The Fibonacci Sequence can be explained and explored with various approaches using a number of physical contexts. The models we have built offer a multifaceted approach by giving you options. Some of these will be very familiar and others less well known

We have available:

1) Just a plain presentation of the numbers. (whiteboard)
2) Liber Aci, Fibonacci’s Original Book (we can send you one).
3) Wooden pieces etched with pairs of Adult and Baby Bunny’s and a “Field” to model the Rabbit Problem
4) Wooden Spiral Puzzle that builds from a 1×1 inch square. Inches are noted with etched lines.
5) Wood Etched with a Fibonacci Tree and enhanced with hooks
6) Reversible Bunny’s and Bees that can hang on the tree.
7) Doormat Roman Numeral Stepping Stones to play the stepping stone game

1) Take a moment to look at the sequence and ask how the numbers build upon one another.
1 1 2 3 5 8 13 21 34 55 89….

Observe the sequence, make conjectures on the pattern. As an optional challenge try to generalize the sequence: 1, x, 1 + x, 1 + 2x, 2 + 3x, 3 + 5x, 5 + 8x, 8 + 13x and so on. Make observations here as well. In conjunction with the stepping stones (e) or separately you can compare this sequence to a geometric sequence(see sheet). This comparison reveals a connection to the Golden Ratio.

2) Challenge visitors to read the original text.
Liber Aci
3) Or model the problem using counters, chits or paper and pencil. You can offer them something like this velvet field and pairs of baby and adult Rabbits.

Fibonacci Rabbits

Once the Fibonacci rule is stated we can offer opportunities to see that rule in action.

4) Grow a Shell. Imagine you are a sea creature that grows its own home in the form of a shell. You start out very small in a space represented by this 1 x 1 square. As you grow, you keep your existing space and extend the walls on one side to double your living area to include another 1×1. In your next phase of growth you use your longest side which is now length 2 to construct a new chamber that itself is 2×2. Building on this growth as you spiral around, your next chamber is of length 3 and total area 9 and so on. This sheet models the building of the spiral and asks the student to find the sizes based on the first block and the pattern. It goes with the puzzle and asks students to consider the ratios of increasing Fibonacci numbers which approach phi, the Golden Ratio which is an irrational number: 1/2 + sqrt(5)/2.
Fibonacci Spiral
Students may also want to compare the blocks in creative ways.
Fibonacci Diagonal Puzzle

5) Grow a Tree? Now imagine you are a tree that grows by a rule. Would you look natural, or would you look too balanced, too regular and therefore artificial? Start with a stem growing out of the ground. Like the baby rabbits, it has to mature into a trunk before it can split. After one time cycle the trunk sprouts a branch. That branch needs to grow and get sturdier for one cycle till it too can be like a trunk. The trunks split every growth cycle and the branches have to grow one cycle before they too can split.

Take a look at this tree and decide if it follows the Fibonacci Rabbit Rule. Count the number of branches at each time cycle (horizontal line). Also consider whether it looks natural to you and whether you could find this kind of growth in a tree. (One fun activity is to have students draw a tree first – just branches with no other instructions. Then “build” this tree and compare them.)
Empty Fibonacci Tree

6) Compare the Genealogical Trees of Humans and Bees. This is an interesting phenomenon that can validate this notion of the tree analogy and also provide a context for thinking about the Fibonacci Tree as one of several kinds of trees, and the Fibonacci sequence as just one of many kinds of sequences. This is a sheet that will guide this activity.

*You can put several models together and combine the Bunny Problem together with the tree model and the Bee genetics.

Tree with Bee Ancestry

7) Another way to consider the Fibonacci Sequence is through the stepping stone game. We can use a small picture or do a whole body experience by using the “stones” we designed out of doormats. Would love to have real stones of course. The activity sheet linked here explains the rules of the game and in following those rules the players can see the Fibonacci numbers emerge.

Roman Numeral Stones