Pascal's Triangle Patterns (Natural, Triangular, Tetrahedral, Square, Cubic, Powers of 11, Powers of 2, Fibonacci, Hockey Sticking, and Multiplying Seeds > 1

Last week the children were introduced to Pascal’s Triangle, its history and its properties.  They created their own Pascal’s Triangle and experimented with different seed numbers (the seed number of a traditional Pascal’s Triangle is 1; 1 is placed on the left and right diagonals of the triangle). You fill each cell with the sum of the two cells above it. 

What Pascal didn’t realize until he created the triangle was that it generates hundred of patterns, namely my favorite sequences of numbers (incidentally, these patterns are the most famous mathematical sequences).

Natural numbers starting with 1,2,3,4… is found in the first element and second to last element of each row. The elements are the diagonals in Pascal’s Triangle; the zeroth element and last element of each row is a 1, so 1111111111111…; the first element is the next diagonal in each direction. What is interesting about this sequence is that we cannot call it the whole numbers because that sequence also includes the number 0. The are the multiples of 1.

The algebraic formula for the series of consecutive natural numbers is n, n+1, n+2 … starting with n=1

Triangular numbers starting with 1,3,6,10,15… is found in the second element and third to last element of each row. They actually form the shape of a triangle by adding each successive Natural number; +2 +3 +4 … The algebraic formula for the series of consecutive triangular numbers is n(n+1)/2.

Tetrahedral numbers starting with 1,4,10,20,35 …  in the third element and fourth to last element of each row. They actually form the shape of a triangular pyramid (a tetrahedron because it has four faces) by adding each successive triangular number; +3 +6 +10 … These numbers were used during warfare when armies stacked cannonballs on triangular bases. Anytime spherical objects are stacked using triangular bases, we have tetrahedral numbers (golfballs, tennis balls, tomatoes, etc.). The algebraic formula for the series of consecutive tetrahedral numbers is n(n+1)(n+2/6.

Square Numbers starting with 1,4,9,16,25,36,49 … is the sum of any two consecutive Triangular numbers: 1,3,6,10,15 … so 0+1=1, 1+3=4, 3+6=9, 6+10=16. They actually form the shape of a square. As shown on page 3 of the pdf. you can see how a 4x4 square number of 16 is broken up into two consecutive triangular numbers 6 and 10. The algebraic formula for the series of consecutive square numbers is n^2 where the exponent of 2 means that you multiply the nth number by itself; so 5^2=5x5=25. Square numbers is how we measure two dimensional area. All flat surface are measured in square units.

Cubic Numbers starting with 1,8,27,64,125, 216 … is found by locating three consecutive tetrahedral numbers in the third element of Pascal’s Triangle; you take one times the first and third of these consecutive tetrahedral numbers and four times the middle tetrahedral number (if you start with 1,4,and 10 you get 1 + 16+10 =27 which is 3 cubed or 3 to the third power or 3x3x3 or 3^3). The algebraic formula for the series of consecutive square numbers is n^3 where the exponent of 3 means that you multiply the nth number by itself three times; so 5^3=5x5x5=125. Cubic numbers is how we measure three dimensional space. All solids are measured in cubic units.

Powers of 11 starting with 1, 11, 121, 1331, 14641 … can be found in the first five rows of Pascal’s Triangle. This even continues past the 5th row but you have to use regrouping since there are multi-digit cells (see page 2 of the pdf). The algebraic formula for the series of powers of 11 is 11^n means 11 times itself the number of times in the number n; so, 11^4 means 11x11x11x11=14,641. We even looked at my algorithm for multiplying any number by 11 which is to add that number to itself but just moved over one place value (in other words, 1 of those numbers plus 10 of those numbers equals 11 of those numbers.

Powers of 2 starting with 1,2,4,8,16,32,64,128… can be found by adding all of the numbers horizontally in a row of Pascal’s Triangle. So, if you add each of the numbers in the 10th row, the sum of 1024 is the 10th power of 2. The algebraic formula for the series of powers of 2 is 2^n means 2 times itself the number of times in the number n; so, 2^10 = 2x2x2x2x2x2x2x2x2x2=1024. I showed the children how I use my fingers to count by powers of 2 by starting with the first finger as 2^1=2, then each finger gives a new power of 2 so the second finger is 4, then 8, then 16, then 32; so if we stop at 32 with five fingers up, we see that 32=2^5=2x2x2x2x2.

Hockey Sticking method was not named by Pascal but its properties were discovered by him. We know this because hockey was not invented until the late 1700s over 100 years after Pascal. If you start with any zeroth cell number 1 and add the numbers in the diagonal element as far as you want, then turn soft to the left or right down, that final cell is the sum of all of the previous numbers. For example, if you start on the left diagonal with 1,4,10,20,35 which are the first 5 tetrahedral numbers and you make a soft turn to the left, you will end up on a cell with 70, the sum of 1+4+10+20+35. This works on any element starting with 1., See page 4 of the pdf.

Fibonacci Sequence starting with 1,1,2,3,5,8,13,21,34,55,89,144 … can be found on Pascal’s Triangle by starting with any number in the zeroth element, “1”, and adding all of the numbers in the soft diagonal as show on page 5 of the pdf. These numbers are found by adding the previous two numbers. We will explore these properties in future Mathletes classes. The formula for Fibonacci numbers is listed on page 7 of the pdf but really is too difficult below the 5th grade level. Fibonacci was credited with discovering the Arabic numerals (0 1 2 3 4 5 6 7 8 9) in Africa in the 13th century and brought Egyptian mathematics back to Europe.

Multiplying Seed Numbers in Pascal’s Triangle generate powers of that seed that are identical to the traditional Pascal’s Triangle 1, 11, 121, 1331, 14641. See page 6 of the pdf. 

Of course, I anticipated that students would want to continue to invent their own Pascalian Triangles so page 8 of the pdf is a blank triangle for further exploration.

Page 7 of the pdf has all 8 sequences organized from 1-20 so the students could practice their skill. They can check with Pascal’s Triangle for the solutions or use the formulas if 4th grade and above.

K-2nd grade were given just the triangular numbers (adding the next natural number to build the triangle), the square numbers (adding two consecutive triangular numbers), and the powers of 2 (add the full rows of Pascal’s) to work with. They were also given a page with all 8 sequences and were introduced to Fibonacci numbers as well.

These are the most fascinating patterns in mathematics and all are found in this one magical triangle of numbers. I hope they enjoy exploring these patterns and find new ones.