# Fibonacci fun

As environmental inquiry is a large focus of the summer numeracy program, it seemed logical to spend some time working on number patterns- more specifically, identifying number patterns in nature. **When thinking about math and nature, the Fibonacci sequence immediately comes to mind**. In case you’re not familiar with it, the Fibonacci sequence is as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Quite simply, it is a **series of numbers where each number is the sum of the two numbers before it**. For example, the 3 (5th term) is found by adding 1+2. The 5 (6th term) is found by adding 2+3, and so on. The TED Talk below is a great summary of the importance of number patterns in mathematics and highlights some of the wonders of Fibonacci’s sequence (which is FUN and BEAUTIFUL).

Mathematics is not just solving for x, it’s also figuring out WHY. (Arthur Benjamin)

I had previously seen a video of a lesson that centred on the Fibonacci numbers, so I revisited the lesson (here) and was inspired to do something similar with our Mathletes. We carried out a series of activities that formed a week of Fibonacci fun, and some rich learning experiences for our Mathletes (with a little bit of added mystery along the way!).

**Activity #1: Fibonacci number sequence**

The students worked in pairs for the first activity. Each pair was given an envelope (see picture), but the contents were left a mystery until they opened them up. Inside, they found cue cards with numbers written on them (Fibonacci sequence numbers up to 144). Since they are so accustomed to math centres, some of our Mathletes thought it was an adding game at first (not a bad idea!). Most had the instinct to order them from smallest to largest. Some pairs required extra guidance to get on the right track, and once they had them in order they were instructed to start looking for a pattern. With some prompting for the younger ones, they discovered the pattern rule for the sequence of numbers, and tested it to make sure it applied to the entire set of numbers they were given. Once they were confident in the pattern rule, they could come to me for envelope #2…

**Activity #2: Square tiles**

For the second activity, the students worked in the same pairs as activity #1. They were given a second envelope that contained square tissue paper tiles in five different colours, and were instructed to see how this could relate to the pattern they just discovered. There were a different number of tiles depending on the colour, but this was not readily apparent for some students. They required prompting to grasp that they had to make different sized squares, and *with some guidance they soon discovered that the side length of the squares corresponded with the Fibonacci numbers!* For most pairs, they got as far as making squares with the different side lengths (e.g. 1 x1, 1×1, 2×2, 3×3, 5×5). As the various teaching hands were circulating throughout the room, we were able to guide some pairs to assemble their squares into a rectangle.

**Extension**

As some of our older students were finished their squares early on (and even assembled them into a rectangle quite quickly), I challenged them with the following:

Based on the pattern rule that you discovered, what would be the next few numbers in the sequence? How do you know?

**Consolidation**

We briefly discussed our Fibonacci discovery experience as a whole group, and we discussed the different strategies that the students used to find the pattern rule and create squares of different sizes. As we were continuing with Fibonacci the next day, we left it at that and praised our Mathletes for their excellent inquiry skills!

**Activity #3: Spiral **

The following day, we started off the lesson by watching a video created by Jo Boaler and her team at YouCubed. This is an *excellent video about patterns, the Fibonacci sequence, and where we see patterns in nature* (it can be found in the YouCubed week of iMath Day 4 section). The students were very engaged by the examples given in the video (*many ooo’s and ahhh’s*), and it served as a good consolidation of the activities we had already completed, as well as leading in nicely to our construction of a spiral of squares. I created an exemplar for the students to refer to, and we provided each student with:

- Large construction paper
- Glue
- Assorted pre-cut tissue paper square tiles

We encouraged the students to first pick out the appropriate number of tiles in different colours for each square, and then arrange them on their page before applying glue. Some of our younger Mathletes required a little guidance with creating the pattern, but *every student was on task and excited to create their own tribute to Fibonacci!* The results were impressive, and we sent our happy campers home with their very own representation of the Fibonacci sequence.

**Activity #4: Patterning centres and beading**

On our final Fibonacci day, we decided to host a celebration of patterns! We kicked the day off with a variety of patterning math centres (see pictures). One of these centres gave the students a chance to explore pinecones and their associated patterns. Thanks to a gracious donation of a large number of pinecones, our Mathletes were able to work as detectives looking at small, medium and large pinecones. I guided some students in** ***counting the spirals in a clockwise and counter-clockwise direction, trying to find evidence of the Fibonacci sequence*. Many were excited for the chance to just hold and manipulate the different types of pinecones, and we were all amazed by the patterns found within!

As our last patterning activity, the students were instructed to create a beaded bracelet that represented the Fibonacci series of numbers. This was the capstone to our week of patterning, and the students were equipped with:

- Plastic beads of different colours
- White cotton string
- Scissors

The results were beautiful, and many of our Mathletes knew the first several numbers of the Fibonacci sequence by heart after creating colour patterns according to these numbers.

Another week of math camp comes to a close, and with it come a few last thoughts about our Fibonacci activities…

Things I would do differently:

- Use a
**sturdier material for square tiles**( I used tissue paper because we had some pre-cut, but it was prone to being accidentally blown around and thus tricky to work with). - Make
**extension activity clear**from the get-go. - Give more
**guidance for Activity #2**(i.e. tell students they need to use tiles to make squares that have side lengths related to Fibonacci sequence)

Things that worked well:

- Working in
**pairs or small groups (3)**for Activities #1 and 2. - Preparing Fibonacci numbers on
**separate number cards****for Activity #1**so students could manipulate and move them around. - We had a couple visiting teachers in the room on the day that we carried out the activities with the envelopes, so there were
**many teaching hands**available to guide groups and prompt as needed.

The curious and inquisitive nature of our Mathletes continues to astound me, and their willingness to think outside the box is motivating me to bring my A-game for the remaining week of summer numeracy camp. I can’t wait to see what our final week has in store for us!

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