Algebra 2 – Pascal’s Triangle Project -Page 1- ... Suppose you are standing at the top left corner of...

Algebra 2 – Pascal’s Triangle Project -Page 1- Due: 11/7/14
Suppose you are standing at the top left corner of the grid (point A1). You are allowed to
travel down and to the right only. The only way to get to point A2 is to travel down one
unit. To get to point C2, you could travel two right and then one down. You could travel
one down and then two right. Or you could travel one right, then one down, then one
right. That makes three possible ways to get to C2. The number of possible ways to get to
A1, A2, B1, and C2 are written on the grid.
1
2
A
B
1
1
1
C
D
E
F
G
H
I
J
K
3
3
4
5
6
7
8
9
10
11
1.
In how many ways can you get from A1 to A3?
_______
2.
In how many ways can you get from A1 to E2?
_______
Place page 2 facing up as the first page of you project. Attach, in this order, colored grid,
Pascal’s Triangle, pattern explanations, and Sierpinski Triangle. (that’s 4 additional
pages).
Name _____________________________________ Per ____
Pascal’s Triangle Project – Page 2
3.
On the grid, mark the number of ways you can get from A1 to each point. Just as the
number of ways has been written for A1, A2, B1, and C2, write the number of ways
for each point not shaded. For example you will write a number at J2, but not at K2.
4.
For each of the points that are shaded, calculate the number of ways to get that point
and write an “E” if it is an even number and “O” if it is an odd number.
5.
Make a copy of your completed grid (from #4). Imagine each number is inside a
square. Imagine each square has a diagonal drawn from the lower left corner to the
upper right corner. Color the right triangle of every number that is a multiple of 2
and every “E.”
example:
1
2
3
4
6
6.
The completed grid is called Pascal’s Triangle. Draw a light line from A11 to K1.
Turn your grid so that the line is horizontal with A1 at the top. On a separate piece of
paper, write the numbers in rows as they appear on your grid. These must be hand
written. If you don’t understand what do to do, either ask Mr. Fitzpatrick (showing
your completed grid), or research Pascal’s Triangle.
7.
On a separate piece of paper, describe any patterns you see in the numbers on the
Pascal’s Triangle. You must give examples of each pattern. Find at least five patterns
other than 1’s down each side or consecutive numbers (i.e. 1, 2, 3).
8.
The pattern you see on the shaded grid is called the Sierpinski triangle. Do research
and find another way to create the Sierpinski triangle. On a separate piece of paper,
create a Sierpinski triangle. This must be neat and use must use a straightedge to
draw the lines. Below your triangle, describe how you created it.
Scoring: Check off the list below to make sure you earn the maximum points possible.
___Turned in on time.
___Grid on page one is NEAT and completed.
___Grid is correct.
_____/4
_____/3
_____/4
___You have a copy of the grid with the even numbers colored (#5).
_____/4
___Have neatly, clearly and correctly written the numbers for
_____/10
Pascal’s triangle in rows (#6).
___You have described and demonstrated at least 5 patterns.
___You have a paper on which you have created a Sierpinski triangle.
_____/10
_____/7
___You have correctly described how you created the Sierpinski triangle.
_____/3
___Your work is neat, organized, and well presented.
_____/5
Total _________/50