# Leslie Lewis

## Pascal's Triangle and Fractals

• Pascal's triangle is often used in algebra classes to simplify finding the coefficients in binomial expansions.

For instance, (X + Y)³ =1 X³+ 3 X² Y + 3 X Y² + 1

• Pascal's triangle is also used when calculating the probability of events.

When I toss 5 coins, in how many ways could I get 3 heads and 2 tails?

From Pascal's triangle we can see that there is 1 way to get all heads; 5 ways to get 4 heads and 1 tail; 10 ways to get 3 heads and 2 tails; 10 ways to get 2 heads and 3 tails; 5 ways to get 1 head and 4 tails; and 1 way to get all tails. Therefore there are 5 ways to get 3 heads and 2 tails.

A fun excursion is to challenge students to find multiples of 2, 3, 4, etc in Pascal's triangle and color them. This is also an opportunity to introduce modular math. If a student is trying to find the multiples of 7 in Pascal's triangle, it is less laborious to use the numbers 1, 2, 3, 4, 5, 6, and 0 and do modular addition then to use the actual values of Pascal's triangle. Multiples of 2 colored into Pascal's Triangle Multiples of 7 Multiples of 4

Fractals and Chaos theory are relatively new topics in mathematics. However, students easily appreciate the meaningfulness of chaos study in our world. A simple introduction to these concepts uses fractal art. Koch's Snowflake Fractal Fern Sierpenski's Gasket