Counting Triangles

Mamie P. Scott                    Coles School        
                                  8440 South Phillips
                                  Chicago, Illinois 60617
                                  312-933-6550
                           
Objectives:

  The sixth grade student will:
  1.  Organize, summarize, and record information.
  2.  Discover a rule or formula that will find the total number of triangles in 
      a figure.        

Equipment and Materials:
  
  1. Overhead projector, transparencies, and marker. 

  2. Transparency displaying one large triangle divided into six small 
     triangles.  Make a 1-part triangle, a 2-part triangle and a 3-part triangle 
     to be used as overlays on the large triangle. 
  
  3. Teacher-prepared worksheet with six triangles arranged in numerical order 
     from 1 to 6.  Each triangle should be equally divided to represent the 
     number to be counted.  
                                                   
Recommended Strategies:
  
  A. Establish a meaningful definition of a triangle.
  B. Form small groups and distribute worksheets.
  C. Use overhead projector to demonstrate how to establish a counting 
     arrangement by writing a letter name starting clockwise in the triangle 
     (a, b, c, d, e, f).  Name all 1-part triangles, name all 2-part triangles 
     and name all 3-part triangles. 
  D. Summarize by organizing all the data into one table like this:
  
                             Triangle Table
     __________________________________________________________________
      Type of triangle       Listing by letter     Number of triangles
     ___________________________________________________________________
         
         1-part              a, b, c, d, e, f             6
         2-part              bc, ed, af,                  3
         3-part              abc, bcd, cde, fed           6
                               afe, fab
         6-part              abcdef                       1
     ___________________________________________________________________
                       Total number of triangles         16

  E. Use the second worksheet to count total number of triangles numbered 1-6. 
     In counting the triangles, a pattern should be discovered by students. The 
     pattern is that the total number of triangles is the sum of all counting 
     numbers from 1 to the number of small triangles in the figure. 
           Example: A triangle equally divided into three small triangles is:
                             1 + 2 + 3 = 6
                                  
Evaluation:

     Provide additional opportunities for reinforcement of this concept by using
     worksheets. 
                                                             
References:

     Lenchner, George. Creative Problem Solving in School Mathematics. Boston:
     Houghton Mifflin Company, 1983.
     
     Seymour, Dale. Favorite Problems. Palo Alto, Calif.: Dale Seymour 
     Publications, 1982.