• The Mathematical Education
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• v.56 no.3
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• pp.235-255
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• 2017

This study compared and contrasted the topics related to congruence and symmetry in the elementary mathematics textbooks series of Korea, Japan, Hong Kong, Finland, and Singapore in three aspects: (a) when to teach, (b) what to teach, and (c) how to teach. Firstly, the results of when to teach showed differences across the countries with a variation of teaching the topics among grades from 3 to 6. Secondly, the results of what to teach revealed subtle but significant differences. Regarding congruence, Korea and Japan deal with congruence in a systematic manner, while Finland tends to address the brief definition of congruence, and Hong Kong and Singapore focus on teaching tessellation which implies congruence. Regarding symmetry, Korea and Japan deal only with a symmetric figure for a line and that for a point, while Hong Kong includes a rotational symmetry and Finland extends further to cover the figures positioned in a symmetry both for a line and for a point. Lastly, the results of how to teach demonstrated that Korea tends to focus on the procedure of drawing both triangles to be congruent and symmetric figures. This implies that we need to consider alternative methods such as using various instructional materials and making an explicit connection among mathematical concepts in teaching congruence and symmetry.

• The Mathematical Education
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• v.46 no.3
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• pp.273-292
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• 2007

The purpose of this study was to find out how second, fourth and sixth graders understood the main contents related to spatial sense in the Seventh National Mathematics Curriculum. For this purpose, this study examined students' understanding of the main contents of congruence transformation (slide, flip, turn), mirror symmetry, cubes, congruence and symmetry. An investigation was conducted and the subjects included 483 students. The main results are as follows. First, with regards to congruence transformation, whereas students had high percentages of correct answers on questions concerning slide, they had lower percentages on questions concerning turn. Percentages of correct answers on flip questions had significant differences among the three grades. In addition, most students experienced difficulties in describing the changes of shapes. Second, students understood the fact that the right and the left of an image in a mirror are exchanged, but they had poor overall understanding of mirror symmetry. The more complicated the cubes, the lower percentages of correct answers. Third, students had a good understanding of congruences, but they had difficulties in finding out congruent figures. Lastly, they had a poor understanding of symmetry and, in particular, didn't distinguish a symmetric figure of a line from a symmetric figure of a point.

• Communications of Mathematical Education
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• v.38 no.2
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• pp.215-230
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• 2024

Congruences and symmetry are familiar concepts that can be encountered in everyday life. In order to effectively understand and acquire these concepts, the role of appropriate visual examples is important. This analysis examined various visual examples used in the process of learning the concepts of congruence and symmetry in elementary mathematics textbooks and focused on the use of convex polygons and concave polygons. As a result of the analysis, various types of polygons were used as visual examples for teaching and learning of congruence and symmetry in textbooks. The frequency of use of concave polygons was higher in the order of congruence, line symmetry, and point symmetry, and it was confirmed that it was used more frequently in the process of exploring properties than in the introduction of the concept. Based on these results, a plan to utilize concave polygons in teaching and learning of congruence and symmetry was sought.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.1 s.173
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• pp.110-117
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• 2000

Vibration modes obtained from a modal analysis can be better explained from a screw theoretical standpoint. A vibration mode can be geometrically interpreted as a pure rotation about the vibration center in a plane and as the twisting motion on a screw in a three dimensional space. This paper, presents the method to diagonalize a spatial stiffness matrix by use of a parallel axis congruence transformation. It also describes that the stiffness matrix diagonalized by a congruence transformation, can have the planes of symmetry depending on the location of the center of elasticity. For a plane of symmetry, any vibration mode can be expressed by the axis of vibration. Analytical solutions for the axis of vibration has been derived.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.3
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• pp.267-282
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• 2018

The purpose of this study is to develop and apply a Convergence program for teaching of congruence and symmetry and to investigate the effects of the mathematical creativity and convergence talent. For these purposes, research questions were set up as follows: 1. How is a Convergence program for teaching of congruence and symmetry developed? 2. How does a Convergence program affect the mathematics creativity and convergence talent of fifth grade student in elementary school? The subjects in this study were 16 students in fifth-grade class in elementary school located in Songpa-gu, Seoul. A Convergence program was developed using the integrated unit design process chose the concept of congruence and symmetryas its topic. The developed program consisted of a total 12 class activities plan, lesson plans for 5 activities. Mathematics creativity test, a test on affective domain related with convergence talent measurement were carried out before and after the application of the developed program so as to analyze the its effects. In addition, students' satisfaction for the developed program was investigated by a questionnaire. The results of this study were as follows: First, A convergence program should be developed using the integrated unit design process to avoid focusing on the content of any one subject area. The program for teaching of congruence and symmetry should be considered students' learning style and their preferences for media. Second, the convergence program improved the students' mathematical creativity and convergence talent. Among the sub-factors of mathematical creativity, originality was especially improved by this program. Students thought that the program is good for their creativity. Plus, this program use two subject class, Math and Art, so student do not think about one subject but focus on topic 'congruence and symmetry'. It help students to develop their convergence talent.

• Communications of Mathematical Education
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• v.37 no.2
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• pp.299-327
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• 2023

In this study, we tried to find out the level of inductive reasoning ability and the aspects of visualization components shown in students in the class using exploratory software for the 'congruence and symmetry' unit in the second semester of the 5th grade of elementary school. To this end, classes using GeoGebra, one of the exploratory software, were conducted for a total of 19 students in one class of fifth graders in elementary school, and the results of the students' activities were analyzed. As a result of this study, the level of inductive reasoning ability of students remained at a similar level or developed, and it was shown that students inferred new properties of shapes using various functions of software inductively. In addition, in terms of visualization, students were able to quickly and easily draw shapes that met the conditions, and unlike the paper-and-pencil environment, using the 'measurement' and 'symmetry' functions, they transformed and manipulated complex yet precisely congruent and symmetrical external representations. Based on these analysis results, implications for the use of exploratory software in the area of figures were derived.

• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.479-495
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• 2013

Taking the weighted geometric mean [11] on the cone of positive definite matrix, we propose an iterative mean algorithm involving weighted arithmetic and geometric means of $n$-positive definite matrices which is a weighted version of Carlson mean presented by Lee and Lim [13]. We show that each sequence of the weigthed Carlson iterative mean algorithm has a common limit and the common limit of satisfies weighted multidimensional versions of all properties like permutation symmetry, concavity, monotonicity, homogeneity, congruence invariancy, duality, mean inequalities.

• Journal for History of Mathematics
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• v.34 no.2
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• pp.39-54
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• 2021

One of the topics in school mathematics is the relation between the roots and the coefficients of equations. It deals with the way to find the roots out of the coefficients of equations. One of the concepts derived from the theory of equations is symmetric functions. Symmetry is a kind of functionality of human cognition. It is, in mathematics, geometrically related to the congruence and the similarity of figures, and algebraically a kind of invariants. We look at stories on the appearance of symmetric functions through the development of the theory of equations.

• Progress in Medical Physics
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• v.9 no.2
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• pp.105-113
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• 1998

Accurate radiation dosimetric characters is very important to determine of dose to a radiotherapeutic patient. Medical linear accelerators have been developed not only its new quality of convenient operation but also electric moderation. It is reliable to measure more detail physical parameter that linac's internal ability. Typically, radiation dosimetric tool is classified ionization chamber, film, thermoluminescence dosimeter, etc. Nowaday, Electronic Portal Imaging Device is smeared in radiation field to verification of treatment region. EPID's image was focused that using both on-line image verification and absolutely minimum absorbed dose during radiotherapy. So, Electronic Portal Imaging was tested for quality evaluation of medical linear accelerator had its pure conditional flash. This study has performed symmetry, Light/Radiation field congruence, and energy check, geometry difference on wedge filter using a liquid filled ion chamber (EPID). Prior to irradiated on EPID, high energy photon beam is checked with ion chamber. Using these results more convenient dosimetric method is accomplished by EPID that taken digital image. Medical image is acquired with EPID too. Therefore, EPID can be analyzed by numerical information for what want to see or get more knowledge for natural human condition.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.2 s.173
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• pp.362-369
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• 2000

Via screw theory, a vibration mode can be geometrically interpreted as a pure rotation about the vibration center in a plane and as a twisting motion on a screw in a three dimensional space. In thi s paper, applying the conditions that can be used to diagonalize the stiffness matrix by a parallel axis congruence transformation, the vibration modes and frequency response of an elastically suspended optical disc drive have been analyzed. It is first shown that the system has one plane of symmetry, which enables one to decouple the complicated vibration modes into two sets of modes independent of each other. Having obtained the analytical solutions for the axes of vibrations, the frequency response for a given applied input force has been demonstrated. Most importantly, it has been explained that this research result could be used in the synthesis process of a linear vibration system in order to improve the frequency response.