Honors Precalculus Pascal S Triangle Investigation
Honors Precalculus
Pascal’s Triangle Investigation
You should have another sheet of paper with Pascal’s Triangle on it. First, we have to fill in the blanks. You fill in a number based on the sum from the numbers directly above it. See the example to the right.
Fill out the rest of the triangle. Be careful with your arithmetic! Check yourself! The bottom row should be:
1 / 15 / 105 / 455 / 1365 / 3003 / 5005 / 6435 / 6435 / 5005 / 3003 / 1365 / 455 / 105 / 15 / 1We label each row starting with 0. So, the top “1” is Row 0, the row “1 1” is Row 1. Label your triangle. The last row should be Row #15.
Now, add up each horizontal row. Write the sums in the appropriate place. What do you notice about the sums?
We can create a recursive (repeated) Sierpinski Triangle. To see what the Sierpenski triangle looks like, color ALL the odd numbered spots in Pascal’s Triangle. Use a highlighter or color lightly, so you can still read the numbers. What do you notice?
Now, take a look at the diagonals. Notice the first diagonal side of the triangle is all 1s. The next diagonal is 1, 2, 3, 4, …. What do we call these numbers?
The next diagonal is 1, 3, 6, 10, 15, …. You may not recognize these numbers. They are called the triangular numbers - how many dots it takes to form a triangle. See the chart below to help you better understand.
What is the next value in the table?
There is a great sequence called the Fibonacci Sequence. It is 1, 1, 2, 3, 5, … It starts with 1, 1 and then the next term is formed by the sum of the previous two terms. (1 + 1 = 2, 1 + 2 = 3, 3 + 5 = 8). What are the next three terms of the sequence?
You can find Fibonacci’s Sequence all over the world in the number of petals on a flower, the number of spirals on a pinecone and the galaxies. It’s in the Golden Ratio which is used to build great buildings such as the Parthenon.
We can even find the Fibonacci Sequence in Pascal’s Triangle! It’s taking “diagonals” or think of it as going up and over repeatedly. See the picture and colors to help you understand. What are the next terms of the Fibonacci Sequence? How can you get them from Pascal’s Triangle?
We also have the hockey stick pattern. If a diagonal of numbers of any length is selected starting at any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself. If you don't understand that, look at the drawing. Once you understand, find four more examples of hockey stick patterns.
Also, take a look at each row that starts (after the 1) with a prime number. So, the row with 1 3 3 1 would be your first one. Look at the other rows starting with a prime number. What do you notice?
Look at this ancient Chinese drawing entitled "The Old Method Chart of the Seven Multiplying Squares". They even knew about Pascal’s Triangle! What do each of the symbols stand for?
Lastly, we do actually use this triangle for math! To get started, I want you to find the following:
· (a + b)2
Compare your result to Row 2. What do you notice?
· (a + b)3
Compare your result to Row 3. What do you notice?
· (a + b)4
Compare your result to Row 4. What do you notice?
The image above is probably the most famous image of Pascal’s Triangle in math history (Yang Hui’s triangle, from the 13th century).
However, the diagram contains a mistake. Where is it?
Go back to (a + b)2, (a + b)3, and (a + b)4.
1. Compare the number of terms you get with the power used. What do you notice?
2. What patterns do you see in the exponents?
3. What patterns do you see in the coefficients?
4. Write (a + b)5 as a polynomial in standard form without multiplying / showing work.
5. Write (a + b)10 as a polynomial in standard form.
6. Expand (q + 5)7.
7. Expand (3x – y)4.
8. Expand (2m + 3n)6.
This is probably the most famous image of Pascal’s Triangle in math history (Yang Hui’s triangle, from the 13th century).
However, the diagram contains a mistake. Where is it?