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Classification of Quadrilaterals

Date: 02/14/2008 at 09:14:20
From: Pat
Subject: Classification of Quadrilaterals

I teach quadrilaterals to 5th graders as a large group of 4-sided 
figures that contains two sub-groups: parallelograms and trapezoids 
(includes right trapezoids).   Under parallelograms, we have two sub-
groups: rectangles and rhombi.  The square is a sub-group of both 
because it has properties of both rectangle and rhombus.            
                    
This is very different from the answer I saw in your archives.  You 
have quadrilaterals divided into kites and trapezoids and 
parallelograms as a sub-group of trapezoids.  Have I been doing it 
all wrong?  I am so upset.  Is the kite group for quads with NO 
parallel sides?

                          QUADRILATERALS
         Parallelograms                         Trapezoids
 (two pairs of parallel sides)           (one pair of parallel sides)

Rectangle                 Rhombus
(parallelogram         (parallelogram with
with 4 right angles)   all sides congruent)

               Square
        (rectangle with
        all sides congruent)



Date: 02/14/2008 at 10:28:36
From: Doctor Ian
Subject: Re: Classification of Quadrilaterals


Hi Pat,

First, relax, because these kinds of taxonomies are neither "right"
nor "wrong".  As long as the kids get the individual definitions
correct, how they're grouped is just a matter of convenience. 

>I am so upset.  Is the kite group for quads with NO parallel sides?

It's not a group, it's one type of quadrilateral.  Note that you can
form a quadrilateral with NO congruent or parallel sides, e.g., 

        ..............     
       .             .
      .              . 
     .               . 
         .           .
             .       .
                 .   .
                     .

All that's really going on with taxonomies like the ones you can
construct for quadrilaterals is that they let you refer to certain
items as special cases of more general items.  For example, I could
think of a square as a rectangle with congruent sides... or as a
parallelogram with congruent sides and angles... or just as a "regular
quadrilateral".  They're all equivalent definitions. 

One way to think about any taxonomy is this.  Suppose I start with a
big bag of all possible quadrilaterals.  I ask a question, like:  Do I
have any parallel sides?  I pull each quadrilateral out of the bag and
look at it, and throw it into one pile if none of the sides are
parallel, and another pile if at least one pair of sides is parallel.

That second pile can be given a name: "trapezoids". 

Now, I can ask another question about the pile of trapezoids, like: 
Is the other set of sides also parallel?  Again, I look at each guy in
the pile, and if it has two sets of parallel sides, I throw it into a
new pile: "parallelograms".

In this way of looking at the world, a parallelogram is a special case
of a trapezoid.  (It just has an extra set of parallel sides, in the
same way that a square is a rectangle with an extra set of congruent
sides.)

This isn't as strange as it may seem.  For one thing, it reduces the
number of formulas you have to learn for the areas of quadrilaterals:

  General Area Formula
    http://mathforum.org/library/drmath/view/54685.html 

This, in fact, illustrates an important use of taxonomies.  If I think
of a parallelogram as a special case of a trapezoid, then if I can
show that something is true of trapezoids, I know automatically that
it must also be true of parallelograms.  That kind of thinking can
save a lot of work... and it also helps explain why we don't want to
lock ourselves into one "correct" taxonomy.  Different taxonomies are
useful in different situations--just as a screwdriver is useful for
some tasks, and a hammer is useful for others.  You don't want to just
have a single tool that you try to use for everything. 

The point is, the order of the questions you ask determines the
taxonomy you end up with... in much the same way that the choices you
make when constructing a factor tree give you different trees:

       36
     /   \
    6     6
   / \   / \
  2   3 2   3


      36
     /  \
    2    18
        /  \
       2    9
           / \
          3   3

      36
     /  \
    3    12
        /  \
       2    6
           / \
          2   3

Does this help?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 02/14/2008 at 11:02:19
From: Pat
Subject: Thank you (Classification of Quadrilaterals)

Thank you so much!  This explanation really helps me--it'll help in
other areas, too.  You're awesome!  Pat
Associated Topics:
High School Triangles and Other Polygons

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