Classification of QuadrilateralsDate: 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 |
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