Teacher2Teacher 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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