Transcript Functions - Newmarket High School

Pascal’s
Triangle
Working with a
partner complete
Pascal’ triangle.
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.
0
1
1
1
1
1
5
6
7
3
6
10
15
21
3
35
10
35
Row 3
1
4
20
Row 2
0
1
2
4
0 Row 1
1
1
0
And the third:
0+1=1; 1+2=3;
2+1=3; 1+0=1.
1
1 Row 0
0
1
5
15
21
1
6
1
7
1
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.
This is item 3, row 4
0
1
2
3
4 5
6 7
This is item 8, row 9
Etc.
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-toright 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
2
1
3
1
4
1
5
1
1
3
4
6
10
1
10
1
5
1
15 20 15 6
1
7 21 35 35 21 7
1
1
8 28 56 70 56 28 8
1
1
1
6
1
The sums of the rows in
Pascal's triangle are
equal to the powers of 2:
20
21
22
23
24
25
26
27
= 1
= 2
= 4
= 8
= 16
= 32
= 64
= 128
1
1
1
1
1
1
1
1
1
6
15
56
1
3
10
21
28
2
4
6
1
3
5
7
8
Pascal’s
Triangle
4
10
20
35
70
1
1
5
15
35
56
1
6
21
28
1
7
1
8
1
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
What other great and amazing
patterns exist with Pascal’s
Triangle?
Homework
Page 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 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
3
1
1
2
6
4
1
3
4
10
10
20
20
20
20
40
60
20
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?
Determine the number of possible routes from point A to point B if
you travel only south and east.