Math Through the Year
Comp Book Review for Chapter 6
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| I am proud of my work with vectors and the laws of trig because it is a very complicated and challenging subject, and yet I was able to complete the task accurately without any questions about the processes of solving a problem that dealt with vectors and trigonometry. In fact, the highest score I got on a test was a 96%, and that test was about vectors and radians. I have always had a hard time with word problems, but when it came to this area in math, I found it was much easier to pull out numbers and reference points than it usually has been for me, so I was really proud of how well I was able to take on such a challenging subject and execute it. |
Logarithm as an Exponent
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| Logarithms presented one of the greatest challenges to me because I didn't even understand the basic concepts of what made up a logarithm, so of course I wouldn't understand how to take these concepts further in order to solve an equation or turn it into an exponential function. After practicing and applying logarithms to several practice questions and tests, I was able to finally get a grasp on the problems, and now, I even feel confident with the properties. |
Binomial Expansion
Math Connection
Pascal’s Triangle has introduced to me a pattern in math where the numbers have both a simple and logical pattern in certain ways as well as a more complicated pattern that explains the more unconnected numbers. This “complicated pattern” is called binomial expansion. If we were looking at the 5th, 6th, and 7th rows in Pascal’s Triangle, we would see numbers in the following sequence:
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
When you first look at these numbers, you may notice that the first and last numbers are ones. Upon further observation, you may even notice that the second and second to last columns increase by one. The most basic patterns of Pascal’s triangle include these, as well as the concept that a number is determined by the sum of the two numbers directly above it. But if you don’t want to work out all the addition problems to figure out what the next number will be, you can figure it out using combinations, which is a probability concept that predicts any number given the row and the element.
In binomial expansion, however, you were on row 6, and you needed to find out what equation fits this particular row, you would take (a+b)^6, which expands to be a long equation consisting of terms and the coefficients are in the 6th row of Pascal’s triangle. The coefficients of the expansion would be the numbers in the 6th row. Binomial expansion is a way to find a form of powers of a binomial without repeated foiling. The Binomial Expansion also allows you to associate an equation with the values in Pascal’s Triangle.
Personal Connection
I chose to do Binomial Expansion in Pascal’s Triangle because I thought it was interesting how something could have such a seemingly simple and easy pattern, but is actually explained by an equation that is much more complicated and can be related to other ideas such as the basic application of cellular automata to a numerical pattern. I was actually inspired to base my project on binomial expansion and it’s relation to Pascal’ Triangle because of my original interest in cellular automata. The basic theory of cellular automata is that a pattern can be found or applied to almost any growth factor. The way I have colored the triangle relates to this basic theory, as I have followed a set of rules that determines the color of the cell. Because I have colored the cells according to odd or even value, they also depict Sierpinski’s Triangle, a direct example of cellular automata. This project also allowed me to review basic binomial properties as well as look into extended binomial ideas. This project definitely made me curious about combinations, and how they relate to binomial expansion, especially in the triangle. What really interests me about combinations is how they can be used to calculate something, especially a coefficient in a specific pattern.
