rechenaufgabe titel

by Toni Chehlarova and Evgenia Sendova, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, and Elisaveta Stefanova, General High School Vladislav Gramatik, Sofia

How can we feed curiosity, wonder and excitement in mathematics and informatics classes? What kind of scenarios can keep up students' attention for a longer period; how can we provoke the exploration and discussion of the process of development? In the nineties, we wrote a textbook in which we invited ninth and tenth graders to design wallpapers and tessellations by using the so called turtle corners. Traditionally, symmetry is expressed through xy-coordinates of specific points (for example 10 , 5 and -10 , 5). In turtle geometry, however, symmetry is defined as certain points are reached by a turtle guided by the following commands: right, left and forward (for example right 45 degrees 100 steps forward, left 45 degrees 100 steps forward). In this case, the turtle is a metaphor of a robot leaving a trace when executing these commands and thereby forming geometric figures.

To design a wallpaper, one has to organize the movement of the turtle making it draw figures in the vertices of a rectangular grid. When the distances are appropriately chosen, you get tessellations, in which the motifs interlock perfectly to fill the plane without gaps or overlapping. The tessellations proved to be an object of exploration with a great appeal to the students. The tasks in the context of tessellations can be organized so that the students combine skills in mathematics and informatics of different levels: for example to find all regular polygons tessellating the plane, to generate the polygon-tiles simultaneously by means of multiple turtles (Blaho & Kalas 1998), recursive procedures (Clayson 1988), and to modify the tessellating regular polygon by implementing geometric transformations such as congruences (translations, rotations, reflections and compositions of those) so as to obtain a tile with a new shape (Sendova & Grkovska 2005; Chehlarova & Sendova 2010).

The idea presented in our textbook can easily be implemented by means of dynamic software as we did in two European Projects dealing with the inquiry based learning (IBL) in science and mathematics. The projects under discussion are InnoMathEd (Innovations in Mathematics Education on European Level) and Fibonacci (Disseminating inquiry-based science and mathematics education in Europe) (Kenderov 2010; Chehlarova et. al. 2011). What follows is an excerpt of a Fibonacci learning environment in the context of tessellations and its implementation in a class setting.

How to transform dynamically a square in a new tessellating tile

Creating a grid of tessellating figures like rectangles, hexagons or triangles, which are connected by means of geometric transformations - translation and or rotation - you could manipulate a tile so as to get another tile, thus imitating the style of Escher.

Let us illustrate the idea of dynamic tessellations by transforming a square tile in a tessellation tile of a new shape. We construct the square as a partial case of the polygon tool, select a point E on its side AB and a point F - on the segment EB. Then we construct an arbitrary point M and the images of E, F and M under translation by vector AD. Connecting the points as shown in the third picture of Figure 1 we get a newly shaped tessellation tile.

rechenaufgabe 1

It is possible to transform the square in another way - let us construct now the images of E, F and M under rotation with center B and angle -90°. Connecting the points as shown in the fourth picture of Figure 1 we get another tessellating tile. Next we can get a module of four tiles by means of translation (Figure 2 upper left) or rotation (Figure 2 bottom left).

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The next steps could be carried out in various modes (by applying a translation, a central symmetry or reflection) which assures a great variety even with such a simple starting shape. Furthermore, the variety of tessellating tiles could be achieved by free movements of the points M, E, F (Figure 3)

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Since M was chosen arbitrarily such a free movement would allow for self-intersection of the contour of the original tile (Figure 3 - the second picture) which contradicts to the idea of keeping its property of being a tessellating tile. To avoid this problem we shall choose point M on a preliminary constructed object, viz. a circle s with diameter close to (but smaller than) AB. We construct scrollers o and a to automise the movement of the points E, F. For the purpose we re-define point E as intersection point of the segment a = AB and a circle with center A and radius o (varied by the scroller o). In order to be sure that such a point exists we fix A and B and assign an appropriate value for the upper boundary of o depending on the length of a = AB. Next we construct point M as intersection point of the circle s with the second leg of the angle with a vertex the midpoint of AB, a first leg passing through B and a measure in grades a (varied by the scroller a). What is left is to hide the unnecessary elements and start the animation mode of the scrollers. These are only some of the ideas implemented in the context of the tessellation scenario but even they gave the impulse for working in exploratory style to teachers and students alike.

This scenario was presented (by the first two authors) with detailed instructions in a Bulgarian journal in mathematics and informatics. The third author (a teacher within the Fibonacci project) took the gauntlet and implemented it with 7-graders in IT classes. Here is what she shared at the bi-weekly seminar of the Fibonacci project: The students started with the regular polygons tessellating the plane and followed the ideas of transforming a tile by means of dynamic constructions as presented in the scenario above. Soon they realised that they had discovered their own land for explorations - playing in the style of Escher by adding new points on the initial tessellating tile (square, triangle, hexagon, rhombus) and modifying them under various congruences so as to get beautiful tiling shapes - flowers, animals, traditional martenitsa' figures, small pieces of adornment, made of white and red yarn, and warn in March (see: Figure 4).

rechenaufgabe 4

The truth is that the students discovered things that were new to me and we shared the joy. The temporary failures didn't discourage us. Sometimes we were looking for one thing and we discovered another, or changed the direction of our explorations. It was with a great pleasure for me to realize that students who thought they didn't like mathematics all of a sudden became very active. In addition, I felt the support of colleagues the school management and the parents. As for our future plans, we already started a deeper inquiry on Escher and found that not only his tessellations but also his metamorphoses are inspirational. Here are their first attempts in this direction: (Figure 4 - bottom-right). The inquiry-based approach to learning bridged the usual generation gap between teachers and students - not only do they learn from us, but for sure we can learn a lot of new things from them and about them.


The best works of the students were published on the Fibonacci website and later presented in the form of book markers, greeting cards and framed paintings at a seminar within the 41st Spring Conference of the Union of Bulgarian Mathematicians (Borovets, April 9-12, 2012). This was one more evidence that even within different computer environments the key factor for good learning outcomes is a didactic scenario tuned to the students' interests. Furthermore, we expect these outcomes to have a long-term effect since the students have worked in inquiry-based style and have constructed a public entity.


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