Mary Margaret Nee              Von Stueben Science Center
                               5039 N. Kimball Ave.
                               Chicago, IL 60625

      Throughout this unit the students are: to construct equilateral triangles on 
cardboard; to develop the transformation concept with a one to one correspondence; 
to generate the rotation and reflection table; and to deduce from the model the 
abstract axioms of a group. 

Behavioral Objectives:
     The students will achieve practical knowledge of abstract concept of a group 
and will verify this through testing with a desired goal of 85% success. 

         A  Overhead Projector
                  1.  Large Plastic Model
                          a.  equilateral triangle with circumscribed circle
                          b.  square with circumscribed circle
                  2.  Transparencies
                          a.  Transformation
                          b.  Rotation and Reflection Table
                          c.  Algebraic axioms for a Group
          B  Each Student needs
                  1.  Paper with a reference circle
                  2.  Cardboard Equilateral triangle
                  3.  Cardboard Square

     A Background
There are many subject areas in Mathematics which would be enriched with this 
presentation.  This should be introduced to a class where student cooperation 
has been well established. The teacher must be extremely careful to instruct the 
class in the correct way of formulating the transformations. 

     B Activity Outline
 1.  pass out model triangles to students
 2.  label the triangles in a clockwise way  A,B,C,
 3.  model transformations on the overhead
 4.  discover individually the finite elements under the definition
 5.  create table of rotations and reflections
 6.  observe patterns within the table
 7.  name these patterns or laws
 8.  define algebraicly the group
 9.  verify the model with its laws as a group

     C  General Plan
     With the large equilateral triangle on the overhead  demonstrate the concepts 
of rotation and reflection. Then allow the student to discover all finite 
     Once the elements are named start gnerating the table with the six elements.  
From the observed data verify that closure, identity element, inverse for each 
element and the associative axions or laws hold for the newly created table. 
Continue to extend the binary operation to three elements by using the associative 
law and then to any number of elements operating together. 
     Show on the overhead the list of definitions for an abstract group.  Again 
affirm that the patterns created by the transformations of the equilateral 
triangle form a set of elements which is well structured set of axioms with the 
the binary operations of the transformation necessary for a group 

     D.  Overhead Transparencies
         1.  One to One transformation of Vertices
                 Name of Vertex          Transformed to Position
                                            held by former Vertex

                               A  goes to  B
                               B    " "    C
                               C    " "    A         call this R1

                               A  goes to   C
                               B   "  "     A
                               C   "  "     B         call this R2

                               A  goes to  A
                               B   "  "    B
                               C   "  "    C          call this R3

                               A  goes to  A
                               B   "  "    C
                               C   "  "    B          call this ra

                               A   goes to C
                               B    "  "   B
                               C    "  "   A         call this rb

                               A  goes to  B
                               B    "  "   A
                               C    "  "   C         call this rc

         2.  Table of Rotations and Reflections of a
                 Equilateral Triangle

             "O"   R1   R2   R3   ra   rb   rc   
           R1]     R2   R3   R1   rc   ra   rb 
           R2]     R3   R1   R2   rb   rc   ra
           R3]     R1   R2   R3   ra   rb   rc
           ra]     rc   rb   ra   R3   R1   R2
           rb]     ra   rc   rb   R2   R3   R1
           rc]     rb   ra   rc   R1   R2   R3

         3.  Definitions
              A.   Group "G" is a set of elements (a,b,c,d,e,...)
                   and a binary operation called  product "O" such
                   that the following axioms hold:
                  1.  Closure:    Form every ordered pair a,b of 
                      elements of G the product a"O"b = c exists
                      and c is an elements of G.
                  2.  Associative Law: (a"O"b)"O"c  =  a"O"(b"O"c)
                  3.  Existence of Unity of the Identity element
                        An element "E" such that a"O"E = E"O"a =a
                        for every a an element of G
                  4.  Existence of Inverse of any element
                         For every a of G there exists any element
                          -1                    -1            -1
                         a   of G   such that  a  "O"a = a"O"a  =E

               B.  Dihedral Group
                      The one to one mappings of a set onto itself
                      which preserves a property usually form a 

                      Symmetries of a geometric figure are of this
                      kind.  These are the congruent(distance-
                      preserving) mappings of the figure onto itself.

                      The symmetries of a regular polygon of sides
                      larger than or equal to  3  form a group 
                      called  a Dihedral Groups of order 2n.

                      It can be shown within these dihedral groups
                          (Ri) =E  where E is the identity element
                                    and i = 1,2,3,..(9n-1)
                               2          2         2
                           (ra) =E ,  (rb) =E,  (rc) =E or for any
                            reflection of the dihedral group squared

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