Fibonacci Trees

For two weeks in a row, in our joint 1st-5th grade math classes, a certain famous sequence made its appearance.  The activities were seemingly very different: in the first one we were climbing stairs one or two steps at a time and in the second one we were growing magical trees according to certain rules.

I was very happy when the sequence started showing up for the second time and the students immediately exclaimed, “It’s the same sequence as last week!”

This post will focus on the second of the two activities: drawing Fibonacci trees.  Here are the rules:

1. Start with one branch.
2. Branches never die and every year a branch grows by the same amount.
3. The first year of its existence, a branch is green and has no offspring.
4. After the first year, a branch becomes brown and sprouts one offspring branch per year.

We used lined paper (about an inch between consecutive lines) to keep track of the years.  The students also had to record the number of branches they had after each year.  The sequence that emerged was 1, 1, 2, 3, 5, 8, 13, 21, 34…  As I mentioned, this was not their first encounter with the Fibonacci sequence and the students recognized it after about the first five elements.

Here are some pictures of the students working.

 

Some of the younger students had a bit of trouble at first keeping track of which branches were supposed to be green and which ones were supposed to have offspring.  However, by the end, just about everyone had at least 7-8 correct levels/years.

Some students insisted on continuing while they could still fit the branches on the paper.  Keeping track of the branches beyond the year with 34 proved to be quite tricky.  However, several students made it as far as 89!

While some students focused on drawing as many levels of branches as possible, others tried to make theirs as aesthetically pleasing as they could (although the rule was that they had to at least get to the year with 21 branches before decorating).

Here are some of the beautiful mathematical works of art that the students created!