Fibonacci Fun with Sunflowers

Supplies:

Fibonacci numbers and sequences are commonly found in nature. For example, you can figure out how many rotations or spirals of seeds are in sunflower or on a pineapple, how many bracts (or rows of petals) are on a pinecone, or how many petals are in each row of petals on a sunflower or another flower such as a dandelion or mum with many rows or layers of petals? You can even determine the distance between planets starting at the sun by using the Fibonacci sequence. The sequence of numbers and the golden ratio formed by these numbers were identified by a mathematician, Leonardo of Pisa, who was the son of Bonacci, and used the name Fibonacci around A.D. 1200, more than 800 years ago.

The first 10 Fibonacci numbers are listed below. Copy them for the students to all read. Have the students try to identify how the sequence is figured or built. They can work in groups and see who can come up with the formula first.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Once they identify how to figure the Fibonacci sequence, ask them to work in groups to figure the next 10 numbers in the sequence. Record those numbers on the board.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Try it out by counting the sunflower, mum, or dandelion petals.

Find one spiral of seeds (starting from the center and arcing out to the edge of the seed head). Mark that spiral with a marker. Count the number of spirals in both directions. Try it out. The sequence for sunflowers might be 34 and 55 or 55 and 89. In other words, you may find 55 spirals with either 34 or 89 on either side going in an anti-clockwise direction. Locate those Fibonacci numbers in the list. Pinecone bracts are generally 5 and 8 and 8 and 13. Pineapples may be 13, 21, and 34.

You might want to check out the Golden Mean, which is the ratio between the Fibonacci numbers. Once you get a few numbers into the Fibonacci sequence, you can determine the ratio between two consecutive numbers and discover it always comes out to be around 1.618. Try it out with the first ten numbers in the Fibonacci sequence. Designers and artists sometimes consciously and many times unconsciously use the Golden Mean to determine pleasing shapes, sizes, and dimensions.

Try this and see if you come up with a number close to the Golden Mean. Measure your friend's height from the top of their head to the bottom of their feet. Then measure the top of their head to their waist. Divide their total height by their head to waist height and you should come up with a number close to the Golden Mean (1.6). Many pleasing things in nature and design are based on the Golden Mean. Test it out.

Supplies:

- Sunflower blooms, mums, dandelions
- Pinecones, pineapples (optional)
- Overhead transparency, marker or chalk board and markers or
- Paper and pencil

Fibonacci numbers and sequences are commonly found in nature. For example, you can figure out how many rotations or spirals of seeds are in sunflower or on a pineapple, how many bracts (or rows of petals) are on a pinecone, or how many petals are in each row of petals on a sunflower or another flower such as a dandelion or mum with many rows or layers of petals? You can even determine the distance between planets starting at the sun by using the Fibonacci sequence. The sequence of numbers and the golden ratio formed by these numbers were identified by a mathematician, Leonardo of Pisa, who was the son of Bonacci, and used the name Fibonacci around A.D. 1200, more than 800 years ago.

The first 10 Fibonacci numbers are listed below. Copy them for the students to all read. Have the students try to identify how the sequence is figured or built. They can work in groups and see who can come up with the formula first.

1.

**1**2.

**1**3.

**2**4.

**3**5.

**5**6.

**8**7.

**13**8.

**21**9.

**34**10.

**56**Once they identify how to figure the Fibonacci sequence, ask them to work in groups to figure the next 10 numbers in the sequence. Record those numbers on the board.

11.

**89**12.

**144**13.

**233**14.

**377**15.

**610**16.

**987**17.

**1,597**18.

**2,584**19.

**4,181**20.

**6,765****Besides how to figure out the Fibonacci sequence, what else is unique about the numbers?**

Look at the Fibonacci numbers for 5 and 12. What makes them unique?

How many odd numbers to even numbers are there?Look at the Fibonacci numbers for 5 and 12. What makes them unique?

How many odd numbers to even numbers are there?

**If you wanted to know the number of petals in each row of a sunflower, mum, or dandelion bloom, how might you use the Fibonacci sequence?**Try it out by counting the sunflower, mum, or dandelion petals.

**What were the number of petals in each row?****Are the number of petals in each row Fibonacci numbers?****How did they become Fibonacci numbers?****If you added the number of the petals in the first inner circle plus the number in the second circle, is the sum the number of petals in the third row?****Is the number of petals in the fourth row equivalent to the sum of the petals in the second and third row?****If you wanted to know the number of seed spirals in a sunflower seed how might you use the Fibonacci sequence?**Find one spiral of seeds (starting from the center and arcing out to the edge of the seed head). Mark that spiral with a marker. Count the number of spirals in both directions. Try it out. The sequence for sunflowers might be 34 and 55 or 55 and 89. In other words, you may find 55 spirals with either 34 or 89 on either side going in an anti-clockwise direction. Locate those Fibonacci numbers in the list. Pinecone bracts are generally 5 and 8 and 8 and 13. Pineapples may be 13, 21, and 34.

You might want to check out the Golden Mean, which is the ratio between the Fibonacci numbers. Once you get a few numbers into the Fibonacci sequence, you can determine the ratio between two consecutive numbers and discover it always comes out to be around 1.618. Try it out with the first ten numbers in the Fibonacci sequence. Designers and artists sometimes consciously and many times unconsciously use the Golden Mean to determine pleasing shapes, sizes, and dimensions.

Try this and see if you come up with a number close to the Golden Mean. Measure your friend's height from the top of their head to the bottom of their feet. Then measure the top of their head to their waist. Divide their total height by their head to waist height and you should come up with a number close to the Golden Mean (1.6). Many pleasing things in nature and design are based on the Golden Mean. Test it out.