**Date and Time ** : 14-06-2020, 10:00-11:20 am

**Facilitator:** Mrs. Purna J. Patil, Govt. Girls’ Ashram school, Bhaliwadi, Karjat

**Participants:**

- From HBCSE: Anish Parmar, Sushma Rawool, Mayuri Twade, Harita Raval, Shweta Naik, Sushant Pawar, Narendra Deshmukh, Mayuri Pawar, Pranav Khot, Megha Chougule, Vinod Sonawane.
- Outside HBCSE: Satyawati Rawool, Sandhya Thakur, Kunda Ma’am, Purna Patil, Shashikala Birajdar, Shaila Thakur, Yogini Chavan,

This webinar was conducted by Dr. Aaloka Kanhere from HBCSE- TIFR. She has been working in the field of mathematics education for more than ten years. Purpose of the webinar to solve problems and exploring the pattern of Fibonacci numbers. Also to see how fibonacci numbers occur in nature very frequently. Also Fibonacci numbers are a very nice example of mathematics in nature. Most of the time the question is asked “where is mathematics in the natural world” and these numbers can be one of the answers. Dr. Aaloka shared a worksheet with the teacher participants on the whatsapp group to solve and understand Fibonacci numbers. Following are the key points from the webinar.

- Narendra Deshmukh sir introduced Dr. Aaloka and her work in education. Her work with ashram schools and writing in some magazines like Sandarbha.
- Dr. Aaloka then started the webinar and she explained the purpose of that webinar. While doing so she pointed out how problem-solving is not most of the time taken seriously and how it is an important problem solving is to understand mathematics.
- She also added how she will try to make this session interactive through problem-solving.
- Then Dr. Aaloka Kanhere started discussing the problems in her worksheet. She started with the problem of coins.
- Then she discussed how coins of a particular amount with the help of coins of one and two rupees can be formed. And in how many ways it can be formed.
- Then from that one sequence of numbers arose which was 1, 2, 3, 5, 8, 13, She asked participants whether they are able to see any pattern in these numbers.
- Then she tried to bring a generalized pattern for the amount “n”. She assumed a number of ways in which amount “n” is formed can be called Cn
- Then by using the principle of mathematical induction she showed how Cn can be obtained from Cn-1 and Cn-2.
- Then the pattern turns out to be Cn = Cn-1 + Cn-2.
- Then Dr. Aaloka discussed the problem of Honey Bees. Which involved finding parents, grandparents, great grandparents, and so on of male and female Honey-bees.
- While solving that problem a similar sequence of numbers that arose during the coin problem arose once again.
- Then she told participants that these numbers are called Fibonacci numbers. Who is named after Fibonacci? She also told the participants that one early Indian mathematician named Pingala and then after him, another mathematician named Virahanka wrote about these numbers from the perspective of music.
- While participants were sharing their experience of solving these questions. Pranav Khot mentioned it was very interesting to solve the first question of the worksheet.
- Then Dr. Aaloka showed one video which showed how the Fibonacci spiral is formed. The same video also involved examples from nature like the number of petals in the flowers, how the shape of waves and shell of nautilus also have a Fibonacci spiral in them.
- After that participants and the resource person together explored the pattern in the squares of Fibonacci numbers. An interesting part of such a pattern was how it also leads to a Fibonacci number sequence with one number skipped.
- Then another pattern was discussed which involved the summation of Fibonacci numbers. Where the difference between the consecutive terms in the summation table was also leading to the Fibonacci sequence.
- After this interesting topic of the Golden-ratio was discussed with participants. How the Golden ratio was obtained from the ratio of consecutive Fibonacci numbers was also shown.
- Dr. Aaloka showed a video which showed where we find the Golden ratio in many things like the proportion of the length of body organs like fingers, hands, etc.
- Dr. Aaloka did one activity with participants which was a mind-reader. Where participants were asked to consider one number from 1 to 20 and then some cards were shown on screen and participants were asked about in which card their number appears.
- The interesting fact from this activity arose which was that any natural number can be written as a summation of Fibonacci numbers.
- Teacher participants Purna ma’am and Shashikala ma’am did this activity and verified it. Also, Shaila Thakur ma’am guessed correctly the initial number on the next card.
- After that Dr. Aaloka discussed one pattern about Fibonacci numbers which says if any natural number n divided another natural number m then the Fibonacci number Fn will divide Fm.
- After these questions and queries from the participants were addressed by Dr. Aaloka. Participants expressed how they felt about the session and after that, the session ended.

**Report prepared by: **Sushant Pawar