## Fibonacci Numbers and the Golden Section

This excellent site contains a large and interesting selection of material on Fibonacci Numbers and their myriad related concepts. It includes web and text references to the rich literature of Fibonacci numbers. It also features a variety of images, graphics and animations. Many of the topics include student investigations for NCTM standard type explorations. The content includes standard motivational examples and applications in nature, but also mathematical explanations and puzzle pages. The...
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Material Type:
Reference Material

Technical Format:
HTML/Text

Date Added to MERLOT:
May 10, 2001

Date Modified in MERLOT:
January 28, 2013

Submitter:
James
Rutledge

Keywords:
golden string,
Egyptian fractions,
Fibonacci and Phi in the Arts,
Farey fractions,
continued fractions,
Fibonacci bases,
phi,
Lucas numbers,
Fibonacci numbers in nature

### Quality

- Peer Reviews:
- User Rating:
- Discussion (57 Comments)
- Learning Exercises (none)
- Personal Collections (321)
- Accessibility Info (none)

### About

Primary Audience:
High School,
College General Ed

Mobile Compatibility:
Not specified at this time

Technical Requirements:
Web browser. Links are provided to download plug-ins such as Quicktime for animations.

Language:
English

Cost Involved:
no

Source Code Available:
no

Accessiblity Information Available:
no

Copyright:
yes

Creative Commons:
unsure

### Connections

### Browse in Categories

### Discussion

## Discussion for Fibonacci Numbers and the Golden Section

Log in to participate in the discussions or Register if you are not already a MERLOT member.

4 years ago

### Evan Rogers (Student)

The content on the site is very helpful and entertaining. The site describes Fibonacci's numbers simply with a variety of useful visual aids.Some of the examples used to describe how Fibonacci's sequence works is a little bit confusing but most of them get the point across well. The examples used to describe the golden number was also fascinating. I found that using the examples was not difficult and the questions did not require too much critical thinking. It was an easy to learn system.

Time spent reviewing site:
56 minutes

4 years ago

### Darla Cooley (Student)

I found this content educational and interesting
I had read about this in my math book but this helped me understand better because of the visual aid.The material was easy to understand,pleasurable to use. Great for the first time user.I thought it presented the content accurately.It made the concept easy to grasp because of the way information was related to every day items.

Time spent reviewing site:
45 minutes

4 years ago

### Rebecca Sarkissian (Student)

I spent about 7 mintues looking over the page, and basically I was looking over for titles that striked my interest. I think the one that I liked alot was about Fibonacci and who he was. I think its great when learning math to understand the history too, that way you can pick up interests to students who love history and can relate math through it. I think the quality of the content is great, its extremely informative, and looks like it is updated consistently. I think it could enhance teaching because of all the information it contains...but, I personally wish there was better visuals ( like maybe pictures of rabbits to replace/represent numbers) that were interesting besides the use of numbers as the visual aid. That would probably keep me more engaged when viewing this site. It was really easy to view this website, especially when it provided a Content Box at the top of the website so I can see a preview of the remaining website and it categories without have to look through the whole thing. Overall I think its great!

Technical Remarks:

Better Visuals

Time spent reviewing site:
Seven Minutes

4 years ago

### Kris Mims (Student)

The website is pretty useful, I spent about 10 minutes trying to navigate to this page, a few times it didnt load right away (might be my computer) As for the content, I enjoyed it. I like Fibonacci numbers and think that material that is at our disposal like this should be used and brought to attention more.

Time spent reviewing site:
20 minutes.

4 years ago

### pati brown (Student)

This Web site is full of information. I spent about an hour browsing the links and looking at it from a teaching perspective as well as a practical perspective. I found that there was a mathematical explanation for a graceful spiral. Who knew! Yes, I'm new to Fibonacci and mathematical theories, but what I found in the art and architecture links opened up a new world to me. As a newbee, the puzzles were a bit over my head, as were some of the explanations. But, new to Fibonacci as I am, the concept clicked when I could link a string of numbers to an aesthetic in art. I recommend checking out Brian Knott's ray tracing in the art section... and in the architecture section, the Cal Poly Engineering Plaza is stunning.
Unfortunately I found many broken or outdated links which was a bit frustrating. However, I still recommend this site for the novice or advanced mathematician as well as those who teach art or math. This is a very complete Web site.

Technical Remarks:

Many broken or outdated links.

Time spent reviewing site:
One hour.

Used in course

5 years ago

### dean levita (Student)

this is very detailed and interesting... it helped me understand fibonaccia numbers a lot more
-dean levita

Time spent reviewing site:
45min

5 years ago

### brenda pinal (Student)

really good links i reviewed alot of fibonocci numbers and phi good refresher

Time spent reviewing site:
20min

5 years ago

### bernadette wren (Student)

I like the fact that we did cover Fibonacci numbers in my class. What really impressed me about this website is that the information was useful if you just wanted to get a quick overview but if you a more clear understanding in depth that is also available to you.

Time spent reviewing site:
10-15 min.

Used in course

5 years ago

### Michelle Davis (Student)

I reviewed Fibonacci and we have also covered it in class, it was interesting to learn about in class and I loved the way the Golden Rule plays many roles in everyday life. The content in this description is very helpful in a way that it helps to describe things a little more clear. I think more information will enhance student and staff learning on this certain mathematical concept.

Used in course

5 years ago

### Mona Franciscus (Student)

I reviewed the contents of Fibonacci to see what kind of information it contained. I was surprised to see such good detail and explainations. There is enough detail to write a paper and give examples for someone to understand. I will tag this site as reference for future papers.

Technical Remarks:

Very effective explainations and lots of reference material.

Time spent reviewing site:
I spent about 5 minutes reviewing it.

5 years ago

### Clarissa Magno Cua (Student)

This is a very informative site. It provides a lot of materials about Fibonacci Numbers and the Golden section. It shows the application of Fibonacci numbers and the golden rule in nature as well as, the mathematical side of these. The content was written in an easily understandable format

Technical Remarks:

The site can easily be navigated by the use of links. The materials are organized into well defined sections.

Time spent reviewing site:
One can grasp the mathematical concepts in 15-30 minutes.

Used in course

5 years ago

### Lora Thompson (Student)

I spent about 45 min on this site! I loove the math involved in fibonacci! (Mostly because it has to do with nature) I think this site does an awesome job explaining everything involved with fibonacci. It's easy to understand and definately will be of help when I need to do further research on this subject! :)

5 years ago

### allie fiorentino (Student)

This section is great for people who have studied Fibonacci numbers in the past, or even if it is their first time. It explains first what they are and what they require, secondly, places you can find them (i.e. nature) and also talks about the golden section. It gives you realistic applications of the Fibonacci numbers which usually helps people when dealing with math. A practical application is always helpful. It also gives art references which would help someone connect to the lesson. The quality of the page and information is decent, accurate and easy to follow.

Technical Remarks:

It it organized very well and easily understood from start to finish.

Time spent reviewing site:
I spent a few minutes reading over the general information of the page.

5 years ago

### Mallory Walker (Student)

I spent about Ten minutes reviewing the material. It was interesting to me because I have seen the Fibonacci numbers and golden section in class before. It was interesting to see a different outlook on the material. The material was put together very well and was very organized. It was easy to understand and read. It presented the material accurately and showed great detail. I feel that if a student needed to access material about Fibonacci numbers and the golden section, this material presented would be a good source of information. It would be good because it is easy to understand, but all has all the information needed to know everything you would want to know about this topic. I this material would be easy to use for first times user. I am a first time user and was glad to find a description of material I had seen before, in a different way. It is good to have a different perspective on material that you have seen before. If I wanted to know more about this topic I would use this website.

Used in course

5 years ago

### Brittany Rediger (Student)

The website is defiantly packed with information. I've done research on Fibonacci numbers before and this website would have been helpful and more trustworthy then some of the other sites I visited. The information is very thorough and would make an excellent resource to teaching.

Time spent reviewing site:
15 minutes

5 years ago

### Jennifer Brown (Student)

I browsed around the site, clicked on links, and surveyed it from the perspective of both a math student and a content judge; read through various portions of the site and subsequent links. Content of the site is excellent, filled with extensive information on the Fibonacci numbers and its related topics; plenty of information along with numerous references and external sites to explore. There is information on who Fibonacci is (personal information, biography) along with what the Sequence is, how it is used, and its history. Very relevant to learning math/Fibonacci sequence and would most definitely be useful for anyone researching this topic, as the information is extensive and well-researched, along with being very well-represented. Site is easy to navigate, links are clearly marked, and would be great for students to use as well as teachers.

Technical Remarks:

Site is easy to navigate, links are clearly marked with no 404 errors or anything similar. Easy to use, easy to navigate forward and backwards within the site, and well thought out layout.

Time spent reviewing site:
10 minutes. Not much time is needed as layout is clear and concise, with no obvious errors.

Used in course

5 years ago

### Sally McMullen (Student)

I think this material is *very* well structured and packed with just about all that one could imagine about Fibonacci Numbers and The Golden Section and more! I spent about a good 45mins to an hour on this website, reading up how Fibonacci # can be found in various ways. The Rabbit Sequencing was very interesting and I liked the activities that went along with it too! I enjoyed reading about the history behind the man, and controversy over the theory. The quality of the materials is outstanding, very well thought out and clearly represented in a manner that is easy for all levels to appreciate. I have no doubt that students would benefit from using these materials to build a foundation of knowledge on this subject and explore new things too.

Technical Remarks:

I believe that first time users those less 'tech-savvy' would have no problem using this site. It's straightforward, point and click!

Time spent reviewing site:
45mins-an hour

5 years ago

### Esther Michael (Student)

This site gives the material type, technical format,location, author and submitter. The material was excellent. There are innumerable concepts for our information. The site is very useful and interesting for new learners like me. Being an art student, I was very impressed by the details of Fibonacci's rabbits, Dudedeney's cows and honey bees and family trees. I was excited to see Fibonacci numbers in so many fruits and vegetables. It was very informative. Dr Knott's Fibonacci and Phi was very impressive. The latest being the Pen rose tilings has exhibited five-fold symmetry yet which do not repeat themselves.

Technical Remarks:

It was quite easy to browse this site. I was able to browse according to my Professor's instructions. And here I am with a little effort.

Time spent reviewing site:
The website 'Merlot' was so fascinating that at first I thought I would just do my assignment and stop. But as I started browsing it took me to innumerable sites which are of great interest not only to teachers and students but also to all people of all walks of life. I spent nearly 2 hourse browsing 'Merlot.'

Used in course

5 years ago

### Vince Cracchiolo

This is a great website in aiding as a learning tool. I browsed the website, and hands down it deffiently is the go to website when searching for answers. I have studdied about Fibonacci's number and this website does a great job on covering the material. They cover the material in every aspect. It gives great examples, what is it, where it's found, where did it come from.

Time spent reviewing site:
40 mins reviewing THIS SUBJECT.

5 years ago

### latisha mills (Student)

When I first clicked on this link, i expected the site to list a bunch of facts on Fibonacci numbers and the golden section. But as i explored the sight further, i found the material to be very helpful and interesting. take for example how the site explained, fibonacci and the original problem about rabbits where the series first came up. I thought that this site would be helpful for someone who wants to learn about Fibonacci numbers and get an overview of some problems. It was pretty self explanitory on how to veiw and use the sight.

Time spent reviewing site:
I spent about 20-25 minutes looking over this site.

5 years ago

### Dannah Sanchez (Student)

Thank you for that. It took me a while to digest what I was reading, but I appreciate the details that went into creating this.
It took about twenty minutes for me to just browse lightly over the website. All in all, I thought that it was very impressive and easy to navigate around.
The content was great!The fact that it was broken down into sections with links below made it a lot easier. I have read lightly on Fibonacci's Numbers and I wish that someone had told me about your work earlier, because the explanation is very detailed and organized--the topic was very well covered. I would definitely visit this site again next time I have to write a paper.
The site is great for students who are looking for more information on Fibonacci Numbers, and it's also good to read for fun and curiosity.

Time spent reviewing site:
20 minutes

Used in course

5 years ago

### Nathanael DeVine (Student)

I think that this topic, the Fibonacci sequence in particular, is very interesting. I enjoy the materials that were used to further explain and illustrate the concept of the Fibonacci numbers. The visual aides that were used were very effective, in fact the material relating to the Fibonacci square I found personally to be quite interesting. I had not heard about the Fibonacci square and with the examples that are used in this material, it makes it very clear and strikes in me a new found intrigue.

5 years ago

### vishal sharda (Student)

the site is really interesting and very exciting. its is very enjoyable somewhat is was kind of hard. if you dont know what Fibonacci means set counts 1 and 2 equals 3. 3 and 2 equal 5. etc

Used in course

5 years ago

### Laura Gossman

i spend a while looking over this website. the website is very detailed. however, it semms like a lot all at once. a little crowded and confusing.

5 years ago

### Ashley Ramirez (Student)

I spent 15 minutes exploring this website. I feel this site has a lot of great material and number patterns tat are interesting and fun. This definently can help students by not only teaching them but giving examples and material to look at. This site is fairly easy to navigate I got mixed up a little.

Time spent reviewing site:
15 minutes

5 years ago

### Jessica Lynn (Other)

I browsed through all the links and used the applets. This site is very helpful, it has a broad list of explanations, pictures, and visualizations to help the learner. It is well organized and had lists of references and further things to do that could be helpful for continued learning.

5 years ago

### Andy Luong (Student)

Hi, my name is Andy, i am a student at Mission College. There is a lot of great stuff on this website, i have spent all most 1 hour and 300 minutes on this website because it is so interesting. This site is very useful because everything is there, we don't need to spend 3 hours looking for the materials. For example, when i first click on this website i found the Fibonacci, so i click on it and it gave me a lot of informations. This section is really clear because it has all the information that a student would be able to do research on.

Used in course

5 years ago

### Ruben Bravo (Student)

This site was a quick find. It took quite sometime to read through the page and get a good understanding of the information. The site is useful for people who are looking to teach and learn about Fibonacci numbers. The site contains great, quality information. There are many links to all the different aspects of this material, making it easier to follow along. For the most part the site is clear and helpful for anyone looking to learn about Fibonacci and the golden selection. The materials are simply written without too many distracting fonts, colors and pictures.

5 years ago

### Farah Tala (Student)

I spent 30 minutes reviewing material related to the Fibonacci website. I spent about half the time looking at the different number patterns and sequences. The other half of my time was spent exploring the many links. In addition, I learned about the Golden String also by Fibonacci which reminded me of modern computer binary code. The website presents many useful teaching instruments, this is done mainly by visually showing the way in which the numbers are sequenced. I would say the website was great and would help people that were interested in learning more about Fibonacci Numbers.

Used in course

5 years ago

### jennifer cancilla (Student)

The Fibonacci Numbers and the Golden Section was an amazing learning material. I spent a few days exploring this information to better further my knowlege about the Fibonacci sequence. I was really happy to find this information because it helped me to write my midterm paper. The material presented showed many concepts that pertain to math. Not only did the inforamtion pertain to math the Fibonacci sequence has connections with nature, sexual reproduction, and the human body. Students will learn a whole new persepctive on concepts that i mentioned above. learning about the fibonacci sequence gave me the opportunity to enhance other students in my class about nature and the golden spiral and how they followed the fibonacci sequence exactly. If you are a first time user using this website you should have no problem trying to access information to help futher better their knowlege on topics of interest.

Used in course

6 years ago

### tammy gorman (Student)

I would like to say thank you to all who have taken the time in creating this web page. I really benefit a lot from the learning material. I have gotten some great ideas for extra math problems for my kids to work on over the summer.

6 years ago

### Pavel Shokun (Student)

After reading this great resource, I learned a lot of new interesting information to me from this site. I have never heard about Fibonacci numbers and the Golden section. I spent more than one hour reading and looking over all the material on this web site. I liked the graphics and images on the web site, because they were engaging. The Fibonacci Rabbit Sequence was one of the most fascinating concepts. I also found the plant example to be interesting, because as it appears that the amount of petals on plants is a Fibonacci number. The information was very informative and descriptive. I had to read a lot to understand the concept but the material was very clear and precise. It is very useful and helpful information. Teachers can use this as an aid to teach this concept. The site is very effective and will enhance a studentsÂ’ knowledge. This site provides a lot of useful information which can be linked to many math courses and a personÂ’s life.

Technical Remarks:

I am a first time user of this web site and I found it very easy to follow. I found a lot of helpful, clearly described content. There are many areas which contain visuals which makes it even more interesting. This site was developed in a very good way. The layout is perfect and the content is easy to understand if you read it carefully.

Used in course

6 years ago

### shanmugasundaram karuppannagounder (Student)

I spent almost 45 minutes to familiarize my self with this webpage. I found it really useful because it is an excellent resourse to aid teachers in teaching this subject and also it enhances learning capabilities. I think the material provided is very useful for Math students to help understand them the course better. I found that this website was very well organized and very easy to read. I spent some of the time on the fibonacciÂ’s numbers, specially the rabbits and reproduction. The visual contents made it very interesting and easy to understand. The potential effectiveness for enhancing for both students and teachers is very high. I think the website is very easy to use and understand.

Used in course

6 years ago

### Asim Pasha (Student)

This is a great example of a well organized informative site. Be prepared to spend a few hours to really get into the material provided. The examples, reasoningÂ’s, even the formulas are well organized. The Visual aids involved is very attractive, and keeps ones attention. ItÂ’s easy to read and we can easy find the value in the information provided. Again I canÂ’t stress the amount of material available, and for those who are new to this material, donÂ’t stress. The way itÂ’s laid out, its presents itself to the highly experience to the inexperience.

6 years ago

### Rosie Berdejo (Student)

I spent one hour going through all the different links at this site. Before I went to this site I had no idea what a Fibonacci number and the Golden Section was. But now after this I have learn a lot. The quality of the material was great and well organized. It was easy to read and learn. I liked the example of plants. On many plants, the number of petals is a Fibonacci number. For example, Daisies can be found with 34, 55 or even 89 petals. Also, Pine cones show the Fibonacci spirals clearly.

Technical Remarks:

This site contains a variety of images and graphics. I found this site to be really easy to go through and very easy to understand.

6 years ago

### varun gandhi (Student)

I spent about two hours and so, I found this website to be really easy to go through and very easy to understand. This section and its visual content made it interesting and easy to understand. The examples and found them fun as well as useful in math. It helped me to grasp a better understanding of the concepts,and the quality of the material is great and written very deeply. I think this is a good site, and very good developed.

8 years ago

### stephanie whitfield (Student)

I spent hours reviweing this site and learning about Fibonacci numbers. The

quality of the material is great and written so you can understand it. This is

an excellent resource to aid teacher's in teaching this subject and it does

enhance learning capablities.

quality of the material is great and written so you can understand it. This is

an excellent resource to aid teacher's in teaching this subject and it does

enhance learning capablities.

Technical Remarks:

I think the material provided was great and well organized. It was easy to read

and learn. With Merlot and a regular search engine the Fibonacci site was easy

to access for me and I'm sure it would be for first time users as well.

and learn. With Merlot and a regular search engine the Fibonacci site was easy

to access for me and I'm sure it would be for first time users as well.

Used in course

10 years ago

### Nicky Thortis (Student)

It was really intresting to come across this topic, as I just recently finished

reading The Davinci Code, by Dan Brown and the Fibonacci Numbers were used as a

cypher in two different situations. This topic is quite fascinating and I

really like how this website depicted feasible ideas of this concept. The

Golden Number was also discussed, which I found equally as intriguing. Again it

was great to see that almost everything in nature correlates with the golden

number.

I spent about 30 minutes browsing this topic and think it has given me a great

topic for my final research paper!

reading The Davinci Code, by Dan Brown and the Fibonacci Numbers were used as a

cypher in two different situations. This topic is quite fascinating and I

really like how this website depicted feasible ideas of this concept. The

Golden Number was also discussed, which I found equally as intriguing. Again it

was great to see that almost everything in nature correlates with the golden

number.

I spent about 30 minutes browsing this topic and think it has given me a great

topic for my final research paper!

Technical Remarks:

The content on this page is quite helpful. Everything is explicitly described

and clearly depicted so it is very easy to follow along. I think everyone

should easily be able to grasp this concept as described on this web page. It

was very effective.

and clearly depicted so it is very easy to follow along. I think everyone

should easily be able to grasp this concept as described on this web page. It

was very effective.

Used in course

10 years ago

### Steven Vidal (Student)

After reading through this great information, it took me about ten to fifteen

minutes looking over and reading it. I thought that all of the information was

very useful and very descriptive as well. I tried taking down some important

notes and began writing down some of the bold letter words.

As I was reading all of the material that was given to me I realized that

most of the information helped me to understand what a real Fibonacci number was

and what it represents. All of the concepts that were brought up in this site

were very descriptive and very educational as well. I thought that all of the

information was educationally significant because it gave certain models and

explinations of what a student was trying to look at or find.

I think that the students learning will be enhanced if they choose to look

at this site because of the simple fact that it gives a great meaning to what

students are trying to find. This site also gives great understanding to all

the material as well. Once a student takes a look at what they are trying to

study, they will find that these sites will be very helpful to them no matter

what.

I also believe that first time users will find this technique very easy to

use and go by because it gives so many different examples and notes to

understand the material.

minutes looking over and reading it. I thought that all of the information was

very useful and very descriptive as well. I tried taking down some important

notes and began writing down some of the bold letter words.

As I was reading all of the material that was given to me I realized that

most of the information helped me to understand what a real Fibonacci number was

and what it represents. All of the concepts that were brought up in this site

were very descriptive and very educational as well. I thought that all of the

information was educationally significant because it gave certain models and

explinations of what a student was trying to look at or find.

I think that the students learning will be enhanced if they choose to look

at this site because of the simple fact that it gives a great meaning to what

students are trying to find. This site also gives great understanding to all

the material as well. Once a student takes a look at what they are trying to

study, they will find that these sites will be very helpful to them no matter

what.

I also believe that first time users will find this technique very easy to

use and go by because it gives so many different examples and notes to

understand the material.

Used in course

10 years ago

### michael bertoldo (Student)

I spent aproximately two hours fumbling through the site. this is no fault of

the site, but the inexperience of the user. i did spend some time on the

fibonacci numbers, specificaly the the rabbits and reproduction. this section

and its visual content made it interesting and easy to understand. i feel the

general effectiveness is good and that the overall learning potential for both

students and teachers is high.

the site, but the inexperience of the user. i did spend some time on the

fibonacci numbers, specificaly the the rabbits and reproduction. this section

and its visual content made it interesting and easy to understand. i feel the

general effectiveness is good and that the overall learning potential for both

students and teachers is high.

Technical Remarks:

the content quality is good at least from the areas i attended. i especially

liked the areas i viewed because the were visual and easy to understand. if

fiist time users are proficient on the computer and the internet

liked the areas i viewed because the were visual and easy to understand. if

fiist time users are proficient on the computer and the internet

Used in course

10 years ago

### Ryan Hendry (Student)

I found this website to be really easy to go through and very easy to

understand. I went through it for about 15 mins and looked at some of the

puzzles and listened to an audio play on the golden section.

The material seemed to be pretty accurate on the goldon section, at least

from what I have learned in my math class. It helped me to grasp a better

understanding of the concepts.

This site seemed like a pretty legitimate site for use as a teaching tool.

I think it could be a good tool.

The materials were very user friendly.

understand. I went through it for about 15 mins and looked at some of the

puzzles and listened to an audio play on the golden section.

The material seemed to be pretty accurate on the goldon section, at least

from what I have learned in my math class. It helped me to grasp a better

understanding of the concepts.

This site seemed like a pretty legitimate site for use as a teaching tool.

I think it could be a good tool.

The materials were very user friendly.

Used in course

10 years ago

### Vincent Klumper (Student)

I learned a lot from the sit, and before I went to this sit I had no idea what a

Fibonacci number was, and had no idea how the golden setion worked. This sit I

feel was very educational, and I also have to say it was very easy to follow.

So I enjoyed this sit a lot.

Fibonacci number was, and had no idea how the golden setion worked. This sit I

feel was very educational, and I also have to say it was very easy to follow.

So I enjoyed this sit a lot.

Technical Remarks:

Try to not be so confusing on the golden section. I understand it might be very

hard not to be confusing, but maybe there can be a more fun approuch to the

material.

hard not to be confusing, but maybe there can be a more fun approuch to the

material.

Used in course

10 years ago

### Matt Imwalle (Student)

I spent a while going through this site and looking at all the different links.

Before I read this page I never even heard of a Fibonacci number or sequence.

But now after this I have an extremely good awareness of what they are now. I

liked the examples that they used on this page, how many bunny rabits can be

produced in a given time and the whole bumble bee thing. I never knew that

rabbits can reproduce after a month. Then the examples with the bee's the male's

compared to the female's were very cool. I tried the Fibonacci sequence for

myself by doing the diagram and following all the directions it says. This

material accurately presents concepts and models that are very good for

educational purposes. They break it down and explain it with out confusing you.

The models are good and also are not confusing. After reading this I now know

what the Fibonacci numbers are and how the Fibonacci Sequence appears in nature.

Before I read this page I never even heard of a Fibonacci number or sequence.

But now after this I have an extremely good awareness of what they are now. I

liked the examples that they used on this page, how many bunny rabits can be

produced in a given time and the whole bumble bee thing. I never knew that

rabbits can reproduce after a month. Then the examples with the bee's the male's

compared to the female's were very cool. I tried the Fibonacci sequence for

myself by doing the diagram and following all the directions it says. This

material accurately presents concepts and models that are very good for

educational purposes. They break it down and explain it with out confusing you.

The models are good and also are not confusing. After reading this I now know

what the Fibonacci numbers are and how the Fibonacci Sequence appears in nature.

Used in course

10 years ago

### Wahl (Student)

I spent a fair amount of time reviewing the material and trying to grasp the

concept. I tried to follow along, rereading certain sections, but had very

little luck understanding.

concept. I tried to follow along, rereading certain sections, but had very

little luck understanding.

Technical Remarks:

I think that the page was laid out well and I found it appropriate to have as

many pictures and examples as it did. Even though I couldn't understand it I

still gave it a four: I think that the material was well organized and I'm sure

would make perfect sense to a more mathematically inclined individual. The

pages go into great detail about the subject and present the material in a

number of different ways. I found the section on the Golden Section in Nature

to be interesting and some of that made sense but apparently, Fibonacci numbers

are not for me. I am sure that the material was presented accurately and that

this is educationally significant. The author gave examples and backed them up

with proofs and pictures/diagrams. I don't know if this would be the best

"candidate" for enhancing learning and teaching, but that is strictly based off

of my own experience with the material and this is not to say that others

wouldn't benefit from it. I suppose it would be fine to use if the words were

put a little more simply. The software was easy to use and the links were easy

to access.

many pictures and examples as it did. Even though I couldn't understand it I

still gave it a four: I think that the material was well organized and I'm sure

would make perfect sense to a more mathematically inclined individual. The

pages go into great detail about the subject and present the material in a

number of different ways. I found the section on the Golden Section in Nature

to be interesting and some of that made sense but apparently, Fibonacci numbers

are not for me. I am sure that the material was presented accurately and that

this is educationally significant. The author gave examples and backed them up

with proofs and pictures/diagrams. I don't know if this would be the best

"candidate" for enhancing learning and teaching, but that is strictly based off

of my own experience with the material and this is not to say that others

wouldn't benefit from it. I suppose it would be fine to use if the words were

put a little more simply. The software was easy to use and the links were easy

to access.

Used in course

10 years ago

### Marcela Alcorn (Student)

After spending 10 minutes on how to figure out how to get to the web page, I'd

say I spent about 20 minutes really grasping all the information I could take

in. This web site was extremely easy to use and to get around. I loved

everybit of this site. The thing that struck me most was not necessarily the

math, but being a visual person, the pictures showed how it could be used in a

different scenario other then numbers and art. The golden section was already

fascinating for me, but to see how they provided different images and

explanations, this made my experience even better than originally expected.

Overall to get to the site was a bit difficult but after a couple of minutes

fooling around, you'd get the hang of it.

say I spent about 20 minutes really grasping all the information I could take

in. This web site was extremely easy to use and to get around. I loved

everybit of this site. The thing that struck me most was not necessarily the

math, but being a visual person, the pictures showed how it could be used in a

different scenario other then numbers and art. The golden section was already

fascinating for me, but to see how they provided different images and

explanations, this made my experience even better than originally expected.

Overall to get to the site was a bit difficult but after a couple of minutes

fooling around, you'd get the hang of it.

10 years ago

### galvan (Student)

I thought the site gave a lot of intresting ideas in fibonacci numbers. They

gave good examples of concepts to this type of math, making it easy to

understand how the math can be used. There was not alot of applications on the

site. It had educational purpose on puting projects togeather.

gave good examples of concepts to this type of math, making it easy to

understand how the math can be used. There was not alot of applications on the

site. It had educational purpose on puting projects togeather.

Used in course

10 years ago

### amy konsterlie (Student)

I enjoyed this facinating site...so many links and so little time! I especially

liked the animation regarding the circular spiral moving within the daisey-like

shape. So much of nature relates to the Fabonacci Numbers and the Golden Section

liked the animation regarding the circular spiral moving within the daisey-like

shape. So much of nature relates to the Fabonacci Numbers and the Golden Section

Technical Remarks:

This site was easy to navigate thru.

Used in course

10 years ago

### Kathy Phelan (Student)

I love this site! I am a first time user and yes, it was very easy to use the

software. I'm embarrassed to admit I did not know what Fibonacci numbers are.

I just like the name, Fibonacci. Once I arrived, the simple explanation of what

Finonacci numbers are fascinated me (I guess it's easy to fascinate me). I

immediately thought, "how cool!" I went to the link that was mentioned in terms

of introducing one to the site and was drawn to the animals. I could spend

hours on this site. In fact, I may go back. Who or what is Fibonacci? I

actually figured out the rabbit problem. The visuals were great and having the

answer available is even better. Thank you.

software. I'm embarrassed to admit I did not know what Fibonacci numbers are.

I just like the name, Fibonacci. Once I arrived, the simple explanation of what

Finonacci numbers are fascinated me (I guess it's easy to fascinate me). I

immediately thought, "how cool!" I went to the link that was mentioned in terms

of introducing one to the site and was drawn to the animals. I could spend

hours on this site. In fact, I may go back. Who or what is Fibonacci? I

actually figured out the rabbit problem. The visuals were great and having the

answer available is even better. Thank you.

Technical Remarks:

Very simple, user friendly. Well done and so easy to understand.

Used in course

10 years ago

### Terence Telkamp (Student)

I spent a few minutes looking over this site. I found that it was organized

fairly well. I liked that right away it showed what the Fibonacci secuence,

golden section numbers, and the golden ratio all were. Then I told where to go

to get a basic understanding of the ideas.

fairly well. I liked that right away it showed what the Fibonacci secuence,

golden section numbers, and the golden ratio all were. Then I told where to go

to get a basic understanding of the ideas.

Technical Remarks:

I have had a very small amount of experience with these concepts. I think this

site does a good job at showing even someone like me how these concepts might be

useful or applied.

ex. rabbit family trees, or packing to save on wasted space

site does a good job at showing even someone like me how these concepts might be

useful or applied.

ex. rabbit family trees, or packing to save on wasted space

11 years ago

### Nguyen (Student)

i thought that this section on fibonacci numbers was very interesting. i spent

a few minutes looking through a few of the examples and found them fun as well

as useful in math. this was on way to find patterns in sets of numbers. the

brink example and the honeybee demonstration was very cool because they had

visual that made it much easier for me to understand. i think this is a good

site for some to look at in order to make some aspects of this math simpler.

a few minutes looking through a few of the examples and found them fun as well

as useful in math. this was on way to find patterns in sets of numbers. the

brink example and the honeybee demonstration was very cool because they had

visual that made it much easier for me to understand. i think this is a good

site for some to look at in order to make some aspects of this math simpler.

11 years ago

### Barbra Bied Sperling (Staff)

If you are someone who enjoys numbers or working with patterns then you will

like this site. This site deals with other kinds of Math. For instance,many

puzzles in dealing with nature and animals to try like the honeybees where

numbers add together to make a certain group. The number Trick is a must to try

under Mathematical Magic Fibonacci Numbers. It gives a very good explanation and

a formula on how to do it. So you pick any two numbers add plus the new number

ten times and try it with your friends. Also, working with triangles in making a

number pattern inside the triangle. And,you can try doing a family tree.

Also,numbers dealing with prime numbers and factors I found that this part was

difficult to understand the pattern of numbers. However, there was a table of

numbers was easier to see the process.As well as towards the end of each puzzle

it gives an option if you like for Things To Do so you can try it yourself. And,

they are also great pictures included for each puzzle. So, if you like doing

puzzles and nature then you would enjoy this site. Also,any of you teachers out

there it is great tool because they can work with their students in groups

dealing with patterns and numbers. I enjoyed viewing this site for about an hour

in solving these puzzles.

like this site. This site deals with other kinds of Math. For instance,many

puzzles in dealing with nature and animals to try like the honeybees where

numbers add together to make a certain group. The number Trick is a must to try

under Mathematical Magic Fibonacci Numbers. It gives a very good explanation and

a formula on how to do it. So you pick any two numbers add plus the new number

ten times and try it with your friends. Also, working with triangles in making a

number pattern inside the triangle. And,you can try doing a family tree.

Also,numbers dealing with prime numbers and factors I found that this part was

difficult to understand the pattern of numbers. However, there was a table of

numbers was easier to see the process.As well as towards the end of each puzzle

it gives an option if you like for Things To Do so you can try it yourself. And,

they are also great pictures included for each puzzle. So, if you like doing

puzzles and nature then you would enjoy this site. Also,any of you teachers out

there it is great tool because they can work with their students in groups

dealing with patterns and numbers. I enjoyed viewing this site for about an hour

in solving these puzzles.

Technical Remarks:

While I was on this site there wasn't any problems while I was looking over this

site. While I was trying the problems it was easy to get to the next page and

move back to the begining. There was no conflict like the site dissapeared or I

couldn't get the page up on screen.

site. While I was trying the problems it was easy to get to the next page and

move back to the begining. There was no conflict like the site dissapeared or I

couldn't get the page up on screen.

11 years ago

### R. Lynn Sanitate (Student)

Fibonacci Numbers and the Golden Section is quite intriguing to me. Not

necessarily the history behind Fibonacci but the section on Numbers and Nature.

Since I have an appreciation for nature I am more inclined to gather my

information here. Mathematics? surrounds me and I haven?t been able to recognize

it. Fibonacci has my attention.

This site offers plenty of resources ? enough to make your head spin. My second

time exploring this site kept me busy for a couple of hours. If you have

difficulty understanding math and love nature this is the place to come.

necessarily the history behind Fibonacci but the section on Numbers and Nature.

Since I have an appreciation for nature I am more inclined to gather my

information here. Mathematics? surrounds me and I haven?t been able to recognize

it. Fibonacci has my attention.

This site offers plenty of resources ? enough to make your head spin. My second

time exploring this site kept me busy for a couple of hours. If you have

difficulty understanding math and love nature this is the place to come.

Technical Remarks:

This site offers students and teachers a like a great variety of topics and

helps. There are convent tools such as calculators, charts and graphs that

assist in understanding mathematical principals.

helps. There are convent tools such as calculators, charts and graphs that

assist in understanding mathematical principals.

11 years ago

### Beatriz Solorio (Student)

Hi, my name is Beatriz Solorio, and I'm a student at Mission college in Santa

Clara California. When visiting the Fibbonacci web site it took me

approximately an hour to read. This was due to the fact that I took time in

reading what I thought was interesting. I gave this site the highest score

because I thought that it really was informative and actually fun to learn.

This site focused on a series of numbers called Fibbonacci. It was interesting

to find out were this idea came from, and how these numbers are found in nature.

It also introduced the idea of "phi" the Greek representation of 1.61804....

The pictures were one of the best parts of this site; they really helped in

understanding the concept.

Clara California. When visiting the Fibbonacci web site it took me

approximately an hour to read. This was due to the fact that I took time in

reading what I thought was interesting. I gave this site the highest score

because I thought that it really was informative and actually fun to learn.

This site focused on a series of numbers called Fibbonacci. It was interesting

to find out were this idea came from, and how these numbers are found in nature.

It also introduced the idea of "phi" the Greek representation of 1.61804....

The pictures were one of the best parts of this site; they really helped in

understanding the concept.

12 years ago

### Tatiana Sviridovsky (Faculty)

Lovely materials! They can certainly brighten up a math class!

12 years ago

### Brian Doore (Student)

Wow! This site is VERY rich in content and quality. I spent over one hour here

just the first time. I'm sure I'll be back many times. It is like math

play-land. The first time I heard of Fibonacci numbers and the golden section

was when I saw a Disney producion called "Donald Duck in Mathamagic land" or

something like that. It was a long time ago. The program and its treatment of

these topics has made a lasting impression on me. I was very pleased to see

those two topics together on one website. I know I only saw the tip of the

iceberg here. The site and its links seem endless. I especially enjoyed the

Fibonacci puzzle section, although I have only visited the simple games thus

far. I can see why this is an award winning site.

just the first time. I'm sure I'll be back many times. It is like math

play-land. The first time I heard of Fibonacci numbers and the golden section

was when I saw a Disney producion called "Donald Duck in Mathamagic land" or

something like that. It was a long time ago. The program and its treatment of

these topics has made a lasting impression on me. I was very pleased to see

those two topics together on one website. I know I only saw the tip of the

iceberg here. The site and its links seem endless. I especially enjoyed the

Fibonacci puzzle section, although I have only visited the simple games thus

far. I can see why this is an award winning site.

Technical Remarks:

Everything I visited on this site worked flawlessly. I am not qualified to rate

it technically, except that as a learning tool, it is very effective.

it technically, except that as a learning tool, it is very effective.

12 years ago

### Thuy Pham (Student)

It took me about couple minutes to get to the site

and more than 2hrs to review the materials on the site. The site is so large. It will take alot of time to cover every single information. I think this web-site is very useful for enhancing teaching and learning. The quality of material content is perfect. We can find most of valued informations we need about Fibonacci Numbers and The Golden section. I love the way they arranged the pages. The informations, formulars, examples, explainations, sources,etc are conected. Web- links and references are powerful. Especially, the website was designed very organization with interested graphics; thus,that makes the users are quite persuaded and actracted. The direction is clear and helpful for users easy to enter the entire pages. The materials are written simple, easy to read and understandable. It is excelent site, and suitable for different levels of background knowledge.

and more than 2hrs to review the materials on the site. The site is so large. It will take alot of time to cover every single information. I think this web-site is very useful for enhancing teaching and learning. The quality of material content is perfect. We can find most of valued informations we need about Fibonacci Numbers and The Golden section. I love the way they arranged the pages. The informations, formulars, examples, explainations, sources,etc are conected. Web- links and references are powerful. Especially, the website was designed very organization with interested graphics; thus,that makes the users are quite persuaded and actracted. The direction is clear and helpful for users easy to enter the entire pages. The materials are written simple, easy to read and understandable. It is excelent site, and suitable for different levels of background knowledge.

12 years ago

### Christopher Black (Student)

This site is rich in information on the Fibonacci numbers and their mystery.

I've spent 15 minutes following the links and the trail of information. Feasting

on examples and charts. The links to outside sites are just as good. I think

this site will do well to instruct the expert as well as the intermediate. There

is such a wealth of information that you'd be hard pressed to not go away withs

some deeper understanding.

I've spent 15 minutes following the links and the trail of information. Feasting

on examples and charts. The links to outside sites are just as good. I think

this site will do well to instruct the expert as well as the intermediate. There

is such a wealth of information that you'd be hard pressed to not go away withs

some deeper understanding.