Today my class decided that the “foil” method of multiplying two binomials is overrated. The following diagram contains the series of steps that I took them through.
In step one I asked them to multiple 78 by 26 by hand – no calculators allowed. After the expected moans, I found that about half of them could multiply as shown in step 1. This is typical for an elementary algebra course taught at the college level. The students wouldn’t be there if they hadn’t trouble with math in their public school career.
I then multiplied 78 by 26 as in step two. Notice the crisscross and add that produced the 58. I do this to show them that there is more than one algorithm for multiplying numbers. I then produce step 3. Some of them are following along well and some are getting an inkling of a pattern. Here I strongly emphasize that no matter what, four multiplications need to be done.
If they haven’t gotten the first three steps, step four brings them back on board. They can see where each number comes from and the reason for lining up the numbers in columns as shown. Step five follows naturally and ninety percent “get it”. They can reproduce the pattern immediately. Some are struggling to remember the “foil” method. I had many students agree that that the foil method as shown in this diagram is overrated.
This method is inferior. I think it was invented to save paper and typesetting costs kind of like the way arithmetic algorithms in the European 1400’s were chosen for how little of the then expensive paper was used. There are several advantages to learning to multiply polynomials in a vertical format. One is that process looks more orderly and keeps like terms in separate columns. Another is that the process mirrors the process that they supposedly learned in elementary school. As one of my students said today, “Instead of a zero it’s just an x.” Exactly it’s the same algorithm. Just think of the x as a ten. Another advantage is that the process will help them recall how to multiply multidigit numbers by hand. And finally the process extends to polynomials with many terms. Try multiplying straight across without using the vertical format. The bookkeeping will drive you crazy.
This term I am emphasizing multiplying polynomials in vertical format and introducing the “foil” method only to justify the multiplication process as multiple applications of the distribution property.