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FOIL - It Doesn't Always Work!

Using FOIL
In more advanced math classes, many instructors happen to hate "FOIL" (including me) because it only provides confusion for the students. Unfortunately, FOIL (an acronym for first, outer, inner and last) tends to be taught as THE way to multiply all polynomials, which is certainly not true. As soon as either one of the polynomials has more than a "first" and "last" term in its parentheses, the students are puzzled as well as off course if they attempt to use FOIL. If students want to use FOIL, they need to be forewarned: You can ONLY use it for the specific case of multiplying two binomials. You can NOT use it at ANY other time!

When multiplying larger polynomials, most students switch
to vertical multiplication, because it is much easier to use, but there is another way. It is called the clam method. (An instructor at the college where I teach says that each set of arcs reminds her of a clam. She’s even named the clam Clarence; so, at our college, this is the Clarence the Clam method.)

Let’s say we have the following problem:

(x + 2) (2x + 3x – 4)

Simply multiply each term in the second parenthesis by the first term in the first parenthesis. Then multiply each term in the second parenthesis by the second term in the first parenthesis.

I have my students draw arcs as they multiply. Notice below that the arcs are drawn so they connect to one another to designate that this is a continuous process. Begin with the first term and times each term in the second parenthesis by that first term until each term has been multiplied.

When they are ready to work with the second term, I have the students use a different color.  This time they multiply each term in the second parenthesis by the second term in the first while drawing an arc below each term just as they did before.  The different colors help to distinguish which terms have been multiplied, and they serve as a check point to make sure no term has been missed in the process.

As they multiply, I have my students write the answers horizontally, lining up the like terms and placing them one under the other as seen below. This makes it so much easier for them to add the like terms:

This "clam" method works every time a student multiplies polynomials, no matter how many terms are involved.

Let me restate what I said at the start of this post: "FOIL" only works for the special case of a two-term polynomial multiplied by another two-term polynomial. It does NOT apply to in ANY other case; therefore, students should not depend on FOIL for general multiplication. In addition, they should never assume it will "work" for every multiplication of polynomials or even for most multiplications. If math students only know FOIL, they have not learned all they need to know, and this will cause them great difficulties and heartaches as they move up in math.

Personally, I have observed too many students who are greatly hindered in mathematics by an over reliance on the FOIL method. Often their instructors have been guilty of never teaching or introducing any other method other than FOIL for multiplying polynomials. Take the time to show your students how to multiply polynomials properly, avoid FOIL, if possible, and consider Clarence the Clam as one of the methods to teach. 

Tally Marks - Different Ways to make them

Tally marks are the quickest way of keeping track of a group of five. One vertical line is made for each of the first four numbers; the fifth number is denoted by a diagonal line drawn across the previous four (i.e., from the top of the first line to the bottom of the fourth line). The diagonal fifth line cancels out the other four vertical lines making the entire set represent five.

Tally marks are also known as hash marks and can be defined in the unary numeral system. (A unary operation in a mathematical system is one element used to yield a single result, in this case a vertical line.) These marks are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. They also make it simple to add up the results by simply counting by 5’s. Here is an illustration of what I mean.

  • The value 1 is represented by | tally marks.
  • The value 2 is represented by | | tally marks.
  • The value 3 is represented by | | | tally marks.
  • The value 4 is denoted by |||| tally marks.
  • The value five is not denoted by | | | | | tally marks. For the number 5, draw four vertical lines (||||) with a diagonal (\) line through them.
I have seen many interesting ways to teach tally marks to younger children. Many teachers will use Popsicle sticks so that the students have a concrete hands-on way of making tally marks. Some have even tried pretzel sticks although there is a good chance some will disappear during the lesson. 

But have you ever seen these kind of tally marks?

My husband, who teaches science, received this data collection paper from a student. The students were tossing coins marked TT, Tt, and tt to determine different genetic traits and tallying the results. The ones seen above are Japanese tally marks. (The student lived in Japan.) I was fascinated about how they were made so I asked him to have this student show me the sequence of how to draw the marks.

I'm not sure what they mean or why they are made this way, but if you look at the 2nd mark you will notice that it looks like a "T" for two. The fourth mark sort of looks like an "F" for four, but so does the third one. As you can see, each complete character uses 5 strokes; so, a series of would each represent 5, just like the English ones. However, to be honest, I am at a total lost to what this really means; so, I resorted to the internet. Here is what I learned. 

Instead of lines, a certain Kanji character is used. In Japan, this mark reminds people of a sign for “masu” which was originally a square wooden box used to measure rice in Japan during the feudal period. Here is what the tally marks would look like if we compared the two systems.

The successive strokes of () are used in China, Japan and Korea to designate tallies in votes, scores, points, sushi orders, and the like, much as is used in Europe, Africa, Australia and North America. Tallies beyond five are written like this  with a line drawn underneath each group of five, followed by the remainder. For example, a tally of twelve is written as 正正丅. 

So the next time your visit Japan or go to a Japenese restaurant to order Sushi, look for the tally marks as the waiter takes your order.