Math teachers please help!!?

Question

I would prefer a Math teacher to answer this question but anyone who understands how to answer my question is welcomed to answer it as well. I am in eighth grade and I have a research paper due in two weeks. The following are the topics:
-The Monte Carlo Method, Buffon's Needle, and other methods of estimating pi
-Conic Sections and their Equations
-Pascal's Triangle and the Binomial Expansion
-Newton's Second Law and Gravity
-Phi and the Golden Ratio
Please explain each topic in 5 or 6 sentences. I have to choose one of them to do my paper on. So far, it seems to me that Pascal's Triangle is the easiest to understand, but I would like a Mathematician's opinion on this. Thanks!

Answer 1

TOPIC 1: The Monte Carlo Method, Buffon's Needle, and other methods of estimating pi

The Monte Carlo Method relies on random chance to estimate pi. Basically if you draw a circle inside a square, then randomly pick points and see if they are inside or outside. The distribution of points inside the circle (compared to the total number of points picked) should come close to the area of the circle. Then using the formula for area, you can get a close approximation of pi.

Buffon's Needle is a similar way of estimating pi using chance. You draw a grid of squares. Each square must be the exact length of a needle. Then you drop the needle on the grid. You count the number of times the needle touches the grid compared to when it lands completely inside a square. Again, this probability is related to pi and can be used to estimate its value.

TOPIC 2: Conic Sections and their Equations
This will get you into formulas for circles, ellipses, parabolas and hyperbolas.
http://mathworld.wolfram.com/ConicSection.html

TOPIC 3: Pascal's Triangle and Binomial Expansion
The rows of Pascal's triangle relate directly to the binomial expansion of (1+x)^k. It also gets into combinatorics and the meaning of C(n, k) --> ways from n items to choose k of them.

It's straightforward, but it might also be a popular topic.

TOPIC 4: Newton's Second Law and Gravity
I believe this is the formula F=ma that relates force to mass times acceleration. In the case of gravity, acceleration is the gravitational constant g. You could figure out the weight of various objects on various planets and moons based on this formula...

TOPIC 5: Phi and the Golden Ratio
This would be my pick for a topic because there is a lot to the golden ratio.

On a basic level, imagine a rectangular piece of paper that was approximately 1618 x 1000 cm. If you cut off a square of 1000 x 1000, you are left with a rectangle that is 618 cm x 1000.

The ratio of the first rectangle is: 1618/1000 = 1.618.
The ratio of the new rectangle is 1000/618 = 1.618.

And if you continue taking away a square of 618 x 618 you get a new rectangle of 618 by 382.
The ratio is 618/382 = 1.618.
And the pattern continues. The actual ratio is (1+√5)/2, but the estimate is 1.618.

This ratio can be seen in nature, with a relationship between diameter of a trunk, compared to the branch, to the twigs. It can be seen in architecture in the Greek temples. It is related to the ratio between numbers in the Fibonacci sequence. It has ties to fractals. It's a very broad, but interesting topic.

Quoting from Wikipedia:
“Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.”
— Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number

http://en.wikipedia.org/wiki/Golden_ratio

Answer 2

In the order you listed them:

1) this has to do with probability analysis techniques and how they can be used to estimate the value of pi (pi is the constant ~3.1415 which is the fixed ration of a circle's circumference to its diameter). probably some interesting websites out there on these.

2) imagine two cones stacked point to point. then find all the ways a plane can intersect them. the resulting intersections are the conic sections: a point, a line, a parabola, a circle, a hyperbola, an ellipse (i think that's all of them -- do the research and find out). this should be easy to research.

3) you seem to already have an idea about this.

4) Newton's 2nd law relates force, mass, and acceleration. in free fall, that acceleration would be gravity.

5) from wikipedia: "two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887." (hey i'm lazy like you, so i'll cut and paste).
the interesting thing is the diverse places you can find this.

Answer 3

the first one is about probability and algorithms which to me sounds a bit boring. It does require harder studying to understand than other topics.

conic sections are about parabolas, hyperbolas and ellipses. It is relatively simple to write a paper on their equations and shapes. You need to understand the focus points and how they are used in conic sections.

Pascal's triangle, it is easy to understand. There is no huge complication behind it. Binomial expansion is an easy concept so your paper is likely to end up short. Although you can add more about other properties of pascal's triangle (there are many) which can make your paper longer.

Newton's Second Law and Gravity is more about history of physics than math.

phi, the golden ratio, is a very interesting and broad topic. phi has many interesting and fun applications in fractals and series. I would recommend this one, but it does require a bit of studing and research so you are clear what phi is and why it is important.

Answer 4

sorry have no clue not a math teacher