FOIL in a Box (algebra!)

Some of us have a ‘thing’ about FOIL as a topic in an algebra class; there are concerns about emphasizing the FOIL process as it can submerge the real algebra going on. Some (perhaps the majority) are not significantly handicapped by being “FOILed”. This post is not about FOIL itself … it’s “FOIL in a Box”.

Okay, so this is what I am talking about. The problem given to the student is to multiply two binomials, such as (2x – 3) and (3x +4). Here is the “FOIL in a Box”:

Some students like this approach, and I think this is because the box lets them focus on one small part of the problem. The overall process is submerged, and the format does all of the work. Of course, this is exactly what many procedures in arithmetic do. The FOIL in a Box method is much like column multiplication, where partial products are arranged in a mechanical way to produce the correct place value. If correct answers to multiplying were the primary goal, there would be nothing wrong with either FOIL in a Box or partial products in arithmetic.

My observation has been that almost all students who use FOIL in a Box are handicapped in working with polynomials. Students have trouble integrating the Box into longer problems. And, though they may have some ‘right answers’ for factoring trinomials, the transition to other types is more difficult.

What should we do instead? My own conclusion is that we need to keep emphasizing the entire idea involved. FOIL is used for “distributing when both factors have two terms”, and “distributing is used to multiply when one factor has two or more terms”. We too often assume that students will keep information connected to the correct context … they don’t automatically know that distributing does not apply to 3 monomial factors [3(2y)4z ≠72yz], nor to a power of a binomial [(x + 3)² ≠x² + 9].

The achievement of correct answers in the short-term should not come at the price of handicapping the student’s future learning. All learning should be connected to good prior learning, and imbedded within the basic ideas of the discipline. We need to be comfortable articulating the full name of what we are doing (multiplying two factors each with two terms), and not use a mnemonic such as FOIL as a container for knowledge of mathematics.

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Content of developmental math courses, Learning math, Uncategorized | Jack Rotman | February 10, 2012 8:58 am

3 Comments

By Bruce Yoshiwara, February 10, 2012 @ 1:58 pm

Hi Jack,

I don’t like FOIL, but I do like the area model of multiplication.

If you show the (binomial) factors as the dimensions of a box, rather than as entries inside a box, then the diagram can be used to emphasize why the distributive law works.

For instance, the product 3(4+5) can be visualized by showing a 3 by 9 box, where the length of 9 is subdivided into two pieces of length 4 and 5. Thus the 3 by 9 box is divided into two sub-boxes of areas 3*4 and 3*5. The fact that the total area 3*9 of the 3 by 9 box can be computed by adding the areas of the two-sub-boxes is a geometric confirmation of the distributive law.

Or (1+2)(4+5) can be visualized by taking the previous diagram and subdividing the side of length 3 into two pieces of length 1 and 2, thus subdividing the original 3 by 9 box into four sub-boxes. The fact that the area of the 3 by 9 box can be computed by adding up the areas of the four sub-boxes is a geometric confirmation of FOIL.

The diagrams generalize easily to the product of any two polynomials. If you wish to multiply a binomial and trinomial, you could draw a rectangle, divide the length into into three pieces and the width into two so that the large rectangle is subdivided into six sub-rectangles. Label the three pieces of the length with the three terms of the trinomial, label the two pieces of the width with the terms of the binomial. The product of trinomial and binomial can be obtained by adding the areas of the six sub-rectangles.
By Jack Rotman, February 10, 2012 @ 2:14 pm

Bruce:
Thanks for the comment; good points.
One issue here is how these visual aids (the box, area model, etc) impact the learning inside the student. The general research I have read suggests that a visual aid can help the learning at the early stage, but hurt learning at the later stages. In other words, my understanding of the research suggests that a visual aid needs to be followed by a strong transition to verbal and symbolic representations.
It’s not that we should avoid visual aids; in fact, the traditional curriculum tends to de-emphasize visualization which later handicaps students who need to represent other concepts or problems from this perspective. The suggestion is that visual aids without a strong transition to other forms creates weaker understanding, so we need to have a deliberate plan for making the idea abstract after the ‘concrete’ representation.
Jack
By Beth in MN, February 27, 2012 @ 10:08 pm

In my classes, FOIL is a “four-letter F word,” and we don’t use them in my class. We distribute.

The very first example I use of multiplying in which both factors and NOT monomials is a binomial times a trinomial.

I often show the box method of multiplying multi-digit numbers and finding their partial products to my arithmetics students, and I allow and encourage its use for polynomials as well if students desire. With multi-digit numbers, I write 386 as 300+80+6 along the sides of the rectangles with the partial products.

I don’t draw the smaller rectangles around the terms of the polynomials or multi-digit numbers–just around the partial products.
We review the use of the distributive property such as
7x^2(3x+5))

and then
(3x+5)(7x^2).

Then
(3x+5)(something)
=3x(that something) + 5(that same thing)

Then:
(3x+5)(7x^s-6x+4)
= 3x((7x^2-6x+4) + 5(7x^2-6x+4)

Eventually when multipllying binomials, someone mentions FOIL, and that’s when I break out my 4-letter F-word spiel. I also remind them that they already know the Distributive property, so why would they need to memorize something new that only works in a special situation? Then I make a joke about my age and brain capacity for long-term storage.

Beth in MN

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