I recently finished another squidoo lens on working with polynomials. One of the goals of the lens is to teach students how to factor quadratic expressions. I was taught the guess and check method to factor quadratics and have always held the belief this was the best method to factor quadratic expressions. One of my fundamental theories on math education is the importance of teaching the why along with the how. The guess and check method may not always be the easiest "how" but I have always thought it was the best "why".
I will confess I have not worked much with the other more algorithmic methods of factoring quadratic expressions. If you teach/use this method, I would love to hear why you think it is the best way to teach the concept?
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I started by teaching the guess-and-check method, but also showed how to use grouping with trinomials (once we got to grouping!). I told my kids they could use either method. I'd guess that they went half and half... some really liked the grouping (because there really was no guessing involved) and some stuck with the guess-and-check. I've always used g-a-c, but can see the good points to grouping, too (though it is a pain when the numbers are big!).
A colleague of mine also introduced me to the grouping method. I explain to students that my professional opinion is to use the guess and check method as much as possible, especially when the lead coefficient is 1, simply because it's faster and more efficient. However, once showing students both methods, I find it's about 50/50 as to which students choose to use. Many of my students come in below grade level and don't have the fundamental number sense skills to perform the guess and check correctly and efficiently. I also show students how to check their answers and connecting their factors to solutions and their respective graphs using their graphing calculators (or checking of solutions arithmetically). Students seem more likely to use guess and check when they know there's a way to check their solutions quickly using their calculator if they aren't sure about a certain set of factors. As in most mathematics topics, there are often multiple methods of carrying out a problem, and my approach is to expose students to many of them so they can choose the 1 or 2 methods that make the most sense to them, while knowing there are others out there that are potentially easier/more efficient should their skills get better.
I just found this website via NCTM Bulletin...looks good so far and I look forward to visiting again throughout the school year! :)
Derek
I prefer a visual approach first. Have you considered using algebra tiles? x^2+5x+6 can be displayed as a rectangle that is x+3 long and x+2 wide. There's a lot more to it than this, but basically we work with the algebra tiles until the students find their own "shortcuts". Of course I lay the ground work by using the tiles for adding like terms, the distributive property and multiplying binomials prior to this. I find students are much more likely to be visual learners AND that this demonstrates the concept much better than just learning a process.
Concept -- what are we doing when factoring a trinomial into two binomials? Aren't we creating a retangle whose area is the original trinomial? Why can't some trinomials be factored by real numbers? Isn't it because that trinomial can't be made into such a rectangle?
I think this ties in with your quest to get to the "why" behind the algebra you are teaching.
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