Number Patterns in Pascal's Triangle

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

Objectives:

     This lesson is designed to enable students at grade 5 or higher to      
recognize the integers, rows and columns that comprise Pascal's Triangle. 

     The main objective of the lesson is to enable students to reproduce the 
first eleven rows of Pascal's Triangle by recalling number patterns given in the 
lesson without having to look again at the original triangle. 

Materials Needed:

     Overhead projector
     Overhead projection transparency film containing Pascal's Triangle
     Overhead projection transparency film containing only blank cells 
     One photocopy of Pascal's Triangle for each student
     One photocopy of blank cells (to reproduce the triangle) for each student

Strategy:

     Inform the students that the rows and columns of integers that make up the 
triangle known as "Pascal's Triangle" contain many number patterns that they can 
easily recognize and duplicate after participation in this lesson.  Begin the 
lesson by displaying the following rows and columns of numbers via the overhead 
projector. 

                                        1
                                      1   1
                                    1   2   1
                                  1   3   3   1
                                1   4   6   4   1
                              1   5  10  10   5   1
                            1   6  15  20  15   6   1
                          1   7  21  35  35  21   7   1
                        1   8  28  56  70  56  28   8   1
                      1   9  36  84 126 126  84  36   9   1
                    1  10  45 120 210 252 210 120  45  10   1

     Point out to the students that each row in the triangle begins and ends 
with the integer 1.  After the students show an adequate indication that they 
recognize this first pattern, show them that the numbers in alternating rows 
form columns that must be lined up under each other as the triangle is expanded 
one number per row of integers.  Finally, show the students that the sum of each 
two successive integers in the row above it is equal to the integer in the row 
below it and centered between the two integers.  The students can then use this 
information and duplicate Pascal's Triangle on the photocopy of blank cells 
provided for the purpose of each student duplicating the triangle following the 
lesson. 

Performance Assessment:

     Monitor the responses of the students in the class as you point out the 
above patterns to them and have them tell you what integers will follow in the 
rows of Pascal's Triangle.  Use the blank cells photocopy for each student to 
make his/her triangle after the lesson without referring back to the original 
triangle.  Quickly collect and correct each student's duplicate triangle.  
Demonstrate and explain again how to add two consecutive integers to find the 
integers in the succeeding rows of the triangle if more than three students did 
not correctly provide all the integers on the photocopy.  Issue new copies of 
the blank cells to those students who did not perfectly duplicate the triangle.  
Have these students write out the process that produced their incorrect integers 
and resubmit a second completed copy of the triangle. 

Conclusion:

     Students can be shown how to identify some of the patterns in Pascal's 
Triangle and duplicate the triangle in a single lesson.  They can then be 
encouraged to look for some of the many other patterns that exist in the 
triangle.

References:

     Pascal's Triangle:  Green, Thomas M., and Hamberg, Charles L.  Dale
Seymour Publications, P.O. Box 10888, Palo Alto, CA 94303.