key: cord-0057003-m2kpwnlv authors: Roanes-Lozano, Eugenio; Solano-Macías, Carmen title: Using Fractals and Turtle Geometry to Visually Explain the Spread of a Virus to Kids: A STEM Multitarget Activity date: 2021-03-05 journal: Math.Comput.Sci. DOI: 10.1007/s11786-021-00500-9 sha: d3882a2162ee7d1e11f327e2f37cd56c0800207d doc_id: 57003 cord_uid: m2kpwnlv A lockdown was ordered in Spain on March 2020 due to the COVID-19 pandemic. The first author, advised by the second author, developed a tale and video (in Spanish) with a simplified explanation of virus propagation for their teenager son, justifying the need to stay at home. The tale and video relate the spread of the virus to fractal trees and aim to raise awareness about the transmission of the disease. The video is available from the web page of the Instituto de Matemática Interdisciplinar of the first author’s university. The code was implemented in Scratch 3 and takes advantage of the “Turtle Geometry” (there is an ulterior version using Maple, available from Mapleprimes). This article includes the English version of the original tale, describes the Scratch 3 code, and details possible derived STEM activities. We plan to experiment them in the classroom during the 2020–2021 academic year. The first author has taught computational mathematics to students of the School of Education of the (Apple II, PC-AT, PC-386, PC-486,…) and languages: * in the past: Logo, Derive and The Geometer's Sketchpad * now: Scratch, Maple and GeoGebra and also to postgraduates at the School of Mathematics of the same university along these years. Meanwhile, the second author teaches at a School of Information and Communication Sciences of the Universidad de Extremadura and has a long experience in distance learning using different computational resources. We have a 14-year-old son that likes sciences and mathematics. In the beginning of the pandemic a lockdown of citizens was ordered in Spain. It is not easy to explain a kid of this age why he shouldn't meet his friends and relatives for some time and, even more, that he should stay at home. Therefore we developed a simplified model and explanation of virus propagation [1] focused on the mathematical perspective (relating it to fractals) and social conscience. The explanation is made visual by using the Turtle Geometry [2, 3] . We believe that the almost forgotten Turtle Geometry still has many possible applications in different fields [4] . We developed in the 1990's Pascal [5] , Maple [6] and Derive [7] implementations of the Turtle Geometry. The simplified model of virus propagation was initially implemented in the computer language Scratch 3 [8, 9] , that our son knew. It is well-known that Turtle Geometry is very well suited for drawing some kinds of fractals [2, 10] , and it is easy to implement a program that draws a fractal tree with any number of branches and any depth using it. The number of branches can be related to the average number of animals infected by each ill animal and the depth can be related to the time that passes until the animals stop meeting each other. Moreover, the graphic cursor of Scratch 3 (a cat) can be used to simulate the spread of the virus in a colony of cats. This explanation was later recorded in a 5 minutes video and written in a tale (both of them in Spanish). Some colleagues liked the video and it is now available from the web page of the Instituto de Matemática Interdisciplinar (IMI) of the Universidad Complutense de Madrid [11] . The program has also been translated to Maple [12] , using our implementation of Turtle Geometry [6] (that has a very good graphics resolution) and an adaptation of the tale is now available from Mapleprimes' web page [13] . The details and peculiarities of the Maple implementation were discussed after ESCO2020 at the Maple Conference 2020 [14, 15] . The English translation of the original tale is included afterwards. Although there are specialised papers relating virus propagation to fractals [16] , we know of no other divulgation approach oriented to kids (or laymen in the topic). As COVID-19 is a global pandemic, there are hundreds of papers on the topic. We could mention [17] as a summary of the different aspects of this disease. The tale, together with the video are presented below as visual approaches to some STEM topics and a way to foster the awareness of the dangers of virus' spread. Fig. 1 A fractal tree of depth 3 with 3 branches at each level. Note that, formally, the trunk of the tree shouldn't be drawn for the plot to be a fractal We would begin with a simplified example: how to draw a fractal tree with 3 branches at each level. In Fig. 1 you can find a tree of depth 3. The Scratch 3 code is simple (Fig. 2) . The main procedure has two input: -the depth of the tree, -the length of the initial branches of the tree and begins by clearing the screen and initiating the recursive subprocedure Aux. Meanwhile, subprocedure Aux iterates the drawing of branches of the same level. If the depth of the tree hasn't been reached: -moves forward the number of steps given by the second input to Aux, -repeats three times: • turn 120 • = 360 • 3 , • execute Aux with the first input increased in one and the second input (the length of the branches of the previous level) divided by two -moves backwards the number of steps given by the second input to Aux). These procedures are slightly more complicated (Fig. 3) . We have used another auxiliary procedure (CS), that clears the screen and resets the cat. Using Fractals and Turtle Geometry The main procedure has now three input (the average number of cats each ill cat infects by average is a new variable). Scratch's cat is hidden to increase the drawing speed and stamped when it reaches a "leave" of the tree. Aux procedure now includes a new conditional that avoids the cat to draw the "trunk" of the tree. And the angles to be turned are more complicated (360 • divided by the number of branches and half this angle). Finally, the decrease in the length of the branches has been chosen to depend on the number of branches too. The tale and video have multiple STEM applications. Their initial goal was extracurricular: to foster the awareness of the dangers of virus' spread, and it is possibly their most important goal. The implementation of Sect. 3.1 is appropriate to be reproduced by a Secondary School students or first year university students. Possibly, the implementation of Sect. 3.2 can be understood by many of such students. We planned to use this software with Secondary School students during the 2020-2021 academic year. Unfortunately, this hasn't been possible so far due to the restrictions derived from the pandemic. If the restrictions continue we'll use it with university students. The video was shown to a group of students from a pedagogy degree at the end of last academic year and was easily understood and very motivating [18] . The problem addressed has several STEM applications, that can be developed in the classroom: -understanding what a fractal tree is (mathematics, computer science), -relating fractal trees to structures in nature (biology), -working with the Turtle Geometry (mathematics, computer science), -understanding recursion (mathematics, computer science). What began as something "for the family" has turned out to be a tale and video for a wider audience. Thinking more calmly about it, we believe it is eye-catching (and, therefore, motivating) for students of very different levels, and, moreover, has many different STEM applications. Propagation of viruses | Animal Turtle Geometry. The Computers as a Medium for Exploring Mathematics Mindstorms. Children, Computers, and Powerful Ideas Dibujando caleidoscopios con una coreografía de tortugas An implementation of "turtle graphics" in Maple V. The Maple Technical Newsletter. Maple Tech. Special Issue An implementation of "turtle graphics" in derive 3 Scratch: programming for all Turtle graphics" in Maple V A simplified introduction to virus propagation using Maple's Turtle Graphics package Viral epidemics in a cell culture: novel high resolution data and their interpretation by a percolation theory based model The first, holistic immunological model of COVID-19: implications for prevention, diagnosis, and public health measures Reflective discourse analysis of a group of pedagogy students regarding an awareness video on Covid-19 propagation Acknowledgements This work was partially supported by the research project PGC2018-096509-B-100 (Government of Spain).