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Re: I like FOIL (I don't, but...)
Posted:
Apr 18, 2000 5:08 PM


At 2:22 PM 0400 4/18/00, Bonnie Edwards Hall wrote:
> Dick Askey says FOIL is "a mindless phrase which almost everyone >who teaches college math knows causes trouble for many students". > I disagee about FOIL. I think it is very helpful and useful, NOT >"mindless." > I currently teach developmental math at the college level. I teach >Math 10, a noncredit course that is taught in two tracks. Track A is >roughly equivalent to high school Algebra I and Track B is roughly >equivalent to high school Algebra II. > I use the term FOIL when teaching my current students for several >reasons: > 1) Since I learned the term, I found it helpful in the past for >teaching alegbra at the high school level and in my present position. > 2) My college students of the past three years are familiar >with the term and use it very well as an aid in multiplying binomials. >Note: A lot of things in math cause trouble for many of my Math 10 >students; FOIL seems to be one concept that most of my students >understand and use well before it is introduced in class. That is not >true for much of other things that my Math 10 students encounter in >Math 10. > 3) The textbook used in Math 10 explains and uses FOIL. That >text is Intermediate Algebra, Second Edition by Mark Dugopolski, >AddisonWesley, Reading, Massachusetts, 1996. This text was not >selected by me but by the Math 10 coordinator. > > What do others think? I'm interested. >
First, I have to confess that I taught our college high school Algebra I class once, and was so frustrated I never did it again. But I do teach a lot of math for elementary teachers, and I like the method below.
To multiply a polynomial with 2 terms by a polynomial with 3 terms, for example (x2)(2x^2  x + 5), do this:
Make a 2 x 3 grid of boxes. Turn the polynomial into a sum by adding the opposite: (x + (2))(2x^2 + (x) + 5). Write each term of each sum across the top and down the side, as appropriate. Fill in the boxes with products like a multiplication table. For this example, you will get 2x^2 x 5  x  2x^3  x^2  5x   2  4x^2  2x  10  
Then add all the products.
Reasons I like this method: *It works for any polynomials, not just binomials (one of Dick's objections) *It's a more visual/geometric way than algebraic/symbolic, which feels more comfortable and understandable to some people. *The reason it works is deeply related to what addition and multiplication are all about, and is connected to length and area, multiplication of whole numbers, multiplication tables, etc. If your students can get these connections, they will be way ahead.
Here's a quick summary of the connections.
For any two positive (real) numbers a and c, you can represent their product by the area of a rectangle with sides a and c. For students who don't get this connection, use whole numbers and graph paper.
To represent the distributive property (a+b)c = ac + bc in this picture, make an (a+b) by c rectangle, and cut it into two rectangles by splitting a+b into a and b. The left side of the equation is the total area, the right side is the areas of the two subrectangles.
The way we multiply multidigit whole numbers can be shown in this same kind of diagram. Divide the length and width according to place values. For instance, 347 x 26 = (300 + 40 + 7) x (20 + 6) Draw the diagram, which will have 6 boxes. Fill the boxes with the products 300x200, 300x6, etc. (It might be worthwhile for those who haven't caught on to the idea to do this on graph paper with actual lengths, though a number in the 100s is a little mean. In general, use a nottoscale diagram.) Then add all the products.
The socalled "standard algorithm" for multiplication in columns does something more compressed than this: if the 347 is on top, then the partial products (6940 and 2082) that you add to get the product are not immediately visible in the box diagram. But they are there: add the boxes across or down as appropriate: 6000+800+140 = 6940, 1800+240+42 = 2082.
Almost the same picture would multiply 3x^2 + 4x + 7 by 2x + 6. Then if x=10, this is the multiplication you just did. If x represents some negative number, the diagram is no longer geometrically meaningful, but the distributive property works all the same. (The number system is designed so that all the properties you're used to work even with new and strange numbers.)
Variation: Fill in 0 coefficients for any missing terms in the polynomial and give them their own rows and/or columns. Then like terms in the product will end up in diagonal rows, providing an extra check. This is almost the same as the lattice multiplication method for whole numbers.
Susan Addington academic year 19992000: otherwise: LTT Math Department Education Development Center California State University 55 Chapel St. 5500 University Pkwy. Newton, MA 024581060 San Bernardino, CA 92407 (617) 9697100 x2807 (909) 8805362 fax: (617) 9656325



