Математика и Информатика
Вход / Регистрация

2015/3, стр. 261 - 271

COMPARATIVE ANALYSIS REGARDING THE STUDY OF TRANSFORMATIONS IN THE EUCLIDEAN PLANE BY APPLYING COMPLEX NUMBERS

Katerina Anevska
E-mail: anevskak@gmail.com
Faculty for informatics – FON University
BUL Vojvodina bb
1010 Skopje, Macedonia
Sava Grozdev
E-mail: sava.grozdev@gmail.com
Institute of Mathematics and Informatics – BAS
Acad. G. Bonchev Street, block 8
1113 Sofia, Bulgaria
Risto Malcheski
E-mail: risto.malceski@gmail.com
Faculty for informatics – FON University
BUL Vojvodina bb
1010 Skopje, Macedonia

Резюме: Adopting transformations in the Euclidean plane and their application is compulsory for high school students. The practice shows that even advanced students face difficulties when learning such a material, especially the part which refers to group properties. On the other hand, complex numbers are also studied in high school and they are characterized by an analytical apparatus which helps learning the transformations in the Euclidean plane. This paper presents results from a comparative analysis referring to the accomplishments of the students who study transformations in the Euclidean plane and their application by using complex numbers (see (Malcheski et al., 2015)) and in the classical way (see (Mitrović et al., 1998)).

Ключови думи: complex number, transformation, test, assessment

1. Introduction

According to the Lisbon Strategy, knowledge and skills are forms of social capital and the society has to take a permanent care of them. Precisely this is one of the reasons for implementing the lifelong education, as well as continuous reevaluation of education and its adaptation to the needs of the society. The strategy of the European Union also supports this. It aims at emphasizing the importance of education, innovations and research for the development of the individuals and their further preparation for successful professional improvement and involvement in social flows. Clearly, this strategy allows a realization of the individual and also results in enriched knowledge and skills of the society in general, which is necessary for overcoming the challenges that the future may bring. The latter imposes a change of the contents and the methods which are used in the realization of the educational process.

Taking into consideration the previously mentioned, the state authorities, the professional associations which deal with education, and many of the teachers invest serious efforts for a successful integration of the Republic of Macedonia in the frames of the European educational sphere. For a long period of time we have been witnessing numerous reforms in the education sector of the Republic of Macedonia, all with the aim to improve the overall educational system, including mathematics education. Nevertheless, the researches carried out by international professional associations have demonstrated that, apart from the numerous projects for modernization of the education, many of which refer to mathematics education, the results are not satisfactory. This especially refers to the preparation of high school students for a successful inclusion at the technical faculties, as well as the applicability of the acquired knowledge. Hence, there is a need to change the mathematics syllabi in certain segments, i.e. to complement them with the goal to:

– improve inter-subject integration of mathematics instruction,

– enable students to acquire permanent and applicable operative and structural knowledge, and

– improve the preparation of the students for successful inclusion in the higher levels of education.

Accomplishing the mentioned goals is a comprehensive and complex task, which requires analysis of the entire material covered in the high school education, and this cannot be accomplished by a single research. So, having in mind that students have continuously displayed poorest results when dealing with transformations in the Euclidean plane and their application, we developed a syllabus which is based on the application of the complex numbers, which can be listed as an elective subject in comprehensive high school, and is the subject matter of the book (Malcheski et al., 2015). Using both, this syllabus and the existing syllabi (see (Mitrović et al., 1998)), we carried out an experiment, which (was) aimed to answer the following question:

When compared to the existing syllabi, does the acquisition of knowledge and skills about the transformations in the Euclidean plane using complex numbers in the high schools enable acquiring advanced operative and structural knowledge?

2. Research design

The previously mentioned defines the subject of the research, which is the accomplishment of students, regarding the transformations in the Euclidean plane by following the experimental syllabus, compared to the accomplishments when the existing syllabi are followed.

According to the subject of research we create the following hypothesis:

The experimental syllabus results in greater knowledge and skills as opposed to the existing syllabi regarding the transformations in the Euclidean plane, their properties and their algebraic structure.

Since the inability to get a simple random sample, in the period 20 January 20 May 2014, we carried out an experiment with voluntary participation organized in two groups, a control group and an experimental one, each consisting of 25 students with advanced mathematical knowledge and skills from the high school of science - Module \(A\). The experiment included the following stages:

– distribution of the students in a control and an experimental group, the basic criterion being that the students have approximately equal accomplishments in the previous years,

– the students of the control group, using the book (Mitrović et al., 1998) and additional literature, revised the material about the transformations in the Euclidean plane,

– the students of the experimental group, besides using the book (Malcheski et al., 20015), also had lessons which included studying the transformations in the Euclidean plane by using complex numbers,

– after carrying out the lessons, a test was given to the students aiming at assessing the knowledge of the students.

Next in line was analyzing the results, which included:

i) assessment of the validity of the test, i.e. whether the accomplishments of the students from the control and experimental group follow the normal distribution, and ii) comparison of the accomplishments of the students from the control and the experimental group, carried out by testing the hypotheses referring to the comparison of the mathematical expectations and the distributions of the accomplishments of the students of the control and the experimental group.

3. Results of the research

As we have already mentioned, the test was created with the purpose to check the accomplishments of the students from the control and experimental group regarding the transformations in the Euclidean plane, covered by both syllabi. The test contained 8 tasks and the students were given 90 minutes to complete it. The accomplishments of the students were assessed according to a proportional scale, because that is the one that allows application of the Kolmogorov-Smirnov test, that is used to test the hypothesis that the accomplishments of the students have an appropriate normal distribution, i.e. the quality of the test is assessed. Further on, the accomplishments of the students from the control and the experimental group were compared for each task and adequate remarks were given. The test given to the students from the two groups is listed below.

TEST

1. (7 points). Prove that the movement in the plane \(E^{2}\), which has two fixed points, is an identity.

2. (\(\mathbf{8}\) points). If the indirect symmetry in the plane \(E^{2}\) has at least one fixed point, then it is a line symmetry, whose axis contains the fixed point. Prove it!

3.(\(\mathbf{1 0}\) points). The direct isometry in the plane \(E^{2}\) with exactly one fixed point is a rotation. Prove it!

4. (15 points). A composition of line and central symmetry, whose center belongs to the line of symmetry, is a line symmetry. Prove it!

5. (\(\mathbf{1 0}\) points). The direct isometry in the plane \(E^{2}\), which does not have any fixed points is a translation. Prove it!

6. (15 points). The image of a circle under homothety is a circle, and vice versa, every two circles in the plane \(E^{2}\) are homothetic. Prove it!

7. (\(\mathbf{2 0}\) points).Construct a circle that passes through the points \(A\) and \(B\) and touches the given circle \(k\).

8. (15 points). Let \(O, A, B\) be three non collinear points and \(I k\) be an inversion with respect to the circle \(k(O, r)\). If \(A^{\prime}\) and \(B^{\prime}\) are images of the points \(A\) and \(B\) under the inversion \(I k\), then \(\triangle O A B \sim \triangle O B^{\prime} A^{\prime}\). Prove it!

Table 1 presents the results from the accomplishments of the students from the control group. Since we do not have information about the arithmetic mean and the standard deviation, and since this information is required for further analysis, they will be calculated by using the data presented in Table 1. The arithmetic mean, i.e. the average number of points scored by the students is \(\bar{x}_{25}=46,32\),and thus the standard deviationis \(\bar{s}_{25}=14,21\).

Table 1.Accomplishments of the students from the control groupStudentPoints per task123456787840100007760100007860100000061510000701000150070101500000010151000078601006078100806078808080781008080781001008078015100067810150080700151008878101510000
7010151000878100101540781001015607810151001007810150020077615108087801510150877615101508781000152015

Furthermore taking into consideration that the test is valid, objective and reliable, i.e. its measure characteristics are correct, then the accomplishments of the students should have normal distribution \(\mathrm{N}\left(46 ; 14^{2}\right)\). This is an indicator that firstly we should test the hypothesis \(H_{0}\) : the function of distribution \(F_{X}\) of the accomplishments of the students is equal to the normal distribution, i.e. the hypothesis \(H_{0}: F_{X}=\mathrm{N}\left(46 ; 14^{2}\right)\) . For this purpose, as we have already mentioned, we will apply the KolmogorovSmirnov test with a level of significance \(\alpha=0,05\). We apply the same procedures as in the previous analyses, for \(z_{i}=\cfrac{x_{i}-46}{14}\). The calculations are presented in Table 2.

According to the data in Table 2, the maximum value of \(\left|F_{n}(x)-F(x)\right|\) is \(d_{25}=0,09146\) and it is achieved for \(x=39\). Since the level of significance is \(\alpha=0,05\) and the number of data is \(n=25\),using the Kolmogorov’s criterion table, we find that \(d_{25 ; 0,05}=0,2639\). Since

\[ d_{25}=0,09146 \lt 0,2639=d_{25 ; 0,05} \]

we have no reason to dismiss the assumption that the distribution of the accomplishments of the students regarding the first test is \(\mathrm{N}\left(46 ; 14^{2}\right)\).

Table 2.Kolmogorov-Smirnov test of the second test of the control groupixin()niFx4614ixiz−=()iFx|()()|nFxFx−2910,04-1,210,113140,073143010,08-1,140,127140,047143120,16-1,070,142310,017693220,24-1,000,158660,081343510,28-0,790,214760,065243710,32-0,640,261090,058913920,40-0,500,308540,091464110,44-0,360,359420,080584310,48-0,210,416830,063174610,520,000,500000,020004820,600,140,555670,044335020,680,290,612260,067745410,720,570,715660,00434
5610,760,710,761150,001156020,841,000,841340,001346110,881,070,857690,022316310,921,210,886860,033146810,961,570,941790,018217511,002,070,980770,01923

The Kolmogorov-Smirnov test will be also used for the experimental group, as well. Table 3 presents the accomplishments in each task by all of the students from the experimental group separately.

Table 3.Accomplishments of the students from the experimentalgroupStudentPoints per task12345678789510000786510050786510500706151000570105015607810150050081015105007810010544781001056578851001057810085510781051000157701585505786151008570615105887810151006570101510858781061015567810510156678106101596781061015967810151086107761510158107810151015108781058152015

Analogously, as in the previous cases, the data in Table 3 shows that the arithmetic mean, i.e. the average number of points scored by the students is, and the standard deviation. According to this, in order to assess the measuring characteristics of the test, with respect to the students of the experimental group, we need to test the hypothesis: the function of distribution of the accomplishments of the students is equal to the appropriate normal distribution, i.e. the hypothesis. For this purpose we will once again apply the Kolmogorov-Smirnov test with a level of significance , for. The calculations are presented in Table 4.

Table 4.Kolmogorov-Smirnov test of the second test of the experimental groupixin()niFx4614ixiz−=()iFx|()()|nFxFx−3910,04-1,270,102040,062044120,12-1,130,129240,009244320,20-1,000,158660,041344510,24-0,870,192150,047854820,32-0,670,251430,068575110,36-0,470,319180,040825320,44-0,330,370700,069305510,48-0,200,420740,059265710,52-0,070,472100,047905920,600,070,527900,072106110,640,200,579300,060706310,680,330,629300,050706720,760,600,725750,034257120,840,870,807850,032157410,881,070,857690,022317810,921,330,908240,011768310,961,670,952540,007468811,0020,977250,02275

Accordingtothedata, whicharegiveninTable4,the maximumvalueof \(\left|F_{n}(x)-F(x)\right|\) is \(d_{25}=0,0721\) and it is achieved for \(x=59\).Since the level of significance is \(\alpha=0,05\), and the number of data is \(n=25\),using the Kolmogorov's criterion table, we find that \(d_{25 ; 0,05}=0,2639\). Since

\[ d_{25}=0,0721 \lt 0,2639=d_{25 ; 0,05} \]

we have no reason to dismiss the assumption that the distribution of the accomplishments of the students regarding the first test is \(\mathrm{N}\left(58 ; 15^{2}\right)\).

According to what was previously stated, we get the conclusion that the test accomplishments of both groups have normal distribution. The latter allows us to compare them. As we were able to see, the students from the experimental group scored 58.32 points on average, and the students from the control group scored 46.32 points on average. The mean square deviation of the students in experimental group is approximately 15 points, and that of the ones in the control is approximately 14 points. This means that the accomplishments of the students from the experimental group in comparison with the accomplishments of the students from the control group are higher by about \(25,91 \%\)., which indicates that studying these contents by applying the experimental syllabus produces significantly better results. Hence, we can conclude that the application of the experimental syllabus, related to the mentioned material, results in improved intersubject integration of the mathematics instruction in the high schools, as well as better preparation of the students for inclusion in the higher levels of education. We can get more precise confirmation for the above conclusion via the test for the difference of the mathematical expectations when distributions are unknown and large samples. This is possible because in the previous analyses we established that the accomplishments of the students in both groups have the normal distribution. In this case \[ \bar{x}_{25}=58,32, \bar{y}_{25}=46,32, n_{1}=n_{2}=25, \bar{s}_{x}=14,21 \text { and } \bar{s}_{y}=15,13 . \]

We will test the hypothesis \(H_{0}: m_{1} \leq m_{2}\) as opposed to the alternative hypothesis

\(H_{1}: m_{1} \gt m_{2}\), with the level of significance \(\alpha=0,01\)

\[ \cfrac{\bar{x}_{n_{1}}-\bar{y}_{n_{2}}}{\sqrt{n_{2} \bar{s}_{x}^{2}+n_{1} \bar{s}_{y}^{2}}} \sqrt{n_{1} n_{2}}=\cfrac{58,32-46,32}{\sqrt{25 \cdot 15,13^{2}+25 \cdot 14,21^{2}}} \sqrt{25 \cdot 25}=2,85 \]

from the table of normal distribution we find that \(z_{1-\alpha}=2,33\). The final result is

\[ \cfrac{\bar{x}_{n_{1}}-\bar{y}_{n_{2}}}{\sqrt{n_{2} \bar{s}_{x}^{2}+n_{1} \bar{s}_{y}^{2}}} \sqrt{n_{1} n_{2}}=2,85 \gt 2,33=z_{1-\alpha} \]

This means that we should dismiss the hypothesis \(H_{0}\), i.e. at a level of significance \(\alpha=0,01\) we accept that the mathematical expectation related to the accomplishments of the students from the experimental group is higher than the mathematical expectation related to the accomplishments of the students from the control group. Further on, the mean square deviations \(\bar{s}_{x}=14,21\) and \(\bar{s}_{y}=15,13\) differentiate insignificantly. Nevertheless, before making a final decision about whether to accept or dismiss the set hypothesis, we will compare the distributions of the accomplishments of the two groups. For this purpose, we will apply the test for equality of distributions of two independent normally distributed features, i.e. we will test the hypothesis \(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\) as opposed to the alternative hypothesis \(H_{0}: \sigma_{1}^{2} \neq \sigma_{2}^{2}\) with a level of significance \(\alpha=0,10\). The given conditions indicate that \(n_{1}=n_{2}=25, \bar{s}_{x}=14,21\) and \(\bar{s}_{y}=15,13\), therefore

\[ \cfrac{n_{1}\left(n_{2}-1\right) \bar{s}_{x}^{2}}{n_{2}\left(n_{1}-1\right) \bar{s}_{y}^{2}}=0,88085 \]

Further on, from the Fisher’s distribution table we find that

\[ F_{n_{1}-1, n_{2}-1 ; \cfrac{\alpha}{2}}=F_{24,24 ; 0,05}=2,66 \text { и } F_{n_{2}-1, n_{1}-1 ; \cfrac{\alpha}{2}}^{\prime}=F_{24,24 ; 0,05}=2,66, \] which means that \(F_{2}=2,66\) and \(F_{1}=\cfrac{1}{2,66}=0,38\). Therefore,

\[ F_{1}=0,38 \lt \cfrac{n_{1}\left(n_{2}-1\right) \bar{s}_{x}^{2}}{n_{2}\left(n_{1}-1\right) \bar{s}_{y}^{2}}=0,88085 \lt 2,66=F_{2}, \] we conclude that there is no reason to dismiss the hypothesis \(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\).

The above stated allows us to conclude that we should accept the set hypothesis, which means that regarding transformations in the Euclidean plane, their properties and their algebraic structure, the experimental syllabus allows gaining advanced knowledge and skills as opposed to the existing syllabi which do not.

Regarding the accomplishments of the students with regard to the separate tasks,

Tables 5 and 6 indicate that regarding all tasks, the students who studied the experimental syllabus accomplished better results than the students who studied the control syllabus.

Furthermore, regarding tasks 1-5 there is a little improvement in the results, which is due to the nature of the tasks, i.e. the fact that they are tasks which directly check the level of acquired operative and structural knowledge, i.e. the properties of the transformations in the Euclidean plane are directly applied. Namely, we directly check the assumption that studying the transformations in the Euclidean plane by using the complex numbers increases the inter-subject integration of the algebraic and geometrical contents which are intended to be learned by the high school students. The latter means that students are better prepared for inclusion in the higher levels of education

Table 5.Points and average score of points per taskTask12345678Total number ofpoints1611491881951949811261Average bystudent6,445,967,527,807,763,924,482,44Table 6.Points and average score of points per taskTask12345678Total number ofpoints168166213223224171191142Average by student6,726,648,528,928,966,847,645,68

Further on, Tables 5 and 6 allow us to see that the students from the experimental group present significantly higher operative and structural knowledge related to the tasks 6-8. Hence, task 6 was adequately and completely solved by 6 and partially by 1 student from the control group, whereas in the experimental group 8 students solved the task completely and 9 nine students provided partial solution of that task. The above results increased the total result of the students of the experimental group for about \(75 \%\).There is a similar difference in the accomplishments of the students from the experimental and the control group regarding (the) tasks 7 and 8, where the accomplishments of the experimental group are approximately \(70 \%\) higher. Further on, regarding the task 8 we can notice that the accomplishments of the students from the experimental group are 2.32 times higher than the accomplishments of the students from the control group. According to analysis of the solutions we can conclude that the differences in the accomplishments of the students from the two groups are mainly caused by the use of the analytical apparatus of the complex numbers for studying the transformations in the Euclidean plane. This is especially true for the study of the group properties and the classification of the similarities and movements. The latter is especially evident when solving the task 8, where the result gained by the students from the experimental group is mainly due to the use of the analytical apparatus when learning inversion.

4. Conclusion

One of the goals of mathematics instruction is for the students to acquire comprehensive, applicable and permanent knowledge, which should allow successful inclusion of the students in the higher levels of education. In order to accomplish this goal, a complete integration of the syllabi is necessary, but we must also have in mind that this integration should also allow adequate and appropriate differentiation of instruction. During the previous elaborations we proved that the introduction of the elective subject Geometry of a complex number, for which an adequate syllabus was created, and the enabling of the students to elect it along with other mathematical subjects, contribute for accomplishing the set goals. Namely, the research has demonstrated that:

– the experimental syllabus increases the inter-subject integration of the mathematics education, and

– by learning about transformations in the Euclidean plane with the experimental syllabus, students acquire advanced knowledge and skills, which increase their ability for inclusion in the higher levels of education.

Furthermore, this research has been carried out with a goal to figure the answer of the questions what, why and how a certain educational segment should be changed in order for the desired goals to be accomplished. Having in mind the previously mentioned, if any steps are made in the direction to change the curriculum and the syllabi in the education system in the future, it is desirable for the authorities to incorporate the algorithm elaborated in this research.

REFERENCES

Anevska, K., Gogovska, V., Malcheski, R. (2014). The role of complex numbers in interdisciplinary integration in mathematics teaching. Elsevier, Procedia: Social and Behavioral Sciences.

Anevska, K. (2014). Methodical approach for introduction of an exponential entry of complex numbers in secondary education, Proceedings of the Fifth Congress of UMM. Ohrid: UMM. (in print)

Malcheski, R., Grozdev, S., Anevska, K. (2015). Geometry of complex number. Sofia: Arhimed.

Mitrović, M., Ognjanović, S., Veljković, M., Petković, Lj., Lazarević, N. (1998). Geometrija za I razred Matematičke gimnazije. Beograd: Krug.

Карамата, Ј. (1950). Комплексан број, са применом на елементарну геометрију. Београд: Научна књига.

2025 година
Книжка 6
ENHANCING STUDENT MOTIVATION AND ACHIEVEMENT THROUGH DIGITAL MIND MAPPING

Mikloš Kovač, Mirjana Brdar, Goran Radojev, Radivoje Stojković

OPTIMIZATION VS BOOSTING: COMPARISON OF STRATEGIES ON EDUCATIONAL DATASETS TO EXPLORE LOW-PERFORMING AT-RISK AND DROPOUT STUDENTS

Ranjit Paul, Asmaa Mohamed, Peren Jerfi Canatalay, Ashima Kukkar, Sadiq Hussain, Arun K. Baruah, Jiten Hazarika, Silvia Gaftandzhieva, Esraa A. Mahareek, Abeer S. Desuky, Rositsa Doneva

ARTIFICIAL INTELLIGENCE AS A TOOL FOR PEDAGOGICAL INNOVATIONS IN MATHEMATICS EDUCATION

Stanka Hadzhikoleva, Maria Borisova, , Borislava Kirilova

Книжка 4
Книжка 3
МОДЕЛИ НА ВЕРОЯТНОСТНИ ПРОСТРАНСТВА В ОЛИМПИАДНИ ЗАДАЧИ

Драгомир Грозев, Станислав Харизанов

Книжка 1
A NOTE ON A GENERALIZED DYNAMICAL SYSTEM OCCURS IN MODELLING “THE BATTLE OF THE SEXES”: CHAOS IN SOCIOBIOLOGY

Nikolay Kyurkchiev, Anton Iliev, Vesselin Kyurkchiev, Angel Golev, Todorka Terzieva, Asen Rahnev

EDUCATIONAL RESOURCES FOR STUDYING MIDSEGMENTS OF TRIANGLE AND TRAPEZOID

Toni Chehlarova1), Neda Chehlarova2), Georgi Gachev

2024 година
Книжка 6
ВЪЗМОЖНОСТИ ЗА ИЗГРАЖДАНЕ НА МЕЖДУПРЕДМЕТНИ ВРЪЗКИ МАТЕМАТИКА – ИНФОРМАТИКА

Елена Каращранова, Ирена Атанасова, Надежда Борисова

Книжка 5
FRAMEWORK FOR DESIGNING VISUALLY ORIENTATED TOOLS TO SUPPORT PROJECT MANAGEMENT

Dalibor Milev, Nadezhda Borisova, Elena Karashtranova

3D ОБРАЗОВАТЕЛЕН ПОДХОД В ОБУЧЕНИЕТО ПО СТЕРЕОМЕТРИЯ

Пеньо Лебамовски, Марияна Николова

Книжка 4
DYNAMICS OF A NEW CLASS OF OSCILLATORS: MELNIKOV’S APPROACH, POSSIBLE APPLICATION TO ANTENNA ARRAY THEORY

Nikolay Kyurkchiev, Tsvetelin Zaevski, Anton Iliev, Vesselin Kyurkchiev, Asen Rahnev

Книжка 3
РАЗСТОЯНИЯ МЕЖДУ ЗАБЕЛЕЖИТЕЛНИ ТОЧКИ И НЕРАВЕНСТВА В ИЗПЪКНАЛ ЧЕТИРИЪГЪЛНИК

Йордан Табов, Станислав Стефанов, Красимир Кънчев, Хаим Хаимов

USING AI TO IMPROVE ANSWER EVALUATION IN AUTOMATED EXAMS

Georgi Cholakov, Asya Stoyanova-Doycheva

Книжка 2
ON INTEGRATION OF STEM MODULES IN MATHEMATICS EDUCATION

Elena Karashtranova, Aharon Goldreich, Nadezhda Borisova

Книжка 1
STUDENT SATISFACTION WITH THE QUALITY OF A BLENDED LEARNING COURSE

Silvia Gaftandzhieva, Rositsa Doneva, Sadiq Hussain, Ashis Talukder, Gunadeep Chetia, Nisha Gohain

MODERN ROAD SAFETY TRAINING USING GAME-BASED TOOLS

Stefan Stavrev, Ivelina Velcheva

ARTIFICIAL INTELLIGENCE FOR GOOD AND BAD IN CYBER AND INFORMATION SECURITY

Nikolay Kasakliev, Elena Somova, Margarita Gocheva

2023 година
Книжка 6
QUALITY OF BLENDED LEARNING COURSES: STUDENTS’ PERSPECTIVE

Silvia Gaftandzhieva, Rositsa Doneva, Sadiq Hussain, Ashis Talukder, Gunadeep Chetia, Nisha Gohain

МОДЕЛ НА ЛЕОНТИЕВ С MS EXCEL

Велика Кунева, Мариян Милев

Книжка 5
AREAS ASSOCIATED TO A QUADRILATERAL

Oleg Mushkarov, Nikolai Nikolov

ON THE DYNAMICS OF A ClASS OF THIRD-ORDER POLYNOMIAL DIFFERENCE EQUATIONS WITH INFINITE NUMBER OF PERIOD-THREE SOLUTIONS

Jasmin Bektešević, Vahidin Hadžiabdić, Midhat Mehuljić, Sadjit Metović, Haris Lulić

СИСТЕМА ЗА ИЗВЛИЧАНЕ И ВИЗУАЛИЗАЦИЯ НА ДАННИ ОТ ИНТЕРНЕТ

Георги Чолаков, Емил Дойчев, Светла Коева

Книжка 4
MULTIPLE REPRESENTATIONS OF FUNCTIONS IN THE FRAME OF DISTANCE LEARNING

Radoslav Božić, Hajnalka Peics, Aleksandar Milenković

INTEGRATED LESSONS IN CALCULUS USING SOFTWARE

Pohoriliak Oleksandr, Olga Syniavska, Anna Slyvka-Tylyshchak, Antonina Tegza, Alexander Tylyshchak

Книжка 3
ПРИЛОЖЕНИЕ НА ЕЛЕМЕНТИ ОТ ГЕОМЕТРИЯТА НА ЧЕТИРИЪГЪЛНИКА ЗА РЕШАВАНЕ НА НЕСТАНДАРТНИ ЗАДАЧИ

Йордан Табов, Веселин Ненков, Асен Велчев, Станислав Стефанов

Книжка 2
Книжка 1
НОВА ФОРМУЛА ЗА ЛИЦЕ НА ЧЕТИРИЪГЪЛНИК (ЧЕТИВО ЗА VII КЛАС)

Йордан Табов, Асен Велчев, Станислав Стефанов, Хаим Хаимов

2022 година
Книжка 6
MOBILE GAME-BASED MATH LEARNING FOR PRIMARY SCHOOL

Margarita Gocheva, Nikolay Kasakliev, Elena Somova

Книжка 5
SECURITY ANALYSIS ON CONTENT MANAGEMENT SYSTEMS

Lilyana Petkova, Vasilisa Pavlova

MONITORING OF STUDENT ENROLMENT CAMPAIGN THROUGH DATA ANALYTICS TOOLS

Silvia Gaftandzhieva, Rositsa Doneva, Milen Bliznakov

TYPES OF SOLUTIONS IN THE DIDACTIC GAME “LOGIC MONSTERS”

Nataliya Hristova Pavlova, Michaela Savova Toncheva

Книжка 4
PERSONAL DATA PROCESSING IN A DIGITAL EDUCATIONAL ENVIRONMENT

Evgeniya Nikolova, Mariya Monova-Zheleva, Yanislav Zhelev

Книжка 3
Книжка 2
STEM ROBOTICS IN PRIMARY SCHOOL

Tsanko Mihov, Gencho Stoitsov, Ivan Dimitrov

A METAGRAPH MODEL OF CYBER PROTECTION OF AN INFORMATION SYSTEM

Emiliya Koleva, Evgeni Andreev, Mariya Nikolova

Книжка 1
CONVOLUTIONAL NEURAL NETWORKS IN THE TASK OF IMAGE CLASSIFICATION

Larisa Zelenina, Liudmila Khaimina, Evgenii Khaimin, D. Khripunov, Inga Zashikhina

INNOVATIVE PROPOSALS FOR DATABASE STORAGE AND MANAGEMENT

Yulian Ivanov Petkov, Alexandre Ivanov Chikalanov

APPLICATION OF MATHEMATICAL MODELS IN GRAPHIC DESIGN

Ivaylo Staribratov, Nikol Manolova

РЕШЕНИЯ НА КОНКУРСНИ ЗАДАЧИ БРОЙ 6, 2021 Г.

Задача 1. Дадени са различни естествени числа, всяко от които има прос- ти делители, не по-големи от . Докажете, че произведението на някои три от тези числа е точен куб. Решение: числата са представим във вида . Нека разгледаме квадрат

2021 година
Книжка 6
E-LEARNING DURING COVID-19 PANDEMIC: AN EMPIRICAL RESEARCH

Margarita Gocheva, Nikolay Kasakliev, Elena Somova

Книжка 5
ПОДГОТОВКА ЗА XXV МЛАДЕЖКА БАЛКАНИАДА ПО МАТЕМАТИКА 2021

Ивайло Кортезов, Емил Карлов, Мирослав Маринов

EXCEL’S CALCULATION OF BASIC ASSETS AMORTISATION VALUES

Vehbi Ramaj, Sead Rešić, Anes Z. Hadžiomerović

EDUCATIONAL ENVIRONMENT AS A FORM FOR DEVELOPMENT OF MATH TEACHERS METHODOLOGICAL COMPETENCE

Olha Matiash, Liubov Mykhailenko, Vasyl Shvets, Oleksandr Shkolnyi

Книжка 4
LEARNING ANALYTICS TOOL FOR BULGARIAN SCHOOL EDUCATION

Silvia Gaftandzhieva, Rositsa Doneva, George Pashev, Mariya Docheva

Книжка 3
THE PROBLEM OF IMAGES’ CLASSIFICATION: NEURAL NETWORKS

Larisa Zelenina, Liudmila Khaimina, Evgenii Khaimin, D. Khripunov, Inga Zashikhina

MIDLINES OF QUADRILATERAL

Sead Rešić, Maid Omerović, Anes Z. Hadžiomerović, Ahmed Palić

ВИРТУАЛЕН ЧАС ПО МАТЕМАТИКА

Севдалина Георгиева

Книжка 2
MOBILE MATH GAME PROTOTYPE ON THE BASE OF TEMPLATES FOR PRIMARY SCHOOL

Margarita Gocheva, Elena Somova, Nikolay Kasakliev, Vladimira Angelova

КОНКУРСНИ ЗАДАЧИ БРОЙ 2/2021 Г.

Краен срок за изпращане на решения: 0 юни 0 г.

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 1, 2021

Краен срок за изпращане на решения: 0 юни 0 г.

Книжка 1
СЕДЕМНАДЕСЕТА ЖАУТИКОВСКА ОЛИМПИАДА ПО МАТЕМАТИКА, ИНФОРМАТИКА И ФИЗИКА АЛМАТИ, 7-12 ЯНУАРИ 2021

Диян Димитров, Светлин Лалов, Стефан Хаджистойков, Елена Киселова

ОНЛАЙН СЪСТЕЗАНИЕ „VIVA МАТЕМАТИКА С КОМПЮТЪР“

Петър Кендеров, Тони Чехларова, Георги Гачев

2020 година
Книжка 6
ABSTRACT DATA TYPES

Lasko M. Laskov

Книжка 5
GAMIFICATION IN CLOUD-BASED COLLABORATIVE LEARNING

Denitza Charkova, Elena Somova, Maria Gachkova

NEURAL NETWORKS IN A CHARACTER RECOGNITION MOBILE APPLICATION

L.I. Zelenina, L.E. Khaimina, E.S. Khaimin, D.I. Antufiev, I.M. Zashikhina

APPLICATIONS OF ANAGLIFIC IMAGES IN MATHEMATICAL TRAINING

Krasimir Harizanov, Stanislava Ivanova

МЕТОД НА ДЕЦАТА В БЛОКА

Ивайло Кортезов

Книжка 4
TECHNOLOGIES AND TOOLS FOR CREATING ADAPTIVE E-LEARNING CONTENT

Todorka Terzieva, Valya Arnaudova, Asen Rahnev, Vanya Ivanova

Книжка 3
MATHEMATICAL MODELLING IN LEARNING OUTCOMES ASSESSMENT (BINARY MODEL FOR THE ASSESSMMENT OF STUDENT’S COMPETENCES FORMATION)

L. E. Khaimina, E. A. Demenkova, M. E. Demenkov, E. S. Khaimin, L. I. Zelenina, I. M. Zashikhina

PROBLEMS 2 AND 5 ON THE IMO’2019 PAPER

Sava Grozdev, Veselin Nenkov

Книжка 2
ЗА ВЕКТОРНОТО ПРОСТРАНСТВО НА МАГИЧЕСКИТЕ КВАДРАТИ ОТ ТРЕТИ РЕД (В ЗАНИМАТЕЛНАТА МАТЕМАТИКА)

Здравко Лалчев, Маргарита Върбанова, Мирослав Стоимиров, Ирина Вутова

КОНКУРЕНТНИ ПЕРПЕНДИКУЛЯРИ, ОПРЕДЕЛЕНИ ОТ ПРАВИЛНИ МНОГОЪГЪЛНИЦИ

Йоана Христова, Геновева Маринова, Никола Кушев, Светослав Апостолов, Цветомир Иванов

A NEW PROOF OF THE FEUERBACH THEOREM

Sava Grozdev, Hiroshi Okumura, Deko Dekov

PROBLEM 3 ON THE IMO’2019 PAPER

Sava Grozdev, Veselin Nenkov

Книжка 1
GENDER ISSUES IN VIRTUAL TRAINING FOR MATHEMATICAL KANGAROO CONTEST

Mark Applebaum, Erga Heller, Lior Solomovich, Judith Zamir

KLAMKIN’S INEQUALITY AND ITS APPLICATION

Šefket Arslanagić, Daniela Zubović

НЯКОЛКО ПРИЛОЖЕНИЯ НА ВЪРТЯЩАТА ХОМОТЕТИЯ

Сава Гроздев, Веселин Ненков

2019 година
Книжка 6
DISCRETE MATHEMATICS AND PROGRAMMING – TEACHING AND LEARNING APPROACHES

Mariyana Raykova, Hristina Kostadinova, Stoyan Boev

CONVERTER FROM MOODLE LESSONS TO INTERACTIVE EPUB EBOOKS

Martin Takev, Elena Somova, Miguel Rodríguez-Artacho

ЦИКЛОИДА

Аяпбергенов Азамат, Бокаева Молдир, Чурымбаев Бекнур, Калдыбек Жансуйген

КАРДИОИДА

Евгений Воронцов, Никита Платонов

БОЛГАРСКАЯ ОЛИМПИАДА ПО ФИНАНСОВОЙ И АКТУАРНОЙ МАТЕМАТИКЕ В РОССИИ

Росен Николаев, Сава Гроздев, Богдана Конева, Нина Патронова, Мария Шабанова

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се намерят всички полиноми, които за всяка реална стойност на удовлетворяват равенството Татяна Маджарова, Варна Задача 2. Правоъгълният триъгълник има остри ъгли и , а центърът на вписаната му окръжност е . Точката , лежаща в , е такава, че и . Симетралите

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 1, 2019

Задача 1. Да се намерят всички цели числа , за които

Книжка 5
ДЪЛБОКО КОПИЕ В C++ И JAVA

Христина Костадинова, Марияна Райкова

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се намери безкрайно множество от двойки положителни ра- ционални числа Милен Найденов, Варна

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 6, 2018

Задача 1. Точката е левият долен връх на безкрайна шахматна дъска. Една муха тръгва от и се движи само по страните на квадратчетата. Нека е общ връх на някои квадратчета. Казва- ме, че мухата изминава пътя между и , ако се движи само надясно и нагоре. Ако точките и са противоположни върхове на правоъгълник , да се намери броят на пътищата, свърз- ващи точките и , по които мухата може да мине, когато: а) и ; б) и ; в) и

Книжка 4
THE REARRANGEMENT INEQUALITY

Šefket Arslanagić

АСТРОИДА

Борислав Борисов, Деян Димитров, Николай Нинов, Теодор Христов

COMPUTER PROGRAMMING IN MATHEMATICS EDUCATION

Marin Marinov, Lasko Laskov

CREATING INTERACTIVE AND TRACEABLE EPUB LEARNING CONTENT FROM MOODLE COURSES

Martin Takev, Miguel Rodríguez-Artacho, Elena Somova

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се реши уравнението . Христо Лесов, Казанлък Задача 2. Да се докаже, че в четириъгълник с перпендикулярни диагонали съществува точка , за която са изпълнени равенствата , , , . Хаим Хаимов, Варна Задача 3. В правилен 13-ъгълник по произволен начин са избрани два диа- гонала. Каква е вероятността избраните диагонали да не се пресичат? Сава Гроздев, София, и Веселин Ненков, Бели Осъм

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 5, 2018

Задача 1. Ако и са съвършени числа, за които целите части на числата и са равни и различни от нула, да се намери .

Книжка 3
RESULTS OF THE FIRST WEEK OF CYBERSECURITY IN ARKHANGELSK REGION

Olga Troitskaya, Olga Bezumova, Elena Lytkina, Tatyana Shirikova

DIDACTIC POTENTIAL OF REMOTE CONTESTS IN COMPUTER SCIENCE

Natalia Sofronova, Anatoliy Belchusov

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Краен срок за изпращане на решения 30 ноември 2019 г.

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 4, 2018

Задача 1. Да се намерят всички тройки естествени числа е изпълнено равенството: а)

Книжка 2
ЕЛЕКТРОНЕН УЧЕБНИК ПО ОБЗОРНИ ЛЕКЦИИ ЗА ДЪРЖАВЕН ИЗПИТ В СРЕДАТА DISPEL

Асен Рахнев, Боян Златанов, Евгения Ангелова, Ивайло Старибратов, Валя Арнаудова, Слав Чолаков

ГЕОМЕТРИЧНИ МЕСТА, ПОРОДЕНИ ОТ РАВНОСТРАННИ ТРИЪГЪЛНИЦИ С ВЪРХОВЕ ВЪРХУ ОКРЪЖНОСТ

Борислав Борисов, Деян Димитров, Николай Нинов, Теодор Христов

ЕКСТРЕМАЛНИ СВОЙСТВА НА ТОЧКАТА НА ЛЕМОАН В ЧЕТИРИЪГЪЛНИК

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

A TRIANGLE AND A TRAPEZOID WITH A COMMON CONIC

Sava Grozdev, Veselin Nenkov

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Христо Лесов, Казанлък Задача 2. Окръжност с диаметър и правоъгълник с диагонал имат общ център. Да се докаже, че за произволна точка M от е изпълне- но равенството . Милен Найденов, Варна Задача 3. В изпъкналия четириъгълник са изпълнени равенства- та и . Точката е средата на диагонала , а , , и са ортоганалните проекции на съответно върху правите , , и . Ако и са средите съответно на отсечките и , да се докаже, че точките , и лежат на една права.

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 3, 2018

Задача 1. Да се реши уравнението . Росен Николаев, Дико Суружон, Варна Решение. Въвеждаме означението , където . Съгласно това означение разлежданото уравнение придобива вида не е решение на уравнението. Затова са възможни само случаите 1) и 2) . Разглеж- даме двата случая поотделно. Случай 1): при е изпълнено равенството . Тогава имаме:

Книжка 1
PROBLEM 6. FROM IMO’2018

Sava Grozdev, Veselin Nenkov

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 2, 2018

Задача 1. Да се намери най-малкото естествено число , при което куба с целочислени дължини на ръбовете в сантиметри имат сума на обемите, рав- на на Христо Лесов, Казанлък Решение: тъй като , то не е куб на ес- тествено число и затова . Разглеждаме последователно случаите за . 1) При разглеждаме естествени числа и , за които са изпълнени релациите и . Тогава то , т.е. . Освен това откъдето , т.е. .Така получихме, че . Лесно се проверява, че при и няма естествен

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 1. Да се намерят всички цели числа , за които

2018 година
Книжка 6
„ЭНЦИКЛОПЕДИЯ ЗАМЕЧАТЕЛЬНЫХ ПЛОСКИХ КРИВЫХ“ – МЕЖДУНАРОДНЫЙ СЕТЕВОЙ ИССЛЕДОВАТЕЛЬСКИЙ ПРОЕКТ В РАМКАХ MITE

Роза Атамуратова, Михаил Алфёров, Марина Белорукова, Веселин Ненков, Валерий Майер, Генадий Клековкин, Раиса Овчинникова, Мария Шабанова, Александр Ястребов

A NEW MEANING OF THE NOTION “EXPANSION OF A NUMBER”

Rosen Nikolaev, Tanka Milkova, Radan Miryanov

Книжка 5
ИТОГИ ПРОВЕДЕНИЯ ВТОРОЙ МЕЖДУНАРОДНОЙ ОЛИМПИАДЬI ПО ФИНАНСОВОЙ И АКТУАРНОЙ МАТЕМАТИКЕ СРЕДИ ШКОЛЬНИКОВ И СТУДЕНТОВ

Сава Гроздев, Росен Николаев, Мария Шабанова, Лариса Форкунова, Нина Патронова

LEARNING AND ASSESSMENT BASED ON GAMIFIED E-COURSE IN MOODLE

Mariya Gachkova, Martin Takev, Elena Somova

УЛИТКА ПАСКАЛЯ

Дарья Коптева, Ксения Горская

КОМБИНАТОРНИ ЗАДАЧИ, СВЪРЗАНИ С ТРИЪГЪЛНИК

Росен Николаев, Танка Милкова, Катя Чалъкова

Книжка 4
ЗА ПРОСТИТЕ ЧИСЛА

Сава Гроздев, Веселин Ненков

ИНЦЕНТЪР НА ЧЕТИРИЪГЪЛНИК

Станислав Стефанов

ЭПИЦИКЛОИДА

Инкар Аскар, Камила Сарсембаева

ГИПОЦИКЛОИДА

Борислав Борисов, Деян Димитров, Иван Стефанов, Николай Нинов, Теодор Христов

Книжка 3
ПОЛИНОМИ ОТ ТРЕТА СТЕПЕН С КОЛИНЕАРНИ КОРЕНИ

Сава Гроздев, Веселин Ненков

ЧЕТИРИДЕСЕТ И ПЕТА НАЦИОНАЛНА СТУДЕНТСКА ОЛИМПИАДА ПО МАТЕМАТИКА

Сава Гроздев, Росен Николаев, Станислава Стоилова, Веселин Ненков

Книжка 2
TWO INTERESTING INEQUALITIES FOR ACUTE TRIANGLES

Šefket Arslanagić, Amar Bašić

ПЕРФЕКТНА ИЗОГОНАЛНОСТ В ЧЕТИРИЪГЪЛНИК

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

НЯКОИ ТИПОВЕ ЗАДАЧИ СЪС СИМЕТРИЧНИ ЧИСЛА

Росен Николаев, Танка Милкова, Радан Мирянов

Книжка 1
Драги читатели,

където тези проценти са наполовина, в Източна Европа те са около 25%, в

COMPUTER DISCOVERED MATHEMATICS: CONSTRUCTIONS OF MALFATTI SQUARES

Sava Grozdev, Hiroshi Okumura, Deko Dekov

ВРЪЗКИ МЕЖДУ ЗАБЕЛЕЖИТЕЛНИ ТОЧКИ В ЧЕТИРИЪГЪЛНИКА

Станислав Стефанов, Веселин Ненков

КОНКУРСНИ ЗАДАЧИ НА БРОЯ

Задача 2. Да се докаже, че всяка от симедианите в триъгълник с лице разделя триъгълника на два триъгълника, лицата на които са корени на урав- нението където и са дължините на прилежащите на симедианата страни на три- ъгълника. Милен Найденов, Варна Задача 3. Четириъгълникът е описан около окръжност с център , като продълженията на страните му и се пресичат в точка . Ако е втората пресечна точка на описаните окръжности на триъгълниците и , да се докаже, че Хаим Х

РЕШЕНИЯ НА ЗАДАЧИТЕ ОТ БРОЙ 2, 2017

Задача 1. Да се определи дали съществуват естествени числа и , при които стойността на израза е: а) куб на естествено число; б) сбор от кубовете на две естествени числа; в) сбор от кубовете на три естествени числа. Христо Лесов, Казанлък Решение: при и имаме . Следова- телно случай а) има положителен отговор. Тъй като при число- то се дели на , то при и имаме е естестве- но число. Следователно всяко число от разглеждания вид при деление на дава ос

2017 година
Книжка 6
A SURVEY OF MATHEMATICS DISCOVERED BY COMPUTERS. PART 2

Sava Grozdev, Hiroshi Okumura, Deko Dekov

ТРИ ИНВАРИАНТЫ В ОДНУ ЗАДА

Ксения Горская, Дарья Коптева, Асхат Ермекбаев, Арман Жетиру, Азат Бермухамедов, Салтанат Кошер, Лили Стефанова, Ирина Христова, Александра Йовкова

GAMES WITH

Aldiyar Zhumashov

SOME NUMERICAL SQUARE ROOTS (PART TWO)

Rosen Nikolaev, Tanka Milkova, Yordan Petkov

ЗАНИМАТЕЛНИ ЗАДАЧИ ПО ТЕМАТА „КАРТИННА ГАЛЕРИЯ“

Мирослав Стоимиров, Ирина Вутова

Книжка 5
ВТОРОЙ МЕЖДУНАРОДНЫЙ СЕТЕВОЙ ИССЛЕДОВАТЕЛЬСКИЙ ПРОЕКТ УЧАЩИХСЯ В РАМКАХ MITE

Мария Шабанова, Марина Белорукова, Роза Атамуратова, Веселин Ненков

SOME NUMERICAL SEQUENCES CONCERNING SQUARE ROOTS (PART ONE)

Rosen Nikolaev, Tanka Milkova, Yordan Petkov

Книжка 4
ГЕНЕРАТОР НА ТЕСТОВЕ

Ангел Ангелов, Веселин Дзивев

INTERESTING PROOFS OF SOME ALGEBRAIC INEQUALITIES

Šefket Arslanagić, Faruk Zejnulahi

PROBLEMS ON THE BROCARD CIRCLE

Sava Grozdev, Hiroshi Okumura, Deko Dekov

ПРИЛОЖЕНИЕ НА ЛИНЕЙНАТА АЛГЕБРА В ИКОНОМИКАТА

Велика Кунева, Захаринка Ангелова

СКОРОСТТА НА СВЕТЛИНАТА

Сава Гроздев, Веселин Ненков

Книжка 3
НЯКОЛКО ПРИЛОЖЕНИЯ НА ТЕОРЕМАТА НА МЕНЕЛАЙ ЗА ВПИСАНИ ОКРЪЖНОСТИ

Александра Йовкова, Ирина Христова, Лили Стефанова

НАЦИОНАЛНА СТУДЕНТСКА ОЛИМПИАДА ПО МАТЕМАТИКА

Сава Гроздев, Росен Николаев, Веселин Ненков

СПОМЕН ЗА ПРОФЕСОР АНТОН ШОУРЕК

Александра Трифонова

Книжка 2
ИЗКУСТВЕНА ИМУННА СИСТЕМА

Йоанна Илиева, Селин Шемсиева, Светлана Вълчева, Сюзан Феимова

ВТОРИ КОЛЕДЕН ЛИНГВИСТИЧЕН ТУРНИР

Иван Держански, Веселин Златилов

Книжка 1
ГЕОМЕТРИЯ НА ЧЕТИРИЪГЪЛНИКА, ТОЧКА НА МИКЕЛ, ИНВЕРСНА ИЗОГОНАЛНОСТ

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

2016 година
Книжка 6
ПЕРВЫЙ МЕЖДУНАРОДНЫЙ СЕТЕВОЙ ИССЛЕДОВАТЕЛЬСКИЙ ПРОЕКТ УЧАЩИХСЯ В РАМКАХ MITE

Мария Шабанова, Марина Белорукова, Роза Атамуратова, Веселин Ненков

НЕКОТОРЫЕ ТРАЕКТОРИИ, КОТОРЫЕ ОПРЕДЕЛЕНЫ РАВНОБЕДРЕННЫМИ ТРЕУГОЛЬНИКАМИ

Ксения Горская, Дарья Коптева, Даниил Микуров, Еркен Мудебаев, Казбек Мухамбетов, Адилбек Темирханов, Лили Стефанова, Ирина Христова, Радина Иванова

ПСЕВДОЦЕНТЪР И ОРТОЦЕНТЪР – ЗАБЕЛЕЖИТЕЛНИ ТОЧКИ В ЧЕТИРИЪГЪЛНИКА

Веселин Ненков, Станислав Стефанов, Хаим Хаимов

FUZZY LOGIC

Reinhard Magenreuter

GENETIC ALGORITHM

Reinhard Magenreuter

Книжка 5
NEURAL NETWORKS

Reinhard Magenreuter

Книжка 4
АКТИВНО, УЧАСТВАЩО НАБЛЮДЕНИЕ – ТИП ИНТЕРВЮ

Христо Христов, Христо Крушков

ХИПОТЕЗАТА В ОБУЧЕНИЕТО ПО МАТЕМАТИКА

Румяна Маврова, Пенка Рангелова, Елена Тодорова

Книжка 3
ОБОБЩЕНИЕ НА ТЕОРЕМАТА НА ЧЕЗАР КОШНИЦА

Сава Гроздев, Веселин Ненков

Книжка 2
ОЙЛЕР-ВЕН ДИАГРАМИ ИЛИ MZ-КАРТИ В НАЧАЛНАТА УЧИЛИЩНА МАТЕМАТИКА

Здравко Лалчев, Маргарита Върбанова, Ирина Вутова, Иван Душков

ОБВЪРЗВАНЕ НА ОБУЧЕНИЕТО ПО АЛГЕБРА И ГЕОМЕТРИЯ

Румяна Маврова, Пенка Рангелова

Книжка 1
STATIONARY NUMBERS

Smaiyl Makyshov

МЕЖДУНАРОДНА ЖАУТИКОВСКА ОЛИМПИАДА

Сава Гроздев, Веселин Ненков

2015 година
Книжка 6
Книжка 5
Книжка 4
Книжка 3
МОТИВАЦИОННИТЕ ЗАДАЧИ В ОБУЧЕНИЕТО ПО МАТЕМАТИКА

Румяна Маврова, Пенка Рангелова, Зара Данаилова-Стойнова

Книжка 2
САМОСТОЯТЕЛНО РЕШАВАНЕ НА ЗАДАЧИ С EXCEL

Пламен Пенев, Диана Стефанова

Книжка 1
ГЕОМЕТРИЧНА КОНСТРУКЦИЯ НА КРИВА НА ЧЕВА

Сава Гроздев, Веселин Ненков

2014 година
Книжка 6
КОНКУРЕНТНОСТ, ПОРОДЕНА ОТ ТАНГЕНТИ

Сава Гроздев, Веселин Ненков

Книжка 5
ИНФОРМАТИКА В ШКОЛАХ РОССИИ

С. А. Бешенков, Э. В. Миндзаева

ОЩЕ ЕВРИСТИКИ С EXCEL

Пламен Пенев

ДВА ПОДХОДА ЗА ИЗУЧАВАНЕ НА УРАВНЕНИЯ В НАЧАЛНАТА УЧИЛИЩНА МАТЕМАТИКА

Здравко Лалчев, Маргарита Върбанова, Ирина Вутова

Книжка 4
ОБУЧЕНИЕ В СТИЛ EDUTAINMENT С ИЗПОЛЗВАНЕ НА КОМПЮТЪРНА ГРАФИКА

Христо Крушков, Асен Рахнев, Мариана Крушкова

Книжка 3
ИНВЕРСИЯТА – МЕТОД В НАЧАЛНАТА УЧИЛИЩНА МАТЕМАТИКА

Здравко Лалчев, Маргарита Върбанова

СТИМУЛИРАНЕ НА ТВОРЧЕСКА АКТИВНОСТ ПРИ БИЛИНГВИ ЧРЕЗ ДИНАМИЧЕН СОФТУЕР

Сава Гроздев, Диана Стефанова, Калина Василева, Станислава Колева, Радка Тодорова

ПРОГРАМИРАНЕ НА ЧИСЛОВИ РЕДИЦИ

Ивайло Старибратов, Цветана Димитрова

Книжка 2
ФРАКТАЛЬНЫЕ МЕТО

Валерий Секованов, Елена Селезнева, Светлана Шляхтина

Книжка 1
ЕВРИСТИКА С EXCEL

Пламен Пенев

SOME INEQUALITIES IN THE TRIANGLE

Šefket Arslanagić

2013 година
Книжка 6
Книжка 5
МАТЕМАТИЧЕСКИЕ РЕГАТЬI

Александр Блинков

Книжка 4
Книжка 3
АКАДЕМИК ПЕТЪР КЕНДЕРОВ НА 70 ГОДИНИ

чл. кор. Юлиан Ревалски

ОБЛАЧНИ ТЕХНОЛОГИИ И ВЪЗМОЖНОСТИ ЗА ПРИЛОЖЕНИЕ В ОБРАЗОВАНИЕТО

Сава Гроздев, Иванка Марашева, Емил Делинов

СЪСТЕЗАТЕЛНИ ЗАДАЧИ ПО ИНФОРМАТИКА ЗА ГРУПА Е

Ивайло Старибратов, Цветана Димитрова

Книжка 2
ЕКСПЕРИМЕНТАЛНАТА МАТЕМАТИКА В УЧИЛИЩЕ

Сава Гроздев, Борислав Лазаров

МАТЕМАТИКА С КОМПЮТЪР

Сава Гроздев, Деко Деков

ЕЛИПТИЧЕН АРБЕЛОС

Пролет Лазарова

Книжка 1
ФРАГМЕНТИ ОТ ПАМЕТТА

Генчо Скордев

2012 година
Книжка 6
ДВЕ ДИДАКТИЧЕСКИ СТЪЛБИ

Сава Гроздев, Светлозар Дойчев

ТЕОРЕМА НА ПОНСЕЛЕ ЗА ЧЕТИРИЪГЪЛНИЦИ

Сава Гроздев, Веселин Ненков

ИЗЛИЧАНЕ НА ОБЕКТИВНИ ЗНАНИЯ ОТ ИНТЕРНЕТ

Ивайло Пенев, Пламен Пенев

Книжка 5
ДЕСЕТА МЕЖДУНАРОДНА ОЛИМПИАДА ПО ЛИНГВИСТИКА

д–р Иван А. Держански (ИМИ–БАН)

ТЕОРЕМА НА ВАН ОБЕЛ И ПРИЛОЖЕНИЯ

Тодорка Глушкова, Боян Златанов

МАТЕМАТИЧЕСКИ КЛУБ „СИГМА” В СВЕТЛИНАТА НА ПРОЕКТ УСПЕХ

Сава Гроздев, Иванка Марашева, Емил Делинов

I N M E M O R I A M

На 26 септември 2012 г. след продължително боледуване ни напусна проф. дпн Иван Ганчев Донев. Той е първият професор и първият доктор на науките в България по методика на обучението по математика. Роден е на 6 май 1935 г. в с. Страхилово, В. Търновско. След завършване на СУ “Св. Кл. Охридски” става учител по математика в гр. Свищов. Тук той организира първите кръжоци и със- тезания по математика. През 1960 г. Иван Ганчев печели конкурс за асистент в СУ и още през следващата година започ

Книжка 4
Книжка 3
СЛУЧАЙНО СЪРФИРАНЕ В ИНТЕРНЕТ

Евгения Стоименова

Книжка 2
SEEMOUS OLYMPIAD FOR UNIVERSITY STUDENTS

Sava Grozdev, Veselin Nenkov

EUROMATH SCIENTIFIC CONFERENCE

Sava Grozdev, Veselin Nenkov

FIVE WAYS TO SOLVE A PROBLEM FOR A TRIANGLE

Šefket Arslanagić, Dragoljub Milošević

ПРОПОРЦИИ

Валя Георгиева

ПЪТЕШЕСТВИЕ В СВЕТА НА КОМБИНАТОРИКАТА

Росица Керчева, Румяна Иванова

ПОЛЗОТВОРНА ПРОМЯНА

Ивайло Старибратов

Книжка 1
ЗА ЕЛЕКТРОННОТО ОБУЧЕНИЕ

Даниела Дурева (Тупарова)

МАТЕМАТИКАТА E ЗАБАВНА

Веселина Вълканова

СРАВНЯВАНЕ НА ИЗРАЗИ С КВАДРАТНИ КОРЕНИ

Гинка Бизова, Ваня Лалева

©2005 Национално Издателство Аз-Буки ООД. Всички права запазени.