Connections

12 x 23

 When I hit the part of Algebra I and II where I teach multiplying binomials, I often start with a review of the distributive property.  For instance, I set up a problem such as 12 x 23 using grid to show the 10's and 1's as in the picture to the left.  This way of showing the distributive property makes it easy to find the product of 273 by just looking at the squares.  Its a variation on elementary techniques but it shows what is happening quite clearly. 

I often spend a full day or two having my students multiply integers using this grid so they get used to seeing a visual representation. Sometimes it helps students do better with a visualization.  Once they have this down, I repeat the same exercise using a pair of binomials for multiplication.

(x+1)(2x + 3)

They set the binomials up the same way they do for distributive except instead of breaking numbers down to tens and ones, its broken down to x''s and ones.  The process is the same and the results follow the same patterns.

10 * 10 is 10^2 or 100 is the specific example while x * x = x^2 is the generalized form.  I've had students make the connection between the two forms and its helped them do better.  If you like you could do it with say 123 x 23  with a generalized form of x^3. 

Furthermore, when I teach students to multiply binomials, I teach them 4 different ways to do it so they don't always have to rely on the FOIL method.  The FOIL is fine but students often find it trickier than other methods especially when they start multiplying a trinomial by a binomial or a trinomial by a trinomial. 

I do start off with the FOIL but then I show them the distributive method, the vertical method and the lattice method with the same problem so they see the result is the same regardless of the method used.  This is important.  I know when I was in school, the FOIL method was the only one taught.  If you couldn't do it that way, you were out of luck.

By starting with the relationship and then giving students a choice of the method they want to use for multiplication of binomials, they do better and catch on a bit faster.