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Poland, 24-28 april 2012 Comenius Project 2011- 2013 “The two highways of the life: maths and English” About the meaning of angles Introduction All it takes is a quick glance around a room, any room, to find angles everywhere. Every corner in the room makes an angle. Anywhere two lines come together to form a point, an angle is created. θ Introduction In geometry, an angle is the figure formed by two rays (or half-lines), called the sides of the angle, sharing a common endpoint, called the vertex of the angle. (http://en.wikipedia.org/wiki/Angle) Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. Eudemus regarded an angle as a deviation from a straight line; Carpus of Antioch, regarded it as the interval of space between the intersecting lines. Euclid in Raphael's “School of Athens” Introduction The angle is one of the first concepts of geometry that pupils study at school but understanding what angles are is not so simple as we can think. Talking about angles is important to understand that: - the measure of an angle does not vary with the length of the sides; - an angle is an open figure; - the arc and the measure of an angle are differents things. Some questions… Look at these angles: - which of these angles has a greater amplitude? α β The angle α. The amplitude of an angle is a measure of the inclination (or rotation) of a ray respect to the other and it is not a consequence of the length of the sides. Look at the plotted points: - which of these points are drawn inside the angle? - which of these points are drawn outside the angle? .B . A δ .C Points A and C are plotted inside the angle. Point B is plotted outside. The part of the plane between the two sides of an angle is extended infinitely because the sides of the angle are two half-lines. Playing with a cake Julia made a cake for her friends and everyone, except Mark, has already eaten a slice. Can you calculate how many people are invited? First you need to measure the angle of a slice. 30° Then you divide 360° (the whole pie) by the degrees that correspond to a slice. 360°/30° = 12 friends 30° Playing with a clock Look at these clocks. Only one between A and B works correctly, which one? A B The clock B. You can see that in the clock B, while the minute hand marks half an hour, the hour hand arrives in the middle between 4 and 5, and this is correct. The rotation speed of the hands, forming different angles respect to the initial position of 12 o'clock, varies in the case of the minute hand and the hour hand. Playing with a clock Consider different positions that the clock hands can have and the angles that they form. How many degrees do they measure? 90° 30° 180° 120° 270° 360° 210° 330° Playing with a clock Which of these angles are acute? Obtuse? Concave? Convex? Right? Straight? Turn? right straight Acute convex Obtuse convex concave concave turn concave Playing with a clock How many degrees is the angle that the minute hand forms in a minute? 6° Playing with a clock According to this, what time is associated with the angle of 1°? A minute divided by 6 is 10 seconds. But if the rotation speed of the second hand is equal to that of the minute hand we would not be able to see the time because the angle of one degree is very small! In order to distinguish hours, minutes and seconds in a clock it is necessary that the different hands have increasing rotation speed relative to each other. According to this, the minute hand has a rotation speed of 1° every 10 seconds and the second hand has a rotation speed of 60° every 10 seconds. θ Relations between the measure of an angle and the length of the sides An angle of one degree is very small ... which objects may be contained in it? a microbe? a bug? a tennis ball? a house model? a real house? a whole planet? Create an angle of 1° with a goniometer and a wool thread… θ Relations between the measure of an angle and the length of the sides Because the sides of an angle are two rays, the part of the plane between them never ends. Within an angle of one degree can also be a whole planet, but it is necessary get far enough away from the vertex! The arcs that we can draw within the same angle are infinite in number and their length grows going away from the vertex. Relations between the measure of an angle and the length of the arc How long must the radius of a circle be so that the arc of an angle of 1° is 1 m long? C = 2πr The arc of an angle of 360° is the whole circumference. We need to calculate the lenght of the radius of a circumference where an arc of 1 m correspond to an angle of 1°, so the circumference measures 360 m. r = C/2π = 360 m/2*3.14 ≅ 57.4 m δ = 1° Arc length = 1m Radius length = 57,32 m As a consequence, an object, observed by a distance of about 57.4 times its diameter, has an angular size of about 1°. Relations between the measure of an angle and the length of the arc Accidentally, the angular diameter of the sun from the earth is equal to that of the moon. Sun is about 400 times farther than moon from the earth but its diameter is also 400 times bigger. This fact implies that in the sky they appear to have similar dimensions. This particular coincidence makes possible exciting eclipses of the sun. Conclusions Angles are a difficult subject to teach because they contain the infinite. Since we usually teach them as imprisoned in a spreadsheet, it is difficult for a student to reach a complete comprehension about this matter. Maybe it could be better to let angles go out from the sheet, out from the school, far away, opening our imaginations. References Anichini G, Arzarello F., Bartolotti C., Ciarrapico L., Robutti O., 2004. “La matematica per il cittadino”. MIUR, UMI, SIS. Link: http://umi.dm.unibo.it/scuola-99.html Arzarello F., Ciarrapico L., Camizzi L., Mosa E., 2006. “Progetto [email protected] – Matematica. Apprendimenti di base con e-learning”. Documento di base 4 aprile 2006. MIUR. Battisti R., Spinelli F, Brunelli F. “L’orologio”. Attività matabel inserita nella sezione GEOMETRIA del PON 2007-2013 (http://www.indire.it/ponmatematicacorso1/) AAVV, 2011. “Matematica: didattica, esperienze e tecnologie”. RCS Libri Education S.p.A. http://www.educationduepuntozero.it/speciali/pdf/specialemarzo11.pdf Web sites: http://en.wikipedia.org http://www.indire.it/ponmatematicacorso1/ http://library.thinkquest.org/2647/geometry/glossary.htm