Edges vs. Corners

Date: 04/18/2002 at 20:16:31
From: Laura Grapski
Subject: Geometry Vocabulary

Dear Dr. Math,

I am currently teaching my 3rd graders about geometric figures. Every 
time we begin using the words edges, corners, and surfaces to describe 
geometric figures, my students get confused on how to determine an 
edge versus a corner. They also have difficulty counting the number 
of edges on a geometric figure, say a cube. 

Is there some good "not too over the head" vocabulary to help my 
students determine an edge versus a corner? Can you explain how to 
calculate the correct number of edges on a cube? Thanks for your help.

Appreciatively,
Laura

Date: 04/19/2002 at 09:04:29
From: Doctor Peterson
Subject: Re: Geometry Vocabulary

Hi, Laura.

Part of the problem is that the word "corner" is ambiguous; we don't 
use it in math, and kids use it in too many ways to get a clear 
picture. 

The proper terms are vertex, edge, and face, where a vertex is a POINT 
where different faces come together, and an edge is a LINE (segment) 
where different faces come together. I can easily picture calling an 
edge a corner in everyday language, so I'm not surprised they get it 
wrong. 

Probably you have been given the terms you are using because "vertex" 
(with its plural "vertices") seems too hard for young children to 
learn, but sometimes there's a reason for introducing new words. 
("Side" is even more ambiguous, so I'm glad you're using "surface" or 
"face.")

Now, how can we count the edges of a cube? It's just a matter of 
finding an orderly way to keep track of them. You could make a cube 
out of paper, and mark each edge as you count it; or use erasable 
markers on a plastic one. Or you can set the cube on a table and count 
one group of edges at a time: there are four edges on the table, four 
on the top, and four standing vertically, making 12 in all.

The fun way is to use the kind of tricks mathematicians like, which 
save work for large problems, while making use of the orderliness of 
a problem. You can look at each corner (excuse me, vertex) and count 
the number of edges there: 3 come together at each vertex. There are 
8 vertices in all, and 3 times 8 gives 24. But that didn't count the 
number of edges, because we counted each edge twice - we really 
counted the ENDS of edges, and each edge has two ends. So we divide 
24 by 2, and get the right answer.

You can instead count the number of edges on each face; again, each 
edge will be counted twice, since it belongs to two faces.

This kind of thinking is extremely useful, and seeing that all the 
different methods come out the same can be exciting, maybe even 
suspenseful!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/