Fibonacci Numbers

St Paul’s Staircase You can walk up steps by going up one at a time or two at a time. If you have three steps there are 3 ways you can walk up them:

St Paul’s Staircase If there were 4 steps there are 5 ways. Here are just two: Can you find the other 3 ways?????

St Paul’s Staircase Investigate how many ways there are to step up staircases with 1,2, 5 and 6 steps. Record your results in a table. Can you spot a pattern?

Square Spiral 29 little boxes down 1 little square Side 1 1 little square 15 little boxes across

29 little boxes down 15 little boxes across 1 more little square Box Side 1 2 1 more little square 15 little boxes across

29 little boxes down 2 x 2 square 15 little boxes across Box Side 1 2 3 2 x 2 square 15 little boxes across

Box Side 1 2 3 4 3 x 3 square

Now complete the pattern to fill your page as much as possible. Box Side 1 2 3 4 3 x 3 square Now complete the pattern to fill your page as much as possible.

Box Side 1 2 3 4

Box Side 1 2 3 4

Box Side 1 2 3 4

Box Side 1 2 3 4

Draw your own “Nautilus” shell by drawing quarter circles in each square and joining to form the spiral

The Fibonacci Numbers }+ 1 1 2 The number pattern that you have been using is known as the Fibonacci sequence. }+ 1 1 2

The Fibonacci Numbers }+ 1 1 2 3 The number pattern that you have been using is known as the Fibonacci sequence. }+ 1 1 2 3

The Fibonacci Numbers }+ 1 1 2 3 5 8 13 21 34 55 The number pattern that you have been using is known as the Fibonacci sequence. }+ 1 1 2 3 5 8 13 21 34 55 These numbers can be seen in many natural situations

Fibonacci’s sequence… in nature 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

Fibonacci’s sequence… in nature Take a look at a cauliflower. Count the number of florets in the spirals. The number in one direction and in the other will be Fibonacci numbers. If you look closer at a single floret, it is a mini cauliflower with its own little florets all arranged in spirals around a centre. If you can, count the spirals in both directions. How many are there? Now look at the stem. Where the florets are rather like a pinecone or pineapple. The florets were arranged in spirals up the stem. Counting them again shows the Fibonacci numbers. Try the same thing for broccoli. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

Fibonacci’s sequence… in nature On many plants, the number of petals is a Fibonacci number: Buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals. 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family. Some species are very precise about the number of petals they have - eg buttercups, but others have petals that are very near those above, with the average being a Fibonacci number. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

Fibonacci’s sequence… in nature One plant in particular shows the Fibonacci numbers in the number of "growing points" that it has. Suppose that when a plant puts out a new shoot, that shoot has to grow two months before it is strong enough to support branching. If it branches every month after that at the growing point, we get the picture shown here. A plant that grows very much like this is the "sneezewort“. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

OK, OK… Let’s talk rabbits… Fibonacci… and his rabbits Leonardo Pisano Fibonacci is best remembered for his problem about rabbits. The answer – the Fibonacci sequence -- appears naturally throughout nature. But his most important contribution to maths was to bring to Europe the number system we still use today. In 1202 he published his Liber Abaci which introduced Europeans to the numbers first developed in India by the Hindus and then used by the Arabic mathematicians… the decimal numbers. We still use them today. OK, OK… Let’s talk rabbits…

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month. So at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die. And the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that I posed was... How many pairs will there be in one year?

Pairs 1 pair At the end of the first month there is still only one pair

Pairs 1 pair 1 pair 2 pairs End first month… only one pair At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits 2 pairs

Pairs 1 pair 1 pair 2 pairs 3 pairs End first month… only one pair End second month… 2 pairs of rabbits 2 pairs At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. 3 pairs

Pairs 1 pair 1 pair 2 pairs 3 pairs 5 pairs End first month… only one pair 1 pair End second month… 2 pairs of rabbits 2 pairs End third month… 3 pairs 3 pairs 5 pairs At the end of the fourth month, the first pair produces yet another new pair, and the female born two months ago produces her first pair of rabbits also, making 5 pairs.

1 1 2 3 5 8 13 21 34 55

Fibonacci in Music The intervals between keys on a piano are Fibonacci numbers. 5 3 2 8 white 13 w & b

Fibonacci in Nature The lengths of bones in a hand are Fibonacci numbers.

Fibonacci in Nature Spirals, like the one you drew earlier are common in nature.

Patterns in Fibonacci 1+1 = 2 1+2 3 2+3 5 3+5 8 5+8 13 8+13 21 13+21 Complete the ‘Fibonacci Numbers’ sheet using column addition 1+1 = 2 1+2 3 2+3 5 3+5 8 5+8 13 8+13 21 13+21 34 Colour all the even numbers in blue and all the even numbers in red. What do you notice about the patterns in the colouring?

Patterns in Fibonacci On another completed Fibonacci sheet, use your knowledge of divisibility rules to a) colour in blue the multiples of 5 b) colour in red the multiples of 3 c) underline the multiples of 6 What patterns can you see? What if you extend the sequence? Are there any patterns in the multiples of 10?

The Golden Ratio

The Golden Ratio The Golden (or Divine) Ratio has been talked about for thousands of years. People have shown that all things of great beauty have a ratio in their dimensions of a number around 1.618 1.618 1

The ratio of pairs of Fibonacci numbers gets closer to the golden ratio 1.618 1.619 1.615 1.625 1.6 1.666 1.5 55+89 = 144 34+55 89 21+34 55 13+21 34 8+13 21 5+8 13 3+5 8 2+3 5 1+2 3 1+1 2

The Golden Ratio Leonardo da Vinci showed that in a ‘perfect man’ there were lots of measurements that followed the Golden Ratio.

Now measure and record the following information for each of the people in your group. Display your results in a table like this: Name Height (in cm) Height of Naval (in cm) Ratio You could also try this with fingertip to shoulder and fingertip to elbow

Plenary Activity Make up a poster telling someone about the Fibonacci Numbers. You can include any work you have already done Include some pictures to make it attractive Use the Internet to find some more information about the Fibonacci numbers and the Golden Ratio