Fourth Grade Geometry Questions

Date: 24 Apr 1995 23:08:40 -0400
From: B Eggers
Subject: Geometry Question

Dear Dr. Math,

   My group of grade 4 regular ed. students are wondering if curved lines 
can be parallel lines.  For instance, if we drew two s's so they were 
parallel.  Also, is a cylinder a circular prism?  Another question:  Do 
polygons have to have straight lines?  Let's say you have a squarish 
figure with a "bite" taken out of it that is a half circle. (Wish I could 
draw it for you).  Is that called a polygon?  It is a closed figure.  Are 
circles like bull's eyes parallel?
  
   Thank you for your help.  These questions have caused lots of 
discussion around dinner tables of the teachers and educational 
assistants. 

Becky Eggers, Chapter 1
Edison Elementary School
Kennewick, WA 99337

Date: 26 Apr 1995 00:09:52 -0400
From: Dr. Ken
Subject: Re: Geometry Question

Hello there!

     "My group of grade 4 regular ed. students are wondering if curved 
      lines can be paralled lines.  For instance, if we drew two s's so 
      they were parallel." 

Well, the term "parallel" really only applies to actual lines.  What it
means is that the two lines don't intersect anywhere (there are other
definitions of the term like the two lines are the same distance apart
everywhere, but the most commonly accepted definition is that they don't
intersect anywhere).  So I suppose that you could try to generalize the
concept of "parallel" to arbitrary curves, but it probably wouldn't be such
a useful tool, and it's probably better to talk about curves that "don't
intersect" or that are "the same distance apart everywhere."

     "Also, is a cylinder a circular prism?"

Well, technically speaking, no.  An ice-cream cone is a circular prism.  A
prism is what you get when you take a closed figure in a plane and you
connect all its points to a point that's not in the plane.  So perhaps you
could say that a cylinder is a circular prism, but the point that the
circle's points are connected to is infinitely far away.  That's what's
known as a degenerate case of a circular prism.

     "Do polygons have to have straight lines?  Let's say you have a 
      squarish figure with a "bite" taken out of it that is a half circle. 
      (Wish I could draw it for you)  is that called a polygon.  It is a 
      closed figure."

Well, this kind of depends on who you talk to and what context you're in.
In some subjects it's convenient to think about polygons with curved edges.
In other subjects (like Euclidean geometry) it's really not convenient to
think about polygons with curved edges.  For instance, if you allow that
kind of thing, a triangle no longer has to contain 180 degrees; it could
contain more or less.  In general, you only talk about polygons being made
up of "lines," which in Euclidean geometry means "straight lines."

     "Are circles like bull's eyes parallel?"  

The term for circles that share the same center is "concentric."

     "Thank you for your help.  These questions have caused lots of 
      discussion around dinner tables of the teachers and educational 
      assistants." 

Sure, we're glad to help!

-Ken "Dr." Math

Date: Wed, 26 Apr 1995 14:20:46 +0000
From: Dr. Math
Subject: Re: Geometry Question

Hello there!

I'm afraid I said something stupid to you in my last response.  I confused
the terms "prism" and "pyramid."  All that stuff I said before is about
pyramids, and wherever you see the word prism in my message, replace it
with pyramid.  I also found out that the word "cone" is usually used to
refer to pyramids when the base isn't a polygon with straight sides.

The term "prism" is defined as "a polyhedron with two congruent and
parallel faces that are joined by a set of parallelograms."  So strictly
speaking, a cylinder isn't quite a circular prism, but if you take the
limiting case of a regular polygon where the number of sides is going to
infinity, you'll end up with a circle.  If you use that circle to generate
a prism, you'll end up with a cylinder.  So in a sense, a cylinder is the
a limiting case of a prism (called a degenerate case of a prism).  Hope
this clears up any confusion!

-Ken "Dr." Math