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A little bit of History
A little bit of History
The study of conic sections comes from the ancient Greek. The best work from that age was the one from Apollonius of Perga (200 B.C.).
Analytic Geometry was born in 17th century, by Fermat and Descartes. Then conics became also equations, apart from beautiful curves. Kepler discovered that planets turn around the Sun following the path of an ellipse. Galieo, that every object describes a parabola when it's thrown with certain angle with the ground. Those conics were described by the language of Algebra.
Descartes proved that any general second degree equation in two variables like this
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
represents either an ellipse, a parabola or an hyperbola.
A general study of conics was made by Desargues with his book "Projective Geometry", in 17th century.
Euler, in 1748, studied any curve represented by a quadratic equation in his book "Introduction to Analysis". That study is similar than that made in many contemporary books.
Finally, in second half of 19th century, Cayley developed "matrices", a special object in Mathematics. Alongside with Silvestre made an analytic study of conics based on matrices and quadratic forms.
Project
Project
Make a slideshow about conic sections. Choose two of them.
It must contain, between four and eight slides.
Every member of the group must be able to answer the questions of the teacher about his or her team's project.
You shoud describe each curve (in terms of Geometry -definition, features and elements- and Analytic Geometry -equation-).
It must appear pictures or videos of real situations where the conic shows up. At least one of the images or videos per conic must be made by you.
The presentation and the slides must be in English.
The project will be presented in about five minutes.
All the presentations will be on April, 1st. The language assistant will evaluate the students performance in English.
All the presentations should be sent to the teacher in advance, before that day (due date: March, 31st).
The sources of your information should appear in your slideshow.
Some ideas for the pictures
Some ideas for the pictures
Circle: a watch face, a car tyres, the rim of a glass or a hat, the gears in a machine, a tour around the world, the pupil of an eye... There are circles everywhere!
Ellipse: it happens that if you cut a cylinder by any plane not perpendicular to its generatrix, you get an ellipse. Thats why any salami slice has that shape, or the rim of a glass from a non vertical point of view. Most of the circles in real life appear to us as ellipses. Artists use this property to represent circles.
Parabola: a stream of water coming from a hose as the trajectory of any projectile is an arc of a parabola. That phisics law was discovered by Galileo Galilei in 17th century and allow the engineers (or anybody with that mathematical knowledge) to calculate the highest point of that track or the range of the weapon by knowing, only, the initial speed and the angle with the ground.
Hyperbola: any lamp with a conic lampshade project an hiperbola in the wall (a plane parallel to the directrix line of the cone).
Kepler's studies on conics (17th century), along with his countless astronomical observations, led him to postulate his revolutionary theory that the orbits of the planets revolving around the Sun were elliptical, with the Sun at one of their foci (in reality, these ellipses are very similar to circles as their eccentricities are very small; for example, Earth's orbit has an eccentricity of 0.0167). Other celestial bodies (those we see only once, as is the case with some comets) follow parabolic or hyperbolic trajectories. This law of elliptical orbits was the first of his three famous laws on the motion of celestial bodies. With these laws, the importance of conics was reaffirmed. "The ellipse, far from being merely a curiosity of Greek mathematicians, had become the precise path traveled by Earth and all of us upon it" (W. Dunham).
It was Newton who provided the mathematical foundation for Kepler's laws. Furthermore, Newton studied other properties of conics, such as the reflective property of the parabola, whereby a light ray emerging from the focus and reflecting at a point on a parabola will depart in a direction parallel to its axis. This property also works reciprocally; that is, any ray incident on the parabola in a direction parallel to its axis will reflect through its focus. This property is the basis for the construction of reflecting telescopes, like the one invented by Newton. Large searchlights, car headlights, and parabolic antennas are also based on the same property of the parabola.
In the case of the ellipse, it is easily demonstrated that rays originating from one of its foci and reflecting off the ellipse converge at the other focus. This is why, on occasion, in vaults with an elliptical cross-section, it is possible to clearly hear the voice of people who are far apart. In the case of the hyperbola, rays originating from one of its foci reflect off it as if they were coming from the other focus.
The hyperbola also has numerous technical applications. One of the most well-known is the LORAN system (long-distance radio navigation) used in airplanes and ships for localization. It is based on the property that defines the hyperbola as a geometric locus: the difference in distances to two fixed points is constant. The system uses synchronized radio wave pulses transmitted by two stations separated by a great distance. The difference in the time of arrival of these pulses to the ship or airplane is constant, meaning that the ship or airplane is always located on a hyperbola with the two stations as its foci.
Finally, let us say that any physical, economic, or other type of law that can be described through an inverse proportionality relationship can be graphically represented by an equilateral hyperbola.
Bonus advice: avoid the so called death by powerpoint ;-) y algunos consejos sobre presentaciones.
Links
Links
There is only one true parabola (standupmaths by Matt Parker).
Sconic sections (scone is a kind of cake).
What your teacher (probably) never tell you about parabola, hyperbola and ellipse.
How to make a
Criterios de evaluación
Criterios de evaluación
8.1. Mostrar organización al comunicar las ideas matemáticas, empleando el soporte, la terminología y el rigor apropiados.
9.3. Participar en tareas matemáticas de forma activa en equipos heterogéneos, respetando las emociones y experiencias de los demás y escuchando su razonamiento, identificando las habilidades sociales más propicias y fomentando el bienestar grupal y las relaciones saludables.
Saberes básicos implicados en la unidad
Saberes básicos implicados en la unidad
C. Sentido espacial
C. Sentido espacial
MATE.1.C.1. Formas geométricas de dos dimensiones.
MATE.1.C.2.2 Expresiones algebraicas de objetos geométricos en el plano: selección de la más adecuada en función de la situación a resolver.
MATE.1.C.3.2 Modelos matemáticos (geométricos, algebraicos, grafos...) en la resolución de problemas en el plano. Conexiones con otras disciplinas y áreas de interés.
F. Sentido socioafectivo.
F. Sentido socioafectivo.
MATE.1.F.2. Trabajo en equipo y toma de decisiones.
MATE.1.F.2.2 Técnicas y estrategias de trabajo en equipo para la resolución de problemas y tareas matemáticas, en equipos heterogéneos.
MATE.1.F.3. Inclusión, respeto y diversidad.
MATE.1.F.3.1 Destrezas para desarrollar una comunicación efectiva, la escucha activa, la formulación de preguntas o solicitud y prestación de ayuda cuando sea necesario.
MATE.1.F.3.2 Valoración de la contribución de las matemáticas y el papel de matemáticos y matemáticas a lo largo de la historia en el avance de la ciencia y la tecnología.
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