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Coordinate Systems and Vector Algebra
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
Geometry is a branch of mathematics that deals with the measurement, property, and relations of lines, angles, surfaces, and volumes. A practical device or part thereof in which the field distribution is to be studied for its behavior requires representation of its shape and size in such a way that all its physical aspects fully conform to the geometry. The system that deals with geometrical aspects of an object is referred to as a coordinate system. The necessity to enhance these limits has led to the development of number of tools. A coordinate system helps us to visualize relative positions of independent points or those belong to a line, a surface or a volume. Thus, while introducing the coordinate systems, it is presumed that a reader fully understands the meanings of point, line, surface, and volume, and also is aware of normal, tangent, and geometrical properties. Furthermore, as and when any point on, or a segment in terms of line, area, or volume of, the system is to be represented (or identified), it should fit in well in the coordinate system selected. The following subsections describe the types of coordinate systems.
Introduction: Drawing with mind and emotions
Published in Mário S. Ming Kong, Maria do Rosário Monteiro, Maria João Pereira Neto, Creating Through Mind and Emotions, 2022
Considered the simplest visual element, the point becomes somewhat abstract when studied in isolation due to its characteristics that can be defined as both immaterial and versatile. It has the particularity of being an element that can represent the beginning or the end, enhancing the imagination and the association of ideas, abstract or not, and being an expression of sensations and emotions. In geometry, the point is considered to have no dimension, a mere mental construction, the place where two lines intersect. However, considering it as a visual element, the point is material (it has intensity, dimension, …) and can be applied, analyzed, and interpreted under various aspects such as magnitude, visual appearance, possible relation to other points, among many other aspects.
Introduction
Published in Xie Heping, Fractals in Rock Mechanics, 2020
Geometry is a discipline in which the shapes of objects are studied. Classical geometry is used to investigate regular and smooth shapes (lines, curves, circles and so on). However, geometric shapes in classical geometry are approximate descriptions of objects in nature. For example, the rough surface of the earth is regarded as a smooth spherical or ellipsoid surface, and an upland is considered a trapezoidal slope (line) surface. Although these approximations do not prevent us from reaching conclusions that agree with practice, the need to obtain accurate descriptions of matter becomes critical to the development of science.
Investigation of preservice mathematics teachers’ concept definitions of circle, circular region, and sphere
Published in International Journal of Mathematical Education in Science and Technology, 2022
As can be seen in Table 2, according to the correctness criterion, 51.79% of the preservice teachers defined the concept of circle correctly. While one of the preservice teachers (S1) defined the concept of a circle as A set of points located at a certain distance to a specific point on a plane, another (S32) defined it as A geometric shape made up of a set of points located at an equal distance to a point on a plane. In addition to these, definitions such as A set of points located at r units distance to a P0 point on a plane were also accepted as being correct. Examples from the preservice teachers’ drawings that correctly define the concept of circle are given in Figure 1.
Teaching the circle equation using a practical mathematical problem
Published in International Journal of Mathematical Education in Science and Technology, 2022
In different countries, usually the topics on circle equation are taught in the upper secondary school through abstract materials. For example, in the Malaysian curriculum, this topic is taught for foundation or pre-university-level students. A circle is a geometric shape, consisting of all points in a plane that is a given distance (value of the radius) from a given point (centre of the circle). The following equation introduces a circle with the centre and the radius ;
Introducing the concept of angle to young children in a dynamic geometry environment
Published in International Journal of Mathematical Education in Science and Technology, 2020
In the research literature, the concept of angle has been defined in many different ways with the definitions varying significantly in their emphases (Henderson & Taimina, 2005; Keiser, 2004). The concept of angle is shown to have three different perspectives, namely angle as a geometric shape, union of two rays with a common end point (static); angle as movement; angle as rotation (dynamic); angle as measure and amount of turning (Henderson & Taimina, 2005). These angle definitions reflect the duality of dynamic and static conceptions of angle. It seems like no formal definition of angle can capture all aspects of angle conceptions. Highlighting the dynamic aspect of an angle, Freudenthal (1983) describes ‘angle as turn’ as ‘the process of change of direction’ (p. 327). In case of turn angle, it is essential to pay attention to several aspects like whether after a full turn one counts further or starts new, or whether it is a right turn or left turn. As Freudenthal (1983) maintains, the ‘process’ is the most important thing in case of turn angle as one has to pay attention to what happens in the meantime when one side is being turned into the other. It seems like the turning aspect of angle is harder to teach because making the turning aspect of angle explicit seems to be a difficult task. Research has reported about the young children’s difficulties in understanding the turn as an angle as well as connecting static angles to turns (Clements, Battista, Sarama, & Swaminathan, 1996; Mitchelmore, 1998). Clements et al. (1996) in their study with third graders form a hypothesis that students learn about turn measurement by integrating two schemes, turn as body movement and turn as number. They conclude that changes in orientation are harder to understand than changes in position and young children do not naturally connect static angles to turns. I propose that DGEs such as Sketchpad might be helpful in making the turning aspect of angle explicit and hence in developing the angle-as-a-turn conception.