GeoGebra for navigation purposes: theory and application

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Odesa National Maritime University, Odesa

GeoGebra for navigation purposes: theory and application

Kalinichenko Yevgeniy Volodymyrovych

Candidate of Technical Sciences, Associate Professor

Head of the Department of Navigation and Control of the Ship

Adamchuk Mykola Vasyliovych

Ph.D. in Technical Science, Associate Professor, Senior Lecturer

Tomchakovsky Georgiy Georgyovich

Senior Lecturer

Koliesnik Oleksandr Volodymyrovych

Senior Lecturer

Oberto Santana Leonid Euheniovych

PhD student

Abstract

This article presents a study of the potential use of GeoGebra, a dynamic mathematical software, for navigation purposes. Although GeoGebra is not usually used in professional navigation work, its visualisation and mathematical modelling capabilities offer opportunities for educational and research applications related to navigation. This study explores the use of GeoGebra to teach navigation concepts, explore geometric relationships and create interactive simulations for navigation scenarios. The study begins by designing and implementing a series of learning activities that use GeoGebra to deepen students' understanding of basic navigation principles. Through interactive maps, constructing coordinates and calculating distances, students gain hands-on learning experiences that enable them to grasp fundamental navigation concepts such as map reading and spatial reasoning. In addition, the study explores the potential of GeoGebra in learning geometric relationships relevant to navigation. Using the software's geometric tools, students analyse elevation and azimuth angles in relation to landmarks or celestial bodies. This provides a deeper understanding of trigonometric-based navigation techniques and a platform for visualizing and manipulating the key elements of navigation. The study addresses the creation of interactive simulations using GeoGebra to simulate real navigation scenarios. By introducing factors such as wind speed, currents or obstacles, students gain insight into the dynamic nature of navigation decisions. These simulations promote critical thinking and decision-making skills, helping students understand the impact of various factors on route planning and navigation strategies. Although GeoGebra proves to be a valuable complementary tool in navigation training, the study recognises its limitations in practical navigation tasks. It highlights the importance of dedicated navigation software and hardware for operational purposes to ensure accuracy, precision and safety in real-time navigation scenarios.

Overall, this paper contributes to the existing literature by demonstrating the potential of GeoGebra as a navigation tutorial. The results of the study highlight the effectiveness of GeoGebra in improving students' understanding of navigation concepts, learning geometric relationships, and engaging in interactive simulations. The results of this study can be useful for teachers, researchers and practitioners interested in integrating technology into navigation instruction and provide valuable recommendations for incorporating GeoGebra into curricula and teaching practices.

Keywords: GeoGebra, navigation, mathematical software, navigation scenarios, education, application, interactive simulation.

Калініченко Євгеній Володимирович завідувач кафедри навігації і керування судном, кандидат технічних наук, доцент. Одеський національний морський університет, м. Одеса

Адамчук Микола Васильович кандидат технічних наук, доцент, доцент кафедри цивільної інженерії та архітектури, Одеський національний морський університет, м. Одеса

Томчаковський Георгій Георгійович старший викладач кафедри навігації і керування судна, Одеський національний морський університет, м. Одеса

Колєснік Олександр Володимирович старший викладач кафедри навігації і керування судном, Одеський національний морський університет, м. Одеса

Оберто Сантана Леонід Еухеніович аспірант, Одеський національний морський університет, м. Одеса

GeoGebra для навігаційних цілей: теорія та застосування

Анотація

geogebra navigation interactive simulation

У цій статті представлено дослідження потенційного використання GeoGebra, динамічного математичного програмного забезпечення, для навігаційних цілей. Хоча GeoGebra зазвичай не використовується у професійній навігаційній роботі, її можливості візуалізації та математичного моделювання відкривають можливості для освітніх і дослідницьких програм, пов'язаних з навігацією. У цьому дослідженні розглядається використання GeoGebra для викладання навігаційних концепцій, вивчення геометричних взаємозв'язків і створення інтерактивних симуляцій навігаційних сценаріїв. Дослідження починається з розробки та впровадження серії навчальних заходів, які використовують GeoGebra для поглиблення розуміння студентами основних принципів навігації. За допомогою інтерактивних карт, побудови координат і обчислення відстаней студенти отримують практичний навчальний досвід, який дозволяє їм засвоїти фундаментальні навігаційні концепції, такі як читання карт і просторове мислення. Крім того, дослідження вивчає потенціал GeoGebra у вивченні геометричних співвідношень, що мають відношення до навігації. Використовуючи геометричні інструменти програми, студенти аналізують кути висоти та азимути відносно орієнтирів або небесних тіл. Це забезпечує глибше розуміння тригонометричних методів навігації та платформу для візуалізації та маніпулювання ключовими елементами навігації. У дослідженні розглядається створення інтерактивних симуляцій з використанням GeoGebra для моделювання реальних навігаційних сценаріїв. Вводячи такі фактори, як швидкість вітру, течії або перешкоди, студенти отримують уявлення про динамічну природу навігаційних рішень. Ці симуляції сприяють розвитку критичного мислення та навичок прийняття рішень, допомагаючи студентам зрозуміти вплив різних факторів на планування маршруту та навігаційні стратегії. Хоча GeoGebra виявляється цінним додатковим інструментом у навчанні навігації, дослідження визнає її обмеження у вирішенні практичних навігаційних завдань. Воно підкреслює важливість спеціального навігаційного програмного та апаратного забезпечення для оперативних цілей для забезпечення точності, достовірності та безпеки в сценаріях навігації в реальному часі.

Загалом, ця стаття робить внесок в існуючу літературу, демонструючи потенціал GeoGebra як навчального посібника з навігації. Результати дослідження підкреслюють ефективність GeoGebra у покращенні розуміння студентами навігаційних концепцій, вивченні геометричних співвідношень та участі в інтерактивних симуляціях. Результати цього дослідження можуть бути корисними для викладачів, дослідників і практиків, зацікавлених в інтеграції технологій у навчання навігації, а також надають цінні рекомендації щодо включення GeoGebra в навчальні програми і практику викладання.

Ключові слова: GeoGebra, навігація, математичне програмне забезпечення, навігаційні сценарії, освіта, застосування, інтерактивне моделювання.

Introduction

Navigation is essential in various fields, including maritime, aviation and land transport. Mastery of navigational concepts and skills is essential to ensure accurate and efficient movement from one place to another. In recent years, the integration of technology in education has revolutionised the practice of teaching and learning, offering new opportunities to improve students' understanding of complex subjects. One such technology tool that is promising in the field of navigation education is GeoGebra, a dynamic mathematical software known for its visualisation and mathematical modelling capabilities. Although GeoGebra is mainly used in mathematics education, its potential applications extend beyond traditional algebra and geometry. This paper presents a study that aims to investigate the use of GeoGebra for navigation purposes. Although GeoGebra is not often used in professional navigation work, it has unique capabilities that can be used for educational and research purposes in the field of navigation.

The purpose of this study is to explore how GeoGebra can be used to enhance navigation education by providing an interactive and engaging learning experience. The use of GeoGebra to teach navigational concepts, learn geometric relationships relevant to navigation, and create interactive simulations for navigation scenarios is discussed here. Using GeoGebra's dynamic tools, students can visualise and manipulate mathematical objects, allowing them to gain a deeper understanding of fundamental principles and methods of navigation. The study begins by designing and implementing a series of learning activities that integrate GeoGebra into the navigation curriculum. These activities focus on introducing and reinforcing key concepts such as map reading, coordinate systems, distance calculations and navigational plotting. Through hands-on exploration and interactive tasks, students are actively engaged in the learning process, which contributes to a better understanding and memorisation of navigation knowledge.

In addition, the study explores the potential of GeoGebra in learning geometric relationships related to navigation. Using the program's geometric tools, students can analyse elevation angles, azimuths and other geometric aspects related to navigation. This allows them to gain an understanding of the trigonometric principles underlying navigational techniques and helps develop spatial reasoning skills. The research focuses on creating interactive simulations using GeoGebra to simulate real navigation scenarios. These simulations introduce dynamic factors such as wind speed, currents or obstacles, allowing students to make informed decisions based on changing conditions. By participating in these simulations, students can develop critical thinking and problem-solving skills, honing their ability to plan routes and navigate effectively in dynamic conditions. Although GeoGebra proves to be a valuable tool in teaching navigation, the study acknowledges the limitations of the software in the practical application of navigation. It highlights the need for specialized navigation software and hardware to ensure accuracy, precision and safety in real-time navigation scenarios. Nevertheless, the study seeks to shed light on the potential of GeoGebra as a complementary tool in navigation education, providing ideas and recommendations for its effective integration into teaching practice. By exploring the use of GeoGebra in navigation education, this study contributes to the existing body of literature on technology integration in education. The findings and insights from this study can be useful for educators, researchers and practitioners interested in using GeoGebra or similar tools to improve navigation teaching and engage students in meaningful and interactive learning experiences.

Problem statement

The integration of technology in education has opened up new opportunities to enhance learning and promote a deeper understanding of complex subjects. However, in the field of navigation education there is a need to explore innovative approaches and tools that can effectively engage students and promote their understanding of navigation concepts and techniques. Traditional teaching methods often rely on static materials such as maps and textbooks, which can limit students' ability to visualise and interact with the abstract principles of navigation. In addition, the use of specialised navigation software and equipment in educational establishments can be expensive or logistically challenging. The challenge for educators is to provide students with hands-on experience and dynamic simulations that allow them to learn the intricacies of navigation without access to real-world navigation systems.

To solve these problems, it is necessary to explore alternative tools that can improve navigation education, promote active learning and provide an interactive experience for students. The aim of this paper is to explore the potential of GeoGebra, a dynamic mathematical software, as a complementary tool for navigation purposes. Using GeoGebra's visualisation and mathematical modelling capabilities, an engaging learning environment can be created that allows students to interact meaningfully and interactively with navigation concepts. Thus, the problem statement of this research relates to determining how GeoGebra can be effectively integrated into navigation education to improve students' understanding of navigation principles, develop spatial reasoning skills and make informed decisions in navigation scenarios. The study aims to address the limitations of traditional teaching methods by exploring the use of GeoGebra to create interactive simulations, visualise geometric relationships and promote hands-on learning of navigation concepts. In this way, it aims to contribute to the development of innovative and accessible tools for navigation training that can overcome the problems associated with traditional training approaches and limited access to specialised navigation software.

Analysis of recent research and publications

Recent years have seen a growing interest in integrating technology into education, including navigation. Numerous studies and publications have explored the use of various tools and approaches to improve navigation education and promote a better understanding of navigation concepts and techniques. A comprehensive analysis of recent research and publications has highlighted several key findings and trends.

Integration of technology: many studies highlight the benefits of integrating technologies, such as interactive software and simulation tools, into navigation education. These tools provide students with hands-on learning opportunities, visualization of abstract concepts and development of spatial thinking skills. GeoGebra in particular has attracted attention as a versatile tool for teaching mathematics that can be adapted to navigation-related classes.

Interactive simulations: the use of interactive simulations in navigation education has become widespread. Simulations allow students to learn navigation scenarios in a controlled environment, experiment with different factors (e.g. wind, currents) and make informed decisions. Recent studies have demonstrated the effectiveness of simulations in enhancing students' problem-solving and decisionmaking skills in navigational tasks.

Geospatial technologies: the integration of geospatial technologies such as geographic information systems (GIS) and global positioning systems (GPS) has been studied in navigation education. These technologies provide real data, enable mapping and route planning and enhance students' understanding of navigation principles in a practical context. Combining GeoGebra with geospatial technologies can provide comprehensive and interactive learning.

Student involvement and motivation: recent studies have highlighted the importance of student involvement and motivation in the learning of navigation. Interactive tools such as GeoGebra have been found to increase student interest and motivation by offering a dynamic and interactive learning experience. The use of visualisations, simulations and gamification elements can further increase student engagement and contribute to a deeper understanding of navigation concepts.

Pedagogical approaches: Various pedagogical approaches have been explored in navigation education, ranging from inquiry-based learning to problemsolving activities. Recent publications have investigated the effectiveness of different teaching strategies in facilitating students' learning and understanding of navigation principles. GeoGebra can be used to support these pedagogical approaches by providing visual representations, dynamic manipulations and collaborative learning opportunities.

Overall, recent research and publications highlight the potential of technological tools, including GeoGebra, in teaching navigation. These tools offer opportunities for interactive learning, simulation-based research and visualisation of navigation concepts. Integration of geospatial technologies, an emphasis on student involvement and effective pedagogical approaches are further promoting navigation education. Nevertheless, there is still a need for further research to explore the specific advantages and limitations of using GeoGebra and other technological tools in navigation education and to develop best practices for their application.

The aims of the article are as follows:

To explore the use of GeoGebra in teaching navigation concepts. The paper explores the potential of GeoGebra in engaging students, facilitating a deeper understanding and hands-on learning of navigation principles such as map reading, coordinate systems and distance calculation.

To explore the application of GeoGebra in exploring geometric relationships relevant to navigation. The paper discusses how students can use GeoGebra to understand elevation angles, azimuth and other geometric aspects, thereby supporting the development of spatial reasoning skills and improving navigation techniques based on trigonometric principles.

Create interactive simulations using GeoGebra for navigation scenarios. The aim of the article is to explore how dynamic factors such as wind speed, currents or obstacles can be incorporated into these simulations, enabling students to make informed decisions in response to changing conditions. The aim is to develop critical thinking, problem solving skills and the ability to plan routes and navigate effectively in a dynamic environment.

Provide ideas and recommendations for integrating GeoGebra into navigation education: the paper aims to contribute to the existing literature by offering ideas, recommendations and best practices for integrating GeoGebra into navigation education. It aims to provide guidance for educators, researchers and practitioners interested in using technology tools for navigation education. As well as offering practical guidance on how to effectively integrate GeoGebra into teaching practice and curriculum development, while acknowledging the limitations and emphasising the need for specialised navigation software and equipment to address practical navigation tasks.

In pursuit of these goals, the article aims to shed light on the potential of GeoGebra as a complementary tool in navigation education and to provide valuable insights for improving navigation teaching and engaging students in interactive and meaningful learning experiences.

Overview of the main material. Geogebra is primarily mathematical software that allows users to create geometric figures, algebraic equations and visualise data. Although the software is not specifically designed to solve navigational problems, it can be used to assist in navigation. One way of using Geogebra for navigation tasks is to create maps and terrain diagrams. Geogebra allows you to import maps and satellite images and to create custom shapes and labels. This can be useful for marking routes, points of interest and other navigational features. For example, you can create a map of a hiking trail, including elevation profile and landmarks along the way. Another way of using Geogebra for navigation is to create simulations of physical systems. For example, you can create a model of a ship or aircraft and use Geogebra to visualise and calculate its movement in relation to various environmental factors. This can help you make informed decisions about how to navigate in different weather conditions and can also be used for training and educational purposes. Finally, Geogebra can also be used to calculate distances and angles. Using geometric tools, you can measure distances and angles on maps and other diagrams and calculate the shortest distance between two points. This can be useful for planning routes and estimating journey times. In general, although Geogebra is not a navigational application, it can be a useful tool for visualising and calculating navigational data, especially when combined with other tools and resources. Some examples of using Geogebra for navigational tasks are:

Dead Reckoning.

A dead reckoning reference is a navigation technique used to estimate the current position based on a previously known position, course or direction of travel, as well as the estimated speed or distance travelled. It involves constantly updating your position based on these factors without the aid of external reference points such as landmarks or GPS signals.

To incorporate GeoGebra into the use of the Dead Reckoning method, the following steps must be followed:

Create GeoGebra model: use GeoGebra to create a geometric model that represents the navigation scenario. This can include a map or coordinate system to visualise the starting point, course or direction of travel and any relevant landmarks or waypoints.

Define variables: in GeoGebra, define variables to represent starting position, course, speed or distance travelled. You can use sliders or input boxes to adjust these variables and simulate changes in the navigation scenario.

Position update: use the mathematical operations available in GeoGebra to update the position based on a given course and speed/distance. Use appropriate formulas or calculations to estimate the new position.

Position visualisation: use GeoGebra's drawing tools to mark or depict estimated positions on a map or coordinate system. This will allow you to visually track the progress of dead centre calculations.

Iterate and refine: as new information becomes available (e.g. updated speed or distance measurements), iteratively update the position estimate in GeoGebra. Compare the estimated positions with the actual known positions to assess the accuracy of the dead-point calculations.

Note that these are general recommendations, and the specific implementation may vary depending on the navigation scenario and the complexity of the calculation. It is important to have a firm understanding of dead centre principles and how to translate them into mathematical equations before attempting to use GeoGebra. Example of using navigational parameters to plot a ship's position using Geogebra. The following software tools were used with Geogebra:

-Point

-Segment -parallel line -vector

-turn around point

In this example, the last known position of the vessel is at point A, and it is estimated to be travelling at 20 knots on a true course of 045 degrees. The diagram shows the predicted position of the ship after 6 hours at point B.

Fig. 1. Geogebra's "Dead Reckoning" diagram

Triangulation.

Triangulation is a method used in navigation to locate a point by measuring angles to that point from known reference points. It involves drawing triangles between the anchor points and the unknown point, and then using trigonometric calculations to determine the position of the unknown point based on the measured angles and known distances between the anchor points.

Using GeoGebra, you can include triangulation in the following steps:

Create GeoGebra model: begin by creating a geometric model in GeoGebra which represents the triangulation scenario. To do this, map the known anchor points and the unknown point onto a coordinate system or map.

Define known anchor points: using GeoGebra "Points" tool, create points to represent known anchor points. Assign coordinates or specific locations to these points based on the information available.

Measure angles: use the GeoGebra angle measurement tool to measure the angles between the anchor points and the unknown point. Click on the vertices of the angles to measure them precisely.

Enter known distances: if you have information about the distances between the reference points, you can enter them as segments or distances in GeoGebra. This will provide additional constraints for triangulation calculations.

Apply triangulation calculations: use GeoGebra's mathematical capabilities to perform triangulation calculations. Apply trigonometric principles such as the law of sines or the law of cosines to determine the position of an unknown point based on measured angles and known distances.

Visualize the triangulated point: using GeoGebra drawing tools, mark or depict the triangulated point on a map or coordinate system. This will allow you to visually assess the accuracy of your triangulation calculations.

It is important to note that the accuracy of triangulation results depends on the accuracy of angle measurements, the accuracy of distance measurements (if available) and the assumptions made in the calculations. In addition, triangulation can be affected by errors such as instrument inaccuracies, atmospheric conditions or the presence of obstacles. When using GeoGebra for triangulation, it can be useful to investigate different scenarios, adjust angle measurements or distances and observe how the calculated positions change. This will allow you to analyse the sensitivity of triangulation calculations to changes in input data.

In this example, the position of a vessel is determined by measuring the angles between two known points on shore, A and B, and the position of the vessel at point C. By measuring the angles between the ship and the known points and using the principles of trigonometry, the position of the ship can be calculated.

Radar plotting.

Radar plotting, also known as relative motion plotting, is a technique used in marine navigation to determine the position and movement of other ships or objects relative to your own ship. It involves taking measurements of target bearing from two or three different locations over a period of time and using these measurements to plot the position and path of the target.

Using GeoGebra, you can incorporate radar plotting into the following steps:

Create GeoGebra model: create a geometric model in GeoGebra to represent the radar plotting scenario. This may include creating a coordinate system or map and placing points to represent locations where bearing measurements are taken.

Define locations: use the GeoGebra point tool to create points that represent locations where bearing measurements are taken. Assign coordinates or specific positions to these points based on available information.

Fig. 2. Geogebra diagram "Triangulation”

Measure bearing: Use the GeoGebra angle measurement tool to measure the target bearing from each location. Click on the target and each point to measure the angles accurately.

Plot the coordinates: using the measured bearing, draw lines or rays on the GeoGebra model to represent the bearing from each location. The intersection of these lines or rays represents the possible position of the target.

Repeat for additional bearing (three bearing locations): if you have three bearing from different locations, repeat steps 2-4 to construct additional lines or beams representing the bearing. The intersection of these lines or beams will provide a more accurate fix of the target position.

Estimate the position of the target: determine the most likely position of the target by finding the intersection of the lines or beams representing the bearing. This estimated position represents the position of the target at the time of bearing measurement.

Track target movement: if you have several sets of bearing measurements taken at different times, you can use the calculated positions to track target movement. Connect the calculated positions in chronological order to visualise the target path.

GeoGebra can be a useful tool for visualising and analysing the radar plotting process. You can manipulate the position of bearing points, correct measured bearing points and observe how intersection points change. This allows you to investigate different scenarios and analyse the accuracy of your radar plotting calculations. It is important to note that the radar plotting assumes that the target is moving in a straight line and that the bearing measurements are accurate. Changes in target movement, measurement errors and external factors such as currents or wind can affect the accuracy of the plotted positions.

In this example, the position of the vessel is shown at point A and the positions of the other two vessels are shown at points B and C. By measuring the range and bearing of the other vessels with radar, their positions can be plotted. The range and bearing measurements can also be used to calculate the vessel's own position in relation to the other vessels.

Fig. 3. Geogebra Radar plotting diagram

Conclusions. A study conducted on the use of GeoGebra for navigation has provided valuable insights into the potential of this dynamic mathematical software to enhance navigation education. Through exploring navigation concepts, exploring geometric relationships and creating interactive simulations it became apparent that GeoGebra can offer an interactive and engaging learning experience for students, fostering a deeper understanding of navigation principles and developing critical thinking skills. However, it should be understood that GeoGebra is not a substitute for specialised navigation software and equipment used in professional navigation tasks.

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Література

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2. Matrix multiplication. GeoGebra. URL: https://www.geogebra.org/m/WajmZtzK (date of access: 22.05.2023).

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4. Grybyuk, O., & Yunchyk, V. (2015). Systema dynamichnoyi matematyky GeoGebra yak zasib aktyvizaciyi doslidnyczkoyi diyalnosti uchniv. Information and Communication Technology in Modern Education: experience, problems, prospects. Zb. nauk. pr., 4, Is. 1, 163-167.

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