# Distributive Property – Multiplying and FOIL

This is part two of my post on the distributive property. I feel kind of guilty calling it part two as it is little more than an extension of my previous post. After all there is a lot to say about the distributive property as it is manifests itself in many places in the curriculum.

One such place is when multiplying a binomial by a binomial. If you think of the old mnemonic FOIL (First Outer Inner Last) to remember the order to multiply the terms it becomes apparent that this is simply an application of the distributive property.

In my mathematical learning I never made the connection with the distributive property immediately. It was not until I graduated high school that I made this connection and saw the relationship to two digit multiplication. I should have been able to connect the dots but to be honest I was more into doing math than understanding the mathematics. I do not recall the teacher making this explicit in my learning. As an educator I now see why this understanding is much more important than simply following an algorithm as it connects to other learning.

My process of making the connection came when someone showed me how to multiply two digit numbers using arrays.

There are tons of videos, pdf files and other resources on-line to demonstrate how this works. I would encourage all educators to search for them and make their own judgements about the one with the best explanation.

The connection that students often bring up in my classes is the similarity between the Punnet Square (genetics) and the array method of multiplication. For students who have learned the traditional algorithm of placing the numerals one over another I like to make the link with partial products prior to “demonstrating” multiplication using arrays to help make the connections.

I think this helps show why the “traditional

method” of multiplication of two digit numerals works.

The important point is to make these connections explicit. After all what may seem obvious to a teacher is not always obvious to a student.

These concepts used in two digit multiplication also apply to the multiplication of binomials. For example the multiplication (10+2)(10+7) is not different from the multiplication of (x+2)(x+7). It is then a natural extension into the multiplication of any two polynomials.

This leads me to the final conclusions:

1) As educators we should make explicit connections within the curriculum

2) It is important to know where math concepts “come from” and “where they are going”

3) When teaching math concepts we need to examine and explain why “shortcuts” like multiplication algorithms work.

I agree whole-heartedly. Making connections explicit for kids (or designing tasks which lead them to make the connections for themselves) is critical in building a breadth of understanding in mathematics.