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Math/Fibonacci TOK

Tok Stuffadoodle!

Angelica Benavides

on 28 May 2014

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Transcript of Math/Fibonacci TOK

How has the history of this area led to its current form?

The Fibonacci sequence was discovered in the early thirteenth century. Since the discovery of this sequence, there have been summation formula that have been created to help quickly add a Fibonacci series mentally. Also, there have been proofs that have been discovered to help get a better understanding of the sequence. The history of the Fibonacci sequence has helped lead to the discovery of many summation formula and proofs. This sequence of events happens with many discoveries in math, like with Pythagoras and other mathematicians. In this century, mathematicians are still adding on and discovering new things about sequences and math discoveries that happened hundreds of years ago.
Historical Development
Links to Personal Knowledge
Knowledge by acquaintance.
What I have learned through my formal education
The results of my personal academic research
-How do Fibonacci’s methods generate knowledge?
-What assumptions are needed to understand Fibonacci’s work?
-How can models aid in our understanding of Fibonacci's concepts?

Number Sense UIL

How do we use language to express the knowledge found within mathematics? To what extent does this differ according to different forms of mathematics? Are there any central concepts for which we need specific language before approaching mathematics?

- Language is a medium to communicate the knowledge or methods in Mathematics. Symbols, definitions, formulas all use the parts of a language to describe itself.

- Language makes math more easily understood

- Math is the only language shared by all humans no matter the culture

- Math is the language of the universe.

The Fibonacci series or Fibonacci sequence are the numbers in mathematical form is as below:
0, 1, 1, 2, 3, 5, 8, 13 ,21, 34, 55, 89, 144, 233, 377, 610 .....
Logic of the Fibonacci series or Fibonacci sequence is in series the third position of the number is the summation of the first number and the second number. After third position the forth position is the same summation of the third position's number and the second position's number. Then same as fifth position's number is the summation of the third position's number and the second position's number. This series is going in the same way. In the mathematical form assume that first position's number is a, and the second position's number is b, initially the a = 0 , and b = 1 and the third position is c. so c = a + b Then the first position is change into second one and the answer c is make the second position for the forth one in the series. so a = b and the b = c; This logic is used in the all language. but the syntax of the language is always different in the all different language.
4 types of Fibonacci sequences.
The sum of first 9 Fibonacci numbers : 11F6 + F1
The sum of first 10 Fibonacci numbers : 11F7
The sum of first 11 fibonacci numbers : 11F8+F1-F2
The sum of any nth number Fibonacci sequences : 2Fn + F(n-1) - F2
1+5+6+11+17+..+73+118 = 2(118) + 73 - 5 = 304.
Take the term and add the previous term.
Plants grow by this aspect by their leaves growing in this fashion. It is not known why that natural occurrence grow like this but Fibonacci did a study on the theoretical birth pattern of bunnies or bees and produce this golden ration .
Sunflowers center grows about like this.
The reproduction process of ants.
Also the reproduction of bees.
Fibonacci number is a type or reasoning why some plants grow like this.
what is the significance of the key points in the historical development of this AOK?

The significance of the historical development of math is that many important math sequences and equations that were discovered years ago are still used today for many important jobs. The Fibonacci sequence can be used to predict shapes and patterns in natural occurrences. Also, photographers and businesses can use the Fibonacci sequence to make their photos and logos. these uses of the Fibonacci spiral show that math has developed into being used in the real world.
How do Fibonacci’s methods generate knowledge?
The Fibonacci number sequence has led to development of further concepts
Sum of squares formula
Fibonacci rectangle
Fibonacci spiral
Understanding patterns in nature
What assumptions are needed to understand Fibonacci’s work?
Proof by mathematical induction
- Must assume true for some nth case to prove true for all cases
Must trust prior mathematical knowledge in order to develop more
- Must trust the sum of squares knowledge before understanding the Fibonacci spiral

Can we know if a concept is really true if we must make assumptions to prove it?
How can models aid in our understanding of Fibonacci's concepts?
Geometric models help to visualize math concepts
- Law of sines/cosines proof
- Proof of sum of squares through Fibonacci rectangle

A Brief Segment of the wonderful Fibonacci numbers
"There is nothing that can be said by mathematical symbols and relations which cannot also be said by words. The converse, however, is false. Much that can be and is said by words cannot successfully be put into equations, because it is nonsense. " -C. Truesdell
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