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Q&A #5391


Factoring quadratics using a box divided in fourths

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From: Jeanne (for Teacher2Teacher Service)
Date: Dec 23, 2000 at 22:31:10
Subject: Re: Factoring quadratics using a box divided in fourths

I'd like to add to what Pat has shared with you by sharing a little of what
I do as I teach factoring of trinomials.  My students call the method you
have described that your son is using "factoring using a generic rectangle"
or "generic rectangle" for short. It is based upon an area model of
multiplication.

Before we get to factoring using generic rectangles, we use algebra tiles. I
have my kids cut out their own tiles, however, I found a site which allows
you to experience the process.

Algebra Tiles Online!
http://168.30.200.21/~bpayne/factor/algtiles.htm

The process is nicely described by Suzanne Alejandre in her webpage on this
topic.  http://mathforum.org/alejandre/algfac1.html

Meanwhile, I'll do the best I can with ascii drawings.

Example:

x^2 + 5x + 6 looks like this with algebra tiles
 _____
|     |         |||||
|     |   and   |||||  and  oooooo
|_____|         |||||

 (x^2)     +    (5x)    +     (6)

When you rearrange the tiles to make a rectangle it would look like this:


 _____
|     | |||
|     | |||           (Note: The top half is x^2 and 3x
|_____| |||
 _____  ooo            and the bottom half is 2x and 6)
 _____  ooo

Notice the o's (the units) form their little rectangle and the x's are split
up into two small rectangles made of 3x and 2x.

Eventually, I give the students trinomials that require large numbers of
tiles to manipulate such as x^2 + 23x + 42 and 2x^2 + 19x + 30.  After some
experience, kids start asking for "an easier way."  This is when I bring in
the "generic rectangle."

Back to x^2 + 5x + 6 ...
                   _____ _____
                  |     |     |  Since the 6 units form a rectangle of their
                  | x^2 |     |  own, the dimensions must be either 1 by 6 or
                  |_____|_____|  2 by 3.
                  |     |     |
                  |     |  6  |
                  |_____|_____|


If the 6 units form a 1 by 6 rectangle, the position of the x tiles are
determined and the generic rectangle looks like.
                   _____ _____        _____ _____
                  |     |     |      |     |     |
                  | x^2 |  1x |      | x^2 |  6x |
                  |_____|_____|  or  |_____|_____|
                  |     |     |      |     |     |
                  | 6x  |  6  |      | 1x  |  6  |
                  |_____|_____|      |_____|_____|

Either way the result is a rectangle that has 7 x's NOT the required 5 x's.
So we must look at the 6 units in a 2 by 3 arrangement.
                   _____ _____        _____ _____
                  |     |     |      |     |     |
                  | x^2 |  2x |      | x^2 |  3x |
                  |_____|_____|  or  |_____|_____|
                  |     |     |      |     |     |
                  | 3x  |  6  |      | 2x  |  6  |
                  |_____|_____|      |_____|_____|

The resulting rectangle makes use of 5 x's as required.  So this is the one
we are looking for.  (Note: if neither the 1 by 6 nor the 2 by 3 works, the
trinomial is not factorable.)

Now for the dimensions (factoring).

                   __x__ __+2_   The dimension of the x^2 tile is x by x.
                  |     |     |  So, the dimension of the top row must
               x  | x^2 |  2x |  be x by (x + 2)
                  |_____|_____|
                  |     |     |
                  | 3x  |  6  |
                  |_____|_____|


                   __x__+__2__
                  |     |     |    In order to maintain the width of the
               x  | x^2 |  2x |    large rectangle to be (x + 2), the
                  |_____|_____|    vertical dimension must be (x + 3).
                  |     |     |
               +3 | 3x  |  6  |
                  |_____|_____|

Therefore, x^2 + 5x + 6 = (x + 3)(x + 2).

The generic rectangle is an efficient way of factoring trinomials with
negative values (as well as trinomials for which a is greater than 1 which
Pat talked about in his response).

For example, x^2 - 2x - 35


                   __x__ _-7__
                  |     |     |
               x  | x^2 | -7x |
                  |_____|_____|   shows  (x - 7)(x + 5)
                  |     |     |
               +5 | 5x  | -35 |
                  |_____|_____|

Hope this helps.

 -Jeanne, for the T2T service

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