An Interesting Triangle

Discovered by a Chinese man in the mid 1200's, the triangle most popularly known as Pascal's Triangle has been the source of answers to many mathematical questions. It is a simple triangle to create, made up of numbers rather than line segments.


Can you see how to create the next few lines? If you do continue the triangle correctly, there are a number of patterns to find. You can start simply by examining the makeup of the triangle and the patterns in the diagonals of the triangle. And here are some other things to try.

• Add up the numbers in each row of the triangle. What pattern do you see in the sums? Make a prediction about the sum of the elements in the 50th row.
• Add up the numbers in each of the colored diagonals, beginning with the first yellow diagonal, then the red diagonal following it, then the green diagonal following that, then the blue diagonal following that, and so on. Do you see a pattern in the numbers you get?
• Continue the triangle for several more rows, maybe up to row 20 or so and make some copies of it. Shade all of the squares which contain odd numbers. What do you notice?
• On a fresh copy of your large triangle, shade in the squares that are multiples of 5. What do you notice? Try other multiples on other copies and see what you notice.
• If you remember some algebra, or if you have algebra students in your household, try figuring out the following binomial expansions: (x + y)⁰; (x + y)¹; (x + y)²; (x + y)³; (x + y)⁴; What do you notice about the coefficients of the terms?

Feel free to post any patterns you see in the comments. Or, give this to your kids to try. Even very young learners can have fun with the triangle, looking for patterns in the numbers.

And there's much more, as well. Perhaps I'll introduce you to some of the more mathematically intense uses of it later on.