Flips, Reflections, Rotation, and Quadrants

Date: 11/10/98 at 23:52:30
From: Roger Johnson
Subject: Quadrants, reflections, flips, and rotation

My daughter is in fifth grade. These terms are being used to describe 
the movement of shapes. I have no idea what they are talking about and 
neither does she. Any ideas?  

Roger L. Johnson

Date: 11/11/98 at 08:53:56
From: Doctor Rick
Subject: Re: Quadrants, reflections, flips, and rotation

Hi, Roger. Do you have a computer drawing program? These terms, except
perhaps for "quadrant," can probably all be found there, and this could 
be a good way to gain an understanding of the concepts.

In many drawing programs, you can create a shape with something like a
"polygon tool." Then, while the shape is selected, you can go to a menu 
and find commands "flip," "reflect," and "rotate." Some programs might 
use different terms, and not every program can do free rotation, so 
I'll describe them.

"Flip" is a (not very technical) term for switching the top and bottom 
of a figure. Imagine attaching a horizontal rod across the middle of 
the shape, then rotating the figure half a turn about the rod. The top 
is now on the bottom and the bottom is now on the top.

"Reflect" refers to what you see when you look in mirror: your left 
hand becomes the right hand of the reflection, and your right hand 
becomes its left hand. So we often use the word "reflect" to mean the 
same thing as "flip" except that the imaginary rod on which you turn 
the shape is vertical this time rather than horizontal.

In more technical use, a "reflection" can be about any axis, horizontal 
or vertical or at any angle.

"Rotate" simply means to turn the shape, just as if you drew the shape 
on paper, put the paper on the desk, and turned the paper.

"Quadrant" is a term that helps us describe where the parts of a shape 
are. If you draw horizontal and vertical lines, they divide the page 
into four parts, each called a quadrant. We number the quadrants like 
this:

          |
          |
   II     |     I
          |
          |
----------+-----------
          |
          |
   III    |     IV
          |
          |

"Flipping" switches quadrant I with quadrant IV and switches quadrant 
II with quadrant III. 

"Reflecting" switches quadrant I with II and IV with III.

Rotation clockwise by 90 degrees moves quadrant I to IV, II to I, III 
to II, and IV to III.

Rotation counterclockwise by 90 degrees moves quadrant I to II, II to 
III, III to IV, and IV to I.

Rotation by 180 degrees (either way) switches quadrant I with III and 
II with IV.

Everything I've said is about two-dimensional shapes (shapes you can 
draw on paper). My description of reflection is wrong in 3 dimensions: 
it isn't rotation on an axis as I described it, but reflection in a 
plane (like a mirror). In two dimensions, these two actions are the 
same.

I hope this has helped. If you have a drawing program that has these
functions, it will be a big help to your daughter - but you can do 
the same things by just cutting a paper shape and turning and flipping 
it.

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