Pascal’s Triangle

Pascal’s Triangle Working with a partner complete Pascal’ triangle.

In this way, the rows of the triangle go on forever. At the tip of Pascal's Triangle is the number 1, which makes up the 0th row. The first row (1 & 1) contains two 1's, both formed by adding the two numbers above them to the left and the right, in this case 1 and 0. All numbers outside the triangle are considered 0's. Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1. Row 0 1 1 1 1 1 1 Row 1 Row 2 1 2 1 1 2 2 1 1 And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1. 1 1 3 3 3 1 Row 3 Add on board notes about tn,n-1 notation 1 4 6 1 5 10 1 6 15 20 1 7 21 35 In this way, the rows of the triangle go on forever.

Notice that the row numbers start at 0. Pascal’s Triangle This is item 0, row 0. Row 0 Row 1 Row 2 This is item 3, row 4 Row 3 This is item 8, row 9 Etc. 0 1 2 3 4 5 6 7 The entries also start at 0.

The natural numbers are also known as the counting numbers. They appear in the second diagonal of Pascal's triangle: Notice that the numbers in the rows of Pascal's triangle read the same left-to-right as right-to-left, so that the counting numbers appear in both the second left and the second right diagonal. Pascal’s Triangle 1 1 1 2 1 3 1 4 6 1 5 10 1 6 15 20 1 7 21 35 1 8 28 56 70

The sums of the rows in Pascal's triangle are equal to the powers of 2: Row # (r) Sum of Row 2r 1 20 2 21 4 22 3 8 23 16 24 5 32 25 6 64 26 7 128 27 256 28 20  =   1 21  =   2 22  =   4 23  =   8 24  =  16 25  =  32 26  =  64 27  = 128 1 1 1 2 1 3 1 4 6 1 5 10 1 6 15 20 1 7 21 35 1 8 28 56 70

What other great and amazing patterns exist with Pascal’s Triangle?

Homework Page 251 #2,3,4,5 That’s Pasc-tastic!

George Polya was a mathematician and mathematics teacher George Polya was a mathematician and mathematics teacher. He was dedicated to the problem-solving approach in the teaching of mathematics. In his words, “There, you may experience the tension and enjoy the triumph of discovery.” In the arrangement of letters given, starting from the top we proceed to the row below by moving diagonally to the immediate right or left. How many different paths will spell the name George Polya?

1 1 1 2 1 1 3 3 1 4 6 4 10 10 20 20 20 20 40 20 60 60 120 Complete the count

A checker is placed on a checkerboard as shown A checker is placed on a checkerboard as shown. The checker may move diagonally upwards. Although it cannot move into a square with an X, the checker may jump over the X into the diagonally opposite square. How many paths are there to the top of the board? X X

Determine the number of possible routes from point A to point B if you travel only south and east.