I am most proud of this last comp book because of my ability to stay on task and focused on the task at hand during the last week of school. I have worked hard to understand the material in this chapter and I feel ready for the test! I have not felt this prepared in Cathy's class since I achieved a relatively high grade on a chapter test according to the class average.
I enjoyed learning about the unit circle. I mostly liked this unit because each topic was a building block for new units that all related back to the same, simple unit circle! It was amazing that we could cover the same topic and relate it to so many different topics throughout an entire semester.
I struggled a lot with finding degrees in equations that represent the unit circle. It was hard for me to grasp that equations and graphs could represent degrees and ratios with endless possibilities and could even be represented on a circle with unit measurements! It was not the process of the subject that I struggled with, but rather its direct relations. I often struggle with topics in math because they are not always fully clear why a tule acts as it does or how an equation relates to separate topics. I overcome these concepts by trying to visualize them in the big picture rather than viewing them through an equation process.
Math Project:
The Handshake Problem and Math
The Handshake Problem like most everything relates to math. You can write an equation for almost everything and analyze it with mathematic terms. With the Handshake Problem made an equation to represent the number of handshakes possible with different numbers of people. We did some calculator work and found some numbers to represent in our equation. From the calculator we figured out that it is a quadratic formula. We were then able to plug in the A, B, and C values along with the X, Y points and get our final equation.
The Handshake Problem relates closely to Pascal’s Triangle. Pascal’s Triangle is triangle that has numbers that are gotten by the previous two. It represents triangular numbers tetrahedral numbers. If you look at it you will see the same numbers in one of the rows that we have for the hand shake problem The Handshake Problem looks at the number of different possibilities of handshakes, you can also use it for different set ups that would include several possibilities. For example if there are two people they can only shake hands once, if there are three people then there are three possibilities to shake hand, it continues by a pattern. With five people it can be written as 5! (1+2+3+4+5) which is 15. There are several patterns that can be followed and analyzed in the Pascal’s triangle. It is really quite unique to the math world.
The Handshake Equation Process:
First we analyzed the graph and determined whether it was Power or Exponential.
To adjust to the curve of our graph we determined we would use a Power Function specifically a Quadratic Equation. This equation is traditionally in an (ax^2+bx+c) form.
X | Y |
1 | 0 |
2 | 1 |
3 | 3 |
4 | 6 |
5 | 10 |
6 | 15 |
7 | 21 |
8 | 28 |
Using the points from the handshake problem found on Pascal’s triangle, we made a matrices table in order to find the “a” “b” and “c” values.
The “a” “b” and “c” values are .5, -.5, and 0 according to our matrices.
We then wrote our equation and made sure it was correct by typing in “x” and “y” in their respected parts of the equation.
.5x^2-.5x+0=y
.5(3)^2-.5(3)=3 –this check helped us prove that this was the correct equation.
We then simplified the equation so we could analyze the equation in accordance with the graph.
((x)(x-1))/2 – this equation is derived from the original equation, yet we were able to visualize its relation to the graph. For example, (x-1) accounts for the handshake problem’s pattern in any number of people where if the number of points decreases by one as you continue to go around the circle of people. This is easily explained using the pentagon shape (see board.) The numbers around the shape represent the amount of possible handshakes per person, (each person is represented by one point.) Notice how the numbers decrease by one per person.
