Genre 3: Letter to Fermat

Monsieur Pierre de Fermat,

I hope that you are in good health and are continuing to make leaps and bounds in your impressive mathematical work. I cannot believe the year 1653 is almost over, it has been a busy few year since the passing of my father. But do not worry, I have not yet given up my work in science and mathematics! In my studies of binomial coefficients I have organized the data in a way that shows some interesting patterns! You can create a triangle by creating rows as follows.

 You now continue down the triangle, with each inside entry being the sum of the two nearest entries above it. If you continue this method for 6 rows you get.

Beyond finding the coefficients of binomials, I find this triangle to have uses in the theories of counting and probability. If you are looking for the number of possibilities to draw objects from a group you can go down to the row of the number you are choosing from (The top row counts as 0) and then go to the right the number you are choosing.

Kindest regards,
Blaise Pascal

Genre 6: Annotated Story Problem

Jane is going to Cedar Point on Friday. Five of her friends want to go, however, she only has room in her car for three. To be fair, she will choose the three that will go with her randomly. Using Pascal's triangle find how many possible ways her friends can go with her (order does not matter).
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Since order does not matter we can use a combination to find the number of ways. She is choosing 3 friends from a group of 5. We can write this in the notation 5 Choose 3. At this point we could either use a formula, or write out all the possible combinations, however, we will use Pascal's triangle to help us.

First, since we are choosing from 5, we will go down to row number 5 (the top row is not counted).

Next, since we are choosing 3, we will count three columns to the right (starting on 5, the first column is not counted)

We can see that this gives us 10. Therefore, there are 10 different combinations of friends she can take to Cedar Point.

Genre 5: Timeline

Timeline of Blaise Pascal
Move cursor over points to see the events.

Timeline Created at http://www.timetoast.com/

Genre 4: Biography

Blaise Pascal was born in France and began his work in mathematics as a child. A famous French mathematician named Descartes read some of the work Pascal published when he was 16. Descartes did not believe that someone so young could have written Pascal's work. Pascal also invented a calculating machine that uses a process similar to computers of today. He also greatly advanced the theory of probability with his work on counting theory. 

To Pascal theology was more important than mathematics. He came to believe that mathematics was not God's plan for him and paused in his mathematical work after having a spiritual experience one night. After he stopped his work in math he focused on philosophy and theology. Pascal was not new to philosophy, he had previously written on the philosophy of math. He believed that the first principles, or truths, of mathematics can never be reached since proving them would require previous truths to back them up. Late in his life, he had a toothache that stopped when he thought about mathematics, he viewed this as a sign from God and returned to his mathematical work for a week and discovered the fundamental properties of the cycloid curve. He died of tuberculosis at the age of 39.

                                                   Works Cited

Clarke, Desmond, "Blaise Pascal", The Stanford Encyclopedia of Philosophy (Fall 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2012/entries/pascal/>.

Dunham, William. "7 A Gem From Isaac Newton." Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, 1990. 157-58. Print.

 Rouse Ball, Walter W. "Blaise Pascal." A Short Account of the History of Mathematics. Blaise Pascal. Trinity College, Dublin. Web. 13 Feb. 2013.

Genre 2: Pascal's Triangle

Blaise Pascal is known by many from learning about Pascal's triangle. It was originally not a triangle that Pascal created, but rather a square table of binomial coefficients. If were to turn his table so that the top left corner is facing up we would get Pascal's triangle (shown below). Pascals triangle has many numerical patterns. An engaging activity for students would be to see how many of these patterns they can find. (http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html)

(This image is licensed under a Creative Commons License in which anyone can share or remix it http://creativecommons.org/licenses/by-sa/3.0/deed.en)

 One pattern can be seen in the way the triangle is created. Each entry of the triangle is the sum of the two entries above it (the outermost diagonals of the triangle are always 1's).

Another pattern that is easy to see is the diagonals that are the natural numbers.

If we look at the powers of 11 we can see that they match up with the rows of the triangle. (At the fifth power you have to manipulate the numbers for the pattern to continue.)

When we add each row we get the powers of 2.

Other famous numbers patterns, such as the triangular numbers can also be seen in the triangle. One of the most useful features of the triangle is finding combinations in counting theory, this will be further explained in the third genre.

Genre 1: Memes

Internet memes are often images with sayings that can become very popular. For this genre I used some of the most well known meme pictures and added text that could help me teach students about combinations, permutations, and Pascal. The memes were generated using the sites www.zipmeme.com and www.quickmeme.com .

In this meme Boromir informs us that one does not simply reach the first principles of mathematics. According to Pascal the first principles, or truths, of mathematics can never be reached since proving them would require previous truths to back them up.

Even the most interesting man in the world uses permutations when he is finding the number of ways things can be chosen from a group when the order matters.

A promising mathematician remembered to use a combination when he was trying to find the number of ways he could choose things from a group when the order did not matter.

After looking at a large formula to find a combination, this angry cat decided that it would use an alternative method of using Pascals triangle to find the answer. (The next few genres will show us how Pascals triangle can be used to find combinations.)