42x43

42x43

The area of a region is the sum of the areas of its non-overlapping parts. How can we use this postulate to demonstrate that this block illustrates the traditional way to multiply 42 by 43? The blue block has dimensions of 40 by 40, making it a square. The green blocks below the 40x40 block are the dimensions 40x1, as are the other green blocks next to the 40x40 block. The yellow blocks represent 1x1 squares. Before demonstrating how the blocks illustrate the equation, take the equation 42 x 43 and multiply it.

First multiply 2 and 3.                           2x3=6

Next multiply 3 by 40.                   3x40= 120

Now multiply 2 by 40.                      2x40=80

Finally, multiply 40 and 40.         40x40= 1600

6+120+80+1600= 1806

This is how one would traditionally multiply 43 by 42. You would line the numbers up and multiply the top by the bottom. Now, multiply the area of the blocks. The area of a rectangle is equal to the base multiplied by the height. Taking this equation, find the areas of each block.

Yellow blocks with 1x1 dimensions:  3 blocks across x 2 blocks down=             2 x 3=6

Green blocks on right with 1x40 dimensions: 3 blocks across=                     3 x 40=120

Green blocks on bottom with 1x40 dimensions: 2 blocks down=                  2 x 40= 80

Blue block with 40x 40 dimension:  1 block=                                          40 x 40= 1600

6+120+80+1600= 1806

Multiplying 42 x 43 and calculating the area of the blocks both yield an answer of 1806. The block method even parallels the multiplication because the exact same equations are performed to find the answer. How does this work? It is important to remember that when you multiply a number, you are saying “it is this number this many times”. The blocks illustrate this idea. You take the block with 1x40 dimensions and you say “it is 1x40 three times because there are three blocks”. The dimensions of the block are 42x43 (the blue block plus two green blocks down and the blue block plus three green blocks across), but you are just dividing the area further so that you can work with more manageable numbers. 

Tom Zhang and Perseverance

What struck me the most about Tom Zhang's interview was what he said about perseverance. " If we count the time I spent thinking about the problem, then it was more than 10 hours a day," he said. 10 hours a day, seven days a week, fifty-two weeks in one year...for four years. In total, Zhang spent about 14,500 hours thinking about one problem. Earlier this summer I read a book that talked about why some people succeeded (Outliers by Malcolm Gladwell). Gladwell talked about how the most successful people all have worked at their goals for about ten thousand hours. Now, I can't be the one to suggest that Zhang reached a solution because he worked for more than ten thousand hours; the point is that he worked for a very long time and was persistent. Math is a very fair system- you get out of it what you put into it. Math is all about hard work and perseverance. Math is somewhat of a safe haven for people who appreciate certainty. There is always a solution in math, and the process is more important than the answer. Zhang's process took time and dedication and in the end he prevailed. There is beauty in math because hard work pays off. 

how to write a term paper

 where do you use math in secret?  Or if you don't use math in secret, where might you start using math in secret or not in secret so you can increase your skills in math?

A lot of students are either math and science oriented or language and history oriented, but I am one of those students who likes all classes. My best class fluctuates and changes every year.There really hasn't been a class that I have never loved.  When I was really young I used to love English, and then in eighth grade I loved math. As a freshman I was extremely interested in history, math and French. Well, last year in the second semester I was taking geometry. The second semester is also when all the freshman write a history term paper. History and geometry were classes that I really enjoyed, and when  term paper time came rolling around, I was able to put them together. The idea behind a term paper revolved around proving a thesis statement. We started our term papers by simply gathering facts about our subject. We wrote each fact on an individual note card. Eventually we started putting facts together to form three arguments that would support our thesis. This is where I began to see a connection between geometry and history. I pretended each fact on a note card was a postulate. A postulate does not need to be proved; it is a fact and we can assume that it is true. The facts that I had gathered on my note cards were just like postulates because I could assume those were true too. I didn't have to prove them in my paper. Theorems are proved by definitions and postulates. My paper needed three arguments to prove my thesis statement. I used my three arguments as theorems. I could prove these three arguments with the facts I had collected on my note cards, just like you could prove a theorem with postulates. Then, my thesis statement was what I was ultimately trying to prove. When I wrote my outline for the essay, I wrote it almost exactly as if I were writing a proof. The only difference was that I also had to prove my three "theorems" (the three arguments that I was using to prove my thesis statement). Next time I am assigned a research paper, I will use this strategy again. 

Grades

Grade is a measure of how inclined something is. The percent grade tells how steep an incline is. The higher the grade, the more inclined it is. Grade is also referred to as slope. Slope (or grade) is measured by rise over run. To find the slope you measure the altitude and the horizontal distance it covers. You divide the altitude by the horizontal distance. www.spinlife.com (and Ms. Mariner’s blog) says that for a handicap ramp, every inch of vertical height should have twelve inches of horizontal length. If a handicap ramp is trying to reach a place somewhere three feet off the ground, the ramp must cover 36 feet vertically. The grade of this ramp is about 8.3% (you find that by dividing 1 by 12 and then multiplying by one hundred to find the percentage). This is where it gets interesting—the slope or grade of a triangle uses the same equation as the tangent. Imagine triangle ABC where angle A is the smallest angle. To find the tangent, you would divide the side opposite angle A (side BC) by the side adjacent to angle A (in this case, side AC). To find the slope, you would also divide side BC by side AC. However, when finding the tangent, you would get an angle measure. The slope however would not equal an angle. It would equal how much horizontal length there is to vertical length. 

Math illiteracy

In high school, the question often comes up, “when are we ever going to use this?” We can argue whether any of the classes we take in school are important to our imminent professional lives, but agree or disagree it is a fact that lawyers and doctors don’t prove two triangles similar on a daily basis. In twenty years I doubt I will remember much about quadrilaterals, or the causes of the Hundred Years War. Everything I pick up in school is important to me now, but it is hard to believe that much would ever apply to my professional career. That is not to say that what we learn in class is not important. When we have math class, not only are we learning geometry, but we are learning life skills. We learn problem-solving, hard work and perseverance. These skills do stick with us for the rest of our lives, and they are something that doctors and lawyers use daily. U.S. citizens aren't mathematically savvy because for the average citizens math hardly applies in their lives. They may have to do simple arithmetic on a daily basis, but besides that, math isn't part of their daily routine. I, (among others I’m sure), wish this not to be the case. This math illiteracy makes for what appears to be a country full of ignoramuses, and is a cause of many stereotypes about our country. While we can hope that lessons from the high school classrooms stick with us throughout our lives, we can revel in the fact that important skills like hard work do stick with us. So while the average U.S. citizen may not be able to prove the Pythagorean Theorem, skills that applied to this proof (like problem-solving) will have stuck with them and become visceral. 

Response and my attitude towards math

a.   a.     This blog entry demonstrates how disinterest turned to passion. When you first moved in, you had no interest whatsoever in the flowers. Your neighbor, her age reflecting her wisdom, sparked your interest in flowers. In this way a teacher, wiser than a student, could spark an interest in math -or any other class- in a student.   When you first attended to your garden, you were trying to have the flowers take up less space- you had no interest in having them bloom. But when your neighbor gave you the correct tools, you used them, and your garden flourished. In this way a student may not have an interest in math, but when given the proper help, their interest sparks. Soon enough their interest will not stop growing. The peaches represent the help students can give to each other. Your friends are your equals, just the way any one student is equal with their peers. When your friends first delivered peaches, you thought it was generous, but you had no interest in them. You soon however learned to love peaches. Similarly, a student may be surrounded by peers who appreciate a subject, and gradually they will come to appreciate it too.

b.      b, When I was younger, I was never too interested in math. Instead of doing basic algebra, I preferred to have my face buried in a book or my hands writing away. My interest in math really sparked in eighth grade. There was no real reason for it- it just happened. It might have had to do with the fact that I had a wonderful teacher, or the fact that I was learning a challenging subject.  I started looking forward to math homework. On those nights with big loads of work, I rewarded myself by doing math last. I can proudly say that I find math to be one of my favorite subjects. I love the challenge. While I appreciate the abstractness of history or English, I realize that routine can be comforting. A math problem will always have a solution, and the challenge of finding the solution is something I love. This summer I read a book about success and how it happens (Outliers by Malcolm Gladwell). One part talked about why some countries are so good at math. Some countries preach hard work, discipline, and never giving up. I love math because I love being challenged, and I don’t like giving up. If I can’t find a solution, I won’t give up. Instead, I am inspired to work harder until I do find it.