area. I loved the way the students kind of went "Oh." When they did their guided practice, most of them just whizzed through the whole set and if they made a mistake, they had no trouble understanding why.
This method is going to show up in my box of teaching tools. Tomorrow, I'm going to be using the same idea to help my Algebra I learn the pattern for binomials squared. Yes!