Pascals Triangle: Diagonals of "1s", "2s", "counting", and "prime" numbers; 12, 20, and 29 Rows

Pascal’s Triangle is named after the French mathematician and philosopher Blaise Pascal (1623-1662). It is a triangular array of counting numbers. “1s” are placed along the diagonals and each other cell is the sum of the two cells above it. The largest number on the 12th row of Pascal’s Triangle is 924. The largest number on the 20th row of Pascal’s Triangle is 184,756. The largest number on the 29th row of Pascal’s Triangle is 77,558760. From seemingly innocent ones, these numbers grow exponentially. 


The most surprising result are the dozens of patterns that emerge from these numbers. We will be exploring these patterns over the next few weeks. Believe or not, this challenge is a risky proposition: usually, we welcome mistakes; they are the only way to develop deep understanding; however, here any mistake no matter how small will effect every number in the triangle below that cell. I explained to the children that I made a mistake by doing these calculations in my head and after I realized my mistake, had to whiteout and recalculate hundreds of numbers. My poor decision was “obtuse” at best. I strongly encouraged them to take the strategic approach by using vertical addition to generate each cell. Of course, the two diagonal rows of counting numbers do not require writing out the calculations, but all sums above two digits should be done using best practices.


The last page of the pdf has all Pascal’s solutions to the 29th row. The children should use this to check their answers only. I gave the children the first 12 rows to being this exploration. Most of the children went beyond the 12th row to the 20th. The attached pdf on 8.5x14 paper allows this additional exploration. The 5th — 7th graders received an 11x17 version of Pascal’s to row 29. They will have to write very small to enter these values past four-digit numbers.


Beyond conventional Pascal’s, I created three additional challenges with (i) diagonals of “2s”, (ii) diagonals of counting numbers, and (iii) diagonals of prime numbers. These additional challenges generate many surprising patterns as well.

Pascals_Triangle_12_rows_traditional_and_variations.pdf1.01 MB
Pascals_Triangle_Rows_13-20_on_8.5x14.pdf429.86 KB
Pascals_Triangle_29_rows_blank_on_11x17.447.01 KB