Definition of Trapezoid
Paul J. Shalonis
discussion here about the definition of a trapezoid. I wasn't
teaching geometry then and didn't follow it. Seemed like you
all had differing opinions whether or not there could be two
pair of parallel sides, i.e. a parallelogram would be a
trapezoid, or just one pair of parallel sides. I've seen both
definitions, but it seems the newer books lean toward the just
one pair of parallel sides.
I'd appreciate any comments on the definition and how you might
discuss this with your students. Thanks.
Paul Shalonis
psha...@pen.k12.va.us
Jeff Miller
>
>I seem to remember about a year or so ago there was a
>discussion here about the definition of a trapezoid. I wasn't
>teaching geometry then and didn't follow it. Seemed like you
>all had differing opinions whether or not there could be two
>pair of parallel sides, i.e. a parallelogram would be a
>trapezoid, or just one pair of parallel sides. I've seen both
It seems that most textbooks require a trapezoid to have exactly
one pair of parallel sides. The UCSMP (University of Chicago)
Geometry textbook, however, allows one or two pairs of parallel
sides, which makes a more interesting hierarchy of quadrilaterals.
We don't use the UCSMP textbooks, but I've told my students
that if they were attending the other high school across town,
a parallelogram would be a trapezoid.
I suppose a similar issue is whether or not an isosceles triangle
can have all three sides congruent.
It would be interesting to see a list of mathematical issues that
might be encountered in high school over which textbooks might
differ. I can only think of a few others. Some textbooks
use the term "reflex angle" for an angle >180, whereas others
apparently do not allow for angles >180. I presume some
textbooks consider an expression with a radical in the
denominator to be simplified, although ours doesn't. And of
course most textbooks consider 0^0 undefined, whereas some authors
argue that it ought to be defined as 1 (which by the way the new
TI-92 does). However this issue has been discussed to death!!
Jeff Miller
Gulf High School
New Port Richey, FL
Je...@sanctum.com
PoHi
is not a type of trapazoid. Both trapazoids and parallelograms are
directly quadrilaterals.
beve
Hi all teachers, Liberians, and other's in which education higher than
mine ( hopefully)
I was wondering when and by who and the schedule that we still fallow
today in school was
invented. I mean its not my favorite schedule to fallow, or to wake up to.
I was just wondering this, one day
( before i fell asleep in first hour ).
Still wondering
Rob (irc beve) be...@usa.pipeline.com
Larry Davidson
This is a reasonable position, but how can you assert it
so dogmatically? In fact, the arguments are almost
certainly stronger for the other definition than they are
for yours. In particular, if you define a trapezoid as a
quadrilateral with at least one pair of parallel sides,
then parallelograms automatically inherit all theorems
about trapezoids. On the other hand, if you require a
trapezoid to have a pair of non-parallel sides, you have
to restate theorems unnecessarily. The only drawback that
I can see to the "at least one pair" definition is that
one has to take additional care in defining an isosceles
trapezoid.
For the skeptical out there, let me make it clear that I
am not making up the lack of consensus on this issue.
There _is_ genuine disagreement, and we are therefore free
to pick whichever definition we prefer. Let me quote three
geometry texts:
"A trapezoid is a quadrilateral which has two parallel
sides. Note that this definition allows the possibility
that _both_ pairs of opposite sides are parallel." --Moise
"A trapezoid is a quadrilateral with one and only one pair
of sides parallel." --Weeks & Adkins
"A quadrilateral is a trapezoid if and only if it has
exactly one pair of parallel sides." --Kalin & Corbitt
While I prefer the less common definition, it _is_ the one
chosen by a professional mathematician and has much to
recommend it.
J. Loomis
Isn't one of the most important parts of geometry learning how to work
with a well defined set of rules and definitions?
One of the nice things about teaching junior high is the rather open
agenda. When I teach quadrilaterals and triangles, I tell the kids that
THEY will decide whether or not parallelograms are a subset of trapezoids
and whether or not equilateral triangles are a subset of isosceles
triangles.
I point out what their USELESS(!) textbook says (and what other resources
say). We discuss what the high school geometry teacher thinks. Next
year, I hope I'll find a debate in this group. (I might even start one.)
Let's try to avoid the unnecessary dogma (there is a little necessary
dogma, but this isn't) and try to help the kids to make rational decisions
instead.
--
We're all in this together.
John Loomis jlo...@erinet.com
*** A lazy man does the most work. ***