Based on the quadrilateral tree

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Based on the quadrilateral tree

Dongwahn Suh
I remember teaching quadrilaterals and creating a tree diagram to differentiate and connect the characteristics of quadrilaterals.  Stemming out of quadrilaterals, the parallelogram then breaks up into rhombii and rectangles, which then combine to form a square. The trapezoid drops down into its own stem and then from the trapezoid was the special isosceles trapezoid.  Since parallelograms must have  two pairs of parallel sides, the trapezoid only has two parallel sides and no more.  Otherwise we would be able to categorize some trapezoids as parallelograms.
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Re: Based on the quadrilateral tree

kirby urner-4
Yes, parallelograms are quadrilaterals with both pairs of sides parallel,
meaning they include rhombi (all sides same length) and rectangles
(all angles same size).

A square is a "rhombic rectangle".

Trapezoids have at most one pair of parallel edges according to the
exclusive definition (versus "at least one").

The trapezoids we don't see so often are the ones where the base
angles are not both < 90 degrees, or both > 90

http://zonalandeducation.com/mmts/geometrySection/commonShapes/trapezoid/trapezoid.html
(second from last example)

This source makes it clear that the inclusive definition is favored in
some circles.

http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html

This inclusive definition does allow us to speak of parallelograms
as a subspecies of trapezoid.

Kirby

On Wed, May 1, 2013 at 1:57 PM, Dongwahn Suh <[hidden email]> wrote:
> I remember teaching quadrilaterals and creating a tree diagram to differentiate and connect the characteristics of quadrilaterals.  Stemming out of quadrilaterals, the parallelogram then breaks up into rhombii and rectangles, which then combine to form a square. The trapezoid drops down into its own stem and then from the trapezoid was the special isosceles trapezoid.  Since parallelograms must have  two pairs of parallel sides, the trapezoid only has two parallel sides and no more.  Otherwise we would be able to categorize some trapezoids as parallelograms.
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Re: Based on the quadrilateral tree

Bishop, Wayne-2
But he is also right.  Although favored in some circles, it is one of
those (rather few) situations where the inclusive definition is not
universal.  In fact, I think the majority of us prefer his, exactly 1
pair of parallel sides.  Moreover, convictions border on the
religious.  You know, God is on our side whichever side that is.

Wayne

At 08:57 PM 5/1/2013, kirby urner wrote:

>Yes, parallelograms are quadrilaterals with both pairs of sides parallel,
>meaning they include rhombi (all sides same length) and rectangles
>(all angles same size).
>
>A square is a "rhombic rectangle".
>
>Trapezoids have at most one pair of parallel edges according to the
>exclusive definition (versus "at least one").
>
>The trapezoids we don't see so often are the ones where the base
>angles are not both < 90 degrees, or both > 90
>
>http://zonalandeducation.com/mmts/geometrySection/commonShapes/trapezoid/trapezoid.html
>(second from last example)
>
>This source makes it clear that the inclusive definition is favored in
>some circles.
>
>http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html
>
>This inclusive definition does allow us to speak of parallelograms
>as a subspecies of trapezoid.
>
>Kirby
>
>On Wed, May 1, 2013 at 1:57 PM, Dongwahn Suh <[hidden email]> wrote:
> > I remember teaching quadrilaterals and creating a tree diagram to
> differentiate and connect the characteristics of
> quadrilaterals.  Stemming out of quadrilaterals, the parallelogram
> then breaks up into rhombii and rectangles, which then combine to
> form a square. The trapezoid drops down into its own stem and then
> from the trapezoid was the special isosceles trapezoid.  Since
> parallelograms must have  two pairs of parallel sides, the
> trapezoid only has two parallel sides and no more.  Otherwise we
> would be able to categorize some trapezoids as parallelograms.
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Re: Based on the quadrilateral tree

Louis Talman
God and John Conway both disagree with you here, Wayne.

But the real lesson here is this:  Always read the definitions that the  
author of the book you're reading uses.

And if you're teaching mathematics below the graduate level, you should  
realize that when you use a definition in the classroom that's different  
 from the textbook's definition you're just asking for trouble.


On Thu, 02 May 2013 14:48:05 -0600, Wayne Bishop <[hidden email]>  
wrote:

> But he is also right.  Although favored in some circles, it is one of  
> those (rather few) situations where the inclusive definition is not  
> universal.  In fact, I think the majority of us prefer his, exactly 1  
> pair of parallel sides.  Moreover, convictions border on the religious.  
> You know, God is on our side whichever side that is.
>


- --Lou Talman
   Department of Mathematical & Computer Sciences
   Metropolitan State University of Denver

   <http://rowdy.msudenver.edu/~talmanl>
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Re: Based on the quadrilateral tree

kirby urner-4
In reply to this post by Bishop, Wayne-2
I think it's healthy and productive in mathematics to find these areas
where people agree to disagree, or else go off the deep end and get
full blown religious, mount some kind of jihad.

The trapezoid example is especially valuable because it's so
accessible.  To find some inclusive / exclusive dialectic in topology
more generally might require too much background to let the reader
even care.

In the programming world I frequent, we have similar vicious debates
between camps which boil down to who is most willing to see it as
either/or.

"Complementary" and "Oppositional" are different concepts for a reason.

Remember "opposites attract": sometimes a divergence is a basis for
strong bonding.  The difference is treasured.

Kirby



On Thu, May 2, 2013 at 1:48 PM, Wayne Bishop <[hidden email]> wrote:

> But he is also right.  Although favored in some circles, it is one of those
> (rather few) situations where the inclusive definition is not universal.  In
> fact, I think the majority of us prefer his, exactly 1 pair of parallel
> sides.  Moreover, convictions border on the religious.  You know, God is on
> our side whichever side that is.
>
> Wayne
>
>
> At 08:57 PM 5/1/2013, kirby urner wrote:
>>
>> Yes, parallelograms are quadrilaterals with both pairs of sides parallel,
>> meaning they include rhombi (all sides same length) and rectangles
>> (all angles same size).
>>
>> A square is a "rhombic rectangle".
>>
>> Trapezoids have at most one pair of parallel edges according to the
>> exclusive definition (versus "at least one").
>>
>> The trapezoids we don't see so often are the ones where the base
>> angles are not both < 90 degrees, or both > 90
>>
>>
>> http://zonalandeducation.com/mmts/geometrySection/commonShapes/trapezoid/trapezoid.html
>> (second from last example)
>>
>> This source makes it clear that the inclusive definition is favored in
>> some circles.
>>
>>
>> http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html
>>
>> This inclusive definition does allow us to speak of parallelograms
>> as a subspecies of trapezoid.
>>
>> Kirby
>>
>> On Wed, May 1, 2013 at 1:57 PM, Dongwahn Suh <[hidden email]>
>> wrote:
>> > I remember teaching quadrilaterals and creating a tree diagram to
>> > differentiate and connect the characteristics of quadrilaterals.  Stemming
>> > out of quadrilaterals, the parallelogram then breaks up into rhombii and
>> > rectangles, which then combine to form a square. The trapezoid drops down
>> > into its own stem and then from the trapezoid was the special isosceles
>> > trapezoid.  Since parallelograms must have  two pairs of parallel sides, the
>> > trapezoid only has two parallel sides and no more.  Otherwise we would be
>> > able to categorize some trapezoids as parallelograms.
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Re: Based on the quadrilateral tree

Gary Tupper
In reply to this post by Bishop, Wayne-2
On 5/2/2013 1:48 PM, Wayne Bishop wrote:
> But he is also right.  Although favored in some circles, it is one of
> those (rather few) situations where the inclusive definition is not
> universal.  In fact, I think the majority of us prefer his, exactly 1
> pair of parallel sides. Moreover, convictions border on the
> religious.  You know, God is on our side whichever side that is.
>
> Wayne
I'm not sure how God fits in here, but:

if we define a trapezoid to be a quadrilateral with 2 (or more) parallel
sides, then any property of a trapezoid will cascade down to
parallograms, rectangles, squares & rhombi. The only such property I am
aware of: the area of a trapezoid is the product of the distance between
parallel sides, and the average length of the parallel sides. And this
formula does indeed cascade down - even to a triangle if we view it as a
degenerate with side length 0 at the vertex.

If there is a comparable advantage to using the restrictive definition,
I do not know it.

Gary Tupper
Terrace BC
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Re: Based on the quadrilateral tree

Louis Talman
On Thu, 02 May 2013 18:01:25 -0600, Gary Tupper <[hidden email]> wrote:

> If there is a comparable advantage to using the restrictive definition,  
> I do not know it.

The restrictive definition allows its advocates to be certain that an  
isosceles trapezoid is not a parallelogram.


- --Louis A. Talman
   Department of Mathematical and Computer Sciences
   Metropolitan State University of Denver

   <http://rowdy.msudenver.edu/~talmanl>
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Re: Based on the quadrilateral tree

Joe Niederberger
In reply to this post by Dongwahn Suh
>The restrictive definition allows its advocates to be certain that an isosceles trapezoid is not a parallelogram.

*Absolutely* certain.

It would be nice if these things could be taught in way that lets the students see when things are, for lack of a better term, "conventional", and when they are not.
This reminds me of the discussion a while back about the conventionality of the area measure of the unit square,
which, it became clear, is not universally recognized as such.

Cheers,
Joe N
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Re: Based on the quadrilateral tree

kirby urner-4
You can kill two virtual birds with one stone if you introduce the
concepts of version control, a branching tree, forking and so on
(Github) -- geek language -- along with Geometry as an example of such
a Github-like tree.

Suppose we add a fork to the tree such that our model of 3rd power
growth 1, 8, 27, 64... is a regular tetrahedron of edges N, rather
than a cube.  Might we develop in this direction to come up with some
interesting geometry?

It's already been done (more pioneers welcome) and put in the language
of version control, it's not even a threat to say so.  Indeed, a
"tree" is a mathematical structure, and it's just as well that we
should see all of mathematics in terms of one (but then it's also a
network).

Kirby



On Tue, May 7, 2013 at 9:46 AM, Joe Niederberger
<[hidden email]> wrote:

>>The restrictive definition allows its advocates to be certain that an isosceles trapezoid is not a parallelogram.
>
> *Absolutely* certain.
>
> It would be nice if these things could be taught in way that lets the students see when things are, for lack of a better term, "conventional", and when they are not.
> This reminds me of the discussion a while back about the conventionality of the area measure of the unit square,
> which, it became clear, is not universally recognized as such.
>
> Cheers,
> Joe N
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Re: Based on the quadrilateral tree

CCSSI Math
In reply to this post by Dongwahn Suh
It's odd how the identical conversation about trapezoids is taking place simultaneously over in the nyshsmath forum, like in a parallel universe, but ne'er the twain shall meet.

Original post:
http://mathforum.org/kb/message.jspa?messageID=8910815

And our response there (among many):
http://mathforum.org/kb/message.jspa?messageID=8911185
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Re: Based on the quadrilateral tree

Joe Niederberger
In reply to this post by Dongwahn Suh
Its not even *just* a matter a definitions, this one seems pretty much a matter of mere labels with little further impact.

On the other hand, we see via:
http://en.wikipedia.org/wiki/.999...
Timothy Gowers argues that even the meaning of .999... is at bottom a matter "of convention" -- "However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."

I happen to agree with Gowers, but also see a pretty big gulf between the two cases, as well as with the "unit square" convention I mentioned earlier.

To me, the upshot, for education, is that these sorts of matters (definitions, mere labels, definitions depending on other definitions, etc.) ought to be understood by those teaching, but apparently, that's not universally the case.

Cheers,
Joe N
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Re: Based on the quadrilateral tree

Robert Hansen
In reply to this post by Joe Niederberger

On May 7, 2013, at 12:46 PM, Joe Niederberger <[hidden email]> wrote:

>> The restrictive definition allows its advocates to be certain that an isosceles trapezoid is not a parallelogram.
>
> *Absolutely* certain.
>
> It would be nice if these things could be taught in way that lets the students see when things are, for lack of a better term, "conventional", and when they are not.
> This reminds me of the discussion a while back about the conventionality of the area measure of the unit square,
> which, it became clear, is not universally recognized as such.
>
> Cheers,
> Joe N


I would definitely place 0.9999... with the "unit square" example.

I would place "whole numbers" with trapezoid v parallelogram. You get around the former lack of accepted strict definition by being explicit, like "a whole number including zero" or "a whole number not including zero". You can also drop "whole" altogether by saying "non negative integer" or "positive integer" although I am not sure younger minds have trained themselves to respect the the subtle difference. In those cases I would stick with whole numbers with or without zero, when such a distinction is even necessary.

Likewise, one can say "a trapezoid that is/is not a parallelogram" or "a quadrilateral with at least one pair of parallel sides" or "a quadrilateral with only one pair of parallel sides".

And if a teacher states that a trapezoid is never a parallelogram, then that is certainly their prerogative, as long as they mention that this is not universally accepted, but is now the norm for this class.

Bob Hansen

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Re: Based on the quadrilateral tree

Joe Niederberger
In reply to this post by Dongwahn Suh
What bugs me about this post is the the taking of the *tree* structure to be fundamentally important, rather than investigating the properties of 4-gons and seeing what structure they naturally lead to, all labels aside.

The notion of a (graph-theory) tree, though, being both mathematical and ubiquitous even through non-math circles as a organizing principle, lends a pseudo-mathematical "rigor" and officious weight to the whole misbegotten proceeding.

Get the (correct, non-tree) structure of properties across, put the labels on afterwards, note the historic confusions for what they are.

Cheers,
Joe N
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Re: Based on the quadrilateral tree

kirby urner-4
The properties of "gons" in general should be reviewed, as students
may forget we're conventionally forcing them to be flat, all vertexes
in one plane.  A necklace of four cylindrical beads, or 4-edged
construction with ball bearing hinges, is going to be floppy in space,
all wobbly, and is not considered a "quadrilateral" except in snap
shot moments when the four edges are "in a plane".  A polygon for
fleeting instants.

What students should be reminded of is the rules of the game (like
chess) narrow the permitted / legal moves to an exponentially tiny
fragment of what's possible, but this strictness is what makes for the
rigorous proofs of Euclidean geometry.  Strict definitions exclude
what's irrelevant.  Besides, we have topology for the more
necklace-like thingamabobs.  It's not like math itself is confined by
Euclidean definitions.

Note that triangles have no choice but to be planar whereas the
quadrilateral is the first n-gon able to "hinge" in a way that
introduces no new vertexes (triangles may be creased, but this adds
new nodes).  A rhombus may be creased along a diagonal to make two
"wings", the tips of which may be connected by another edge of equal
length.  A tetrahedron is born.  It is not floppy either, being made
of triangles.

The wobbly hexahedron frozen in cube moments (unstable) is the hedron
of choice for European volume units, such as grams.  A tetrahedron,
calibrated to standards (of weight, of size), might sit in some
school's museum as an alternative choice.  It's another possible
mathematics and is accessible to all ages.  I'm something of a tour
guide in this area.  I think you understand your own ethnicity better
when you have an opportunity to compare it with something more alien.
Good mental exercise.

Kirby


On Wed, May 8, 2013 at 8:40 AM, Joe Niederberger
<[hidden email]> wrote:
> What bugs me about this post is the the taking of the *tree* structure to be fundamentally important, rather than investigating the properties of 4-gons and seeing what structure they naturally lead to, all labels aside.
>
> The notion of a (graph-theory) tree, though, being both mathematical and ubiquitous even through non-math circles as a organizing principle, lends a pseudo-mathematical "rigor" and officious weight to the whole misbegotten proceeding.
>
> Get the (correct, non-tree) structure of properties across, put the labels on afterwards, note the historic confusions for what they are.
>
> Cheers,
> Joe N
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Re: Based on the quadrilateral tree

Joe Niederberger
In reply to this post by Dongwahn Suh
Kirby says:
>What students should be reminded of is the rules of the game (like chess) narrow the permitted / legal moves to an exponentially tiny fragment of what's possible, but this strictness is what makes for the rigorous proofs of Euclidean geometry.

Not sure what you are getting at. Let's have the 3d necklace thingy - one can still constrain two opposite sides to be coplanar and parallel, or simply lie in parallel planes, etc. and map out the various properties of the so-and-so constrained varieties. That's a nice way to look at the whole game - as one of imposed constraints and degrees of freedom.

My point had to do with a prejudice to see (some) taxonomies as naturally forming a tree, but that is sometimes misleading.

I'm viewing your post as pointing out that there is a prejudice to seeing "stick-gons" as planar -- fair enough.

Cheers,
Joe N