Visuals Patterns in Pascal's Triangle

Ulysses Harrison                          Dunbar Vocational High School
8435 South Wood Street                   3000 South King Drive
Chicago IL 60620                         Chicago IL 60616
(312) 239-6333                           (312) 534-9000

Objectives:

1.  To use a phenomenological approach to impress upon students the many
    visual patterns and number patterns that are present in the sequence
    of numbers that make up Pascal's Triangle.  

2.  To convince students that they are able to construct Pascal's Triangle 
    on their own without use of notes after the conclusion of this presen-
    tation. 

3.  To encourage students to explore on their own patterns that exist in 
    Pascal's Triangle, but which they were previously unaware.


Materials Needed:

    Overhead Projector
    Three (3) prepared overhead displays of Pascal's Triangle:  one (1) 
    display with all the numbers included, one (1) display with numbers 
    included in the first 4 rows only, and one (1) display with no numbers
    included in any of the elements.           

Presentation:

1.  Introduce Pascal's Triangle by showing the completed triangle on the
    overhead.  Explain to class that there are many beautiful patterns in
    Pascal's Triangle of which they are unaware.  Challenge class with 
    promise that each of them can reproduce all the numbers seen in his
    triangle that is displayed on the screen after this presentation.

2.  Remove the display of the completed Pascal Triangle and replace it with
    the triangle that has four (4) lines completed.  Explain that the outer    
    numbers are all "1's" and the inner numbers are always the sum of the 
    two numbers immediately above them.  Armed with this information, each
    student should now be able to complete the numbers in a Pascal Triangle 
    of any size.  Call on various students to supply the values for the 
    various numbers in the partially completed triangle that is displayed.

3.  Project the blank pattern of Pascal's Triangle on the screen and call
    on various students to supply the numbers that make up each cell of the
    triangle.  When it becomes apparent that students are confident of their
    abilities to complete the numbers in each cell of any Pascal Triangle,
    proceed to point out some of the many number patterns and visual patterns
    contained in Pascal's Triangle.

Conclusion:

    One of the first patterns that can be pointed out is the sum of the 
    numbers of any diagonal.  The sum of elements of any diagonal is the 
    number immediately below the last number of the diagonal.  Another 
    easily seen pattern is the sum of the rows for any row in Pascal's 
    Triangle.  For any row, the sum of the numbers in that row is 2 raised
    to the exponent of that row.  The sum of all the elements of of any
    number of rows is 2 raised to 1 more than the number of the row, then
    subtract 1.  

    Pascal's Triangle also contains the "triangular numbers":  

                   Tn = 1/2(n)(n+1) 

    and the Fibonacci numbers.  Isaac Newton, in his "binomial theorem", proved 
    that entries in the Pascal Triangle represent coefficients in the expansion 
    of (x+y)n, where n is any counting number. 

References:

    Visual Patterns in Pascal's Triangle, Dale Seymour Publications (1986).
    P.O. Box 10888, Palo Alto CA 94303.