In this unit we look at Fibonacci numbers and how they occur in nature. We also look at what factors Fibonacci numbers can have.

- understand the concept of Fibonacci numbers and how they are generated
- find factors of a number
- make conjectures and attempt to prove them
- find generalisations

This unit provides an initial exploration of Fibonacci numbers. These numbers are a specific example of sequences that arise from simple recurrence relations that are useful in theory and practice. Their applications usually come as a way to count various situations.

Another example of numbers satisfying a simple recurrence relation is to be found in What Happens on Average? Algebra, Level 5. Such recurrence relations occur often.

In a later unit, Fibonacci II, Level 6, we look more deeply into the Fibonacci numbers to see what other relations hold between terms; what the ratio of two consecutive terms approaches; and what the formula for the n-th Fibonacci number is.

For background reading we suggest the following useful web sites:

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/

http://turnbull.mcs.st-and.ac.uk/~history/HistTopics/

www.mcs.surrey.ac.uk/Personal/R.Knott/

www.mathforum.com/dr.math/

(The St Andrews site (first two links) can be used for general history and the lives and works of a variety of mathematicians.)

The following books may also be useful

Hilton, Pedersen and Holton, Mathematical Reflections, Springer Verlag, New York, 1997

Hilton, Pedersen and Holton, Mathematical Vistas, Springer Verlag, New York, 2002

Copymaster 1B

Fibonacci numbers, conjectures, generalizations, factors, recurrence, sequential patterns, proofs, divisible

#### Session 1

*In this session we look at the classical Fibonacci problem of the rabbits to introduce the Fibonacci sequence of numbers. The students then look at two situations that produce the same sequence and a sequence that is very similar to the Fibonacci sequence.*

#### Teachers’ Notes

The Fibonacci sequence is the sequence of numbers that starts off with 1 and 1, and then after that every new number is found by adding the two previous numbers. So if we start with 1, and have 1 next, then the third number is 1 + 1 = 2, the fourth number is 1 + 2 = 3, the fifth number is 2 + 3 = 5, and so on. The first 20 Fibonacci numbers are listed below

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

233, 377, 610, 987, 1597, 2584, 4181, 6765

In this unit we will use F(n) to represent the n-th Fibonacci number. This means that F(1) = 1, F(2) = 1, F(3) = 2; F(4) = 3 and F(5) = 8.

Using the F notation, the relation that ‘every new number is found by adding the two previous numbers’ is expressed by the recurrence relation

F(n + 2) = F(n + 1) + F(n).

This sequence is named after Fibonacci because he used it in his book Liber Abaci (1202) to show how much easier it was to compute with the ‘new’ Arabic number system compared to the Roman system that was then commonly used in the Western World. (For more on this see John Stillwell’s article in Newsletter No. 26).

The problem that he posed went something like this. Suppose that rabbits mate when they are one month old and when a female gets to be two months old she produces a pair of baby rabbits, one male and one female. If we start off with a new-born male and female rabbit, how many pairs of rabbits will there be in after 12 months? How many will there be after n months?

This first session looks at the Fibonacci numbers and the recurrence relation between them.

The two Copymasters 1A and 1B give slightly more realistic scenarios with bees. Here the different generations of the drone’s family tree follows Fibonacci with D(n) = F(n) and the different generations of the queen’s family tree has Q(n) = F(n + 1).

### Teaching Sequence

Tell the students the story of the rabbits and with their help produce several generations on the board.

Let the class assist you in drawing up the numbers of pairs of rabbits in each generation.

Ask them what patterns that they can see in these numbers. (Look in particular for the fact that the next number is the sum of the two previous numbers.)

Tell them that we call these numbers the Fibonacci numbers and define F(n) to be the n-th Fibonacci number. Check that they are up with the play by asking them to tell you what F(3), F(5), F(6), etc., are. Also ask them if they can find the values of n in the equations F(n) = 8; F(n) = 144; F(n) = 377; etc.

You might want to discuss whether this is a very good model of population growth or not. (Would the world be inundated with rabbits by now?)

Let the students go into their groups and work on Copymaster 1A or 1B. (This is an exercise that parallels the rabbit problem.)

Get the students to discuss the results that they found. (Here we are looking for the fact that D(n) = F(n) and Q(n) is nearly F(n) with Q(n) = F(n + 1).)

Let them write down their conclusions in their note books.

#### Session 2

*This is a data gathering session by the students. It will need to be sited in the school computer centre and/or the library. In groups students will gather information about Fibonacci and places where the Fibonacci numbers appear in nature.*

#### Teacher’s Notes

This session allows student the opportunity to gather information themselves on Fibonacci and on the occurrence of Fibonacci numbers in nature.

For this session, we suggest that you divide the class up into groups of 2 to 4. This number is likely to be dictated by the number of computers that you have available and whether you want to rely solely on the computer information or whether you plan to use the school library.

The kind of information that they should be looking for is listed below.

Fibonacci: What does the name mean?

What other names did he have?

When did he live?

What was the Liber Abaci?

How was it printed?

Why was it written?

What relevant pictures are there?

What else did you find out about Fibonacci or his family?

Nature: Where in nature can you find Fibonacci numbers appearing?

Are there any other occurrences like the rabbits or the bees?

What occurrences are related to flowers?

What relevant pictures can you find?

Where do spirals come into this?

Are any special names given to these spirals?

Can you explain what is going on in these situations?

What else have you discovered about this topic?

It is up to you to decide how much direction that you give the students. For instance, you may decide to give them a list of questions to answer or you may let them decide for themselves what the important issues in each mini-project are. You may also like to specify certain sites or books that they can use to obtain the information or you may just let them undertake the searches using their own initiative. (You might also like to written report.)

For your information, relevant websites are listed above. For more websites, do a search on Fibonacci or Fibonacci Numbers or Fibonacci Sequences.

### Teaching Sequence

Introduce the purpose of today’s session and that is to find out more about Fibonacci and where the Fibonacci numbers occur in nature.

Divide the class into groups and assign the topics.

Tell them that at the end of the session they have to have some pictures and a set of bullet points indicating the main things that they have found.

Let them work together on the computers (probably have at most two students per machine). Walk round and provide help and encouragement as needed.

Ensure that the groups each have at least 6 bullet points and one picture at the end of the session.

#### Session 3

*In this session, the class will discuss the notes that the groups produced in Session 2.*

#### Teaching Notes

This session can be handled in a couple of ways. The whole class could be given a copy of each group’s notes from the last session and be asked to produce a combined set of notes. This set might then be discussed in class. Or the approach of the Teaching Sequence could be taken. Which ever route is taken, the end result should be a combined set of notes plus pictures that the class understands and can explain.

### Teaching Sequence

Remind the students of what happened in the last session. Make sure that the students can see a copy of their group’s work from that session. (This may require the original copy to be photocopied.)

Have different groups talk about what they have done and add their notes to a white board collation of the class’ work. (Take the two group topics separately.)

When it comes to the counting of spirals in a flower or pineapple, the students should be able to explain exactly what is going on.

Make the combined notes into two wall posters with suitable decorations and display them on the classroom walls.

#### Session 4

*We now look at some factors of Fibonacci numbers and see what patterns are produced. *

#### Teachers’ Notes

In the unit Guzzinta, Level 6 we look at simple ways to tell if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, or 11. It might be useful (but not completely necessary) for the class to have looked at these ideas before they tackle this session.

The basic idea here is that only certain of the Fibonacci numbers are divisible by 2 but that we are able to tell exactly which numbers these are. Let’s have a look at the first 20 Fibonacci numbers to see what we can see. Below the even Fibonacci numbers are shown in bold.

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

233, 377, **610**, 987, 1597, **2584**, 4181, 6765

The Fibonacci numbers seem to go odd, odd, even, odd, odd, even, odd, … So if n is a multiple of 3 the number is even and if it is not a multiple of 3 it is odd. We

**Conjecture 1**: F(n) is even if and only if n = 3k.

Let’s work with e for even and o for odd. The Fibonacci sequence starts off with o, o. The next number is the sum of two odds and so is even e. OK so so far we have o, o, e.

But then the next number is o + e = o to give o, o, e, o, so far.

But then the next number is e + o = o to give o, o, e, o, o, so far.

But then the next number is o + o = e to give o, o, e, o, o, e, so far.

But now we are going around the argument again. Once we get to o, o we have to have e. So the pattern o, o, e repeats indefinitely. So every third Fibonacci is even and the rest are odd. Good Conjecture!

Flushed with that success maybe we should look at numbers that are divisible by 3. We have gone through the first 20 and written the numbers divisible by 3 in bold.

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

233, 377, 610, **987**, 1597, 2584, 4181, **6765**

It’s at this point, when finding the bold numbers, that the Guzinta unit may come in handy and save some time. Clearly we have the

**Conjecture 2**: F(n) is divisible by 3 if and only if n = 4k.

(Note that if this is correct, then we may have a bigger conjecture that covers both Conjecture 1 and 2.)

We can prove this in the same way that we did Conjecture 1, but we need a slight variation. If a number is divisible by 3 we’ll replace it by 0; if has a remainder of 1 after dividing by 3 we’ll replace it by 1; if has a remainder of 2 after dividing by 3 we’ll replace it by 2. This means that the first 20 Fibonacci numbers look like

1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0,

2, 2, 1, 0, 1, 1, 2, 0

In Conjecture 1 we found the pattern o, o, e repeated. Here it looks as if the pattern we want is 1, 1, 2, 0, 2, 2, 1, 0. Just by adding together consecutive terms and throwing away multiples of 3, we can show that this pattern holds. Clearly the fourth and eighth terms are the only ones that are divisible by 3 and so we have another good conjecture.

How far can we go with this? Is it true that

**Conjecture 3**: F(n) is divisible by d if and only if n = (d + 1)k?

### Teaching Sequence

Remind the students of the tests for even numbers and numbers that are divisible by 3. Let them tell you what happens if you add two even numbers, two odd numbers and an even and an odd number.

Get the students to list the first 20 Fibonacci numbers.

*Which of these numbers are even?**Is there any pattern here?*

Can you express this pattern using the F notation? Can you phrase this in terms of a conjecture? (Try to steer them to Conjecture 1 above.)

How do we work with conjectures? (Find a proof or counter example.)In their groups give them a chance to look at the following questions.

*Are you happy with the conjecture in F notation? Does it work for all 20 numbers we’ve listed above?**Can you show that it is true for all Fibonacci numbers? Is it at least true for the first 30 Fibonacci numbers?*Get the groups to report back by saying that they support (i), (ii) or (iii). (Where we have (i) the conjecture is false; (ii) it seems to be true for the first 30 numbers; or (iii) that it is true.) Don’t ask them yet to say why they support (i), (ii) or (iii) but take a vote to see what the predominant view is.

In group discussion see if anyone has a counter example to the Conjecture; check that the first 30 Fibonacci numbers satisfy the conjecture.

*Is there a neat way to check the next 10 numbers? Do you have to actually find out the numbers or is there a quicker way to see if they are even or odd? (If necessary lead them to the o and e approach.)*Lead the class, if necessary to a proof of the conjecture. Give them time to write up a version of the conjecture and its proof.

Repeat with Fibonacci numbers that are divisible by 3. Pose the question in a whole class setting but let them work on it in groups and report back later.

Let the groups report back and justify their results. Give all of the students a chance to take part in the discussion.

Allow the students the time to write up their conclusions. Check what they have written.

#### Session 5

*In this session we see if the results of the last session can be generalised. This is done by looking at other factors of Fibonacci numbers.*

#### Teacher’s Notes

Clearly it is starting to look as if we can

**Conjecture 3**: F(n) is divisible by d if and only if n = (d + 1)k.

Can we prove that?

It’s hard to work with d because it’s a general number so we’ll look at a specific number. Since we know what happens for d = 2 and 3, we’ll try the next number, 4. Following the approach in Teacher’s Notes to Session 2, we’ll go through the first 20 Fibonacci numbers and write in bold the ones that are divisible by 4.

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

233, 377, 610, 987, 1597, **2584**, 4181, 6765

Now that’s a nuisance. If d = 4, Conjecture 3 becomes

**Conjecture 3´**: F(n) is divisible by 4 if and only if n = 5k.

But the bold numbers occur every sixth number! And if we replace the numbers by their remainder after dividing by 4 the first 20 numbers look like

1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0,

1, 1, 2, 3, 1, 0, 1, 1

Because the sequence 1, 1, 2, 3, 1, 0, repeats, we can prove that

**F(n) is divisible by 4 if and only if n = 6k.**

But that is not what we were trying to prove.

Conjecture 3 is certainly false but is that because it only works for 2 and 3 or because 4 is an awkward number?

What happens with 5, 6, 8, 9, 10? The old argument can be used to show that

F(n) is divisible by 5 if and only if n = 5k.

F(n) is divisible by 6 if and only if n = 12k.

F(n) is divisible by 8 if and only if n = 6k.

F(n) is divisible by 9 if and only if n = 12k.

F(n) is divisible by 10 if and only if n = 15k.

Can we conclude from this that

**Conjecture 4**: F(n) is divisible by d if and only if n = rk for some r depending on k.

It certainly looks as if, given any factor we like, there is a Fibonacci number that is divisible by that number. What’s more, there are an infinite number of Fibonacci numbers that are divisible by that factor and that they are all equally spaced along the Fibonacci sequence.

### Teaching Sequence

Remind the students of the test for divisibility by 4, 5, 6, 8, 9, and 10.

Recall the results that we have for d = 2, 3.

I*s there a more general pattern here? (Conjecture 3.)*

How could we tackle a question like this?At this point you might like to think about how much time you want to spend on this question. You could either send them into their groups to work on just the case d = 4 or you could divide the numbers 4, 5, 6, 8, 9, 10 up among the groups so that the various cases are covered more quickly.

Walk round the groups helping where necessary and asking searching questions.

Let the various groups report on their work. Give as many people as possible a chance to suggest a result and/or explain why their argument is correct. If you have only covered the case d = 4 you will need to go back to step 4 and let the groups work with d = 6, etc.

Give the students a chance to write up the results that have been obtained along with an outline of the proofs that have been found.

Hold a whole class session to consider these results and to see if there may be a more general result.

#### Dear Parents and Whanau,

This week we have been looking at Fibonacci numbers. Ask your child to tell you what Fibonacci numbers are.

We have produced a poster about these numbers. Your child can show you the poster and explain how these numbers occur in nature. Together you might like to further research Fibonacci numbers. They are fascinating. You may be surprised by what you discover together.