Pascal’s Triangle

Pascal’s Triangle

Working with a partner complete Pascal’ triangle.

1 7 21 35 35 21 7 1

1 6 15 20 15 6 1

1 5 10 10 5 1

1 4 6 4 1

1 3 3 1

1 2 1

1 1

1

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.

10

1

Do the same to create the 2nd row: 0+1=1; 1+1=2; 1+0=1.

And the third: 0+1=1; 1+2=3; 2+1=3; 1+0=1.

In this way, the rows of the triangle go on forever.

0

1

1

2

0

1

1

0

1

1

3

2

3

1

1

0

Row 0

Row 1

Row 2

Row 3

Pascal’s Triangle

This is item 0, row 0.

This is item 3, row 4

This is item 8, row 9Etc.

Notice that the row numbers

start at 0. Row 0

Row 1

Row 2

Row 3

0 1 2 3 4 5 6 7

The entries also start at 0.

1 7 21 35 35 21 7 1

1 6 15 20 15 6 1

1 5 10 10 5 1

1 4 6 4 1

1 3 3 1

1 2 1

1 1

1

1 8 28 56 70 56 28 8 1

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 7 21 35 35 21 7 1

1 6 15 20 15 6 1

1 5 10 10 5 1

1 4 6 4 1

The sums of the rows in Pascal's triangle are

equal to the powers of 2:Pascal’s Triangle Row

# (r)

Sum of

Row 2r

0 1 20

1 2 21

2 4 22

3 8 23

4 16 24

5 32 25

6 64 26

7 128 27

8 256 28

1 3 3 1

1 2 1

1 1

1

1 8 28 56 70 56 28 8 1

20  =   121  =   222  =   423  =   824  =  1625  =  3226  =  6427  = 128

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

HomeworkPage 251 #2,3,4,5

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 proceedto 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

1

1

1

2

3 3

4 46

10 10

20

20 20

2020 40

60 60

120

Complete the count

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?

XX

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

A

B

A

B