• East Asian mathematical journal
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• v.31 no.2
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• pp.167-188
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• 2015

Formula for the area of a trapezoid is an educational material that can handle algebraic and geometric perspectives simultaneously. In this note, we will make up the expression equivalent algebraically to the formula for the area of a trapezoid, and deal with the conversion of a geometric point of view, in algebraic terms of translating and interpreting the expression geometrically. As a result, the geometric conversion model, the first algebraic model, the second algebraic model are obtained. Therefore, this problem is a good material to understand the advantages and disadvantages of the algebraic and geometric perspectives and to improve the mathematical insight through complementary activity. In addition, these activities can be used as material for enrichment and gifted education, because it helps cultivate a rich perspective on diverse and creative thinking and mathematical concepts.

• Education of Primary School Mathematics
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• v.24 no.4
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• pp.247-264
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• 2021

The area of a trapezoid is an important concept to develop mathematical thinking and competency, but many students tend to understand the formula for the area of a trapezoid instrumentally. A clue to solving these problems could be found in Dienes' theory of learning mathematics and Watson and Mason' concept of example spaces. The purpose of this study is to obtain implications for the teaching and learning of the area of the trapezoid. This study analyzed the example spaces constructed by students in learning the area of a trapezoid based on Dienes' theory of learning mathematics. As a result of the analysis, the example spaces for each stage of math learning constructed by the students were a trapezoidal variation example spaces in the play stage, a common representation example spaces in the comparison-representation stage, and a trapezoidal area formula example spaces in the symbolization-formalization stage. The type, generation, extent, and relevance of examples constituting example spaces were analyzed, and the structure of the example spaces was presented as a map. This study also analyzed general examples, special examples, conventional examples of example spaces, and discussed how to utilize examples and example spaces in teaching and learning the area of a trapezoid. Through this study, it was found that it is appropriate to apply Dienes' theory of learning mathematics to learning the are of a trapezoid, and this study can be a model for learning the area of the trapezoid.

• School Mathematics
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• v.8 no.3
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• pp.327-339
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• 2006

In this study, a teaching units for teaching and learning of secondary preservice teachers' mathematising is designed, focusing on reinvention of Bretschneider's formula. preservice teachers can obtain the following through this teaching units. First, preservice teachers can experience mathematising which invent a noumenon which organize a phenomenon, They can make an experience to invent Bretscheider's formula as if they invent mathematics really. Second, preservice teachers can understand one of the mechanisms of mathematics knowledge invention. As they reinvent Brahmagupta's formula and Bretschneider's formula, they understand a mechanism that new knowledge is invented Iron already known knowledge by analogy. Third, preservice teachers can understand connection between school mathematics and academic mathematics. They can understand how the school mathematics can connect academic mathematics, through the relation between the school mathematics like formulas for calculating areas of rectangle, square, rhombus, parallelogram, trapezoid and Heron's formula, and academic mathematics like Brahmagupta's formula and Bretschneider's formula.

• Journal of Internet Computing and Services
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• v.4 no.6
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• pp.57-67
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• 2003

The recent rapid development of multimedia and optical technologies brings great attention to application systems to process facial Image features. The previous research efforts in facial image processing have been mainly focused on the recognition of human face and facial expression analysis, using front face images. Not much research has been carried out Into image-based detection of face direction. Moreover, the existing approaches to detect face direction, which normally use the sequential Images captured by a single camera, have limitations that the frontal image must be given first before any other images. In this paper, we propose a method to detect face direction by using facial features such as facial trapezoid which is defined by two eyes and the lower lip. Specifically, the proposed method forms a facial direction formula, which is defined with statistical data about the ratio of the right and left area in the facial trapezoid, to identify whether the face is directed toward the right or the left. The proposed method can be effectively used for automatic photo arrangement systems that will often need to set the different left or right margin of a photo according to the face direction of a person in the photo.

• Journal of Information Technology Applications and Management
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• v.10 no.4
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• pp.135-147
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• 2003

With the development of multimedia and optical technologies, application systems with facial features hare been increased the interests of researchers, recently. The previous research efforts in face processing mainly use the frontal images in order to recognize human face visually and to extract the facial expression. However, applications, such as image database systems which support queries based on the facial direction and image arrangement systems which place facial images automatically on digital albums, deal with the directional characteristics of a face. In this paper, we propose a method to detect facial directions by using facial features. In the proposed method, the facial trapezoid is defined by detecting points for eyes and a lower lip. Then, the facial direction formula, which calculates the right and left facial direction, is defined by the statistical data about the ratio of the right and left area in facial trapezoids. The proposed method can give an accurate estimate of horizontal rotation of a face within an error tolerance of $\pm1.31$ degree and takes an average execution time of 3.16 sec.

• School Mathematics
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• v.3 no.2
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• pp.447-473
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• 2001

Since the greater part of mathematical concepts have been developed in the direction of “from the concrete and realistic aspects to the abstract level”, children should be secured to learn mathematics genetically with various manipulative materials. The aim of this study is to instigate the active use of geoboards in mathematics classroom. To achieve this arm, we first embodied the several significances on the use of geoboards in mathematics instruction. And we then performed an instruction that children discover and justify the formula related to the area of trapezoid by exploring with geoboards, and analyzed the instructional episode to support our assertion about some secure merit accompanied by using geoboards. From this study, we obtained the conclusion that geoboard activity contains many significances such as children can explore congruence, symmetry, similarity, fundamental properties of figures, and pattern. Futhermore, geoboard activity enable children to transform a figure into other equivalently, develop spatial sense, have basic experiences for coordinate geometry, build a concrete model to explain abstract ideas, and foster the ability of problem solving and mathematical thinking.