# Presentation

**Improving the efficiency of geometry teaching by means of didactic models.**

I, SamvelMovsisyan, top-rank teacher, am the author of the invention "Geometric teaching guide NANE" and the manual "Mathematics in the weight balance". In 1999 I patented the "Geometric teaching guide NANE". In 2006 I patented the manual "Mathematics in the weight balance".

The launching of distribution of these inventions has started since 1999 in Armenia, where the schoolscontinuallyacquired them as training aids for mathematics cabinets and pre-school institutionsobtained the "Mathematics in the weight balance"for pre-school room groups. At the same time Moscow Uchcollector№1 since 2001 collaborating with me realized/sold huge number of models sets.

Since 2009Iamdealingwithdisseminationoftheproductinlotsofregions/provincesandtownsofRussia, suchasfollows: Moscow, Kiev, Kislovodsk, Stavropol, Pyatigorsk, Yessentuki, Cherkessk, Rostov-on-Don, Bataysk, Azov, Orenburg, Taganrog, Chaltyr, NizhnyNovgorod, Chapaevsk, Kinel, Zhigulevsk, Dzerzhinsk, Novoshakhtinsk, Samara, Toljatti, Novokuibyshevsk, Kazan, NaberezhnyeChelny, Nizhnekamsk, Almetyevsk, Bugulma,Zelenodolsk, Cheboksary, Novocheboksary, YoshkarOla, Yerevan, Gyumri, Vanadzor, RepublicofNagorno-Karabakh, etc. Almost at all schools the majority purchased/acquired it with great enthusiasm and delight. The fact that there are no stereometric models at schools, the teachers perceive them not only as a visual teaching aid, but also as a "new method" of geometry teaching. This is particularly obvious/evident when teachers give me the opportunity to demonstrate the model for students in the class. And a profound interest arises at students to demonstrate all sorts of conversions/transfigurations of these models. And the experience shows that this is interesting for everyone. To demonstrate the capabilities of the model the seminars are organized for teachers of mathematics, which show solutions to non-standard, complex tasks.

Geometry teaching efficiency improvement by means of didactic models.

Over 10 years ago the geometric models appeared at schools, which are made on the principle of telescopic rods. The link up points/butt joins of the edges that are on the top and bottom of the geometrical objects are connected by the original ring, which, combined with unlimited number of freedom degrees, allows to change the lengths of the sides and rotate the rods.

The essence of these models is transformation, through which one geometrical figure is converted to another one, and it in its turn to the next one, etc. The planar figure is transformed into the stereometric figure and back. With the aid of these models the solutions of the problems, the known theorems and the creation/construction of three-dimensional figures intersection are visually demonstrated. As a result it is possible to get a set of polygons and polyhedral. The simplest of these models is the pyramid. Using this model, one can obtain all types of triangles and quadrilaterals, including spatial/three-dimensional quadrangles. This model with its versatility/generality allows the pupils to master in geometry not through overlearning or abstract representation, but more visually and in the dynamic.

The study of the topic Quadrangles/tetragonis entirely based on the logic of transformation.

Convex Quadrangles/tetragon-------->Trapezoid ------->Quadrat

ConvexQuadrangles/tetragon->Parallelogram/Rhomboid ->Rectangle ->Rhombus ->Quadrat

Previously, these changes could be only imagined, and with the help of this model, the teacher will be able to visually/obviously demonstrate all. On the board it is impossible to imagine this chain of transformations. It turns out that the study of this topic has a discrete character/quantum nature, and the study of this model becomes continuous. In geometry there is a notion of "Spatial/three-dimensional quadrangles".The usage of the model makes it possible to show in the dynamic how one of the nodes/points starts to move away from the plane of the other nodes, and at that time the triangular pyramid is being formed. When the teacher for the first time on the board draws a triangular pyramid, he/she produces serious and hard work, trying to explain the essence of these drawings to the pupils, but with the help of this model, it can be done very shortly and efficiently. The white color of links/verges of the model is not chosen by chance: it repeats the color of chalk on a blackboard. When the teacher holds the model appropriately at the blackboard, the pupils draw in the exercise book what they see.

Studying the theory of three perpendiculars, the pupils are asked to convert 4 triangles intothe right triangles, what, as it turns out, is not so simple. When the multiple trials of the pupils are not producing the results, the teacher who is inferior in resourcefulness against the pupils, but superior them in his/her knowledge, shows the miracle of the theorem of three perpendiculars. That'showisthelearningefficiency.

For many years I was giving my geometry lessons with the help of these models. Each pupil having on the table this model expresses the great interest in the subject and is actively involved in the lesson trying independently to get different/diverse geometric figures. To show the signs of equality of triangles, many teachers are making the corresponding triangles from paper, and putting on each other they prove their equality. And with the help of this model it is possible to show it easily and beautifully. Red ends of the telescopic rods remind the arrows of vectors, allowing to visually/obviously demonstrate the operations on vectors (adding/subtracting). Stereo-metric tasks on the topic of "Triangular Pyramid" are diverse. The construction/structure of cross-sections, the distance between crossing straight lines and angles formed between them –this model demonstrates all this obviously and clearly.

Objective: Given: a pyramid with links/edgesa, b, c, which are mutually perpendicular. Find the volume of the pyramid.

To solve this kind of problem, the teacher has to make such figures on his one, and using this model it will be possible to achieve the goal in just a few seconds.

Often teachers give a practical task - to prepare various models of geometric figures. If the pupil has such a model, he can not only simply copy the geometric figures, but he will also discover the interesting and complex world of geometry using infinitely/continuously transformable universal models. Therefore, they should be on the table during the geometry lesson along with a ruler, protractor and divider.