Download Great Lesson Idea - Fibonacci Numbers

Great Lesson Ideas - Secondary Maths
Fibonacci numbers
Teachers pack
Asnat Doza, Comberton Village College
This lesson is about Fibonacci numbers,
but my year 7 pupils don’t know this to
start.
I introduce the lesson as a challenge, for
them to work in groups through a set of
mysterious puzzles in envelopes I’ve put
on their desks.
Each numbered envelope contains one puzzle, and pupils must do them in
order. Each puzzle has a clue, but the group must agree that they’re stuck
before they can look at it. When they’ve completed a puzzle, they check
with me before moving on to the next.
I build up the challenge by explaining that, if any group completes all five
puzzles, they’ll get a special sixth envelope with a problem that’s never
been solved. It’s a version of the ‘Hailstone Problem’, which is so
deceptively simple that even some of my year 8 pupils have a go at it, even
though it’s has yet to be solved!
[NB. Not all this introduction is in the video]
This pack has a pupil guide and sheets and clues for each of the puzzles,
which need to be printed, laminated and copied.
1. Fibonacci numbers - this sheet has a series of Fibonacci numbers which
need to be cut up before putting them in the first envelope. There are no
instructions, so pupils need to work out what’s going on and recognise the
pattern and explain it. The background picture might help some pupils
construct the jigsaw. I explain that these are called Fibonacci numbers and
they are very special, while not giving away anything in later puzzles.
2. Fibonacci squares - has a series of squares which need to be cut up before
putting them in the envelope, or you can give pupils a photocopy of the
random squares to cut up for themselves. Pupils should recognise the
pattern, build up the shape and explain what’s going on. There’s also a
spiral to help explore how this relates to the sequence of squares. Don’t
provide the complete or combined image in the envelope - you decide when
to show it or demonstrate how the separate images overlay. You could
revisit this later to discuss Fibonacci numbers in nature.
3. Fibonacci rabbits - based on the problem Fibonacci was asked to solve.
4. Pascal triangle - for pupils to recognise various number sequences.
5. The Golden ratio and phi - for more advanced pupils to work out the
connection between phi and Fibonacci numbers.
6. The Hailstone Problem - for those who make it that far!
There’s a website about every aspect of Fibonacci numbers at
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/
- and more about the ‘Hailstone Problem’ at
http://blog.functionalfun.net/2008/07/project-euler-problem-14-hailstone.html
You will have:
6 mathematical puzzles - one in each envelope
The envelopes are numbered.
You have 5 envelopes are on your desk - one is missing!
In each envelope you will find:
1 x master colour copy of the question
4 x black and white copies to write on if you need to.
What to do:
Work through each envelope in order and solve each puzzle.
When you’ve finished one puzzle, check with a teacher
before going on to the next.
You also have:
Clues for each puzzle.
Only open a clue when the whole group has run out of ideas!
Pack away each puzzle when you have finished with it.
Remember - talk amongst yourselves
BUT ONLY ABOUT THE MATHS!
Puzzle 1
1
1
2
3
5
8
13 21 34
55 89 144
Puzzle 2:
1. What do these squares have to do with Fibonacci
numbers?
2. Starting from one of the small squares, can you put
them together so that every time you add a square you
can create a rectangle?
3. How do these squares help you draw a spiral?
Puzzle 3
Fibonacci rabbits
A pair of baby rabbits are put in an enclosed garden.
Each pair of rabbits produces a new pair of rabbits
every month.
From the second month on, these rabbits also become
productive.
How many pairs of rabbits will there be in the garden
after one year?
Continue the diagram and find out the number of pairs
of rabbits in the Fibonacci garden after one year.
Puzzle 4
Pascal Triangle
1. Complete the next 2 rows of this Pascal triangle.
2. Write down as many sequences as you can find in the
triangle.
3. Can you find a Fibonacci sequence in the Pascal
triangle?
Puzzle 5
Phi
The Golden Number
1.61803398874989...
Φ = 1.618033988749894842.....
The Golden Ratio is a special number that goes on
forever, without repeating, like Pi (π),
It is often written as the Greek letter Phi (Φ)

Use a calculator to find a link between the Fibonacci
sequence and Phi.
Explain what you found
Clue to puzzle 1:
Put the numbered cards in order - the picture might help.
Look at the sequence you get.
Can you work out the rule that links the numbers?
_________________________________________
Clue to puzzle 2:
Look at the side length of each square.
Can you work out the rule that links them?
How does it compare with the number sequences in puzzle 1?
_________________________________________
Clue to puzzle 3:
Consider the number of pairs of rabbits each month.
How many are there at the beginning?
How many after a month?
After 2 months?
After 3 months?
How do the pairs link to Fibonacci?
_________________________________________
Clue to puzzle 4:
Consider drawing lines across the hive.
Is there a link between each number
and the ones above it?
_________________________________________
Clue to puzzle 5:
Use your calculator to divide each Fibonacci term by the previous one:
1 2 3 5 8
, , , , , and so on…. How does this relate to the value of phi?
1 1 2 3 5
Puzzle 6
Mathematical mysteries: Hailstone sequences
This problem is easy to describe but it is one of mathematics'
unsolved problems.
Starting with any positive integer n, form a sequence in the following
way:
If n is even, divide it by 2 to give n' = n/2.
If n is odd, multiply it by 3 and add 1 to give n' = 3n + 1.
Then take n' as the new starting number and repeat the process. For
example:
n = 5 gives the sequence 5, 16, 8, 4, 2, 1, 4, 2, 1,...
n = 11 gives the sequence
11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...
These are sometimes called "Hailstone sequences" because they go up
and down just like a hailstone in a cloud before crashing to Earth −
the endless cycle 4, 2, 1, 4, 2, 1.
It seems from experiment that such a sequence will always eventually
end in this repeating cycle 4, 2, 1, 4, 2, 1,... and so on.
But some values for N generate many values before the repeating
cycle begins. For example, try starting with n = 27. See if you can
find starting values that generate even longer sequences.
The unsolved problem is to prove that every starting value will
generate a sequence that eventually settles to 4, 2, 1, 4, 2, 1,...?
Could there be a sequence that never settles down to a repeating
cycle at all?
Good luck!