>someone explain how to factor Y = X**2 + 2X + 1 using "concrete pacifiers?"
I assume that you mean x^2+2x+1? This can be multiplied with algebra tiles
(including completing the square) by using a very easy process. It goes
back to the historical origins of how "completing the square" was originally
derived. That is, using pictures, blocks, and building an actual square.
By the way, this process indirectly lead to the discovery of imaginary
numbers, a nice discussion tool in a beginning algebra class. Students can
learn to factor and multiply binomials with this method. They are not tied
to the FOIL or IFOL or PAAU acronyms because they understand the
distributive rule that is taking place.
The process: start with a square block that is X by X units, include two X
by 1 blocks, and include one 1 by 1 square. The beautiful part is that the
length of X can be anything (in fact it is unknown) as long as it is held
constant throughout the problem. If you arrange all of these items into a
"perfect square" the length of one side will be X+1 and the length of the
other side will be X+1, thus giving you the two factors.
>
>After that, how about (-1)(-2) using manipulatives?
This can also be accomplished with colored beans. Take one set of the
opposite of two green beans (I sprayed pinto beans green for negative
integers). This process leads nicely into solving linear
equations-including fractional answers.
>
>How can one "complete the square" using manipulatives?
>
see above
Before the masses stone me for swinging too far into the direction of the
Standards, I will freely admit that my students have a difficult time making
the transition from the concrete, hands-on solving equations and combining
integers to the abstract notion of a variable on paper. I am sure "The Old
Pro" can offer some words of wisdom here. My students do not, however, have
difficulty understanding the factoring/multiplying process with binomials.
================================================================
Kevin Trutna, Ed.D. | P.O. Box 929
Professor of Mathematics | Yuma, AZ 85366-0929
Director Honors Program | (520) 344-7772 Office
Arizona Western College | (520) 344-7685 Honors Center
|
================================================================
****************************************************************************
* To post to the list: email mathedcc@archives.math.utk.edu *
* To unsubscribe, send mail to: majordomo@archives.math.utk.edu *
* In the mail message, enter ONLY the words: unsubscribe mathedcc *
* Words in the Subject: line are NOT processed! *
* Archives at http://archives.math.utk.edu/hypermail/mathedcc/ *
****************************************************************************