Number Patterns in Pascal's Triangle

Ulysses Harrison Dunbar Vocational High School
3000 S. King Drive
Chicago IL 60616


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


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

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

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.


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


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

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