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