Visuals Patterns in Pascal's Triangle
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Ulysses Harrison Dunbar Vocational High School
3000 South King Drive
Chicago IL 60616
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-
3. To encourage students to explore on their own patterns that exist in
Pascal's Triangle, but which they were previously unaware.
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.
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.
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
Pascal's Triangle also contains the "triangular numbers":
Tn = 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.
Visual Patterns in Pascal's Triangle, Dale Seymour Publications (1986).
P.O. Box 10888, Palo Alto CA 94303.