Date: 11/21/2002 at 20:04:30 From: Stephanie Subject: Why foil? Hi - I am getting ready to student teach and am preparing lesson plans. I am doing one on FOILing, i.e. (2x + y)(x +2y) first, inside, outside, last. I'm wondering how to explain to the class the importance of doing this. I realize that the goal is to get them to be able to later factor expressions like x^2 + 4x + 3 (and I'm not sure the significance of being able to do that either except for help graphing and finding zeros). But the students want to know the reason for doing FOILing now. Please help me get them interested! Thanks, Stephanie
Date: 11/21/2002 at 23:18:13 From: Doctor Ian Subject: Re: Why foil? Hi Stephanie, If you want to do your students a favor, skip FOIL altogether, and make sure they understand the distributive property: When FOIL Fails http://mathforum.org/library/drmath/view/53241.html To answer your question, the reason you go from a nice, tidy representation like (2x + y)(x + 2y) to a messier one like 2x^2 + 5xy + 2y^2 is usually so that you can add some other terms. Then you convert back to the tidy representation again, e.g., (x + 2)(x - 3) = 24 x^2 - x - 6 = 24 x^2 - x - 30 = 0 (x + 5)(x - 6) = 0 Standard polynomial form makes addition easy, and most other things difficult. Factored form makes addition impossible, but lots of other things easy. Part of the confusion that surrounds FOIL is treating it as if it's something new or special, when it's not. It's just the typical use of the distributive property, Distributive Property, Illustrated http://mathforum.org/library/drmath/view/52842.html i.e., you un-factor so that you can combine like terms, and then turn around and factor again. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 11/21/2002 at 23:28:53 From: Doctor Peterson Subject: Re: Why foil? Hi, Stephanie. Let me first state that I dislike the FOIL approach. It's a crutch, and sometimes crutches are useful, but I'd much rather teach a method that is not restricted to multiplying two binomials. Here is what I have said in this area: Explaining Algebra Concepts and FOIL http://mathforum.org/library/drmath/view/52844.html But that wasn't your question; you want to know how to motivate students to care; perhaps you want to motivate yourself as well. I suppose that's actually another reason to avoid FOIL: it's just another formulaic method that has value only as a tool for one job. If you're not going to be doing that particular job, you don't need the tool. Here are some reasons for learning to multiply polynomials, which is the real task that FOIL helps with only in special cases: 1. We're learning about polynomials, and to do that, we need to know how to do things with them. Unless you can add, multiply, and so on, what good are they? 2. We'll see how polynomials are just an extension of numbers, because the method I teach looks exactly like multiplying two numbers. One of the key principles in math is to relate new ideas to old ones, and look for the familiar in the unfamiliar. As you learn this, be looking for connections to what you already know, and you will get some interesting surprises - both in things that look familiar, and aspects that are different from what you've seen before. 3. One side effect of that connection is that you will gain a better understanding of the arithmetic you've already learned, by seeing it in a different context. 4. It's an important application of the distributive property, so as you learn it you will be getting a stronger understanding of how numbers work. 5. Working with polynomials is an important aspect of algebra; if you go on to any kind of further math, you will have to do this task of multiplying over and over. It's not an end in itself, but an important basic tool of algebra. 6. There are many other areas of life where you will need to find a way to organize a task like this that involves finding all the ways to combine things (in this case, multiplying every term of one polynomial by every term of the other). Polynomial multiplication serves as a model that you can use elsewhere in problem solving. In fact, FOIL is an alternative method for the same thing, and makes a good demonstration of how there can be two ways to accomplish the task. Whatever method you teach, it will probably help if you start by showing the task the hard way (applying the distributive property over and over), and then say that there is a way to organize the work so that it is so easy it becomes almost automatic. Then it becomes self-motivating - at least if they think the task of multiplying has any value in the first place. Perhaps you can look for a realistic problem in your text in which this task plays a part, to start the whole discussion. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 11/25/2002 at 23:33:40 From: Stephanie Subject: Thank you (Why foil?) Hi - Thanks for responding. I really appreciate it and your response triggered some thoughts to improve my lesson. Thanks.
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