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Fibonacci Numbers and The Golden Ratio

Even if you were taught about the Fibonacci number sequence in school, you probably don’t remember much about it. As with other higher levels of math, many aren’t sure how Fibonacci could possibly be relevant to their real lives; so, why should they even attempt to remember him or his sequence? In reality, Fibonacci numbers are something you come across practically every day. Even so, let’s go back and start at the beginning.

The Fibonacci number sequence is named after Leonardo of Pisa (1175-1240), who was known as Fibonacci. (I love to say that name because it sounds like I know a foreign language.) In mathematics, Fibonacci numbers are this sequence of numbers:
As you can see, it is a pattern, (all math is based on patterns). Can you figure out the number that follows 89? Okay, let's pretend I waited for at least 60 seconds before giving you the answer….144. By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. For those who are still having difficulty (like my daughter who is sitting here), it is like this.

The next number is found by adding up the two numbers that precede it.
  • The 8 is found by adding the two numbers before it (3 + 5)
  • Similarly, 13 is found by adding the two numbers before it (5 + 8),
  • And the 21 is (8 + 13), and so on!
It is that simple! For those who just love patterns, here is a longer list:


Can you figure out the next few numbers?

The Fibonacci sequence can be written as a "Rule “which is:   xn = xn-1 + xn-2   The terms are numbered from 0 forwards as seen in the chart below.   xn is the term number n.   xn-1 is the previous term (n-1) and xn-2 is the term before that (n-2)

Sometimes scientists and mathematicians enjoy studying patterns and relationships because they are interesting, but frequently it's because they help to solve practical problems. Number patterns are regularly studied in connection to the world we live in so we can better understand it. As mathematical connections are uncovered, math ideas are developed to help us be aware of the relationship between math and the natural world. 

As stated previously, we come across Fibonacci numbers almost every day in real life. For instance, my husband and I were at the Wonders of Wildlife Aquarium in Springfield, Missouri. (If you haven't been, you should go because it is spectacular.) He was noticing how the herrings were swimming counter clockwise and discussing the Coriolis effect with the guide. When we got to the lower levels, where the sharks were, they were all swimming in a counterclockwise direction as well. I asked my rocket scientist husband why this was and again he said, with a straight face, "The Coriolis Effect."

Inside of a Nautilus Shell
I then spied seashells and started talking about Fibonacci numbers and the Golden Ratio. (I know the visitors around us were wondering just who we were!) On the right, you will see a picture of the inside of a Nautilus Shell taken by me! It clearly shows the Golden Ratio. (The Golden Ratio is a special number equal to about 1.6180339887498948482. The Greek letter Phi is used to refer to this ratio. Like Pi, the digits of the Golden Ratio go on forever without repeating.) Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral.

Are you still not sure what I am talking about? Have you ever watched the Disney movie entitled Donald in Mathmagic Land? (It's an old one that
The Golden Ratio
you can find on You Tube.) Well, in the movie they talk about the Golden ratio. This is a proportion that is found in nature and in architecture. The proportion creates beauty. And that proportion is the Fibonacci sequence! If you divide consecutive Fibonacci numbers you will always get the Golden ratio. Try it! Start with the big numbers. If you divide 89 by 55, you get 1.61. If you divide 55 by 34, you get 1.61. If you divide 34 by 21, you get 1.61, and so on. You can look up the Golden Ratio and explore it more. It’s fun!

As I close, here are four questions to think about:
  1. How might knowing this number pattern be useful? 
  2. What kinds of things can the numbers in the Fibonacci sequence represent?
  3. Where is the Golden Ratio found in the human body?
  4. Why is the golden rectangle important in architecture and art? 

A Go Figure Debut for an Australian Who Is New!

Australia with Queensland Highlighted
Our featured teacher lives “down under” in Queensland, Australia which is situated in the north-east of the country. His Teachers Pay Teachers store is Creating Light Bulbs and that is what he loves most about teaching - that light bulb moment. When you teach a child a new concept or a concept that they are finding difficult, and you see the light bulb turn on when they understand it, this is the reason why he teaches. Mr. Light Bulb has been teaching for five years and describes his classroom as fun, innovative with an emphasis on a growth mindset. 

Mr. Light Bulb enjoys running and is currently training to run a half marathon. At times he dabbles in video games and computer coding. He enjoys cooking, particularly desserts and sweets (YUMMY!) such as sticky date pudding or salted caramel tarts. Also, he is 90% confident that he is addicted to coffee.

The majority of his store is based around mathematical activities including open ended questions, games and drills. He also has a range of STEM and project based learning activities. Currently, Mr. Light Bulb has 49 products in his store; six of them are free.

One of his free resources is entitled Free Fact Families – All Operations and is appropriate for grades 3-5. This sample pack provides the teacher and students with the opportunity to try two high quality products covering all four math operations. The full products includes all of the facts from 1-12, rainbow facts, leveled extension activities and blank sheets to create your own. As one person commented when giving this resource a 4.0 rating, it is also “great for stations and interactive notebooks.”

In addition, Mr. Light Bulb has a $10.00 bundle (you save $5.00 by purchasing the bundle) that is called Open Ended Math Questions – MEGA Bundle – Over 65 Different Questions.  Check out the preview of this resource to view some of the questions contained in this bundle. It combines all of his opened ended math questions products and targets number, fractions and decimals, money, chance, data, probability and word problems. Open ended questions are ones that require more than one word answers. The answers could come in the form of a list, a few sentences or something longer such as a speech, paragraph or essay. They provide an entry point for all students and allow the teacher to quickly gauge the varying levels of knowledge and understanding in the classroom. As one of his buyers expressed, This resource is “outstanding value! My students will love these.”  

I am especially partial to this resource because of the importance of asking open ended questions, especially in mathematics. I teach on the college level, and many of my students cannot think beyond a yes/no or a couple of words answer. Open ended questions allow me to analyze a student’s response to a question giving me an opportunity to learn how they think. The students’ responses reveal what they know and how they apply that knowledge. Having this type of resource in the elementary grades would introduce students to this type of questioning early and prepare them better for college.

So head on over to Creating Light Bulbs and take a look at his resources. I am sure you will find many items that you can use in your classroom. And while you are there, take the time to download his free item and leave a rating. He would love to know what you think.

20 Study Tips You Won't Forget! Tips on How to Be Successful in School

Students struggle with many difficulties and setbacks in their lives, and because of all of the competing things vying for their attention, it is hard for them to concentrate on studying. And yet if you are a student, you have to do at least a little bit of studying in order to progress from year to year.

Effective studying may not seem like the most exciting topic for anyone, but think of the big picture. The better a student's study skills are, the better the student will do in school, plus mastering effective study habits will make it easier to learn. Also effective studying can lead to better grades (in high school and college) and doing better on standardized tests.  Because I teach on the college level, I encounter many students who lack effective study skills or even habits, but no matter what study skills a student presently has, I know they can learn new strategies that can assist them in the future. 

For example, better time management and note-taking skills are important for many jobs. Being able to break down tasks into more manageable steps can help a student get things done in less time; thus, having more free time for themselves. Being able to handle test anxiety may help a student deal with other stressful situations such as an interview, a speech or oral exam.

FREE Resource
One of my freshmen classes I teach is called "Conquering College" where we discuss useful strategies for effective studying. For this class, I developed a list of 20 study skills or tips that can help students succeed. Since the key to effective studying is studying smarter, not longer, have your students begin studying smarter with these 20 helpful and effective Study Tips You Won't Forget!  It is a free resource in my store on Teachers Pay Teachers.