Each problem that I solved became a rule, which served afterwards to solve other problems

Hello Class!

We have made it to the last blog post for this month’s mathematician! You all have had some wonderful thoughts about both the life of Descartes and his work in mathematics. Much of Descartes work that we have discussed so far connected topics you have learned in previous math classes to problems in this class. This week, we are going to use Descartes to introduce a major concept in this class, proofs. I am sure that many of you have heard about proofs from your friends who have already taken geometry, but before we get started I want you to think about and respond in the comments below to the following “I Wonder” questions and then read more about how the idea of proofs came to be.

I Wonder
1. What have you heard about proofs before this class?

2. Do you think you have ever used the concept of proof before geometry? If so, how? If not, why not?

While Descartes studied at La Fleche, he had the opportunity to work with many other famous mathematicians who contributed greatly to the field. During this time he made great contributions to the concept of geometric proofs, especially when it came to identifying curves. The addition that Descartes was able to provide to geometric proof was the intense need to justify and explain what was happening in the math. For those of you who admitted to using the concept of proof before in you life, my guess would be that you provided an example in which you had an opinion or stance on an issue and then proved that your opinion was valid. In order to do this, you had to explain the situation, support your stance with facts, and present the material in a clear and orderly way. Descartes used this same process many years ago. He had a deep understanding of both geometry and algebra and sought to connect the two by using one to justify things that were happening in the other. To get a better understanding of this, try to reason through the algebraic proof below:

This idea of having to justify every step of the problem is something that Descartes believed in. He spent the later part of his life unmarried and uninvolved in his one daughters life because he was dedicated to understanding and explaining why things were occurring in the then booming subject of geometry. Despite Descartes early death due to health complications from his difficult birth, he spent his life trying to apply this idea of proving things to mathematics and the world around him.  

Keeping in mind the spirit of Descartes desire to find explanations for things occurring in the world around him, I would like you to pick one of the topics below and prepare a convincing argument, or proof, for your stance on the topic for class tomorrow. You should be prepared to present your argument to the class.

Topics:

1. Apple vs Android
2. The legal driving age (should it be 16 or 18)
3. Twitter vs Facebook
4. Climate Change

Remember that Descartes worked hard to apply factual information to his opinions and explanations on topics. Your arguments may require some research and should be delivered in a clear and concise way!

Domski, M. (2015, December 11). Descartes' Mathematics. Retrieved March 01, 2018, from https://plato.stanford.edu/entries/descartes-mathematics/

Perfect Numbers like Perfect Men are Very Rare

Hello Class!

Thank you for all of your insightful comments last week about how someone’s culture can impact the work that they do. We had such a great conversation about how everything going on in our lives impacts what we do at school everyday, how we perform in our sports and clubs, and even what we want to do after high school. Hopefully we can keep those things in mind as we continue through this class.

Now we have been using a lot of basic algebra in class recently as we factor expressions for various reasons. Many of the polynomials that we have factored so far have been quadratics. Have you ever wondered who decided we would write a little 2 in above, and slightly to the right of a number or variable to signify that we wanted to multiply it by itself twice? I would imagine that many of you never thought to questions why we write things that way. If your teacher told you thats how its done, then that’s all that matters right? Well, not anymore! This week we are going to talk some about how the mathematician we are studying impacted exponent notation. Before reading the rest of the blog post about Descartes, please answer the following “I Wonder” question in the comments below.

I Wonder

If we did not have the exponent notation () that we are used to today, how would you denote the following:

Create your own exponent notation for the expression and place it in the comments below. It MUST be different than the notation that is widely used today.

Before we discuss Descartes contributions to exponent notation, let's discuss other mathematicians that impacted this area of mathematics. There are records that trace the use of exponents all the way back to Euclid (our first mathematician that we studied in September). This shows that the idea of exponents was not new to math when Descartes started studying math. In fact, Nicolas Chuquet used a raised number to represent an exponent in the early 1500s. Several other mathematicians made contributions to the topic over the next 100 years until Descartes added to their work in 1637 by providing the notation of superscript that we are used to seeing today.

Your homework for class tomorrow is to answer the following question and provide two citations to support your claims. Be prepared to discuss your responses with the class.

1. In the blog post this week, I mentioned that several other mathematicians made contributions to the topic of exponents and their notation over a couple hundred year time period. Pick on mathematician that made a contribution to the topic and explain why you this that contribution was significant to how we use exponents today.

**You may not use Descartes or Chuquet because I already told you their
contributions.

Hay, K. (n.d.). History of Exponents. Retrieved February 26, 2018, from https://www.sutori.com/story/history-of-exponents

It is not enough to have a good mind; the main thing is to use it well.

Hello Class!

Thank you all for your hard work and thoughtful comments on last weeks post.
It was so interesting to hear how you all were able to relate your life to the life of
Descartes. Even though he lived many years ago, we can find quite a few similarities to the struggles
he faced and those that we all experience today.

For your “I Wonder” question, please respond to the following prompt in the comments below.

What are three real world applications of the coordinate plane? Feel free to list ideas that you
come up with or ones that you have researched. Instead of simply providing a list, explain how
each example uses the coordinate plane.

The next area of our mathematician’s life that we are going to focus on this week is the era in
which he lived and how that impacted his work. We have spent quite a bit of time over the past
week talking about the recent events that occured at Marjory Stoneman Douglas High School in
Florida. As a school, we have mourned along with those in Florida and we have had many
conversations about classroom and building safety. The reality of those events have directly
impacted what we have done in school this week. While Descartes did not experience the same
situations we currently are, there were any things in his life and the culture that surrounded him
that might have impacted his work.

Descartes lived during the reign of King Henry IV and an era of religious wars and architectural
growth. As a young man, he enrolled in the army and was placed in a division that worked to
design technologies that would aid and protect soldiers in battle. This division allowed Descartes to
improve his applied math skills and form a close relationship with Isaac Beeckman. Beeckman
served as a role model for Descartes and inspired him to study more in the hard math and science
areas. As a soldier stationed in a division that used math to solve the everyday problems of soldiers,
Descartes was able to see how math could be used in many areas! This resulted in Descartes
continuing his studies in the maths and sciences when he left the army. The culture that Descartes
would have entered into after his time in the army was one of growth and prosperity for the wealthy.
Various cities in France were being seeing large amounts of new buildings and developments.
This was a time where the wealthy began to congregate in the city and at events held by the king.

Last week I asked you to consider how Descartes might have come up with the idea to use variables
in problems to represent unknown values if he didn’t take a large number of math classes. It very well
could have been his experiences in real life that allowed him to see how crucial math can be.  
Variables were not the only thing that Descartes was able to discover during his career. He is also
thought to have contributed to the creation of the Coordinate Plane or otherwise known as Cartesian
Coordinates. Knowing a little bit more about the culture surrounding Descartes during his life, answer
the following questions for class tomorrow:

1. How do you think the development of the Coordinate Plane might have helped with events that were transpiring during Descartes’s lifetime?
2. What is something that is going on is our culture (something from the news, in the economy, in politics, or socially) that you think could be solved using math? Explain why you think math could help with this problem.
   

Smith, K. (2001, April 09). Descartes' Life and Works. Retrieved February 25, 2018, from https://plato.stanford.edu/entries/descartes-works/

Trueman, C. N. (2015, March 17). France in the Seventeenth Century. Retrieved February 25, 2018, from https://www.historylearningsite.co.uk/france-in-the-seventeenth-century/france-in-the-seventeenth-century/

I Think, Therefore I Am

Hello Class!

Before you come to class tomorrow I would like you to solve the story problem below. Don’t worry too much! The story problem is one that you probably learned how to solve in middle school. Instead of focusing on the computations, I would like you to answer the two “I Wonder” questions that follow by commenting on the blog and then read some more about our mathematician of the month: Rene Descartes.

Let’s Take a Field Trip

331 students went on a field trip. Six buses were filled and seven students traveled in cars. How many students were on each bus?
I Wonder?
1. Were you able to do this question in your head or did you set up some sort of an equation?
2. Even if you did not set up an equation, list all of the components that would be included in one to represent this problem (i.e. what number, operations, etc).

“I think, therefore I am”

I am sure that many of you listed a variable as a component that should be included in the equation representing the “Let’s Take a Field Trip” story problem above. Have you ever thought about where this idea to use a letter to represent an unknown value came from? While your middle school teachers are brilliant people, they are not the ones who invented this idea! In fact, our mathematician that we are studying this month was one of the main influences in impacting how we use variables today.

Before we dive too deep into all of the great things that Rene Descartes discovered in mathematics, let’s make sure we know a little bit about where he came from. Descartes was born on March 31st, 1596 in La Hayne en Touraine, France. (That was four hundred and twenty-two years ago and over four thousand miles away!!) Some reports indicate that Decartes’ mother died in a complicated childbirth and others state that she had passed away by the time he was one. Without a mother to raise him, and a father that was a council member in the parliament, Rene found himself being cared for by his maternal grandmother (the grandma on his mom’s side of the family). Despite his unconventional upbringing, Decartes’s father valued a good education and paid for him to attend boarding school at Jesuit College of Henri the IV in La Fleche.

While in school, Descartes was a model student. Despite health complications that he faced due to his complicated childbirth, he was able to attend classes in the afternoon while spending his mornings in bed studying. He studied music, math, metaphysics, philosophy, and ethics during his time at Jesuit College. After completing boarding school (what would be the equivalent of you graduating high school) he was then accepted to a four year college where he studied law. Can you believe that somehow out of learning all of those different subjects he was able to come up with the idea of variables? I wonder which class he took in high school helped him with that!

As many of you can probably guess, in the years after Descartes graduated from college he went on to discover a multitude of things in math and in other fields. One of Descartes accomplishments was publishing the book Discourse de la methode ("Discourse on Method), in which he used variables for the first time. Throughout the rest of the month we will take a closer look at how his contributions affect the math we do in class and at what the world looked like when Descartes was alive. For class tomorrow please be prepared to talk about these two questions before we get started:

1. What part of Descartes story did you find relatable to your own life? (You are encouraged to research him some more if you don’t think your life is anything like his!)
2. What do you think led to Decartes idea of using variables to represent unknown values when solving problems?

Resources:

7TH CENTURY MATHEMATICS - DESCARTES. (2010). Retrieved February 14, 2018, from http://www.storyofmathematics.com/17th_descartes.html

René Descartes Biography. (2016, December 23). Retrieved February 14, 2018, from https://www.biography.com/people/ren-descartes-37613#later-life-death-and-legacy