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2018/5, стр. 444 - 454

LEARNING AND ASSESSMENT BASED ON GAMIFIED E-COURSE IN MOODLE

Mariya Gachkova
E-mail: mariq.gachkova@gmail.com
Faculty of Mathematics and Informatics
University of Plovdiv “Paisii Hilendarski”
236, Bulgaria Blvd.
4003 Plovdiv, Bulgaria
Martin Takev
E-mail: takevm@gmail.com
Faculty of Mathematics and Informatics
University of Plovdiv “Paisii Hilendarski”
236, Bulgaria Blvd.
4003 Plovdiv, Bulgaria
Elena Somova
E-mail: eledel@uni-plovdiv.bg
Faculty of Mathematics and Informatics
Plovdiv University “Paisii Hilendarski”
236, Bulgaria Blvd.
4003 Plovdiv, Bulgaria

Резюме: The paper presents game-like elements suitable for the realization of a gamified e-course and their representation as components in the e-learning environment Moodle. An example of the design of a gamified e-learning course in Moodle is shown. The paper contains comparison charts for the assessment of the course participants. Feedback of the effectiveness of certain aspects of the gamified e-course is collected by a survey.

Ключови думи: gamification; e-learning course design; e-learning

I. Introduction

In accordance with the development of technology and the evolution of the learning environment, the needs of modern learners have changed. Educators often use games and game-like environments (Gachkova & Somova, 2016) to attract learner’s attention during the instructional process (Kocadere & Caglar, 2018). Gamification, the use of game elements in non-game contexts, has become a popular technique to enhance instructional outcomes in both education and organizational learning, and many recommendations have appeared regarding such learning environments1) (Kapp, 2012), (Landers & Callan, 2012).

Gamification of learning is an educational approach to motivate students to learn by using video game design and game elements in learning environments (Kapp, 2012). Gamification, broadly defined, is the process of defining the elements, comprising games, make those games fun and motivate the players to play them continually, and then using those same elements in a non-game context to influence behavior (Deterding, Dixon & Khaled, 2011).

Gamification can be provided in two variants (Kapp, 2012): structural gamification (without any changes to the learning content) and content gamification (consisting of altering the learning content to make it more game-like).

Some Learning Management Systems are using methods and ideas from games and propose gamification, like GENIE, TalentLMS, Moodle, Frog and Expertus One.

For example, GENIE 2 implements the following game-like elements: rewarding with badges and points for achieving learning goals, preparation of leaderboards for stimulating the competitive spirit, putting deadlines to tasks, and learning through gradually passing through levels.

The cloud-based platform for e-learning TalentLMS4) supports gamification through: giving points for performed actions (going through learning resources for example), collecting various badges for passed tests, receiving certificates and awards for finishing a course, re-certificate on certain time periods, leaderboards – displayed by charts and diagrams and passing the course by levels.

Although Moodle is not a platform designed for gamification, it has most of the popular game-like elements, which were explained in (Somova & Gachkova, 2016) and we will use in the current study.

The aim of the paper is to make a research on learner’s opinion about gamified approach in learning and the obtained impact in learning quality and efficiency, according to students. For this purpose, a gamified e-course has to be created.

Section 2 presents the used methodology approach in this case study. Section 3 shows the design of our e-course, including game-like elements and assignments. Participants in the course and their user types are determined in Section 4. Analysis of the of assignments’ assessment is presented in Section 5. A COLLES survey with participants and an analysis of the survey results are given in Section 6. The paper ends with a conclusion summarizing the contributions of the authors as well as with some ideas for future works.

II. Methodological Approach

The study is designed as a case study based on structural gamification (Kapp, 2012) of a standard learning environment. (Kapp, 2012) describes structural gamification as the application of game elements to propel a learner through content with no changes to the content itself. The content does not become gamelike, only the structure around the content. The primary focus behind this type of gamification is to motivate the learners to go through the content and to engage them in the process of learning through rewards (for example: gaining points for watching a video or completing an assignment).

Structural gamification in our study is achieved by including some gamebased elements in the e-course sources, that are explained in the next section. Participants use the gamified learning environment for ten weeks. Based on weekly assignments, the participants are evaluated. The research process consists of 4 phases:

– The design of the gamified e-course in e-learning environment;

– Determination of participant types in the case study;

– Analysis of the assessment of assignments included in the gamified e-course;

– Analysis of the answers collected by COLLES Survey included in the gamified e-course.

III. The Design of Gamified e-Learning Course

A gamified e-learning course is designed for use in this study. The course supports the discipline “Modeling and management of business processes” in the bachelor’s degree program “Business information technologies” at University of Plovdiv “Paisii Hilendarski”, Bulgaria. The e-course is realized in the learning management system Moodle3). In Table 1 gamification elements are presented as Moodle components.

The e-learning course is arranged so that each learning week corresponds to one level (totally 10 levels). The course uses 4 levels with entry requirements and rules.

Levels include quests (assignments), suitable to the current week’s subject. There are various quests, such as preparing a project, building a wiki, taking a test or reading an article. Some of the assignments are set as individual tasks and others have to be finished as a team work.

During the learning process, learners are awarded points for completing assignments. Only those with a certain score are able to pass onto the next level. The learners without sufficient score to unlock the next content, could re-send corrected assignments a second time (game-based learning model uses game cycle by (Garris, Ahlers & Driskell, 2002)).

8 assignments are set in the current gamified e-course (see Table 2), and 7 of them need assessment. 7 of the assignments are individual (marked with I in Table 2) and 1 is a group one (marked with G in Table 2). Levels 2, 4, 8 and 9 have entry requirements – a set of nested restrictions to apply complex logic, every level can start only if the following 3 conditions are met:

– The previous assignment(s) is finished;

– The previous assignment(s) is finished with the minimum required points in assessment (e.g. equal or greater than 30%);

– The deadline is met (timelines are set).

Table 1. Gamification elements used during the design of gamified e-course in Moodle

Gamication elementMoodle realizationLevelto unlock the levels)QuestIndividual or Group assignmentBadgeMoodle custom badgeLeader boardVisible report with the assignment’s points and ranksBonusPoints for nishing an assignmentRewardAdditional recourses with interesting facts, etc., unexpectedlyreceived after completing some work in the courseComboAdditional instructions/help for the next assignment or forreaching special requirements
Game RulesLearning process rulesStory/historyDescription of the learning resources and activitiesGame progressCourse progressStatusDi󰀨erent character is attached to each level (section)TeamMoodle GroupTime frameTime restrictions for tests and assignments

Out of the levels (marked with level 0 in Table 2), the participants have a group assignment to build a wiki resource. In the last level, learners have to answer the questions in a COLLES Survey.

Table 2. Gamified e-course components (by levels)

Level012345678910EntryRequirements––yes–yes–––yesyes–Assignment (I/G)GII–III–I–IForumyesyesyesyesyesyesyesyesyesyesyesBadges(B1/B2/B3)B3B1,B2B1,B2–B1,B2B1,B2B1,B2–B1,B2––Point Bonusesyesyesyes–yesyesyes–yes––Combos–yesyes–yesyesyes––––Rewardsyesyesyes–yesyesyes–yes–yes

The gamified e-course is designed to give badges for a set of requirements. On Table 3 the used three types of badges (B1/B2/B3) are presented, and each one of them has different requirements. There are two methods to achieve the badges in the course – automatically, when the user meets some certain criteria and manually – by the teacher, who can decide to give the badge to the student.

Students can receive different prizes: bonuses like points for finishing an assignment, special combos like additional instructions and help for the next assignment, or unexpected rewards like additional recourses with interesting facts, etc.

To provoke competitive spirit, learners’ scores (ordered from highest to lowest) are placed on a leader board, visible for all learners, with the points they earn throughout the course.

As a social component, students have forums in each level where they can share their accomplishments with other learners, share how they have achieved them or discuss the problems in quests.

Table 3. Types of badges set in gamified e-course

Type of BadgeDescriptionAchievement TypeFirst Finished (B1)The learner should be the rst that nishedand sent the assignment.AutomatedBest result (B2)The learner should have more than 80% fromthe maximum points.AutomatedBest group work (B3)The team should have the best result in theassignment. Each learner from the awardedteam receives the badge.Manual

Students receive different character (status) on each level (for example: Worker – level 1, Team Leader – level 2, …, President – level 10). Some story/history is added in the description of the learning resources and activities. For example, the story of assignment 2 is: To obtain the higher position “Team Leader” in the company “Progress Ltd” you have to model a diagram for the business process “Purchasing a property” with following activities and requirements: ….

At any time, students can see their course progress (current position in the course, to where are checked passed resources and activities by teacher or students themselves).

All the students, who have started with the gamified e-course and did not pass some of the levels, are left to continue with the standard e-course without gamified elements.

IV. Participants and User Types

1. Participants in the case study

For the analysis of the results and statistics, data is obtained from 113 undergraduate students, enrolled in the education program “Business information technologies” at the University of Plovdiv “Paisii Hilendarski”, Faculty of Mathematics and Informatics. Opportunity to choose between standard and gamified e-learning environment is given to the students. A total of 41 students have preferred to participate in the designed gamified e-course and 27 of them have completed the full list of assignments. There are three reasons for the remaining 14 participants to discontinue the gamified e-course:

– The participants did not meet the time criteria – the time limits are set for 4 assignments in the e-course;

– The participants did not meet the points criteria – for some of the assignments the participants could not meet the target for minimum points required;

– The participants prefer to continue learning in a standard e-course.

2. Determination of user types

One of the theoretical models outside the field of gamification suggests that the use of new technologies in training, including gamification, may lead to poor instructional outcomes when people are not comfortable and experienced with them (Landers & Callan, 2012). (Landers & Armstrong, 2017) concludes that the learner’s lack of experience with virtual worlds and poor attitude towards the value of virtual worlds for training would minimize any benefit from using virtual worlds.

Based on these studies, at the beginning of the course students have the ability to choose between gamified or standard e-learning environment. 41 of 113 students have preferred to participate in the gamified variant of the course.

(Bartle, 1996) examined players expectations and produced the first effort at player categorization, classifying players using their act/interact preferences and orientation (world/player). He considers 4 categories (killer, achiever, explorer and socializer) described in Table 4.

Table 4. Game elements according to the user type

User typeDenition (Bartle, 1996)Game elementsKillerTries to dominate other players by actingon the people in the environmentLeader board, Points, Courseprogress, Status, CombosAchieverActs in the world and care about theassignments in the environment in orderto winBadges, Levels, Courseprogress, BonusesExplorerInteracts with the world. They want toexplore the environment and discover asmany new things as possibleQuests (Individualassignments), Rewards,HistorySocializerInteracts with the people in an environmentand usually takes advantage of thecommunication function to socializeForums, Quests (GroupAssignments), History

According to (Kocadere & Caglar, 2018), learning environments should be designed to incorporate a variety of elements so that each player type is able to encounter those that attract them. We suggest some appropriate game elements in

Table 4 to attract each learning (player) type.

V. Analysis of the Assessment of Assignments

The main assessment of the students in the gamified e-course is done by the first six assignments in levels 1, 2, 4, 5, 6 and 8. At the end all students (from both variants of the course) fill in a test. The total number of students that passed the exam (in their first attempt) is 59, where 27 of them have finished the gamified course and 32 – the standard e-course.

Fig. 1 shows the extended chart with average students’ scores (in points, where the maximal points per assignment are 100) for each assignment and the count of participating students.

Figure 1. Average students’ scores vs. count of participating students

Fig. 2 shows the average scores of only those students, who have successfully completed the whole gamified course. It is obvious that the lower scores have been achieved for the first three assignments. We think this has mainly happened, because the previous students’ experience with gamification is poor or is missing.

Figure 2. Average assessment of students that entirely completed the gamified e-course (by levels)

This is also one of the reasons that many students discontinue the gamified course. The other reasons are lack of time to follow time requirements, preferences for easier learning process without many activities (regardless of the learning quality), not attracted of the game approach, etc. The percentages of discontinued students are as follows: after level 1 – 50%, level 2 – 29%, level 4 – 0%, level 5 – 14%, level 6 – 7% and level 8 – 0%. Thus, most of the participants (79%) discontinue the course after the first two assignments.

VI. Analysis of the Colles Survey

The gamified e-course is designed including COLLES (The Constructive On-Line Learning Environment Survey)5). The survey provides feedback on the effectiveness of certain aspects of online learning. By definition COLLES survey is carried out twice with the same questions; the first time, in the beginning of the course – to find out what the student wants (so called “Preferred”) and the second time, at the end of the learning – to find out what actually happened (so called “Actual”).

Current survey is divided into six categories: Practical importance, Self-criticism, Dialogic, Teaching help, Mutual assistance and Understandability. Students can choose from five-points Likert scale to answer the questions in COLLES survey. Possible values in the scale are scored as follows: 1 – Almost Never, 2 – Seldom, 3 – Sometimes, 4 – Often, 5 – Almost Always.

Figure 3 shows summarized results, divided into the six categories of the conducted COLLES survey.

According to students, the achieved “Teaching help” is higher than expected. In four categories (” Practical importance”, “Self-criticism”, “Dialogic” and “Mutual assistance”), students’ “preferred” answer is scored higher than the “actual” answer. But we have to take into account that the maximum difference calculated between both answers for “Practical importance” category is 0,91 (total of 5).

In category “Understandability” of the gamified course, the “actual” average score is equal to the “preferred” score of the involved students. This means that the participants are satisfied with the understandability of the course.

Figure 4 presents in details the results of COLLES survey for the category “Understandability”. The students’ opinions are shown in 4 categories: “understanding other students”, “other students understand me”, “understanding teacher” and “teacher understands me”.

Figure 3. COLLES survey results for gamified e-course by categories

The average results for “Actual” and “Preferred” answers in each category is more than 3 (Sometimes) total of 5. Presented as a percentage, average “Understandability” is more than 75%.

Figure 4. COLLES survey results for the category “Understandability” of the gamified e-course

Students give higher assessment to “Preferred” according to “Actual” state for the categories “teacher understands me” and “understanding other students” and lower assessment to others. It is obvious that students have obtained more understanding from teachers in the learning process than they expected. The same situation is with “other students understand me”. Something more, these values strongly incline to the answer “Almost Always”.

VII. Conclusions

The presented gamified e-learning course is proposed together with a standard e-learning course to give students options to choose the more appropriate environment for their learning. In the game-like environment there are 4 types of learners (players): killer, achiever, explorer and socializer, which are attracted to different activities and environment components. Our gamified learning course is intended for all learning (player) types, because the course design incorporates a variety of elements, so that each learning type is able to encounter those that attract them.

In conclusion, the current study is thought to provide further clarity about the result of designing and applying gamified learning courses. Due to the given analysis in Section 5 and 6, the results of this study can be summarized in 4 main points:

– Students, that participate in the gamified e-course, spend more time working on assignments in the learning environment, than the students who preferred the standard e-course;

– Students in the gamified e-course have higher grades than the students, attending the standard e-course;

– Students in the gamified e-course are searching for less explanations than students in the standard e-course;

– The average understandability of students in the gamified course is equal to their preferences and higher than 75%.

Our future work in this field continues with design and realization of a Moodle module for gamified learning, which will make the realization of a gamified course in Moodle with all game-like elements (some of them now missing) easier.

NOTES

1. Brousell, L. (2013). How gamification reshapes corporate training. http://www. cio.com/article/728268/How_Gamification_Reshapes_Corporate_Training. Accessed 09.07.2018

2. GENIE. http://www.growthengineering.co.uk/genie-content-authoring-tool/. Accessed 09.07.2018

3. Moodle. https://moodle.org. Accessed 09.07.2018

4. TalentLMS. http://www.talentlms.com/. Accessed 09.07.2018

5. The Constructivist On-Line Learning Environment Survey (COLLES). https:// surveylearning.moodle.com/colles/. Accessed 09.07.2018

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Gachkova, M. & Somova, E. (2016). Game-based approach in E-learning, IX National Conference „Education and Researches in Information Society”, 26 – 27.05.2016, Plovdiv, 2016.

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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

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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

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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

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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 .

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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)

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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

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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

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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

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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
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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

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Веселин Ненков, Станислав Стефанов, Хаим Хаимов

FUZZY LOGIC

Reinhard Magenreuter

GENETIC ALGORITHM

Reinhard Magenreuter

Книжка 5
NEURAL NETWORKS

Reinhard Magenreuter

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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
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