Have you ever seen a triangular arrangement of numbers that starts with a 1 and gets wider and wider as it goes down? Each row's numbers are made by adding the two numbers just above them — and this is exactly Pascal's triangle. It's remarkable that such a wide variety of mathematical patterns pour out of one simple addition rule.
Building it is very simple. Place a 1 at the very top, and in every row below, always put a 1 at both ends, and fill in the numbers in between by adding the two numbers just above them. The third row becomes 1, 2, 1, and the fourth row becomes 1, 3, 3, 1. Keep going, and the fifth row becomes 1, 4, 6, 4, 1.
Astonishing patterns are hiding inside this triangle. Add up all the numbers in any row, and you get exactly a power of 2 (1, 2, 4, 8, 16, ...); color the even and odd numbers differently, and a self-repeating pattern called a fractal appears. You can also discover that the Fibonacci sequence is hiding along the diagonals if you add numbers that way. The fact that so many mathematical treasures are hidden inside a single triangle is exactly why Pascal's triangle has been loved for so long.
In practice, Pascal's triangle is used directly in probability and combination calculations. When you're calculating the probability of getting a certain number of heads after flipping a coin several times, or the number of ways to choose a certain number of items from a group, each row of Pascal's triangle gives you the exact answer. The coefficients that appear when you expand a power of (x+y) in algebra also match Pascal's triangle exactly.
Pascal's triangle is also a table representing the number of combinations. The r-th number in row n is exactly equal to "the number of ways to choose r items out of n." For example, if you want to know the number of ways to choose two people out of five, find row five of Pascal's triangle and look at the third number, and you get the answer (10 ways) right away. Being able to solve complicated combination formulas in your head, without a calculator, is one of this triangle's biggest charms. And if you keep zooming in on the pattern created by coloring the even and odd numbers, a shape strikingly similar to a fractal figure called the "Sierpiński triangle" appears. The fact that one triangle of numbers connects probability, combinations, and fractals is a great illustration of why mathematicians have kept studying Pascal's triangle over and over.
When working through this with kids, it helps to have them draw the triangle themselves and color the even numbers yellow and the odd numbers blue. Their eyes tend to light up watching the fascinating pattern appear as the number of rows grows. On our activity page, changing the number of rows with a slider draws the triangle automatically, and one button lets you check the even/odd pattern in color. Increase the number of rows to 10, then 20, then 30, and watch the pattern get denser and more intricate — kids will be amazed that one simple addition rule can create such a beautiful pattern. Some kids notice, while coloring it in, that smaller triangles keep appearing again and again inside the big triangle — and that moment is exactly when they first encounter the concept of a fractal, hands-on, before they ever hear the word.