1. Pascal’s Arithmetic Triangle
Kelly Shattuck
MAT 2009

2. Pascal’s Triangle

3. Triangle Terminology
Elements
Rows
Diagonals

4. Patterns in the Rows Sum of the Rows Powers of 11’s
The sum of the numbers in each row is equal to a power of 2 where n is the row number.
Powers of 11’s
If a row is made into a single number by using each element as a digit, the number is equal to a power of 11 where the power is the row number.
20 = 1 21 = 1+1 = 2 22 = = 4 23 = = 8 24 = = 16

5. Patterns in the Diagonals
Triangular Numbers
Triangular numbers can be found on the diagonal starting with row 3.
where stands for the term and
1, 1+2, 1+2+3, , , etc

6. Hockey Stick Pattern
The diagonal of numbers of any length starting with any of the 1s bordering the side of the triangle and ending on any element inside the triangle is equal to the number below the last element of the diagonal not on the diagonal

7. Now…
Let’s Color!!

8. Coloring Multiples
Even Numbers

9. Coloring Multiples
Multiples of 3

10. Coloring Multiples
Multiples of 4

11. Coloring Multiples
Multiples of 7

12. What is the probability of tossing 2 Heads if you toss 4 fair coins?

13. Example: Toss a fair coin 4 times.
Applications
It shows you the results of heads and tails when a fair, 2-sided coin is tossed
Example: Toss a fair coin 4 times.
0H 1H 2H 3H 4H
TTTT HTTT HHTT THHH HHHH
THTT HTHT HTHH
TTHT HTTH HHTH
TTTH THHT HHHT
THTH
TTHH

14. Applications
Pascal’s Triangle saves the trouble of using this tedious formula
Example:
Pascal’s Triangle Video

16. Applications
The numbers in each row of the triangle are precisely the same numbers that are the coefficients of binomial expansions.
Example: Expand

17. Lessons and Activities
Pattern Exploration
Middle School level exploration of the triangle
Coloring Multiples Exploration
Coin Tossing Activity
Exploring theoretical and experimental probability
Pizza Problem
Discovering the number of combinations of pizza topping
Binomial Coefficients
Relates the triangle to the Binomial Theorem