Teacher: Anna Wasiuta
The topic of the lesson: Fibonacci’s extraordinary numbers. The beauty of nature and the power of mathematics
The duration of the lesson: 45 min
Curriculum: in accordance with the curriculum approved by MEN (The Ministry of National Education)
(It is assumed that the students:
- already know the notion of number sequence – both the finite and infinite one
Metoda pracy: Metoda ćwiczeniowa, komunikacyjna, podająca, problemowa.
The method: the practical method, the communicative method, the problem method.
Form of work: together with the entire class
Teaching aids: a blackboard, an interactive whiteboard, a computer, multimedia, tablets, smartphones, a quiz (QUIZIZZ.com, learningaps.org), computer games.
Coursebook: Helena Lewicka, Marianna Kowalczyk „Matematyka wokół nas. Klasa 8”
TOK LEKCJI (Lesson Procedures)
|Czas (time)||Czynności prowadzącego (Lesson Procedures)||Przewidywane czynności ucznia Students’ actions||Uwagi po realizacji lekcji (Remarks after the lesson)|
|I start a presentation in Power Point and present the topic of the lesson and display it on an interactive whiteboard. SLIDE 1: Extraordinary Fibonacci numbers. The beauty of nature and the power of Mathematics. I tell my students that today we will discover a new, world – famous number sequence. You probably don’t know that you may see it every day. This number sequence, know as the Fibonacci number sequence, will be the subject of our lesson. I tell my students that they will be rewarded for their engagement in the lesson.||Students write down the topic of the lesson in their notebooks|
| SLIDE 2: I present this number plate: This number plate consists of numbers and letters. What is the first number or letter on this plate? What is the fifth one?|
Let’s have a look at the next example: SLIDE 3-4 Here you can see a computer password matematyka 16# (Mathematics 16#). The password consists of letters, numbers and a special character #. Which position in this password does the letter ‘t’ have? This type of assignment which assigns certain values to consecutive natural positive numbers is called a number sequence. SLIDE 5 Examples SLIDE 6 Today we will focus on number sequences which consist of only numbers. Each person has got a unique number sequence assigned to him/herself. It’s called the PESEL identification number. It consists of 11 elements. SLIDE 7 Another example of a number sequence is a number sequence of the positive multiples of 2 What are these numbers? Is it possible to write all the numbers belonging to this number sequence? We would have to do it infinitely as this is an infinite sequence. What about PESEL? In order to indicate that a sequence is infinite we use three dots after a comma. SLIDE 8 (TASK 1) COMPLETE THIS NUMBER SEQUENCE These are the initial values of the number sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21 … Figure out how are the consecutive numbers created in this sequence. (give at least five of them) SLIDE 9 Teacher presents students’ answers on the board and rewards them. Teacher presents the solution to the task and the curiosities connected with the Fibonacci sequence that was used in the task. The creator of this sequence is Leonardo Fibonacci – an Italian mathematician who lived at the turn of XII and XIII century. He created a formula which determines the consecutive numbers of this mathematical sequence. The first two elements of this sequence are 0 and 1 (sometimes 1 and 1), the next number, however, is the sum of the previous two. 0 + 1 = 1 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8 5 + 8 = 13 8 + 13 = 21 13 + 21 = 34 21 + 34 = 55 34 + 55 = 89 The results of these calculations (additions), so the elements of the sequence, are called the Fibonacci numbers.0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377… SLIDE 10 A very special kind of a number sequence is the Fibonacci sequence. Leonardo Fibonacci – an Italian mathematician who lived in Italy in 1175-1250. The founder of the Fibonacci sequence that we can observe in nature. SLIDE 11 Attention! Count off! Maybe we will try differently? Instead of adding two 1 to the previous number all the time, let’s add two previous numbers. This way we will create a new sequence: (I read only the elements marked in yellow colour) SLIDE 12 So this is how we get the Fibonacci Numbers SLIDE 13 The Fibonacci sequence has got many interesting features. If we take any two consecutives numbers of the Fibonacci sequence and divide them by themselves, e.g. 987 : 610; 89 : 55, the sum of these calculation will be always the same and amount to approximately 1.618. The bigger the numbers in the sequence that we divide are, the more exact approximation of this number (1.618) we get. This number is called “ the golden number” and is often marked with a Greek letter φ (pronounced as „ /faɪ/”). This ratio is often called „the golden ratio” or „divine proportion”. Another interesting phenomenon is the Fibonacci spiral – a special type of a so-called „golden spiral” . The width of the spiral increases (or decreases) by 90°, exactly φ times. It looks like this: SLIDE 14 The golden spiral is a graphic interpretation of the Fibonacci sequence. The two numbers (1 and1) will be represented by two squares with the dimensions of 1 unit by 1 unit. The next element of the sequence is the number 2, so it is represented by a square with the dimensions of 2 units by 2 units. Let’s attach it to the longer side. The next element is the number 3, so we get 3×3 square which we attach to the longer side. Then we repeat the same procedure with 5 and 8,13,21,34 Pay attention to the fact that the ratio of the longer sides of the squares amounts to phi (φ) and is getting more and more exact as the numbers in the sequence grow. SLIDE 15 Now, we will get the golden spiral by inscribing it in the points of osculation of the segments with the ratio of phi. A spiral that keeps this ratio is the so-called golden spiral. Its shape is made up by the Fibonacci numbers and the golden ratio. SLIDE 16 As we have mentioned earlier, the Fibonacci sequence can be found everywhere. We will prove it now! The world of nature and the Fibonacci sequence. Have you ever wondered why it is so difficult to find a four-leaf clover? The secret of the four-leaf clover is hidden in Mathematics. SLIDE 17 Maths and nature I would like to draw your attention to the phenomena which exist in the world of nature that are connected with Maths. We can see them on a daily basis though we are usually not aware of them. SLIDE 18 The Fibonacci sequence can be found nearly everyday, even while admiring flowers in a garden. We can see it when we look at: – monopetalous Zantedeschia
– Trillium with three petals
– Aquilegia with five petals And many other flowers, with the numbers of petals corresponding to the consecutive numbers of Fibonacci sequence.
SLIDE 19 A flower that is in full bloom (without any deformations or mutations) should always have the number of petals which is a Fibonacci number. Therefore, finding a four-leaf clover is so difficult. SLIDE 20 Let’s look at the stem and leaves of a sunflower from above. Now, concentrate on one particular leaf and count the leaves clockwise. You will find out that the numbers of leaves that we pass and the number of encirclements around the stem that one has to do to be under this one particular leave are the consecutive Fibonacci numbers. This allows each leaf to have the largest amount of sunlight. SLIDE 21 The growth of fruit gives them a mathematically predictable number of sections. A banana has got 3 sectionsAn apple has got 5 sectionsA grapefruit has got 13 sections SLIDE 22 Also the process of growth of a plant takes place according to the rule of the golden ratio. Therefore, we are able to find the golden spiral in most of plants that we know: in sunflowers,in cones,in daisies,in pineapples,in broccoli,in cauliflowers,in cabbages etc. SLIDE 23 In the photo you can see the golden ratio in a kiwi fruit. SLAJD 24 The Fibonacci sequence can be also found in the breeding pattern of bees. In every succeeding generation of bees the number of their ancestors reflects the rules of the Fibonacci sequence. 1 parent, 2 grandparents, 3 great-grandparents, 5 great great-grandparents. SLIDE 25 Every scale in a cone also reflect the Fibonacci sequence. The relationship between every one of them represent this sequence. Extraordinary, isn’t it? SLIDE 26 – A SHELL This is probably the most well-known example of the Fibonacci sequence in nature. The shell of an animal called “łowik” is based on the spiral made up by the numbers of the Fibonacci sequence. This phenomenon fascinated even such people as Leonardo Da Vinci who devoted many of his works to it. SLIDE 27 Hurricanes and spiral galaxies are also created according to the rules of the golden ratio. Spiral galaxies, the most common types of galaxies in our universe, are characterised by many bent arms. Even the Milky Way is a spiral with the angle of 12 degrees. What is even more interesting, galaxies defy the rules of Newtonian physics in order to act according to the rules of the Fibonacci sequence. They have greater rotational speed near their edges than in their centres. SLIDE 28 The arrangement of eyes, fins and the tail of a dolphin also reflects the Fibonacci sequence. The diameter of the tail part of a dolphin is in the golden ratio with its upper part of the body. SLIDE 29 – GAMES Students log into a website shown by the teacher and use tablet computers connected to the Internet to solve a quiz and play educational games. The results are shown on the board.The winners get small prizes. SLIDE 30 Teacher asks students to watch a short film presenting the Fibonacci sequence in nature. https://www.youtube.com/watch?v=M16GINf8A50
made by Anna Wasiuta
|S Students answer: The first one is number 1. The fifth one is the letter B. It is in position 3 and 7. Students answer: 2,4,6,8,10,12… It is not possible PESEL has always got 11 numbers so it is finite. Students search for five consecutive numbers of the sequence Students together with the teacher think about the way the sequence should be written Students take notes Students log into a website shown by the teacher and use tablet computers connected to the Internet to solve a quiz and play educational games. The winners get small prizes.||I give my students some hints I monitor the work of my students. Students were able to understand the topic without major problems. They dealt with tasks quite well.|
|9.40||Homework: Find examples of the Fibonacci sequence in nature. Make a poster to present them.||Teacher explains the homework|