Playing with Pascal
October 31, 2009
Looking at Pascal’s triangle, note that the outer edge always starts with 1. That is because the numbers that form the triangle are equal to the sum of the two numbers above it – and of course the apex of the triangle is 1. For the outer edge, there is only one number above it: 1. And as we know from the identity property of addition, any number plus zero, is equal to that number (1+0=1). Now look at the third row which 1, 2, 1. There are two ones above the 2 … 1+1=2. The fourth row follows the same pattern; above the 3 is a 1 and a 2 … 1+2=3. And so the triangle expands infinitely.
Once you have built the triangle to your self-determined length, you can identify numerous other number patterns, for example, the edge right underneath the outer edge increases sequentially by 1. Notice too, that if you exclude the middle row of numbers (perpendicular to the bottom edge of the triangle), that the sides on each side of the middle row are mirror images of each other. Also notice, the sum of each diagonal row forms the set of Fibonacci numbers. If one highlights all of the number divisible by 5, inverted equilateral triangles are formed with each side 4 numbers long. Having students play with this triangle would be a great exploration activity.
