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Topic: FOIL
Replies: 10   Last Post: Apr 20, 2000 2:45 PM

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Susan Addington

Posts: 28
Registered: 12/6/04
Re: I like FOIL (I don't, but...)
Posted: Apr 18, 2000 5:08 PM
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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 non-credit 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,
>Addison-Wesley, 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 (x-2)(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 multi-digit 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 not-to-scale
diagram.)
Then add all the products.

The so-called "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 1999-2000: otherwise:
LTT Math Department
Education Development Center California State University
55 Chapel St. 5500 University Pkwy.
Newton, MA 02458-1060 San Bernardino, CA 92407
(617) 969-7100 x2807 (909) 880-5362
fax: (617) 965-6325






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