Students will be able to identify occurrences of the Fibonacci sequence.

Fibonacci Sequences

15 minutes

I have this written on the board:

“Do you notice anything special about this plant?”

Students enter write down their observations about the plant that I sketch on the board. Students usually start by sketching out the plant and are often stuck on the next move, but I usually help them by announcing key observations and writing them out for others to see:

Lucy is searching for a pattern in the leaves.

Franc is searching for a pattern in the stems.

Ted is splitting the plant into levels.

Each announcement will encourage the group to hunt for a pattern. Some students seem to see the Fibonacci sequence immediately. I encourage them draw a taller version of the plant. “What is the plant didn’t flower until it grew higher? What would it look like?” I rephrase this in many ways, “can you extend the pattern to fit the entire page?” Sometimes students are so excited by the discovery of the pattern that they only need some color to fill in the leaves and outline the paths formed by the splitting of the stems. I also give struggling students a handout that splits the plant into the levels needed to find the Fibonacci sequence:

If most of the students are struggling with the pattern, I display the flower on the projector with the above image to get their ideas going. I finish this pattern hunt by displaying the above image so that students can explain their findings. I circulate as a student shares their findings. If students have a good energy around their discoveries, I usually let many students share. Otherwise there is always one student who was able to discover different occurrences of the Fibonacci sequence.

“What did you notice about the leaves?”

“I counted the number of leaves in total and then counted the number of leaves at the same height. I wrote listed these out and noticed that they pretty much matched the Fibonacci sequence from yesterday.”

“Could you show us that process?”

I run a discussion around the leaves and make sure the students color or highlight the leaves that share the same heights. It often isn’t enough to say an observation. I want the whole class to take part in the conversation so I make sure they *show* their observation. The same conversation works with the stem pattern.

“Can you explain your observations around the stem pattern in the plant?”

“I counted the paths in the stem and amount of times they split.”

“Can you show us how you counted the stems and splits?”

“There is one stem in the first region and then in splits into two and by the time you get into the third region there is another split and three stems and then another split and then five stems and so forth. This is like the Fibonacci sequence, but it goes 1,2,3,5 instead of 1,1,2,3,5.”

“How could we extend the stem so the pattern fit the Fibonacci sequence exactly?”

“I guess we could extend the stem at the beginning to that it went across two regions.”

“Would it surprise you that this a diagram for a real plant called sneezewort?” I continue to talk to the class about the idea that the Fibonacci sequence models different growth patterns and processes in nature. Its not always exact and there are many plants and occurrences in nature that certainly don’t follow the Fibonacci pattern, but its amazing that so many things fit the pattern perfectly. We are going to explore some of those connections today. And don’t worry, we will talk about the other non Fibonacci occurrences and explain the bigger picture. Right now we are going to spend some time hunting for patterns!

30 minutes

I have several boxes and bags in the room. In each box and bag is a different “secret” Fibonacci item. Students will spend some time doing some station investigation, but I don’t want them spending time arguing over the “best” station. So I hide the contents of the station in the boxes and bags and ask one student from each table to get their Fibonacci item and return to their station. Their goal is to spend 15 minutes finding as many examples of the Fibonacci pattern as they can.

I include two pineapples with three colors of electrical tape. I also warn students to be careful of the “sharp” objects inside. I also include packages with pinecones, flowers and vertical plants. Two groups also get a photocopied chapter from the book the Number Devil on the reproduction cycle of Bunny Rabbits, but there are so many versions of this online that I sometimes just do a quick search and print out the friendliest looking version of Bunny Reproduction cycles.

Students open their containers and start. Their first reaction is usually a mixture of laughter and confusion. It seems that the last thing the expect to work on is fruit, plants and bunnies. I circulate and ask them to read the simple prompts in each package. A pineapple or pinecone hint might read, “count my spirals.” The bunny station would be prompted to “read the chapter and list out the bunny population.” The flower station is a bit trickier, but one flower station usually contains several small annuals and a group of photos. These flowers have petals numbers that match the Fibonacci sequence, but the prompt needs to keep them engaged without giving this away. If we simply point to the petals they will lose interest, so I usually write something like, “don’t these flowers look nice? What do they have in common?” The other flower station is given a protractor and some type of common vertically growing plant. I shop for these with a protractor and look for one in which the rotation of the leaves is related to the golden ratio and the amount of leaves that separate two leaves directly above each other matches elements of the Fibonacci sequence. Shopping for plants in this way leads to some strange looks, but might also lead to some interesting conversations at the store. People are always fascinated to know that a math sequence can help us better understand the design of plants. This station is usually the toughest, so I give them the most direct prompts, “what is the angle between leaves?” “How many leaves separate leaves that are directly above one another?”

Students will switch stations when another table is ready to switch. But obviously each group has a different challenge and might not finish at the same time. Its important that each group spends time recording their observations with sketches. So I ask a group that is done early to take detailed notes and think about how they will share their observations. If each group is take the full time to work, then nobody switches. But this is fine since each group will have a chance to share.

15 minutes

I give each group time to share their findings. The pineapple tables usually surprise each other, since there are three common spiral sets that match the Fibonacci numbers and usually one group finds one and the other finds the other two. The pinecone groups have similar stories to tell. The flower petal groups are usually shocked simply because they never noticed this obvious categorization of flowers into Fibonacci groups. The plant group usually goes last and struggles a bit, but I give them a chance to start the discussion around their findings. I often display the bunny groups findings on a document camera and have them explain the concept. The important thing is to finish this conversation with the “so what?” Many students are still struggling with the concept of aesthetics in mathematics. They want the math to have an application and are resistant to the idea that math can be beautiful in its own way.

“These discoveries are only a start. The Fibonacci sequence helps us understand the nature and design of these plants and animals. But we need to remember that there are plenty of flowers with 4 petals and plenty of pinecones with non-Fibonacci spirals. Tomorrow we will discuss what this means and how this investigation helps us better understand the nature of the world around us!”