Return to Mathematics IndexNumber Patterns in Pascal's Triangle

Ulysses 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. Dale

Seymour Publications, P.O. Box 10888, Palo Alto, CA 94303.