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Tuesday, December 10, 2013

SP #6: Unit K Concept 10: Writing Decimals as Rational Numers


In this section, we now get a number of repeating decimals and in order to find the sum of the infinite geometric numbers, we have to do a series of steps. First, we ignore the whole number in front, so we can use it later on. We work with the repeating numbers by using summation notation. This means that we first have to change our first time into a fraction and the ratio also needs to be included. Then, we do some division with these and make sure to cancel off properly. Always be on the look out because you can some across a problem that needs to be simplified and this problem requires of that. After that, we bring back the whole number and add it to the fraction. Be careful with your multiplication and addition. Other than that, this is the answer to your problem!




Sunday, December 1, 2013

Fibonacci Beauty Ratio

          The Fibonacci "Beauty Ratio" Actvity was performed in order to figure out which person from our class was the the most mathematically beautiful. I recorded Jorge, Vanessa, Judith, Ana, and Victoria and out of all of them, Victoria came out to be the most beautiful. According to the Fibonacci Golden Ratio, she got an aveage of 1.620. I got the ratio by taking everyone's meansurements from different parts of their body, such as the distance from thier foot to naval, navel to top of head, navel to chin, chin to top of head, knee to navel, and foot to knee. Altough the Golden Ratio is amazing since it is found all around nature and in a lot of beautiful art work, it might not be completely true for my measurements. I could have easily made mistakes in my measurements since the rulers were not long enough. Also, I do not agree that the Golden Ratio determines someone's beauty. Sure, it might just be mathematical, but I feel that simply having a face structure based on the Golden Ratio is not sufficient enough to determine beauty. Regardless of my opinion, I do beieve that the Golde Ratio and the math behind it all is amazing. 

Fibonacci: Reflective Essay

Word Count: 791

Golden Ration in Human Body
There far more similarities within this world's nature and our own bodies than we ever thought about. The similarities between the pyramids in Egypt, Leonardo da Vinci's Mona Lisa, snails, etc. all boils down to one mathematical ratio known as the Golden Ratio, which approximates to 1.168. This numbers begins to appear in many measurements, but begins to reoccur after the thirteenth number in the series. This series is based on the sum of two preceding numbers and when they are divided, they follow the Golden Ratio. In fact, some designers have used the Golden Ratio without realizing that they were following nature's common ratio. This ratio is so common that it even appears in the average human body as the ideal proportional ratio that allows one to be aesthetically pleasing. We can find the Golden Ration between the height of the navel and the foot, distance between the fingertip and the elbow, the distance between the shoulder line and the top of the head, the distance between the navel and the knee, etc. The Golden Ration even appears in our teeth's measurements and the length of the face over the width of the face. A study in 1985-1987 revealed that the Golden Ratio was in the structure of the lung, which is asymmetrical in the ratio of 1.618.

The Beauty of the Golden Ratio
As previously mentioned, the Golden Ratio is a repetitive number that was found in the famous Egyptian pyramids in Giza. It has also been found in the Greek Parthenon, which was constructed in 447 B.C. to 472 B.C. This was far before the Italian mathematician Fibonacci discovered the Fibonacci sequence terms (makes up the Golden Ratio). Even though the Golden Ratio does appear in very ancient or somewhat older buildings, it still appears in modern architecture. The Golden Ratio was used all throughout the Renaissance years when Renaissance artist designed Norte Dane between the 12th and 15th century. The UN Building also has the Golden Ratio in its structure since the width of the building to the height of the every ten floors leads to the Golden Ratio.

Le Blog: The Golden Ratio Revisited
The Golden Ratio has also been known as the "Divine Proportion" as you have seen since after all, it does lead to perfect structures of buildings such as the ones I previously stated. Corbusier's designs were all great examples of the architectural beauty based on these mathematical proportions. When Leonardo da Vinci drew the "Vitruvian Man", he had created an ideal human body that was completely based on the Golden Ratio. The distances and ratios between his body proportions come to make up the Golden Ratio. He also represents the Golden Ratio in his other masterpieces, such as the Mona Lisa and "De Divine Proportione". Other than the architecture and the paintings, the Golden Ratio appears on simply Mother Nature. The Golden Ratio allows plants like flowers and pineapples to be aesthetically pleasing to the eye. It is no doubt that the Golden Ratio appears on many objects and justifies the reason why some objects are more beautiful than others.

Nature's Number: 1.6180033988...
There have been many fascinating numbers that often intrigue people, such as the famous Pi or the concept of zero. There are many numbers and patterns to pay attention to and even though the Golden Ratio appears everywhere, it is often ignored. It is nature's number that has been forgotten. One can figure out how "phi" come to be by breaking down a square into part rectangles. After breaking down rectangles into smaller sections, one will get a series of small squares and rectangles that are "just right". When the bottom right corners are connected, they create a spiral that appears in nature, oh, so many times. The Golden Ratio has allowed nature to be placed into a prefect perspective.

Personal Reflection
Although I must agree that I am not a mathematical genius, I am greatly fascinated by the Golden Ratio. Yes, I do believe that the Golden Ratio does exist in many parts of nature's beauty, but I must admit that I do not see it much in the human body. It is hardly impossible for a person to be "perfect", so maybe this beautiful perfection only exists in architecture and other plants. Wherever the Golden Ratio exists, it is truly amazing to find such patterns in completely odd or different things. Whether it comes from an ancient architectural masterpiece to someone's naval and knee, or to a simple snail, the Golden Ratio has made a lot of contribution to the way we see our surroundings. The Golden Ratio allows us to admire beauty at its very best.




Sunday, November 24, 2013

Fibonacci Haiku: Black Friday

 Dude! 
 Run
 Black Friday 
  That crazy day  
 People will knock you down 
 Just make sure you grab that blue jeans 

Monday, November 18, 2013

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








In this Student Problem, the same rules from Concept 6 apply, except that in this problem, we find repeated factors, which basically means that there are two or more factors, such as in this example (x+1)^3. There are still tricky sections in this problem though because there are parts that we have to include even though I did not go over it on the last blog post. For starters, it is important to remember that we need to count up to the exponent that the factor x+1 says, which is 3. Then we, have to make sure we factor out before distributing. In this problem, I got the resulting system and combined the first two problems and then I solved the last two through the elimination process. After that, I then combined those resulting individual equations and solved them by the elimination process. It is helpful to use the original equations, so make sure you do not leave them out!




Sunday, November 17, 2013

SP #4: Unit J Concept 5: Partial Fraction Decomposition with Distinct Factors

                               

 In this student problem, I go over over partial fraction decomposition. We will break down the problem and then we will use our algebraic skills to finish solving the problem. Keep in mind that matrices are back, so make sure you remember how to check your problem with rref. When you start to break down the problem and then go on to foiling, remember to be very careful with basic math (distributing, multiplication, etc.) because it can throw you off. Also, leave the denominator factored out, so it is easier for you to break down the fraction. I used A, B, and C as my letters, but it really does not matter which one you decide to use. However, keep in mind you should avoid X because after all it's one of your variables.


Monday, November 11, 2013

SV #5: Unit J Concept 3-4: Solving Three-Variable Systems Using Gaussian Elimination

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In this video, I will be going over the process of using matrices and the Gaussian Elimination process in order to find the correct values for the consistent independent ordered pairs. There are my tricky things in this video that we must pay attention to, so make sure that you pay attention to everything in general. To be specific though, remember that there are four steps you need to take in order to find the correct functions, which means that you must remember them in order to complete the matrix. Along with that, remember how to check your answers with reduced row-echelon form to save you time in case you make a mistake along the way. Yes, fractions are back, but that does not mean that you will get decimals in your values. In fact, make sure you only get whole numbers as we are only working with whole numbers for now. Other than that, thank you for watching! :)

Wednesday, October 30, 2013

WPP #6: Unit I Concept 3-5: Compound Interest, Continuously Compounding Interest, and Investment Application


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This Word Problem Playlist goes over compound interest and how to find the interest or time given certain information. There are some tricky part in trying to solve the problem, such as remembering to keep the quotient a fraction when dividing the total resulting money by the principal. When completing this, remember to get enough numbers so you can get to a rate that has a percent that includes the hundredths as well. It is also significant to remember that the needs to be rounded to be nearest month using proportion of x/12. Lastly, the most important part is to remember all the formulas that accompany each part of the problems, so you do not get confused. Good luck! 

Saturday, October 26, 2013

SV #4: Unit I Concept 2: Graphing Logarithmic Functions

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This video goes over logarithmic functions and also includes the parts that help find the graph to it. There are many key factors that you must pay attention to because this time around there's a log involved. Given this information, you know that for the x-intercept, we have to exponentiate in order to get rid of the log. Also, for the y-intercept, you must remember to change the base in order to figure our the ordered pairs. Also, in order to graph the equation, you need key points of significance. However, in order to get thoe points, you first need to plug it into your calculator the way I described in the video, so please make sure to do this before continuing. Lastly, as a clarification, the graphing calculator will not entirely graph the equation correctly, so the part left of -5, which is the asymptote, will be blank because of that same reason. Don't worry though, you do not need to graph that section. Other than that, thank you for watching :D

Thursday, October 24, 2013

SP# 3: Unit I Concept 1- Graphing the Exponential Function




          This Student Problem goes over functions and explains how to graph one. As you can see, the function is composed of an exponent and is in the form of a parent graph. Using the values, key points, asymptotes, x-intercepts, y-intercepts, domain, range, and range, the image provided showcases a correct graph for the function. You will find that my ordered pair (2,-17) is off the graph, so it is place at the bottom of the graph just as a guide to help draw the graph.
          In order to successfully solve this function, remember that there are some key factors that will determine the graph, such as the x-intercept. To find the x-intercept, start by substituting a 0 for the y-value and continue the process of solving the problem. After some addition (+5 to both sides) and diving (-3 to both sides), you will notice that you get -5/3. Move the exponent (x-1) to make it a coefficient and you will go over natural logs to get rid of the 4. However, the problem is that you cannot divide the logs, as LN -5/3 is negative and one cannot divide negative logs; it would be undefined, so there is no x-intercept. Remember that for the y-intercept, you have to solve the 4^-1 properly by finding its reciprocal of (1/4). For guidance, the "a" value will help you determine whether the asymptote is below or above, but in this case, it's above. For the key points, it's helpful to use ordered pairs that are not all so close together, so you can get a better view of your graph.

Thursday, October 17, 2013

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

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          This is problem goes over how to find logs based on approximations. Before you start remember to add logbB=1 and logb1=0 to your given clues in case you need them. For this particular problem, we will only be using the property that equals 1. Another important thing that you must remember is the difference between product and quotient logs.  Also, just to make this problem easier, divide the log your are looking for  by the given clues so you won't have to take much time trying to figure out which factors will work for the problem Logs that have addition in between them are product (vice versa) and those that have a subtraction sign are quotient (vice versa). When your expanding, you should have one log for each number so you can then substitute in the values or letters that were given at first. I did not mention this is in the video, but implied that each number has to have a log, so I hope that makes sense now,if it didn't before. Thank you for watching! 




Again, I am sorry for the mistake. Basically the only difference is that all the factors on the denominator will have subtraction signs as it is a quotient considering that is underneath the numerator. 

Sunday, October 6, 2013

SV#2: Unit G Concepts 1-7- Finding All Parts and Graphing a Rational Function


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

Click HERE to watch :)


          This video goes over rational functions and as well includes the slant asymptote, vertical asymptote, holes, domain, intervals, and explains how to graph it. This functions does not include a horizontal asymptote because its leading degree on the numerator is bigger than the leading degree on the denominator. Therefore, this graph has a slant asymptote with one hole and two x-intercepts. Along with that, this video goes over notations, equations, and ordered pairs.
          In order to understand this problem better, you must note that the slant asymptote does not require you to continue solving past the needed y=mx+b equation. Along with that, because you have the same factors on the numerator as well as the denominator, the graph is going to have a hole. The graph will then skip the ordered pair as it is a hole. Lastly, it is very important to remember that the graph does not touch the asymptote even though is might seem as it does.




****Correction:
I'm sorry for the mistakes, but here is a correct version of the graph. For the holes, the canceled factor, which leads to the equation x=-3, is the x-intercept for the ordered pair. You just have to plug in the x-value in the equation as shown in order to get the y-value of -3/4. This means that the hole on the graph changes, but you still have to skip it as it does not exist on the graph when you trace it (please see the graph from the picture). As for the y-intercept, you have simply plug in 0 for the x-values in order to find the ordered pair of (0,0). Other than that, the rest is correct.
   
                                                                                                                              


Sunday, September 29, 2013

SV#1: Unit F Concept 10- Finding all zeroes to a 4th degree polynomial

Click HERE to watch video :D

Student Video #1

        This video covers how to find all zeroes to a polynomial of 4th degree. It will start by finding all the possibilities of the coefficient and the leading degree.  Of course, there were a lot of possibilities, so we narrowed down the possibilities by using Descartes Rule of Signs. That narrowed down our possibilities and we then went straight into trial-and-error with the different zeroes using synthetic division. After bringing down the equation to a quadratic equation, we used the quadratic formula to find the remaining zeroes. We carefully solved the equation and set them x-(...) to find the exact zeroes.

        The viewer has to note that the odd degree changes signs, but that the even degree does not, just as it was mentioned in the video. Along with that, one must acknowledge that not all the possibilities work and that a only a process of trial- and-error will help lead to the quadratic equation. Also, remember that the the radical cannot be left as a negative, so there will be imaginary numbers. This goes back to the reason why we tend to count down by 2's during Descartes Rule of Sign. Lastly, the factors can be written as a quadratic equation, but it can also be left as the answer one gets after completing the quadratic formula.



*Here's a correction, sorry about the mistake.





Tuesday, September 17, 2013

SP#2: Unit E Concept 7- Graphing Polynomials and Identifying All Key Parts




   This concept explains the end behavior, the multiplicities that derive from zeroes, y-intercept, and a graph to represent a factored polynomial. The multiplies and the zeroes help describe the shape of the graph and the end behavior describes the limit notation used to graph the polynomial. Meanwhile the y-intercept will simply be plotted on its corresponding ordered pair. This polynomial is a positive, fourth degree function, which means that there will be a total of 4 x-intercepts, or zeroes when the reader adds the multiplicities.
          The reader needs to pay attention to the small details, such as the correct factorization since this eventually leads into our zeroes. In this case, an x^2 can be factored from the polynomial, so the reader must remember that this means that there will be two zeroes since there is a variable. Along with that, label the ordered pairs with T, B, or C because this will determine if the graph will go through, bounce, or curve; in this case, it goes through, bounces, and goes through again. Lastly, remember that the end behavior describes how the graph acts at the extreme, so make sure that the direction are correct.

                                                                                                                       


Monday, September 9, 2013

SP#1: Unit E Concept 1- Graphing Quadratic and Identifying All Key Parts

Identifying x-intercepts, y-intercepts, vertex(max/min),axis of quadratics and graphing them














          This problem represents quadratic equations in standard form. This representation includes the detailed explanation of the steps included in order to identify the x-intercepts, y-intercepts, vertex
(max/min), axis of quadratics and how to graph them. We start by completing the square of the a standard form equation so we can then put in in the parent function form [f(x)= a(x-h)^2 + k]. Next, we will graph the equation by locating the vertex, the y-intercept, and up to two x-intercepts, as well as one dotted line, which will be the axis. This will result in an accurate representation of the function.
          Of course, there will be many aspects to this equation and you must make sure you notice the characteristics that must be included. For example, you must remember to complete the square correctly by first adding 2 and then factoring out the coefficient from "a". Then, you must find the parent function equation by remembering to subtract the 10 so it all equals to y. Also, one of the main points, is to notice that even though the "h" is negative, you have to get its opposite value and that the "k" remains as is. A common mistake that should not be left out is to get a positive and negative square root 5 and not just a positive value. Lastly, always check that there are at least five points in your graph before connecting the parabola's points (unless there's an imaginary x-intercept, which is not the case in this problem.