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listen: Peter Lascoumes about favoritism and corruption in the French


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Pierre

Lascoumes About Favoritism and Corruption in the French Suite in ideas
by Sylvain Bourmeau 18.12.2010
Pierre Lascoumes CNRS research director at the Center for European Studies, Sciences Po

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video: Alain Garrigou in issued this evening or never!

video: Alain Garrigou

, author of Drunkenness polls FRANCE 3 - Tonight or Never! presented by Frederick Taddeï

December 9, 2010 -

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a pretty equal volume calculation of the regular tetrahedron

I got a good thirty journals, and I could edit the last activity. I realized that many students have trouble seeing in space. The manufacture four tetrahedra in the yard and was able to superimpose the levy from the interior volume remained empty. Students have nodes on the edges of the octahedron and could see in particular the lines parallel to each other.
The most amusing is that I felt my first module group needed to move to this construction, while the second, warned that it could be required to make in the yard and in the cold, thought he could be so interesting to make a figure in perspective and are arrived quickly to the result.
fact remains that the calculation is a bit annoying that the volume of the tetrahedron.
Students learned the formula: Volume of a pyramid
= one third of the area of the base times the height
But in a regular tetrahedron, what height, what is the bottom? and how to calculate this? In a

session course (probably a few, prefer a group session in half), I asked three students to construct a regular tetrahedron with my journals two meters (screw eyes at the end of the journals, and to link colson ), after pointing out the number of edges at each vertex.

After two minutes, the tetrahedron was constructed.

(I did not take pictures during my course, I would have liked to have presence of mind to do so in the heat of the moment, so I just took in my backyard, it will not reflect During this lively )

Question: while the height, what is it?

A student stands up, points to the height, took the rule and attempts to measure it.

then I go out a plumb and resumed his idea by attaching it to the top so that the tip touches the ground: here is the height. We can actually measure the rule.

can also be estimated by comparing with my size, because the top comes up to me, at about 1.65mètres. As I let them move around or even in the tetrahedron, the students can get an idea based on their own size.

But is there a way to know the exact value?

And besides, where the falls plummet?

responses flowing:

- in the center of the base (which center of the triangle?)

- the center of the circle circumscribed

- the center of gravity

- the center of the inscribed circle

- the orthocenter

- at the intersection of bisectors (how to draw?)

- at the intersection bisectors (trace how exactly?)

- at the intersection of heights (trace how exactly?)
- at the intersection of the medians (how to draw?)

Who is right among all these proposals?

- the triangle is equilateral, so everyone is right.

OK, we'll draw these lines so remarkable. I marked the middle of each edge. What can we trace exactly?

Some students use strings to represent the medians.

Can you calculate the length of the median?

- yes, because there is a triangle, so we can use the Pythagorean theorem.

With what lengths?

- the median coincides with the height, so one side is 1 meter and the hypotenuse is 2 meters.

Everyone is able to calculate the median?

And the center? Where is he?

Here, I had to remember that the center of gravity is located at 2 / 3 of each center, starting from the top.

And the height?

There I got a surprise. Nobody said he could see to put on a triangle. I pulled out my bracket and I turned around the plumb line, showing that this line was perpendicular to each line of the soil, especially medians plotted. The silence has been attentive to this point showed that something was happening, and that most students did not realize this fact.

Students were then set themselves to work to calculate all values. A figure

was marked with the names of points, they have the same calculation.

Those difficulties may remain around the tetrahedron to take all necessary measures. Those who want to check their calculation can also get up to measure. This course was interesting and lively. I think the students moving around and in a volume to calculate took another consciousness and developed their vision in space, the tetrahedron is to build this and build these lines provided to students. Some very average students first took the measurements used to calculate formulas an approximate value of the volume, and then have it validated their response. I then asked them to correct all calculations, with explanations. They are then returned to a good start in the problem.

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activity on trisectors

Since the time that this blog is dormant, I experimented with other objects. This idea
activity m has been blown by Danielle Salles group geometry IREM de Basse Normandie : Machine trisection angles. Share

an angle into two equal angles is possible with ruler and compass, all students remember the bisector. Cut at an angle into 4 or 8 is not much more complicated and just requires some precision. But cut at any angle into three equal angles is not possible using only ruler and compass. This is an old problem, also known as construction with ruler and compass of squaring the circle and duplicating the cube.
By cons, other technical solutions, using other tools provide solutions to this problem. After this
introdution history, I suggest to my students get in groups of 3 and use the following tools, using the manual to understand how they work. Before the end of the day, students must turn on all the workshops, making the angle trisection in each case, and validated by the teacher the proper use of the object, then choose two of these objects and sketch a figure to illustrate the geometric object. The study of these figures is presented at the session next module, after we had discussed what should be a mathematical figure, as a basis for effective demonstration.

objects trisection are to:
The tomahawk (or trisector Bergeryís)
manufacturing: a board cut with a jigsaw

box camembert:
manufacturing: a curtain rod, a screw that fits into this groove.
The geometric figure is almost visible, but I do not think I could hide it.

The tee:

It should be noted that these first three tools come down to exactly the same geometric figure and same reasoning. Of the two tools that students must study, he should consider a mandatory among these three, the second choice among the three following demonstrations that lead to quite different.

Double bisector:
bars and plastic fasteners. We can operate a slot on the bars of trisector, but can also cope by superimposing two bisectors.

Inspired trisector MacLaurin found on the site Geogebra
Made mecanno bars and elastics.

With bars mécanno:
Technique: two strips revolve around a groove. The angle formed at the end by these two brackets is the third of an angle formed by the triangle built with bars mécanno.

there is also a method of trisection using bends, I did not mention, because it is a little repetition with the first trisectors.


The instructions are on this PDF

After discussing the mathematical figure of it must contain (name of points, equal in length, angles), students needed to demonstrate the operation of two of them (one of the first 3 and the last three).
The results were somewhat disappointing, although there were some very interesting ideas of ownership of the problem. Given some mistakes, I realized that having just the photo does not always suffice to make the figure. So if I start this activity very rich, I will leave these objects available to my students at the bottom of my room, so they can handle it again if they feel the need to verify some properties and they have interviews.

Excerpts Copy:
folds to represent the transformations of the triangles into the tee:

passage of drawing the figure in three stages:

Just a drawing not when it interesting

Ultimately, this work has led students to think and move, to paraphrase Ruben Rodriguez, a passage from one universe to be experienced machines to a formalized world figures, then to a world of ideas and demonstrations.

An example:

Universe experimenting with the machine between the two screws E and D, there is the same number of holes between the two screws D and C.

Universe formalized the mathematical figure:

demonstation of the Universe:
ED = DC so the triangle is isosceles D EDC.


Even if it was rewarding, even if the necessary background to the level of fourth, it was not an exercise easier for students because it requires a lot of hindsight, the more difficult, as the saying one of my very good students, there was no solution on the Internet.

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