12: Finding Fibonacci patterns in nature Adapted from activities and information found at University of Surrey Website http://www.mcs.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.html Curriculum connections Use of this learning and teaching activity may contribute to achievement of the Standards. Indications of relevant Domains and Levels in the Victorian Essential Learning Standards are provided to assist teachers to make decisions about the appropriateness of the activity for their students. Summary Students will explore patterns in nature through looking at the Fibonacci sequence of numbers and applying it to their surroundings. Victorian Essential Learning Standards Domains and (Levels): Mathematics (3,4,5,6) Science (3) Interpersonal Development (3,4,5) Communication (4,5,6) Information and Communication Technology (3,4) Thinking Processes (3,4,5) Duration: 1 lesson to a whole week. Setting: Classroom and/or the garden. Student outcomes Students will be able to: understand that the Fibonacci sequence is reached by computing the next term from the previous two terms (recursion) test generalisations about numbers by investigating patterns in nature apply structures such as Venn diagrams to different sets of numbers develop further understanding of the nature of similarities between things in nature. Background notes for teachers Patterns in nature are a good way of getting students to explore some of their own sequences by looking at familiar everyday objects in nature. Leonardo Fibonacci, an Italian mathematician, was one of the first people to introduce the Hindu-Arabic number system into Europe. This is the number system we use today which is based on ten digits: 1 2 3 4 5 6 7 8 9 0. Fibonacci s book on how to do arithmetic in the decimal system persuaded many European mathematicians of his day to use this "new" system. The book describes the rules we all now learn at primary school for adding, subtracting, multiplying and dividing numbers. In his book Fibonacci recognised mathematical patterns in nature and he wrote about the Fibonacci sequence (although he wasn t the first to describe it) in the early 13 th century. This sequence or pattern occurs in many places in nature, from pinecones to sunflowers and sea shells. Learning in the Garden page 98
This is how it works: A Fibonacci sequence includes the numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on (each number is the sum of the two preceding numbers). In nature, these values often show up in the form of a certain number of spirals around an object. For instance, pinecones usually have 8 spirals going around in one direction and 13 in the other (both Fibonacci numbers). You can share the Fibonacci pattern with students in the following activities. This activity could be used as part of inquiry into mathematical patterns. Materials Closed pine cone A sunflower head with seeds intact A range of fruit and vegetables including: cucumber, citrus fruit, apple, banana, cauliflower, pineapple, capsicum, and any other fruits and vegetables that you would like to investigate. Digital camera and computers if available The activity Level 3 and 4 1. Introduce students to the sequence of numbers and ask them to work out which number comes after 21. (See Worksheet 1) Use a calculator to check your answers. For younger students, counters could be used to represent number patterns. 2. Breaking students into pairs and using a different item (fruit or vegetable) per pair, challenge students to find other Fibonacci numbers. (See Worksheet 2) 3. Split students into groups and give them the digital camera and ask them to explore the garden and take photos of spirals or patterns that link to these numbers. Look for: numbers of petals or stamens number of segments numbers of seeds numbers of leaves and their patterns etc. Students may uncover vein patterns in leaves, a repeated design on a flower petal, a spider's web, or the arrangement of seeds in a sunflower. Once they have photos in hand (or on screen), ask students to observe, compare, and describe different patterns and sort the photos as they see fit (eg. symmetrical vs. asymmetrical patterns). (See Worksheet 3) 4. Investigate where else Fibonacci numbers are found in fruit, vegetables and in nature. Investigate some of the samples bought to class and take digital photos of the Fibonacci findings to use in presentations. 5. Students can develop PowerPoint presentations about the patterns and numbers they have found using their digital photos. Learning in the Garden page 99
Level 4 and 5 1. Introduce the Fibonacci sequence and ask students if they can recognise the pattern and if they are able to write a formula to represent it. 2. Split the class into pairs. 3. Introduce the number of spirals idea by asking students to find the spirals in a pine cone, a sunflower or a pineapple. Ask students to count the two opposite sets of spirals in these examples to find a Fibonacci number. 4. Give each pair a single cauliflower floret (one stem of the original cauliflower) to examine for the Fibonacci numbers. 5. Ask students to look at it: Count the number of spirals on your cauliflower floret. The number in one direction and in the other will be Fibonacci numbers. If you can, count the spirals in both directions. How many are there? Start at the bottom and take off the largest floret, cutting it off parallel to the main "stem". Find the next one up the stem. It'll be about 0 618 (phi) of a turn around (in one direction). Cut it off in the same way (for more information on phi and Phi or the Golden Ratio see number 3 below). Repeat, as far as possible. Now look at the stem. The florets are much like a pinecone or pineapple. Also notice that the florets are arranged in spirals up the stem. Count them again to show the Fibonacci numbers. Cauliflower Cauliflower with one set of spirals Cauliflower with two sets of spirals 6. Lettuce is similar but there is no proper stem for the leaves. Instead, carefully take off the leaves, from the outermost first, noticing that they overlap and there is usually only one that is the outermost each time. You should be able to find some Fibonacci number connections. 7. Ask students to look for Fibonacci in the garden to see if they can find the numbers anywhere else they may be surprised. Students should take notes on what they find to Learning in the Garden page 100
share with the rest of the class. Students may use a digital camera to record and demonstrate their understanding and to develop a series of PowerPoint slides to present their Fibonacci findings to the class. 8. Each group can present their findings to the rest of the class. Level 5 and 6 1. Use Venn diagrams to compare the Fibonacci sequence with other sequences that can be found in nature. One of these is the Lucas numbers (the Lucas numbers are formed in the same way as the Fibonacci numbers - by adding the latest two to get the next, but instead of starting at 0 and 1 [Fibonacci numbers] the Lucas number series starts with 1 and 2). Show the relationships of intersection, union, inclusion (subset) and complement between the sets. 2. Explore the question: does the Fibonacci sequence have a scientific basis? Develop a classroom debate. 3. To discover the relationship between the Fibonacci sequence, Phi and phi complete the Golden Ratio activity at http://www.mcs.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.html#rabeecow Extension activities For levels 5 and 6 Fibonacci sequences can be demonstrated in family trees of bees. Bees: There are over 30,000 species of bees in the world. The bee we know best is the bee that makes honey and lives in a hive. Bees have an unusual family tree. Investigate their family tree using the Fibonacci numbers and the information below. Did you know that not all bees have two parents? In a colony of bees there is a special female called the queen who produces eggs. There are many worker bees that are also female but unlike the queen bee, they produce no eggs. There are some drone bees that are male and do no work. Males are produced by the queen's unfertilised eggs, so male bees have a mother but no father. All the females are produced when the queen has mated with a male and therefore have two parents. Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens who start new colonies when a new nest is needed. So female bees have two parents, a male and a female, whereas male bees have just one parent, a female. Learning in the Garden page 101
If you look at the family tree of a male drone bee: 1. He has 1 parent, a female. 2. He has 2 grandparents, since his mother had two parents, a male and a female. 3. He has 3 great-grandparents: his grandmother had two parents but his grandfather had only one. 4. How many great-great-grand parents did he have? Draw a diagram of your understanding of how bee family trees demonstrate Fibonacci numbers. Complete the following questions: Draw a bee family tree for 6 generations of a male bee and a female bee what is the difference? What pattern do you see in the number of females in each line of the family trees? What about the males? Investigate other animals. Is this pattern observed anywhere else? Do Fibonacci numbers occur in the human body? Related LandLearn activities Agrimaths - relating maths to our food and fibre activity booklet available on LandLearn Resource Booklets CD. All activities are relevant. Learning in the Garden page 102
NAME: Worksheet 1 Finding Fibonacci in nature Fibonacci was a famous mathematician he found mathematical patterns in nature and developed the Fibonacci sequence. This sequence or pattern occurs in many places in nature, from vegetables to sunflowers and sea shells. This is how it works: A Fibonacci pattern includes the numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. Each number is the sum of the two preceding numbers added together. 0,1,1,2,3,5,8,13,21 Just add the two numbers next to each other together to the make Fibonacci pattern. For example 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 Try it for yourself. Write your answers in the box below How far past 21 can you go? Try it in the box below Learning in the Garden page 103
NAME: Worksheet 2 Finding Fibonacci in nature Remembering rules for knife safety carefully cut the following fruit and vegetables to find the Fibonacci sequence. Note: there is a right way and a wrong way to cut the fruit to find Fibonacci. Apples: Cut the fruit in half across the middle. How many seeds or segments do you see? Is there a Fibonacci number? Bananas: Count how many "flat" surfaces it is made up of. Is it 3 or perhaps 5? When you've peeled it, cut a slice and look at the seeds. Is it a Fibonacci number? Pineapple: Count the spirals in the outer scales of the pineapple to find the Fibonacci number. Pine cones have similar patterns in their spirals. Cucumber: Look for the different patterns in a cucumber. Are there any Fibonacci numbers? Onions: Can you find any Fibonacci numbers in onions? Sketch your findings below Draw and label what you find in the boxes below: Fruit: Fruit: Vegetable: Vegetable: Learning in the Garden page 104
NAME: Worksheet 3 Finding Fibonacci in nature Use the digital camera to photograph Fibonacci patterns that your group finds in nature. Explore the vegetable garden or school grounds. What did you find? Draw and label your findings below. Learning in the Garden page 105