`Number Patterns in Pascal's TriangleUlysses Harrison               Dunbar Vocational High School                               3000 S. King  Drive                               Chicago IL 60616                               (312)534-9000Objectives:     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 studentStrategy:     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.  DaleSeymour Publications, P.O. Box 10888, Palo Alto, CA 94303.`