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a theodolite.

Within the scientific methods and practices dealing with astronomy, I have developed several sequences on the triangulation, with objective whether a meteor could result in a meteorite, and if so to estimate its drop point. After a review of documents, we reach the conclusion that a meteor that goes to 25 to 35 km altitude is likely to result in a meteorite. It is also necessary to measure or calculate the altitude.
The first session is to present the problem and measurement tools, simplified to give meaning to the concepts of azimuth and elevation of a star, so that to manipulate the object of measurement gives meaning to this same extent.

At first, students have handled a quadrant to measure angles and heights to infer from buildings or trees. The quadrant is
actually consists of a protractor graduated between 0 and 90 °, with a plumb line. It can carry a viewfinder consists of a stud screw and a nail with a head at the same level as the center of the stud.

After two years on paper, here are the students left in the yard. As I could not recover hectometres sports teachers, who were in an outdoor venue, the measurements were not made. I had to improvise, but I find it quite welcome to manipulate other units of length, provided that students handle them with rigor, to the extent with a steady pace as in the calculation and writing.

After students have a measure, I asked students if they were able to measure the height of a building that exceeded the roof of the school.
-no, because we do not know how far it is ground.
So I tried the following, which I found on this site .
is blindfolded with a student.
asked a student to depart the group and make a noise (whistle, clap). The student blinded points to the direction accurately, but very difficult to estimate the distance. Where is the fellow? On a ray, which can be drawn, but it is unclear at what point.
then asked a second student to blindfold. It puts a little further from the first student, then move the transmitter noise and repeating the experiment. Again, students have difficulty in knowing just how far the sound, but are able to fairly accurately pointing direction. By extending the half-lines, we find the position of the transmitter of sound.
Thus, without knowing the distance, with two measures derived different positions, one can determine precisely the source of sound. And place it on a plane. Must still measure the angles of the triangle.
So I took out my theodolite.
It is quite rudimentary but aims to fix ideas. And then the budget math my school does not buy one.
I have a high shelf with a hole for the bear lying around in the attic. I take.

On two opposite sides of a cardboard box, I operate a center hole with a pen. I drew a reporter from the north facing in the negative sense that I glue on the top of the box, after drilling a hole in the center.
on two ice caps, I operate as a hole in the center. I slip a pin into the box, I maintain inside sliding lids and positioning against the two inner sides of the box.

I fixed a bug or a nail just above the reporter, and a quadrant with a thumbtack. The nail and the quadrant are oriented in the same way.

A compass and a level that can position the instrument in the right position to make a measurement of the azimuth and height of a target in one place, then a little further.

After being refitted bathroom, a point is made on the azimuth and height. And then a year in which, from measurements of azimuth, trace the triangle formed by three students.
one can determine the distances with a scale drawing, which recalls the importance of the north. The next few weeks, even using the sine law and the theorem of Al-Kashi, we will determine the distance calculation, then we will determine the position in the sky a meteor, when it flares up and when he goes out, to get an estimate of where it could fall. Patterns in different planes are required to understand the problem.
the plan and the proposed exercises.
Subsequently, the last Senac cover the earth moon distance or distance from a star to the sun.

On these sessions that I find rich and interesting, I still have some questions: I am not very rigorous on the path that is not straight, I use properties first. Certainly, if I do not use them, I'm not far at all. I am under the MPS?
If you have any comments, criticisms, advice, feel free to make comments.

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comics and mathematics: 1) mathematics as a background

With math, my other passion is comics. Like every year in late January with the festival of Angouleme, the comic is to honor, I thought it was a good time to start talking about comics and mathematics.
In some albums, the mathematical world is at the bottom of the story, in others it is rather the form it provides openings.
For now, I look in my library albums whose story inspired by the mathematical and mathematicians. The list will not be exhaustive, but feel free to notify me of books I'd forgotten.
humor
The obsession of the learned Cosine 1899 (Christopher)

Christopher was a recognized scholar of his time and spent his free time doing small Miquet. His learned Cosine would be inspired by the mathematician Jacques Hadamard and legendary entertainment, or by Henri Poincare. His sense of adventure will not exceed the limits of Paris.

Quadratino 1910 (Antonio Rubino)

Heroes seven short stories, quadratic, a hero to the square head, made a mistake in his head every aventureet systematically to another figure. Fortunately, Mom Geometry, grandmother and aunt Mathematics Algebra arrange everything to end.

The Shadocks 1968 (Jacques Rouxel)
heroes well known television, Shadocks multiply references to logic and mathematics, even if it's for better shape.

at math 1962 (Francis)

resumption of a mini story Spirou, this short comic story of a boy not very intelligent, by bumping, acquires at math and became a genius during his or her hump reabsorbed. Fun.

Cat (Geluck)

Many gags on logic and mathematics that have even led to a study by Daniel Justen.

Exploitable for a course.

Archibald Higgins (JP Small)

Many educational comics that bring humor to math questions. I suggested this source to students of OFWs, who have found material for crafts and reflection. All the bands here is freely available .

Fever Urbicande 1990 (Schuiten Peeters)

A strange illness hits the town Urbicande. A cube placed carelessly on the desk begins to grow and multiply after an inexorable logic, is gradually transforming into a giant octahedron, by the method of the Abbe Rene Just Haüy.

This method can probably lead to years of research in the first S in order to address the consequences.
I could not help but do a little DIY. This comic book invites us to do so much ...
With plastic cubes (called ice cubes) of 3 different colors, and the dough to fix, we can construct the object to the first generation

then the second,

the third.

I have enough cubes to the fourth .

can then ask the question of how many cubic nécéessaires the n ^ th generation, and the number of end of dough needed to be fixed (or at the surface to be painted).

History of mathematics and mathematicians

theorem Morcom (Peeters, Goffin)

In a construction of history that reminds Citizen Kane, the reporter discovers gradually Secrets of a recently deceased mathematical genius who helped decipher the Enigma secret messages. A biography disguised by Alan Turing.


Logicomix (by Apostolos Doxiadis K., Christos Papadimitriou, Alekos Papadatos, Annie Di Donna)

After Uncle Petros and Goldbach's conjecture, the author continues his theme of Doxiadis the search for mathematical truth at the gates of madness. A reading of the history of the crisis of logic in the early 20th century, doubts mathematicians, and the hero, Bertrand Russell, shown first as a man, not just a pure mind, capable of making mistakes in his life. Setting the abyss showing the authors questioning the story they're telling, with their questions and doubts, brings even more depth to this story. Philosophical ideas soar high, often approaching the heroes of vertigo. A beautiful presentation of what happens in mathematics in the early 20th century ...
My bookseller was amazed to have devoured a book about mathematics.

Hypatia 2010 (Pécout, Greiner)

the beautiful Hypatia is now famous for the film released last year. A band that takes the tragic fate of this woman of science. Of moments, as when, resuming the work of Ptolemy, she built an astrolabe.



The math appeared in many successful comics dealing with the child or the world of education: the small Spirou, Titeuf or Kid Paddle or Profs.

Next time, I'll probably level mathematics as the inspiration of the shape and structure.

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around the parable of the plank

two crafts that I used at the beginning of the school year, to ask the question: "what is the plotted curves are parabolas with ways to understand quite different.

Act 1: in module:

The first craft uses the fact that the parabola is the locus of points equidistant from a straight line and a point outside this line.
On a plate, screw a curtain rail "rail" at the top.
Then, cut the hook with a plastic knife to have a flat surface.

On the plate, tape a sheet of paper that will be replaced after each use or Velleda white adhesive film. With four bars
meccano, make a diamond. Two opposite vertices are made with screws and nuts, the other two are manufactured using bugs that are fixed on the hook amended, the other on the board.
Hang a plumb line on the pin of the hook and a rubber band between the two screws.

It will probably add a level to ensure that the wire is perpendicular to the rod.
felt drawn to the point of intersection between the elastic and the plumb.
The pin on the rod is movable. While moving it, find some other points of intersection between the elastic and the plumb.

Once it has few or a dozen, we can ask of the curve formed if we placed all these points of intersection.
is the first time this year that it begs the question of the locus and questions to be answered first: what is fixed in this item? what is mobile? What point does one consider the position?
The shape evokes the parable, but it must justify that indeed obtained a parabola.
Two approaches, one digital, the other geometric we will work on Geogebra. Geometric approach

But before working on geogebra, he should probably just roughing the figure to bring out important points.
A major obstacle to understanding the figure is the fact that bars Meccano almost hide the rest of the figure, as he must first reflect on the nature of the line carried by the elastic. Reflecting together, we arrive at that point M is on the bisector of the segment [HF] between the two pins, then the point M is equidistant from the right and to the point F. We arrive at the definition of the curve and head home.
It then tries to imagine this building without the physical constraints of the length of the rod and the length of the bar meccano. The curve is limited?
Once it laid flat, the students will model the curve using GeoGebra. It is relatively straightforward.
Easier than inserting applets geogebra anyway.

Numerical approach.
How do we define a parabola? For now, students have a digital design is a curve with equation y = a x ² + bx + c. We must find a, b and c. But it only makes sense when viewed in a frame. We must choose.
Fixed objects are fixed in this frame. One can choose according to this benchmark thereof. Posing as the rod axis and the pin as the point coordinates F (0 - 2), for example. Mobile
points are defined from the second bug, since it enables you to find the points of the parabola. Since it belongs to the rail, the ordinate is 0 and its coordinates are H (x, 0), with x real and purpose of the calculation is to find the point of abscissa x M such that HM = FM.

Intermission:
As homework, I asked the FIG to build the simplest possible geogebra and save under their name in the directory of the LCS class, and determine the algebraic equation this curve by placing the x-axis on the curtain rod and the pin at coordinates (0 - 2).
In another exercise of this duty, I asked students to determine the equation of the parabola passing through three given points not aligned.
After rendering duties, we can conclude that by three points not aligned, you can find one that matches trinomen and therefore there is only one dish.

Act 2:
In seeking a halyard between two points at the same height, one obtains a curve. This curve is it a parable? Justify.
A debate is established.
Student responses:
If it is a parable, we can determine the equation had to choose a landmark.
can use a software image processing to be coordinated;
Or geometry software.
Once we can find the coordinates of 3 points, we can get the trinomial. If the other points satisfy the equation found, then it is a parable, or else ...

I took a photo of this coube, and I put it in their directory.
By opening a file and inserting geogebra image, students can place this image in a frame and put some points on this coube in order to determine the coordinates of a rather specific. Students
comfortable chose their mark so that the coordinates are as simple as possible, others have chosen a random marker and had to be faced with a very complicated system of equations.
By moving the benchmark and changing the lines with the wheel, we can get this:

When they found the values of a, b, c for the curve passes through three points, they can noted that a fourth point does not pass by this parable.

In fact, this is called a catenary curve, and the equation used in the functions they will end.

On two occasions, students were asked "is that the curve is a parabola?" and the response method is not the same: in the first case, a parabola, the formula we will find is a trinomial for all points (proved in general) in the second This is not a parable, and there is a cons-example.