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Sunday, December 8, 2013

SP #6: Unit K Concept 10 - Writing repeating number as rational number using geometric series

The viewer needs to pay special attention to see if the number continues on forever. The process of infinite series must be used. All work must be shown, meaning a calculater cannot be used to get the answer, only to check it. If it has a number at the beginning of the problem next to the decimal, you must remember to add it to the sum.

Monday, December 2, 2013

Reflective Essay Assignment

Word Count: 526
Video-Golden Ratio in Human Body
In this first video, I was amazed when I saw what the Golden Ratio was. If you divide a number by the number before it, you get a number that is very close to it and after the 13th number, it is known as the Golden Ratio (1.618). Artists, scientists, and designers take the proportions of the human body
which are set out to the Golden Ratio when conducting research. Leonardo da Vinci and Le Corbusier used this ratio in their designs. The whole body can be measured out to find the Golden Ratio, even the teeth. The structure of our lungs and our DNA can also be measured to get the Golden Ratio. It is incredible all the places the Golden Ratio is found.

Website- The Beauty of the Golden Ratio
The Golden Ratio has also appeared in ancient architecture. One of the Seven Wonders of the World, The Great Pyramid of Giza, has the Golden Ratio. The Greek Parthenon is another example of where the Golden Ratio exists. The UN building had the Golden Ratio when measured height by width for every ten floors. The exterior measures of the Parthenon have the Golden Ratio as well.

Video- Natures Number:1.618...
Fibonacci was a mathematician who created the Fibonacci number. Natures Number is 1.618. The Golden Ratio is a forgotten number. Numbers like pi, infinity, etc. are more famous that the Golden Ratio. Yet, ironically, this number appears in pretty much everything. It is amazing how art like the Mona Lisa have the Golden Ratio as well. Leonardo da Vinci seemed to have thought that 1.618 was a perfect proportion. A DNA structure and the heart have the Golden Ratio. This may be a coincidence, but the precision in everything makes it seem like it was meant to be there.

Website- Golden Ratio in Art and Architecture
Le Corbusier is probably one of the most famous/strongest for the application of the Golden Ratio during the 19th and 20th centuries. Le Corbusier was facinated with Aesthetics and the Golden Ratio. Le Corbusier's search for a standardized proportion led to the creation of the Modulor. The Modulor was a proportioning system bases on the FIbonacci series. Also, it is thought that great musicians like Mozart knew about the Golden Ratio and used it to compose their music. Interestingly, musical scales are based on Fibonacci numbers. Also, musical instruments are sometimes based on phi.

Reflection
All in all, I found it very interesting how the Golden Ratio is incorporated in pretty much everything around us. It is facinating that so many ancient architecture and paintings have the Golden Ratio as their proportion. It is incredible that this ratio can determine if something is beautiful. I think that in some cases the idea that 1.618 means beauty is true. Everyone has a different perspective, so not everyone will think that something is beautiful, even if it has the proportion of the Golden Ratio. I am personally just fascinated that so many things have the same proportion. The Golden Ratio, in my opinion, does not classify something as beautiful because not everyone may feel that way.

Fibonacci Beauty Ratio Activity

Leslie E.
foot to navel: 106 cm
navel to top of head: 61 cm
ratio: 106/61= 1.747 cm

navel to chin: 48 cm
chin to top of head: 17cm
ratio: 48/17=2.823cm

knee to navel: 57 cm
foot to knee:47 cm
ratio: 57/47=1.213cm
Average: 1.927cm

Katie W.
foot to navel: 102 cm
navel to top of head: 65 cm
ratio: 102/65=1.569cm

navel to chin: 48 cm
chin to top of head: 17 cm
ratio: 46/19=2.421cm

knee to navel: 55 cm
foot to knee: 48 cm
ratio: 55/48= 1.145cm
Average: 1.711cm

Daisy L.
foot to navel: 96cm
navel to top of head: 63 cm
ratio: 96/63=1.52cm

navel to chin: 44 cm
chin to top of head: 19 cm
ratio: 44/19=2.31 cm

knee to navel:52 cm
foot to knee: 46 cm
ratio: 52/46=1.13cm
Average:1.653cm

Christine N.
foot to navel: 96 cm
navel to top of head: 59 cm
ratio: 96/59=1.627cm

navel to chin: 40cm
chin to top of head: 22 cm
ratio: 40/22= 1.818cm

knee to navel: 51 cm
foot to knee : 46 cm
ratio: 51/46=1.109cm
Average: 1.518cm

Vivian P.
foot to navel: 101 cm
navel to top of head: 68 cm
ratio: 101/68=1.481

navel to chin:51 cm
chin to top of head: 22cm
ratio: 51/22= 2.318

knee to navel: 55 cm
foot to knee : 51 cm
ratio: 55/51= 1.078 cm
Average=1.629 cm

According to the Beauty Ratio, Vivian is the most beautiful. She was closest to the Golden Ratio of 1.618. I personally feel that the Golden Ratio does not determine if a person is beautiful or not. I feel that a person is beautiful based on their personality and their characteristics. This may be valid to determine the proportion of a person, but not the personality.

Sunday, November 24, 2013

Fibonacci Haiku: Winter Wonderland

Winter
Serene
Beautiful wonderland
 White Crystalline Snowflakes
Snowy terrain and sparkling lights
The coldest season exposes warmth in our hearts

http://wallfoy.com/wp-content/uploads/2013/11/Beautiful-Winter-Wallpaper-115.jpg

Monday, November 18, 2013

SP#5: Unit J Concept 6: Partial Fraction Decomposition with Repeated Factors




The viewer must pay attention to the common factors. You need to pay attention to the factors and count them because that is how many times there will be an exponent. The viewer needs to distribute and combine properly because small mistakes can be made in this area. The viewer needs to look at how many systems there are in order to have the same amount of variables. 


SP#4: Unit J Concept 5: Partial Decomposition w. distinct factors

The first thing we do is compose. This means we multiply by the bottom numbers and then we get a larger fraction. (you first have have to combine like terms)

This part is decomposing. You have to find what multiplied into this large fraction to get there.

Type in what you end up with in the calculator. You should end up with what you started with.

This shows that once I checked it, it was right.
 
The viewer needs to pay attention to combining like terms in order to get the problem right. The viewer must pay attention on the least common denominator. The viewer needs to make sure their answer is correct after checking it with what the problem originally was. This can be checked in the calculator. 

Wednesday, November 13, 2013

SV#5: Unit J Concept 3-4: Solving three-variable systems with Gaussian Elimination


The viewer needs to pay special attention to how to do the whole matrix because it can become very confusing. Also, you should pay attention to how to check it on your calculator. You should also put a - on the z so it won't be confused for a 2. At the beginning, if something can be simplified, then do it because it will make it simpler. Also, if an equation has negatives and you want to make it positive, multiply it by -1 to make everything easier. The viewer must make sure to do every step correctly because if not, then everything can get messed up.

Sunday, October 27, 2013

SV#4: Unit I Concept 2: Graphing logarithmic functions and identifying x-intercepts, y-intercepts, asymptote, domain, range


The viewer needs to pay attention to how the asymptote is found. For this function, x=h, so you need to look for h to find the asymptote. The viewer must also pay attention to how to do the change of base formula to find the y- intercept. The viewer must know how to put these numbers into the calculator. To find the x-intercept, the viewer must know how to exponentiate. Also, they must pick key points that are to the right of the asymptote. Another thing to pay attention to is that the range is always all real numbers and the domain depends on the asymptote.

SP#3: Unit I Concept 1: Graphing exponential functions and identifying x-intercept, y-intercept, asymptotes, domain, and range

The viewer needs to pay special attention to the a. The a tells you if it is above or below. The viewer must also pay atention to the k because that is the asymptote. The viewer must pay attention to y=k, meaning that there are no x-value restrictions, so the domain is all real numbers. Also, the viewer must pay attention to the asymptote because the range depends on that. Another thing is that the viewer must pay attention and understand why there is no x-intercept.

Sunday, October 20, 2013

SV#3: Unit H Concept 7: Finding logs with given approximations



The viewer needs to pay special attention to a couple of things. The viewer should pay attention to the clues given to them because this can make them solve the problem faster. It can be solved faster because many times, the numbers you can simplify with are given. The viewer must also pay attention to the base because the base can possibly give them clues. The viewer must also pay attention to the letters given in the clues. These letters are the last step to the problem, so they are very important.

Monday, October 7, 2013

SV #2: Unit G Concept #1-7: Rational Functions


This problem shows you how to find a rational function when the degree on top is bigger than the degree on bottom. This means that it is a slant asymptote and this means that there is no horizontal asymptote. This problem consists of various steps that are all explained in the video above. To graph this, we must find the x and y intercepts, the domain, and a couple of points to help us plot our lines into the graph. This example contains a numerator with a degree of three, a denominator with a degree of two, one vertical asymptote, one hole, and two x-intercepts.

In this problem, we must focus on the degrees of the beginning of the problem. This is important because this tells us what asymptote it is. When there is a slant asymptote, there is no horizontal assymptote and when there is a horizontal, there is no slant. To find a slant asymptote, we need to use long division. The remainder does not matter in this case. The equation you get is for the line that goes on the graph. The viewer must also pay close attention to finding the correct vertical asymptote. If it is done correctly, then it makes you a step closer in having a correct graph. The viewer must also pay attention to the holes the problem has. The viewer must plot points correctly on the graph (x/y inercepts., holes).

Sunday, September 29, 2013

SV #1: Unit 7 Concept 10: Finding all zeroes (real & complex)


This problem is about solving for the zeroes and finding the factorization of -22x^4-167x^3-103x^2+47x +5. For this problem, you have to find the p's and q's which is the possible real/rational zeroes. Then, you need to find the possible positive/negative real zeroes using Decartes Rule of Signs. Then, you must use synthetic division and plug in numbers form your p/q list. Once you have a quadratic, you can factor or use the quadratic formula to solve and get your final zeroes.

The viewer must pay special attention a couple of things. The viewer must pay attention to finding the possible real/rational zeroes. This way, the viewer eliminates many numbers that they can plug in while doing the synthetic division, The viewer must also pay attention to finding the possible positive/negative real zeroes. This involves Decartes Rule of Signs and if you do this right, then you can eliminate some numbers from your p/q list. The viewer must also pay attention to distributing negatives properly while doing the quadratic formula. This makes is easier for the viewer to do their work.

Monday, September 16, 2013

SP #2: Unit E Concept 7: Graphing Polynomials, uncluding: x-int, y-int, zeroes (with multiplicities), end behavior.



This problem is about solving a polynomial with a fourth degree and then graphing it. To find the x-intercepts, you must factor the polynomial. From here, you can find the multiplicities of it. Once knowing what the multiplicities are, then you will know if it will go through, bounce, or curve.

The viewer needs to pay special attention to the leading coefficient. This is the key to finding out how the graph will look like. The viewer must also pay attention to the multiplicities. The multiplicities will say if it goes through, bounce, or curves.

Monday, September 9, 2013

WPP#3 Unit E Concept 2: Path of Football


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SP #1: Unit E Concept 1: Identifying x-intercepts, y-intercepts, vertex (max/min), axis of quadratics and graphing them.

This problem is about sketching graphs of quadratics using shifts of the parent function accurately with the graphing calculator. The equation begins in standard form, which is f(x)= ax^2+bx+c. To make it simpler, we must complete the square to make it : f(x)=a(x-h)^2+k. The graph will have as much as 4 points which are: the 2 x-intercepts, the y-intercept, and the vertex.The axis will be a dotted line.

The viewer must pay close attention to see if the graph if positive or negative because that will determine if it is a minimum or a maximum. In this example, it is positive, and this makes it a minimum. The h and k values are what make the vertex of your graph. If the h is negative, then the x on the vertex will be positive (and vice-versa). The k stays what it is. Also, the axis is not just a number; it is x=__.