Parabolas

I was thinking about parabolas and when I see them or use them in daily life. It struck me, as I was thinking about the upcoming state tennis tournament, that I see a parabola every time I toss a ball to serve. (That's a lot of parabolas.) Here is what my toss would look like if I graphed it:
I toss the tennis ball about four feet in the air, so the maximum y value would be at 4 on the graph. If we were to graph the height of the toss vs time, the y-axis would be height (h) and the x-axis would be time (t). Assume I released the ball at about nine feet in the air. One x intercept would be at the origin. This makes sense, because at 0 seconds, the ball hasn't gone anywhere.Let's assume that the ball travels up at about 40 ft/second and the velocity coming back down changed by 16 ft/s every second (so v= 16 ft/s^2). So the height for the parabola would be calculated using the equation:
h = -16t^2 + 40 t - 9

with help from http://www.mathsisfun.com/algebra/quadratic-equation-real-world.html

Let's Tessellate

Think about regular polygons for just a minute. A regular polygon is a polygon with sides of equal length and angles of equal measure. There is an infinite number of regular polygons, and the more sides they have, the more they approach looking like a circle. (On a side note, this reminds me a lot of what we were talking about in class about how calculus allows the denominator to be zero because it can be so small that it is almost zero, but not quite zero. Similarly, a polygon can have so many sides that we could almost consider it to be a circle, but it will never truly be a circle. I love the idea of infinity!) So even though the number of regular polygons is infinite, there are only three that tessellate. The fraction of the number of regular polygons that tessellate out of all regular polygons is 3/infinity...so the fraction is infinitely small. That's a weird thought but it's veering off this post's main idea: why do only triangles, squares and hexagons tessellate?

(from mathisfun.com)
The answer isn't too complicated. The equation for finding the measure of an interior angle of a regular n-gon is 360/n. So the measure of an interior angle of a triangle is 60, and that of a square is 90, and that of a hexagon is 120. In a tessellation, the corners of the shape being tessellated must meet. And the golden reason for a regular tessellation is having those corners all add up to 360. When triangles tessellate, six corners meet. 6 corners x 60 degrees= 360 degrees. With a square, it's 4 x 90...which is also 360. And with a hexagon, it's 3 x 120 (yes- 360!). I knew that triangles tessellated (and it was pretty easy to figure out that squares tessellate too). If you remember that six regular triangles make up a hexagon (as you can see in the tessellating triangles picture) it's easy to realize that a hexagon would also tessellate. So to figure out if a regular n-gon would tessellate, you would use the equation:
360/n x number of corners meeting in tessellation
If this again equals 360, then the regular n-gon tessellates. But let me remind you right now--unless its a triangle, square or hexagon, that equation won't work

Pascal's triangle

Pascal's triangle is really cool. It can go on forever. The way it forms is by adding the numbers next to each other and then writing it in the next row down, between the two numbers. This is just its basic function however. There are a lot of things Pascal's triangle reveals about numbers. First, as we have already seen, you can use Pascal's triangle to expand powers like (x + y)^5. You just take row 5 of Pascal's triangle and use it as the coefficients, and then have the power on x decrease as the power on y increases. Another cool thing to note about Pascal's triangle is how it works with the powers of 11. Take 11 to the zero power. That equals 1, row zero of Pascal's triangle.  11 to the first equals 11, row 1 of Pascal's triangle. 11 to the first equals 121, which is, yes, row 2 of Pascal's triangle. Think the pattern holds up all the way? 11 to the ninth power is 2357947691. Row nine of Pascal's triangle is 1 9 36 84 126 126 84 36 9 1. At first, that doesn't appear to follow the pattern at all, but take a closer look. You have to add the double digits to the number to the left of it. So 9+3 (from the 36)= 12. Take the 1 from 12 and add it to the 1. Now you have 2, 2, 84. Add the 8 and 4 and add the tens digit (1) to the 2. Now you have 2, 3 and 2 (remaining from the 12). Now add the numbers to the right of it (1 and 2) and now you have 5. Continuing in this way, you will get 11 to the ninth power, or 2357947691. And yes, by the way, if you add the numbers in the ninth row of Pascal's triangle, you get 512, which is 2 to the 9th-- a much easier pattern to see.
There is a cool connection to how the triangle relates to geometry. If you set up a chart going from point to segment to triangle to quadrilateral, etc (you just add a side to the side of the polygon each time) and you laid out Pascal's triangle under those rows, it would provide you with information about the formations in circles. Taking row three of Pascal's triangle, in a circle with three POINTS, you would have three SEGMENTS, and one TRIANGLE. This ignores the 1 that is always the first digit in Pascal's triangle, but you could think of it being in 1 circle. Here is an image to help visualize (from www.jamesrahn.com)


Blaise Pascal was a Frenchman who lived from 1623 to 1662. He did a lot with math, but he was also known for his work in chemistry. The unit pascal (Pa), a measure of pressure, is named after him.

math humor

1. Rene Descartes went into his favorite bar and the bar tender asked, "would you like your usual drink, Monsieur Descartes? " Descartes replied "I think not" and promptly disappeared.
from http://www.sonoma.edu/math/faculty/falbo/jokes.html
explanation: the key to understanding this joke is that one of Descartes's famous lines is "I think therefore I am." so if he "thinks not", then he "is not"- he doesn't exist. 


2. Q What does trigonometry have in common with a beach?  A: Tan Gents
from  http://www.sonoma.edu/math/faculty/falbo/jokes.html
3.

from mrmcdanielsteacherpage.wikispaces.com

4.

from mathjokes4mathyfolks.wordpress.com
 They set up the proportion "grew" is to "grown" as "flew" is to a variable. This couldn't work in an equation because grew and flew are variables so you can't cancel the letters because they are part of the variable and don't have any value on their own, but I still thought it was clever!
5. 
from https://www.pinterest.com
6. 
math pick-up line from https://www.pinterest.com
7. 
from https://www.pinterest.com
8. Theorem:
All positive integers are interesting.
Proof:
Assume the contrary. Then there is a lowest non-interesting positive integer. But, hey, that's pretty interesting! A contradiction.

from: http://www.sgoc.de/math.html

9. 
4+3+17+8= 32 --> 32/4=8 which is the "average" or "mean"
from https://www.pinterest.com
10. 
from https://www.pinterest.com

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.