Problems In Geometry Research Articles

Learning trajectory of geometry proof construction: Studying the emerging understanding of the structure of Euclidean proof

This paper presents a learning trajectory of geometry proof (LTGP) for Indonesian prospective mathematics teachers (PMTs) during their first year of studies at an Indonesian university. The trajectory aims at PMTs’ progression of their understanding of the structure of proof and their proof construction abilities. We designed and implemented teaching materials with geometry problems based on the use of the flow-chart proof format and the model of understanding of proof structure from Miyazaki et al. (2017). We present an analysis of data from pre- and post-tests of proof construction problems, written answers to proof problems during intervention with 60 PMTs, and individual interviews with eight PMTs. We found that the intervention supports PMTs’ understanding of the structure of proof and their proof construction abilities. Our findings contribute to knowledge about teaching strategies to support students in their understanding and construction of a proof. From our findings, we suggest the use of the flow-chart proof format together with other more formal proof formats in creating, reading, and rewriting proof of geometric propositions and the use of open problems to encourage students to think forward and backwards interactively to help students plan for proof construction.

Analisis Pemahaman Konsep dalam Menyelesaikan Soal Geometri Dimensi Tiga Pada Peserta Didik SMK di Kota Makassar

This type of research is descriptive research. This study aims to determine the understanding of concepts in solving three-dimensional geometry problems in class XI students of SMK Muhammadiyah 3 Makassar. The population in this study were all students of class XI SMK Muhammadiyah 3 Makassar. Furthermore, selecting samples with Cluster Random Sampling technique. The selected sample in this study is class XI Accounting Department. From this sample, interviews will be continued with 9 students consisting of 3 students in the high category, 3 students in the medium category, and 3 students in the low category. The instruments in this study in the form of concept understanding tests and interview guidelines used to determine the understanding of concepts in solving three-dimensional geometry problems in class XI students of SMK Muhammadiyah 3 Makassar. The data analysis technique in this study is descriptive statistical analysis with the aim of describing the understanding of mathematical concepts of students in class XI SMK Muhammadiyah 3 Makassar. Based on the results of data analysis and discussion that has been stated, the conclusion in this study is the understanding of concepts in solving geometry problems of three dimensions in class XI students of SMK Muhammadiyah 3 Makassar is on the indicators: (1) Concept understanding on indicator I, namely giving examples and not examples of a concept by 91%, (2) Concept understanding on indicator II, namely presenting concepts in various forms of mathematical representation by 53%, and (3) Concept understanding on indicator III, namely applying concepts or algorithms in problem solving by 57%.

Solving Geometry Problems by Alternative Methods in Mathematics Education

Solving geometry problems is both difficult and interesting. Difficult because there is no general algorithm to solve more or less non-trivial problems as every single problem requires individual and creative approach. At the same time, this is a very interesting activity, because for almost every problem there are plenty of ways to solve it. In this work, we present the method of auxiliary circle divided into equal parts. This method allows finding solution algorithm for some geometry problems which are hard to solve by the method of additional constructions.

Students’ Errors in Solving Plane Geometry Problems Using E-Learning-based Diagnostic Tests

During the Covid-19 pandemic, the implementation of online and shift learning systems has been identified as one factor contributing to learning loss in mathematics problem-solving skills specifically in plane geometry problems. The primary objective of this research was to investigate the different types of errors committed by junior high school students in Banda Aceh, Indonesia, in solving plane geometry problems using e-learning diagnostic tests. A descriptive qualitative approach was adopted, and 24 Year 8 students were selected as subjects. The data were collected through tests and interviews, and data analysis involved data reduction, data presentation, data verification, and drawing conclusions. The findings revealed that the percentage of students' errors in solving conceptual problems, principal problems, and verbal problems were 76.39%, 68.75%, and 75%, respectively. Subsequently, the results from the second test and interviews with six students showed that students encountered several conceptual errors, including errors in stating angle symbols, errors in identifying the type of shape provided, errors in deducing information regarding properties, perimeter, and area of plane geometry, lack of knowledge regarding the application of the appropriate plane geometry formula, and inability to solve problems using the correct procedure. Principal errors were related to algebraic operations, while verbal errors involved translating problems and improper application of concepts and principles. The understanding of the types of errors will help teachers detect errors made the students early.

A three-dimensional solver for simulating detonation on curvilinear adaptive meshes

A generic solver in a parallel Cartesian adaptive mesh refinement framework is extended to simulate detonations on three-dimensional structured curvilinear meshes. A second-order accurate finite volume method is used with grid-aligned Riemann solvers for thermally perfect gas mixtures. Detailed, multi-step chemical kinetic mechanisms are employed and numerically incorporated with a splitting approach. The adaptive mesh refinement technique is applied to a mapped mesh using modified prolongation and restriction operators. The flux along the coarse-fine interface is considered in a correction procedure to ensure the conservation of the solver. The numerical accuracy, conservation and robustness of the simulations are verified and validated with suitable benchmark tests. The new solver is then used to simulate detonation problems in non-Cartesian geometries. A simulation is conducted of the three-dimensional detonation propagation in a 90-degree pipe bend. A detonation in a round tube is also simulated in a Galilean frame of reference. Both a rectangular mode and a spinning mode are observed in the simulations. In addition, the fundamental problem of detonation wave/boundary layer interaction is studied. The results show that the new solver can simulate high-speed reactive flows efficiently by the combined use of a curvilinear mapping with mesh adaptation.

ANALISIS KEMAMPUAN SPASIAL VISUALIZATION SISWA SEKOLAH DASAR DALAM PEMECAHAN MASALAH GEOMETRI

Spatial Visualization ability is related to geometry and has the ability to visualize spatial displacements and helps students to solve problems correctly. In solving geometric problems is the ability to solve problems by mastering the rules and concepts of geometry. This study aims to identify and analyze the lack of spatial ability which was carried out to analyze how the Spatial Visualization ability is applied to solving geometry problems for fourth grade students at Alam Islam Talasia School, Bogor City. This development uses a qualitative method. with a descriptive approach. This research begins with collecting various information needs, making observation grids, making research instruments, validating by experts, then carrying out data collection, and analyzing the results of the data obtained. The results of the study stated that there were students who did not yet have spatial visualization abilities and there were also students who had spatial visualization abilities, as well as several influencing factors.

Solving some elementary geometrical problems by Euclidean geometry’s methods

The article introduces a method of exploiting the Euclidian geometry to find solutions for elementary geometry problems. By analyzing some special coordinates in the solution, we will detect some extra points or lines to help solve elementary problems.

EFEKTIVITAS PEMBELAJARAN PENDEKATAN OUT SIDE THE BOX DALAM MENYELESAIKAN MASALAH KREATIF MAHASISWA PENDIDIKAN MATEMATIKA

This study examines the problem of learning effectiveness of the out side the box approach in solving creative problems in fifth semester mathematics education students at the University of Semarang in working on geometry problems. The sample technique used is cluster random sampling technique. The hypothesis test used is the one-sided test with the formula t test, completeness test, both tests are used to determine the effectiveness of learning. Based on the results of the study, the researchers draw the following conclusions; 1) Based on the results of this development research, it can be concluded that student learning outcomes in learning the out side the box approach in solving creative problems in mathematics education students in class VA are better than student learning outcomes in conventional learning class VB. 2) Based on the learning outcomes data for class students who use the out side the box learning approach in solving creative problems in mathematics education students seen from the percentage of completeness of learning outcomes achieving completeness

G1 interpolation of v-asymmetric data with arc-length constraints by Pythagorean-hodograph cubic splines

The interpolation problem of a sequence of points with prescribed segment arc-length by PH cubic spline is addressed. It turns out to be a v-asymmetric G1 Hermite interpolation problem with specified arc length. Using rectifying control polygon and reduction to a canonical form, it is shown that the problem can be expressed in terms of finding the real solutions to a system of nonlinear equations in two variables. A detailed and thorough analysis of the resulting system of nonlinear equations and closed form solutions are provided for any given data. It is confirmed that this construction of G1 Hermite interpolants of specified arc length admits at least two formal solutions, both of which have attractive shape properties and some (if any) must be discarded due to undesired looping behavior. The algorithm developed herein offers simple and efficient closed-form solutions to a fundamental constructive geometry problem that avoids the need for iterative numerical methods. Moreover, the algorithm is illustrated by several numerical examples.

A New Algorithm for Euclidean Shortest Paths in the Plane

Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (inSIAM Journal on Computing, 1999) gave an algorithm ofO(nlogn) time andO(nlogn) space, wherenis the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space toO(n)while the runtime of the algorithm is stillO(nlogn). In this article, we present a new algorithm ofO(n+hlogh) time andO(n)space, provided that a triangulation of the free space is given, wherehis the number of obstacles. The algorithm is better than the previous work whenhis relatively small. Our algorithm builds a shortest path map for a source pointsso that given any query pointt, the shortest path length fromstotcan be computed inO(logn) time and a shortests-tpath can be produced in additional time linear in the number of edges of the path.

A nearly throughput-maximum knotted intersection design and control for connected and automated vehicles

Traffic at urban intersections constitute a series of complex conflicting vehicular movements and contribute greatly to the problems of traffic congestion and air pollution. The emerging of connected and automated vehicle (CAV) technologies inspires innovative ideas in traffic management at intersections; it not only enables new control paradigms, but also allows for possibilities to revolutionize the geometric layouts of intersections. This study proposes a novel intersection design called knotted intersection (KI), to resolve the complexity of conflicting relations at intersections in a full CAV environment. This new design is associated with a set of neat control rules for vehicles to realize the smooth traffic operation at these intersections. The design problem of intersection geometry and control strategy is optimized jointly and formulated as a mixed-integer linear program (MILP), and some basic theoretical properties of KI design are derived, including nearly throughput-maximizing property and bounded within-intersection delay of vehicles. Furthermore, the KI design is then extended to a road network composed of multiple intersections to improve the overall efficiency. The optimal design of multiple knotted intersections (MKI) is developed to coordinate the traffic organization methods among intersections and a two-level solution method is proposed to solve the optimization efficiently. Numerical and simulation experiments are conducted at both the intersection level and network level under various cases, and the results validate the superior performance of the proposed methods compared to classical traffic signal control (TSC) strategies.

The Fun of Complex Numbers

By studying the chapter on complex numbers, one found that they are very different from real numbers. Complex numbers can solve problems that cannot be solved by real numbers, such as computation, geometry problems in the complex plane, etc. In this article one will introduce the basic operations of complex numbers, complex conjugates, algebraic and trigonometric forms of complex numbers and their transformations. And how to find the roots of complex numbers is something that one will cover in the text. When one talk about these knowledge points, one will combine the title to help one better explain and readers' understanding. Before writing this article, one read a lot of literature, which helped one learn more knowledge and made one more aware of complex numbers. Therefore, in this article, one will also refer to the literature one refer to and cite them.

Using activity method to address students’ problem-solving difficulties in circle geometry

This action research sought to address the problem-solving difficulties of students in circle geometry problems using activity method of teaching and learning. The study in investigated students’ difficulties in solving circle geometry problems. It also found out how activity method of teaching and learning helped address the students’ difficulties of solving circle geometry problems. Finally, it established the extent to which the activity method of teaching and learning affected the performance of students in circle geometry. The accessible population was all form two students at Zion College in the Volta Region of Ghana. It numbered 499. The sample was general science 2B students. The sample size was 27 students made up of 10 girls and 17 boys. The instruments for data collection were pre- and post-test. The pre-test was conducted to establish the existence of the difficulties of students in solving circle geometry problems. After establishing the problems, activity method of teaching and learning was used as an intervention to remedy the situation. A post-test was administered to measure the improvement. The data was presented using tables and figures. Inferential statistics was also used to analyze the data. From the analysis, activity method help the students solve complex questions, which demand multiple circle theorems to solve. It was recommended that mathematics teachers should always use activity method of teaching and learning in teaching circle geometry.

ГРАФИЧЕСКИЕ ДИСЦИПЛИНЫ И ИНФОРМАТИЗАЦИЯ ИНЖЕНЕРНОГО ОБРАЗОВАНИЯ

The development of information technology has given a significant impetus and progress both for various industries and life, as well as for education. The national program "Digital Economy of the Russian Federation" provides for ensuring the introduction of digital technologies in the economy and social sphere, and is associated with the National Project "Education (2019-2024)", which includes the Federal Project "Digital Educational Environment" based on the introduction of the target model of modern digital technologies into educational programs. Innovative methods and forms of training create a strategy for the professional training of specialists, and computer visibility of the forms of the part gives an idea of the assembly technology, allows you to perform competent drawings. The purpose of the study is to analyze the methods of solving the problems of descriptive geometry and engineering graphics together with the use of methods of surface formation by geometric methods, as well as on the basis of basic CAD operations COMPASS-3D. 
 In modern high school, in most cases, drawing is not studied, or only the most general concepts and definitions are studied. The existing course of school computer science deals only with general issues and practically does not give skills in drawing and modeling. As a result, applicants for the most part come to a technical university unprepared for the normal perception of geometric and graphic disciplines. In this regard, two priority tasks of further informatization of engineering education at the technical university have been formulated. On the one hand, this is the improvement of the methodology for teaching geometro-graphic disciplines, including in distance learning. On the other hand, it is the involvement of schoolchildren in various Olympiads and competitions held on the basis of technical universities to develop the initial skills of drawing and modeling simple objects.

Still Spinning: The Moser Spindle at Sixty

Summary The Moser spindle, a seemingly simple geometric object with seven vertices and 11 line segments of unit distance, was first introduced in the early 1960s. This article shows that the Moser spindle still plays a major role in the search for solutions to some of the most intriguing open problems in discrete geometry. Specifically, the problem of finding the chromatic number of the plane and the problem of finding the fractional chromatic number of the plane.

Improving middle school students' geometry problem solving ability through hands-on experience: An fNIRS study.

Hands-on learning is proposed as a prerequisite for mathematics learning in kindergarten and primary school. However, it remains unclear that whether hands-on experience aids understanding of geometry knowledge for middle school students. We also know little about the neural basis underlying the value of hands-on experience in math education. In this study, 40 right-handed Chinese students (20 boys and 20 girls) with different academic levels were selected from 126 seventh-grade students in the same school, who learnt "Axisymmetric of an Isosceles Triangle" in different learning style (hands-on operation vs. video observation). Half of them operated the concrete manipulatives while the other half watched the instructional videos. The learning-test paradigm and functional near-infrared spectroscopy (fNIRS) technique were used to compare the differences in geometry reasoning involved in solving well-structured problems and ill-structured problems. Behavioral results showed that hands-on experience promoted students' performances of geometry problem-solving. Students with lower academic level were more dependent on hands-on experience than those with higher academic level. The fNIRS results showed that meaningful hands-on experience with concrete manipulatives related to learning contents increased reactivation of the somatosensory association cortex during subsequent reasoning, which helped to improve the problem-solving performance. Hands-on experience also reduced students' cognitive load during the well-structured problem-solving process. These findings contribute to better understand the value of hands-on experience in geometry learning and the implications for future mathematics classroom practices.

An Efficient GNSS Coordinate Recognition Algorithm for Epidemic Management

Many highly contagious infectious diseases, such as COVID-19, monkeypox, chickenpox, influenza, etc., have seriously affected or currently are seriously affecting human health, economic activities, education, sports, and leisure. Many people will be infected or quarantined when an epidemic spreads in specific areas. These people whose activities must be restricted due to the epidemic are represented by targets in the article. Managing targets by using targeted areas is an effective option for slowing the spread. The Centers for Disease Control (CDC) usually determine management strategies by tracking targets in specific areas. A global navigation satellite system (GNSS) that can provide autonomous geospatial positioning of targets by using tiny electronic receivers can assist in recognition. The recognition of targets within a targeted area is a point-in-polygon (PtInPy) problem in computational geometry. Most previous methods used the method of identifying one target at a time, which made them unable to effectively deal with many targets. An earlier method was able to simultaneously recognize several targets but had the problem of the repeated recognition of the same targets. Therefore, we propose a GNSS coordinate recognition algorithm. This algorithm can efficiently recognize a large number of targets within a targeted area, which can provide assistance in epidemic management.

A Generative Approach to Hull Design for a Small Watercraft

Abstract In the field of ocean engineering, the task of spatial hull modelling is one of the most complicated problems in ship design. This study presents a procedure applied as a generative approach to the design problems for the hull geometry of small vessels using elements of concurrent design with multi-criteria optimisation processes. Based upon widely available commercial software, an algorithm for the mathematical formulation of the boundary conditions, the data flow during processing and formulae for the optimisation processes are developed. As an example of the application of this novel approach, the results for the hull design of a sailing yacht are presented.

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most$3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric$3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

Hubungan Kecerdasan Spasial dan Kemampuan Penalaran Matematis Peserta Didik Dalam Menyelesaikan Masalah Geometri

This study aims to find out whether there is a significant relationship between spatial intelligence and the mathematical reasoning ability of class IX MTs Negeri 2 Mataram students in solving geometry problems. This type of research is correlational research. The sample in this study was students of class IX A MTs Negeri 2 Mataram who were taken with purposive sampling techniques. Data collection is carried out using test instruments in the form of description questions about spatial intelligence, reasoning ability, and solving geometric problems. The data were analyzed using descriptive statistics and correlation tests with a quantitative approach. The results of the data analysis showed: 1) there was a significant relationship between spatial intelligence and solving geometry problems of class IX MTs Negeri 2 Mataram students of 0.366 with a weak relationship category; 2) there is a significant relationship between mathematical reasoning ability to solve geometric problems of class IX MTs Negeri 2 Mataram students of 0.386 with a weak relationship category; 3) there is a significant relationship between spatial intelligence and mathematical reasoning ability of class IX MTs Negeri 2 Mataram students in solving geometric problems of 0.532 with sufficient relationship categories.