Date: Mar 7, 2010 7:45 PM
Author: Allan Turton
Subject: Inclusive and exclusive definitions... again!
After wrestling with definitions and hierarchies for quadrilaterals yet again, I'd like to get some other opinions on the matter. I adopt inclusive definitions for the good reasons offered by John Conway, Walter Whiteley, et al. To avoid all the arguments about what is included in an inclusive definition of quadrilaterals, I've included a link to a picture that should show it clearly enough: http://www.origoeducation.com/classification-charts-2d-shapes/ A word of warning: it is written for UK and US audiences so UK trapezium = US trapezoid. As a refresher to the inclusive definition of trapezoid, it means that it has *at least* (not *exactly*) one pair of parallel sides.
What I think compounds the problem of teaching an inclusive approach is what to call the "endpoints". For example, following a line of descent down from trapezoids, through parallelograms then rectangles, we come to two end points - either a "square" or a "non-square rectangle". Now, as John Conway noted, "oblong" is a perfectly acceptable term to use instead of "non-square rectangle". So, we now have two very clear terms to describe related (but different) shapes that we can say are both rectangles.
But what names do we have for the other end points? Looking at the class of shapes we call rhombuses, we have endpoints of "squares" and "non-square rhombuses". The latter term is a bit of a mouthful and the only alternatives I've found are "oblique rhombus", "lozenge" and "diamond". You'll see in the diagram that it is just labelled "other". The term "diamond" has become dreadfully unfashionable for reasons I haven't been able to fathom, yet in everyday use it almost always refers to a non-square rhombus. It seems perfectly suited to the job! Anyone know of the sources that put the death warrant on "diamond"?
Stepping up a level to look at parallelograms, we have ones that are either rhombuses, rectangles or "non-rhombic, non-rectangular parallelograms". Awful. A suitable replacement is "rhomboid" which is underused.
John Conway, in looking at the class of kites, suggested that kites could be divided into the end points of "rhombuses" and "strombuses", "strombus" being used for the toy shape (two adjacent short sides, two adjacent long sides). He mentioned that he hoped to include the term in books he was writing - does anyone know if this happened?
As for trapezoids, well, what should we call "non-isosceles trapezoids"? I know there are "right trapezoids", but what about the ones that are non-right and non-isosceles? "Scalene trapezoid" is about the best I can think of, but even this isn't truly accurate for the whole shape, just the "sides".
Isosceles trapezoids branch into "rectangles" (then onto "squares" or "oblongs") and "non-rectangular isosceles trapezoids". I am at a complete loss as to what to call the latter. This naming problem would provide the only reason I'd have to exclude trapezoids from having more than one set of parallel sides. What do we call it apart from non-rectangular? For such an omnipresent shape (kids see them all the time in pattern blocks) it is no wonder that it is often seen as the only representative of the term "trapezoid" - there simply doesn't appear to be any other name for it!
In outside-school use, and especially in the early years of school, we use the "end points" more often than the "classes" - they have more utility. I think that once there are names established for the end points then the higher classes of shapes can be more easily introduced. For example, using "rhomboid" to label that particular type of shape would have to be better than labelling it with the higher order name of "parallelogram", then trying to convince kids later that sometimes when we talk about "parallelograms" we're actually meaning a whole bunch of shapes, some equiangular, some equilateral, some both and some neither!
I hope I've made sense. Can anyone provide some words of wisdom apart from "relax"?