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Triangles and quadrilaterals
Published in James Kidd, Ian Bell, Maths for the Building Trades, 2014
A triangle is an area, on a flat surface, enclosed by three straight lines that intersect at three points, shown as A, B and C in Figure 10.1. An angle is formed where any two lines meet. The amount of opening between the lines dictates the angles that make up the triangle. The angles of a triangle are not dependent on the lengths of the triangle’s sides or the area it covers.
Measuring
Published in Roger Timings, Engineering Fundamentals, 2007
A right angle is the angle between two surfaces that are at 90° to each other. Such surfaces may also be described as being mutually perpendicular. The use of engineers' try-squares and their use for scribing lines at right angles to the edge of a component will be described in Chapter 7. Figure 6.15(a) shows a typical engineer's try-square.
Measuring and marking out
Published in Roger Timings, Fabrication and Welding Engineering, 2008
A right angle is the angle between two surfaces that are at 90° to each other. Such surfaces may also be described as being mutually perpendicular. The use of engineers’ try-squares and their use for scribing lines at right angles to the edge of a component will be described later in this chapter. Figure 5.8(a) shows a typical engineer’s try-square.
Teaching and learning angles in elementary school: physical versus paper-and-pencil sequences
Published in International Journal of Mathematical Education in Science and Technology, 2022
Valérie Munier, Claude Devichi
This third session is, as in the physical sequence, a decontextualized session, and the activities proposed to the pupils are identical. In both half-classes, the strategies and difficulties that occurred were very similar to those of the physical sequence. The definitions of ‘angle’ that were produced were also very similar, and the teachers rephrased them as previously mentioned (an angle is the degree of openness between two half lines (the sides) that extend from a common point: the vertex). As in the physical sequence, pupils frequently referred to the initial situation they experienced (here, the initial paper-and-pencil exercise). They also discussed the fact that two half-lines delineated two angles: the closed angle corresponding to the area containing all points that can be connected to O while crossing [AB] and the open angle corresponding to the area containing all points that can be connected to O without crossing [AB].
Preparation of stimuli-responsive nanogels based on poly(N,N-diethylaminoethyl methacrylate) by a simple “surfactant-free” methodology
Published in Soft Materials, 2018
Lizbeth A Manzanares-Guevara, Angel Licea-Claverie, Francisco Paraguay-Delgado
The size distribution of the nanogels was obtained by dynamic light scattering (DLS) using a Zetasizer Nano ZS (ZEN3690; Malvern Instruments, Miami, FL, USA) equipped with a red laser of 630 nm. The angle of measurement was 90°. Dialyzed and redispersed samples were analyzed. The hydrodynamic diameter (Dh) and polydispersity index (PDI) were calculated using Malvern Instruments dispersion technology software based on CONTIN analysis and the Stokes–Einstein equation for spheres; for Dh, the average value of three measurements is reported. The effect of the pH and temperature on the particle size of the nanogel products was studied by DLS using the same Zetasizer Nano ZS equipment. For the temperature sensitivity determination, a trend method was edited to go from 20°C to 65°C in two-degree steps, equilibrating for 240 s once the measurement temperature was attained; the transition temperature reported is the minimum value of the first derivative of Dh with respect to temperature. The pH sensitivity of PDEAM is well known; for this reason the Dh of the nanogels was studied only at pH values that are interesting for the biomedical applications, namely pH 5, 6.8, 7.4, and 9. The measurements were carried out at 25°C
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.