I'll try it; and let you know how it works.
What is the problem these days:
Is it that teachers cannot teach? or......
Is it that students cannot learn?
Regards,
The Old Pro
Jim C. Gajniak
Math Teacher
At 12:04 PM 4/23/1997 -0400, you wrote:
>Dear "Old Pro",
>
>>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
> |
>================================================================
>
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