Pascal’s Triangle Patterns and Combinations

Now that the children have discovered the process of building the most famous array of numbers in Pascal’s Triangle, we began to discover its many patterns. 

Counting Numbers: The second diagonal of Pascals deserved a short discussion of what we call the Natural Numbers 1,2,3,4,5 … We contrasted this with the whole numbers which also contain zero.

Triangular Numbers: The third diagonal of Pascals are a list of the consecutive triangular numbers which add one more to each iteration 1,3,6,10,15,21 … These numbers form a triangular array. It is best to view these as circles stacked in triangles. 

Tetrahedral Numbers: The fourth diagonal of Pascals are a list of the consecutive tetrahedral numbers which add one more triangular number to each iteration 1,4,10,20,35 … I explained to the children that these numbers are formed by stacking triangular numbers on top of each other until you reach a triangular pyramid or tetrahedron. These are how cannonballs or spheres can be stacked with a triangular base. So when cannonballs were used to fight wars, when there were 10 cannonballs on the edge of the bottom triangle, there were 220 cannonballs in the stack (the 10th tetrahedral number).

Square Numbers: We then looked at the third diagonal of Pascals again with the consecutive triangular numbers and noticed that the sum of any two consecutive (one right after the other) triangular numbers will give you a square number: 1,4,9,16,25,36….. Incidentally, There are only three tetrahedral numbers that are also square numbers: 1,4, and 19,600 (the 140th square number and the 46th tetrahedral number). 

Powers of Two: Each horizontal row of Pascals adds up to a power of two: 1,2,4,8,16,32,64….

Hockey Sticking: Pascals is also a mechanical calculator adding numbers in any diagonal starting with “1” and adding all of the numbers consecutively (one right after the other) and you can stop at any cell. The sum of those numbers will be the number in the cell down to the left or right from the last cell (if your diagonal was a forward slash, down and to the left, the sum will be one cell down and to the right and vise a versa).

Sierpinski’s Triangle: Lightly color each even number in Pascals and you get a fractal (self-similar object) who's triangles grow by powers of thre: 1,3,9,27,81,243… 

The children should challenge themselves to calculate the greatest numbers of powers of two and hockey sticking sums. The hockey sticking sums are the best because they can check their answers and try to find their mistakes. They should also color the even numbers in one of the triangles to create the Sierpinski’s fractal.

The kindergarteners were given only 12 rows of Pascals to do their calculations but I included a 29 row page for the parents.

The 5th-7th graders also were given the challenge of generating any cell of Pascals using combination notation: nCr. Where “n” is the numbered row and “r” is the element. The element is the cell counting horizontally from “1” being the zeroth element; the numbered row is the first element and so forth. nCr = (n!)/(r!(n-r)!) I taught the children factorial language where n! is n x (n-1) x (n-2) … all they way until you are multiplying by 1. For example 5 factorial or 5! = 5x4x3x2x1. I taught the children how to solve without multiplying these numbers by simply cancelling using division and factoring. These are very satisfying calculations, when they work. An example is provided in the pdf.

Finally, I gave the children a two-sided copy of a 29 row Pascal's Triangle cut in half on each face. One side includes the middle row and the other does not. Of course, these can be put together to make one triangle.