| Curriculum Topics: | Pascal's triangle,
binomial coefficients, patterns of even and odd numbers |
| Fractal Topics: | Sierpinski
triangle, generating fractals by recursion |
| Number of Days: | 1 |
| Grades: |
| PreCalculus Course (high school or college level) |
| Pre-Service or In-Service Mathematics Teachers Course (college level) |
| Liberal Arts Mathematics Course (college level) |
|
| Authors: |
| Virginia R. Jones | Central Connecticut State University |
| Kathleen Stankewicz | |
|
| Objective: | To have the students compare the Pascal Triangle to the
Sierpinski Triangle and discover relationships and patterns that occur within each
mathematical object and between the two mathematical objects.
|
| Resources: |
| Working from handouts of: |
| 1) an equilateral "grid" for displaying the Pascal Triangle which requires the
students to fill in the first 16 rows of numbers in the Pascal triangle.
(This gives the opportunity to review the coefficients of the binomial expansion
for (x + y)n and to discuss the symmetry about the center
of Pascal's triangle.) |
| 2) a display of the Sierpinski Triangle for the first four levels |
| Note: These handouts can be simple paper copies or transparency overlays.
Click for copies of Pascal's triangle
and the Sierpinski triangle in new windows. |
|
| Description of Lesson: |
| Activities: (not an exhaustive list) |
| After the students have entered the values in the Pascal Triangle, have them
highlight the values that are even numbers.
|
| 1) Ask them to describe what happens when they place the Sierpinski
Triangle over the Pascal Triangle and align the two objects. |
| 2) Ask them to conjecture what would happen if both the grids were extended
so the Pascal Triangle had more rows below the given grid and the Sierpinski Triangle
was extended so it covered the new Pascal grid. |
| 3) Have them shift the Sierpinski Triangle so that the triangle of it matches
a value in some other row of the Pascal Triangle grid. Can they see another
relationship between the two objects? Can they write a conjecture about this
relationship in mathematical terms? |
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