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Discussion: Dynamic Geometry Exploration: Properties of the Midsegment of a Trapezoid tool
Topic: Midsegment of a rectangle?
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Subject:   RE: Textbook?
Author: Mathman
Date: Dec 11 2004
On Dec 10 2004, Susan wrote:
> We used to use the Integrated Mathematics series by McDougall
> Littell, and in Integrated II, this exact question is posed to the
> students!  It asks the students to explain why the two definitions
> are different, and then it gives diagrams of 6 quadrilaterals, and
> asks the students to choose which are trapezoids applying defintion
> #1 and which are trapezoids if you apply definition #2. I checked
> the glossary of the textbook though, and they use "A quadrilateral
> with one pair of parallel sides."  Seems like they took the middle
> road, and left it up to interpretation!  I usually discuss both
> definitions in class, but use whichever one their text uses just so
> there isn't any confusion. Guess I'm copping out too!

I somehow doubt that.  You are far to concerned about doing things properly for
your students.  I won't say "I'm right and they're wrong.", but will ask you to
simply consider that the text is just another person's opinion.  You will find
this constantly if you look around trough varoious texts on any particular
subject.  High School texts often have numerous errors in them, and the authors
are not producing bibles.  Neither are they an ultimate authority, but just one
guide.  It would be good for the students to learn to look for resources in many
different ways.  This is precisely what they must do when they attend university
or college, unless things have changed.  I can recall countless hours with not
one, but a pile of texts and references.

Oh well, OK, in this case I think that author is wrong.  It happens.  A simple
trapezoid has only two sides parallel.  But the parallelogram must be a
trapezoid having all of its properties[**].  It is also more than that, and has
"at least" two sides parallel.  At least I'd be forced to tell that to my own
students, then have them discuss the similarities and differences of opinion to
reach another concensus of opinion.

[**]  This is important in formal proof.  When deciding to prove a certain
figure is a trapezoid, it is necessary and sufficient to prove that it has four
sides and has two sides parallel.  In order to prove that a point lies on the
perpendicular bisector of a line segment, it is necessary and sufficient to show
that it is equidistant form the end-points of the line segment.   The
"necessary and sufficient" comes from the definition, being the least that is
required to define it.

I am a Canadian. It is not necessary to state whether I am white, male, height
and weight in order to state that fact.  I am all of that and more, but am
Canadian by right of citizenship.  You may be entirely different in many
aspects, but hold your citizenship by the same right.  Being more does not deny
the fact of your citizenship.  I do not say I am "at least a Canadian".
    
David.

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