Title
Pascal's Patterns


Problem Statement
 

                 1        
              1   1              
            1   2   1            
          1   3   3   1          
        1   4   6   4   1        
      1   5   10   10   5   1      
    1   6   15   20   15   6   1    
  1   7   21   35   35   21   7   1  
1   8   28   56   70   56   28   8   1


What is the sum of each row? Find a way to describe the pattern of sums by comparing each row's sum to the corresponding row number.  

Investigation/Exploration of the Problem

This is very similar to the Spreading Rumors Investigation because we are looking for a pattern and need to explain that pattern algebraically. If we find the sum of each row, we notice that the sum in the next row is twice as big as the sum in the previous row. If we let the sum of the first row be r1, the sum of the second row be r2, and so on, we can develop the equation  rn= 2(rn-1).

 

We can easily see the pattern in a spreadsheet. Click here for the spreadsheet.

We can also see that the sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row. For example:

20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16

 

Extensions of the Problem

What other patterns do you notice in this triangle of numbers?

There are many, many patterns shown in this triangle of numbers. Check out this website to see how cool Pascal's triangle is!!

 

                                    

                        Click here for the website!!

 

 

 

Author & Contact
Laura Thomas

Memorial Middle School

Conyers, GA

Email

Link(s) to resources, references, lesson plans, and/or other materials
Explore Patterns in Pascal's Triangle
Blaise Pascal (1623-1662)