## Fibonacci: Surprise and Pattern in Mathematics

Posted by mark schwartz on June 18, 2016

__Introduction.__

Teaching math involves a lot of things, first of which is “do I have their attention?” and secondly, “how do I keep their attention?” It took me a while to realize that these were the wrong questions. The real questions are “does math have their attention and does it keep their attention?” There is a difference. We’ve all developed presentation modes, perhaps several of them, tailored to the content and the class … we do things that feel comfortable but also things that seem to keep students’ attention. Let me show you one that I’ve used in a developmental class that gets conversation going beyond the actual exercise and leads to two things important to math that I like to emphasize as the semester continues: surprise and pattern.

__The Story.__

An interesting way to demonstrate surprise and pattern is the use the Fibonacci series. Don’t teach it, just use it. In fact, in some instances it’s never mentioned but in most instances, students want to know how the exercise came out the way it did.

The first thing is to have students work in groups of two or three. This group behavior engages each student in the process of deciding who is going to do what part of the activity, and as they work together they realize that working together is actually a fun math experience. I typically randomly assign students to groups in an effort to get them to work with someone new and this involves their moving around to sit with their new partner. This activity alone is typically something that they’ve never experienced in a math class. I give them time to introduce themselves to each other.

Once all the hubbub of moving and introductions is over, I ask them to get out their calculators or phones that have calculators because they’re going to need them for this exercise.

I then ask them to designate a writer and a “calculator” and ask that the writer get out a piece of paper and make a list down the page from 1 to 25. When everyone is ready, I then ask each group, one group at a time, to give me two numbers from 1 to 9, inclusive. Sometimes the word “inclusive” needs a little explanation. The writer is then to put their pair of numbers as the first and second number in the list. If the question is asked about the order of the numbers, tell them that it’s their choice. I then write their choice on the board. I do this because I want each group to have its’ unique set of two numbers and also if they see what’s been selected it will help avoid duplicates. Once every group has done it, I ask them if they’re ready to go on.

The instruction is to add the first and second numbers to get the third number. Then take the 2^{nd} and 3^{rd} numbers to get the fourth and continue this way until they’ve filled in their list to 25 numbers. I take one of the pairs from the board and show them how it works. I again ask if there are any questions before they continue. I point out that as they move down the list the numbers may get big and the job of the calculator is to get the numbers correct, so work slowly. As an aside, this sometimes leads to the group members deciding that they will all do the calculations to check on each other and make sure they’re right. If no more questions, get to work. I roam the room and watch them work and comment occasionally.

After all the groups have their list of 25 numbers, I ask them to slowly and carefully do these things: divide the 25^{th} number by the 24^{th} number, ignore the decimal point, and write down only the first 4 digits from the left (sometimes I have to show them what this means).

After they’re all done, I announce that I will now tell each group what their 4 numbers are. Using the list of their pairs on the board, I start with announcing 1618 for the first two groups and then I quit and announce that they all have the same number.

This is the first surprise. They typically verify with each other that this is true. Usually someone will ask how that happened and this begins the conversation; how can each group start with two different random selected numbers and yet come out with the same number? Enjoy the conversation and let them roam around for a while; they might hit on what happened. If they ask you for the answer, don’t give it but you can provide a hint that the answer is hidden in the process; can they see any patterns?

This first day exercise lays the foundation for students to realize that a lot of the math they’ll be doing will have surprise and pattern and that as we go through the semester, I’ll be referring to this a lot (which I do; after all, rules, formulae and algorithms are “frozen” patterns, aren’t they)?

And finally, when the class is over and they’re gathering their stuff to leave, there are animated conversations about what they just did and that’s very satisfactory to me. You might want to experience it too.

## Stella Schneider said

Mark, I love this post. Of course I use the fibonacci series frequently in making patterns for weaving.

Another thought’: Your type size could be larger to accommodate old eyes.

Thanks,

Stella

## mark schwartz said

hi stella – thanks for the comment … the Fibonacci series and the golden ratio associated with it are two of my favorite mathy things … when you get a moment you might want to look at the “Fibonacci: Surprise and Pattern in Mathematics” posting … I’ll see if I can make the type size bigger … the wordpress web has some constraints on things, but I’ll give it a try … mark