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Secondary Magazine - Issue 45: Focus on


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 14 October 2009 by ncetm_administrator
Updated on 27 October 2009 by ncetm_administrator

Secondary Magazine Issue 45Sierpinski Gasket
 

Focus on...Pascal's Triangle

The rows of Pascal’s Triangle are made from the digits of consecutive powers of 11. 110 = 1, 112 = 121, 113 = 1331, 114 = 14641… They are also the coefficients of a binomial expansion.
 

Pascal’s Triangle was known well before French mathematician Blaise Pascal (1623 – 1662) put his name to it. In 13th Century China, Yang Hui used a triangle which is the same as Pascal’s Triangle (Pascal’s Triangle is known as Yang Hui’s Triangle in China). However, Pascal was the first to draw all the information about it together and developed many applications of the triangle in his 1653 work Traité du triangle arithmétique. The triangle was named after him by the French mathematicians Pierre Raymond de Montmort and Abraham de Moivre in the 18th Century.
 

Taking the first diagonals as those that contain all 1s, the second diagonal contains the counting numbers, the third diagonal is made from the triangle numbers and the fourth of the tetrahedral numbers.


The sequence generated by the sum of the ‘almost diagonals’ is the Fibonacci sequence:
 
illustration showing the sequence generated by the sum of the almost diagonals in Pascal's triangle

As the odd numbers are shaded the triangle tends to the Sierpinski Gasket

Sierpinski Gasket

 
Numbering of the rows traditionally starts with 0, so row 3 is 1 3 3 1.
The sum of the nth row is equal to 2n
The sum of the squares of the nth row is equal to the middle number of the 2n-1th row


 
If the number after 1 in a row is a prime number, then every number in that row will be a multiple of that prime.


 
Start anywhere on the edge of the triangle and, following a diagonal, shade the numbers. The sum of the numbers you’ve shaded will be found at the end of your shaded numbers at a right angle to them. For example,

illustration showing sum of diagonal numbers at right angle in Pascal's triangle


 
Substituting fractions for the integers in Pascal’s Triangle generates an activity for pupils who already have a good understanding of fraction addition and subtraction. Most benefit can be gained if discussion goes beyond 'pattern spotting' to making the connection between the values in the triangle and the underpinning structure.
 
 
 
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