I have big news that I am dying to share, but it will have to wait until all of the involved parties have been brought into the loop. It's scary and exciting, but I don't want to be a tease, so I'm going to move on to something I realized about five minutes ago.
When I first learned to multiply binomials in eighth grade, my Algebra teacher introduced the FOIL method as this mystical set of steps you have to follow in order to simplify an expression when the plain ol' distributive property won't do it. I am a type A personality all the way, so having a set procedure appealed to me immediately, but I'm a little embarrassed to say that the whys didn't matter much to me. I distinctly remember many of my classmates leaving class with a dazed, terrified look on their face and hearing that two of them planned to convince their parents to request placement in standard math (Pre-Algebra).
In high school, our physics teacher reviewed the FOIL method, encouraging us to draw lines between the terms as we multiplied and calling the resulting diagram the "Happy Tomato." He actually told a story about the tomato, which I thought was crazy, but I dutifully drew the lines each time I multiplied binomials. I did notice that my peers who were foiled by FOIL were happy to use the Happy Tomato. (See what I did there?)
I think this was the first time that I actually realized how important it is to offer students alternative ways of doing things. Both teachers taught (reviewed) the same skill and even the same method, but they used completely different explanations, and it made a difference for their students.
A few minutes ago, I saw a product of two binomials in a Google ad (spend much time searching for math activities, and you, too, can have algebra ads framing your windows), and it hit me that multiplying binomials is EXACTLY like multiplying two digit numbers. If a student can multiply two-digit numbers, and they can find the product of a variable and a number and the product of two variables, then multiplying binomials is a no-brainer. FOIL just allows us to multiply with our terms written horizontally.
FOIL just gives us options, so why is it not presented that way? If I were teaching students to multiply binomials, I would start by asking them to multiply two two-digit numbers (written vertically). Then, I would stack the binomials, and show students that the same method is used to multiply binomials. Then, I would write the binomials horizontally and ask students how they would proceed. I suspect they would just rewrite the problem vertically, and here is where I would introduce FOIL to eliminate that step. Instead of rewriting the problem, just "FOIL it out." Everyone loves a shortcut!