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SNAP – An Introduction to Group Theory

This activity is based on an activity found in:

Huetinck, Linda and Munshin, Sara N. Teaching Mathematics for the 21st century: Methods and Activities for Grades 6-12. Prentice Hall. New Jersey: 2004, pp. 287-89.

Audience: 8th – 12th grade

Goals:

  1. To practice the algebraic properties of identity, inverse, closure, commutative property, and associative property.
  2. To define a group.
  3. To find a group isomorphic to the dihedral group of order 6.

Materials: SNAP Board, rubber bands, handouts (Review topics for group theory, see answers to the review, SNAP-Introduction to Group Theory, and SNAP boxes).

Implementation:

  1. Divide the class into groups of 3-4.
  2. Each group must determine how many possible ways exist to connect the dots in the first row to the dots in the second row. Each dot has only one connection and each dot is used only once.  Each figure will then be referred to as an element. Label the elements as follows:

 

  1. The students should now fill in the elements on the table at the top of the “SNAP-Introduction to Group Theory” handout.

 

  1. The SNAP operation begins. Use the “SNAP boxes” handout to sketch the figures before performing the SNAP operation.

Example:

  1. Use 3 rubber bands to construct the same figure on the SNAP Board by stretching the rubber bands between the top and bottom pegs around the middle pegs. Do not twist the bands.
  2. Do the SNAP operation by removing the middle pegs.
  3. Record the figure that remains on the SNAP Board in the SNAP box and label the element

 

Example:

  1. Record the result on the table.

S
N
A
P

A

B

C

D

E

F

A

 

B

 

 

 

 

B

 

 

 

 

 

 

C

 

 

 

 

 

 

D

 

 

 

 

 

 

E

 

 

 

 

 

 

F

 

 

 

 

 

 

                          

  1. Complete the table and discuss observations.
  2. Define a group and identify the SNAP elements as a group
  3. This activity can be continued to show that SNAP is isomorphic to dihedral group of order 6. (see triangle activity)

 

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