Return to Mathematics IndexVisuals Patterns in Pascal's Triangle

Ulysses Harrison Dunbar Vocational High School

3000 South King Drive

Chicago IL 60616

(312) 534-9000Objectives: 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": T_{n}= 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.