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Activity solid geometry (second) of

An activity to introduce the representation of solids in perspective and make volume calculations. Initially, students will construct a tetrahedron origami. For this I will show him this little film


and will have a sheet with pictures to do so at their leisure. In passing it may consider to demonstrate that the faces are obtained by folding equilateral triangles. Once folded, the student must identify and calculate the length of the side, the position of the center of the face and height using the Pythagorean theorem. It can thus calculate the volume of the tetrahedron it comes to manufacture. Once manufactured the tetrahedra, the students instructed to assemble tetrahedrons in order to obtain a tetrahedron with dimensions doubled. we quickly realized that it useless to paste the tetrahedra face against face, edge or even against edge, and in fact they should be put on top summit, even if the resulting solid contains a large hole. But what is the shape of this hole? It's hard to see ... Two solutions to better realize: magnetic rods. 1) We remake the four tetrahedra on each other, then removes what is too ... 2) We drew the four tetrahedra in perspective. In passing, we remember the rules: two parallel lines in space are represented by two parallel lines on the drawing. On parallel lines, proportions in length are retained. In going back to red edges of the hole, we realize we have shown an octahedron, and its parallel edges are found in FIG. We arrive at the almost magical part of the activity. What is the volume of the octahedron obtained? Students, surely a bit chilled by the calculation of the volume of the tetrahedron, will go against the grain. And yet! Obtained as the tetrahedron is an enlargement of the small initial tetrahedron with a coefficient 2, the volume has increased by 8. However, there are only four small tetrahedrons. So the full volume is equal to 4 times the initial volume. So the empty part is also 4 times the initial volume. The octahedron has a volume four times larger than the tetrahedron same side. In passing, we can see that the tetrahedron can not pave the area, but the tetrahedron and octahedron pave all the space, as illustrated beautifully in this print by Escher, "Planar".

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topology superheroes

Topology is a branch of mathematics concerning the study of spatial deformations by continuous transformations (without tearing or gluing of structures). (Wiki) The first
problem that can be connected to the topology of the problem is seven bridges of Königsberg, studied in 1736 by Swiss mathematician Leonard Euler.
Later, in 1895, Henri Poincaré launches first foundations of his work in topology analysis situs and introduces the concept of homology that interests us here. To talk

great popularizer, interest is the number of holes in the volumes. For example, we differentiate the ball (no holes) donut (torus) that has a hole. We will continuously deforms the ball fine, without tearing, it never will obtain the torus, and vice versa. The
topologist will knead, stretch, deform in emulation volumes of rubber to obtain topologically equivalent volumes.

A joke is a topologists topologist confuses a cup with a handle and a donut.


The best teacher of topology that we can dream is Professor Richards, head of the Fantastic Four, also known as Mister Fantastic. A brilliant scientist coupled with a deformable continuous rubber ball is the dream.

In looking for the different manifestations of the powers of the king by Mister Fantastic Jack Kirby, I would argue that Jolly Jack was intuitively understood the concept of topological equivalence. Examples support.

could summarize the power of Reed Richards saying it is topologically equivalent to a ball .

So by continuous deformation, Mister Fantastic can turn into a ball.


He therefore has no hole, and therefore can not be pierced by bullets. In fact, every hole, its topology becomes that of a donut.
And if Red had an ear piercing done before getting his powers, his powers would probably have been changed. He probably could deform and move the hole so that bullets pass through this hole.


So he has two Stategies for an attack by gunfire. Remember bullets continuously deforms its body to accompany their journey or deform so as to avoid (drawing Pollard K)



The ball is topologically equivalent to a block rectangualaire (anyone who has done The pastry can testify) and also bagged (not perforated), which can be handy if you want to capture the Hulk, for example.


To turn it into a torus, Red must pick up its structure, by withholding part of his body with his arms.



In all the episodes of Kirby FF from my collection, I found he respected the continuous deformation and equivalence to the ball. Except for these few exceptions, but I think we can charge them with ink, worse topologists.

In FF11, the two seem to weld arm at the elbow, which would make it equivalent to the torus. This is undoubtedly an error inking, or there was a bubble in place of the error, this error is too coarse to be desired.



In FF14, Mister Fantastic is transformed into a net to capture Namor. The ink seems to give the idea of a volume with many holes:


Fortunately, close up, Kirby explains how to obtain the net while keeping the ball structure. It seems to work better. Would still have to study the net structure using graph theory to be sure.


Even schema in FF17. Kirby seems to drill many holes in the structure of Mr. Fantastic. But it fixes the idea in the following picture showing us his stuff and leaves no doubt: it is a mistake to inking.


I do not think Jack Kirby has held a topology book in his hands, but his intuition of the deformation of a ball was perfectly coherent over the 102 episodes that he has drawn.

But to understand a concept, it may be interesting to have examples-cons, I was told these little monsters multicolored who do not know the topology.

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the magic square (puzzle)

A Homework I gave in 1S, which studies a puzzle sold as magic square. It consists of four parts in a stacked configuration and can give a square of side 10cm and another, giving a square a little larger, in which a square hole of side 2 cm was made in the center. I chose the values to find the calculations in a trinomial that simplifies easy and simple values for construction, but you can choose any values at the start. The construction of this puzzle played on a double-cutting plan. January 1 paving with two squares of different sizes: February 1 with a square tiling with vertices the centers of four squares connected by a small: This construction allows to demonstrate how very pretty the theorem Pythagoras. Perigal Henry, a London stockbroker 19th found that dissection of the square in 1874. Henry Dudeney, the famous creator of mathematical games, popularized in 1917. It should be noted that we can have both solutions in handling, after connecting the parts with a hinge. I do not know if it exists, but I think you can imagine square tables measure that could fit in some maneuvering around a central pillar.

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working on the Bayeux Tapestry Plate

Professor in a school in Bayeux, following discussions with colleagues on math problems on the local heritage, I began searching for monuments may succiter mathematical questions.
I walked the city, and, apart from a curve on the south facade of the hotel's dial
and a tetrahedron from the landing beaches (the same as them)

, I have not found much.
And then I thought of a novel I read two years ago, "Intrigue in the English" by Adrien Goetz, taking place in Bayeux around her embroidery in which one heroes playing with a reproduction of the Bayeux Tapestry, by wrapping around a cylinder. A fun theory, which suggests the Trajan column or columns of the XIth century found in Germany and showing scenes from the life of Christ. A clever and seductive idea, to take as it is. In any case, this raises interesting mathematical questions.
I wrote the novel that the author was delighted and gave me the references of a book with a reproduction of the tapestry very handy for making a model.
Using excerpts from the novel, I created this duty free time and this model.




On this model, with the alignments mentioned in the novel, one can find the diameter of this column there are hypothetical and amusing coincidence.
Almost on the same vertical, the oath of Harold, then, above the finger of God who would testify on oath, and yet above the comet, a sort of divine message.

I have not given the duty, I hope this funny vision of our local treasure will motivate them more, and that the next "read an excerpt from a novel using math" seduce students and offer them openings. A discussion with a student who said 1S literary mathematicians invited me to believe that it is an interesting idea.
The regret I have is that there is no more or column that can stick with the results found. I would have liked to do as an outlet to the cathedral of Bayeux with all my students armed with dressmakers to measure the round cylinders, but no pillar is cylindrical.
Neither the abbey to the ladies or men in the Abbey, though.

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Galton (second, first) Slideshow

This year I am nominated in high school, and my plan to make crafts for the college program is modified. Conduct visual and tactile tools for teaching math in high school seems a little harder, and limit off-topic, the program going towards more abstraction. However, for some lessons, like geometry in space, this approach can be interesting. I made this object a few years ago, but I shall certainly need this year, I recovered (Thanks Jacques). A Galton board, named after its inventor, Sir Francis Galton . Two removable rods up and down to release the ball. I used hex head screws, to have a probability 1 / 2 for each side. The results are correct, I wanted to use barriers to make a circular cross section for another board (the wooden dowels), but the results are very disappointing (the balls are very often on the sides). With the screws, once the board placed on a balanced place, we have results rather correct. is the first object I tinkered, the biggest problem he has is that the equipment cost me quite expensive, especially screws 21 (around 25 €, I remember correctly)



With the new programs of a second, an interesting place is given to algorithms, and I plunged with delight into the lineup of small programs. I used to write software scratch, because the new programs we offer second to discover this program. I was amused to perform a simulation of the Galton board. One of the nice features of Scratch is the opportunity to share our programs to other users, who can download and use and modify if they wish. Here is my simulation (take a value less than 150, please) Activity related to the Galton board: After a run of sixty balls, establish a frequency table for each column. How to explain this bell curve? Throw ball: where the ball will she fall? Why does the ball lands it more often in the middle column? Property: the probability for the ball falling into a column is proportional to the number of paths that lead to this column. We must therefore count for each column the number of paths leading to it. A path notation: For each obstacle, the ball will either left or right. Noting that the G and D to move left or right for each obstacle, draw the path GGDDGGGD. Where the ball lands after she this way? Give another path that leads to the same column. How many are there of road that leads into the right-most column? What is its code? By studying the number of opportunities in each barrier 1) on the edges 2) on an obstacle located on the banks, we come to realize that we can get closer and closer the number of paths leading to each step. We get Pascal's triangle, known as the 14th century Chinese

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displayed in the streets of Marseille ...

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club polyhedra

Here is a slideshow which presents the work of the club in my polyhedra college.

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A pantograph axisymmetric (6th, 5th)


I do not know if the object exists, I know that the pantograph central symmetry where to dilation, but I think it's a beautiful object to illustrate the course of the axial symmetry sixth and to give an application of the properties of the diamond in fifth.
Edit: After two three studies, I realize that not only the object exists, but there are also pantographs for travel, for the similarities and rotations. very clever too.
construction:
On a board, I screwed a rack that will act as rail.
I put two wheels (recovered from an old piece of furniture) that fit the size of the rail and I asked which branch of the same size Meccano.
This gives a diamond. On the other two peaks, I asked two short strips of bent where I felt stuck so they receive the support. It
sellotape two sheets of paper symmetrically relative to the rail.

For now, two drawbacks remain: can not remove the pencil and the support of markers tends to twist and distort the drawing.
To overcome these shortcomings, it takes two to manipulate the object, the first who made the drawing and the second which is guided, the two people paying attention to what are felt in the axis of the diagonal of the diamond. Whoever runs the drawing can also ask his buddy to lift the pen if necessary. My children were happy to try, but it will still have to train in handling.

Fifth, after the course on parallelograms individuals can be asked to demonstrate that the bars form a rhombus and the two markers are always placed symmetrically with respect to the rail.

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The yell of Marseille and Lyon

found in Section 27-05-2009 of the Marseillaise by MICHAEL CAIRO.

Transport. Karim Zeribi (RTM) and Bernard Rivalta (Sytral Lyon) allege the 35 billion given to the subway in the Greater Paris while the rest of the country is 710 million shares.
"If we want to do the Grand Paris and nothing beside it must be said," says Karim Zeribi, Marseille president of RTM, which was yesterday at the side of Lyon Bernard Rivalta, the chairman of the Joint Transport Union for the Rhone and Lyon. They held a joint press conference to denounce the state policy that ignores the larger provincial towns. They denounce the fact that 35 billion euros are planned for 130 km of subway in the Greater Paris while at the same time, the Grenelle Environment Forum provides only 710 million for transport, outside Paris. Karim Zeribi explains the meaning of the initiative: "Faced with the choice of the government for aid to public transport, we wish to unite because we share the same vision. We wish to express our anger at the imbalance that exists between the Grand Paris and the rest of France with only some crumbs. " He added: "The second and third largest cities of France are set to say that our country is not confined to Paris and that ourselves, but Toulouse, Strasbourg and Lille could have been associated with this approach. He considers "Borloo and Sarkozy must hear the message. We are not here to argue, when a decision is good it is, it comes from right or left, then it is wrong. Bernard Rivalta advance: "At the same time we saw the president, whose role it did not announce the plan for Greater Paris, and Jean-Louis Borloo and Fadela Amara, Dominique Bussereau and Hubert Falco announce figures in the Grenelle of the environment, without any measure with those of Greater Paris. Now, whether we build a multipolar France, as may be Germany, Italy and Spain in particular, is constructing a megalopolis and is terminated decentralization. He also criticized the way the money was distributed: "It's sprinkling and we played small towns against big. "While Karim Zeribi still wonders:" What criteria, three of the six projects that I proposed were rejected. I do have no idea. And the elected Marseille concluded: "It is clear that today it lends to banks and it does not help the joint associations of transport. We need a less Jacobin, more sharer, otherwise it will produce frustration in the population that feels forgotten. "

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This shit is a joke? Not even! Poor town on the day, where the Festival of Marseille is drowning, an artist (?) Finds nothing better than to plagiarize the symbol of Hollywood (and even he could have cried GAUDIN or OM)! The worst is that a regional newspaper resumes the info! And yet the worst of the worst is that we are confident that elected officials of this city will find the idea and the funding! Festival de Marseille poor, poor city ... 2013 is still very far!

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Incredible article on the rehabilitation of the center of Marseilles ...

item found in Provence, 11 May 2009, written by Florent Provansal

Rehabilitation of the center of Marseilles, the arm-of-war continues

Jean-Claude Gaudin refused the "ten suggestions" by Philippe Sanmarco to stay ahead of Marseille Development.

The company in charge of the layout of the Marseille area should have a new president on May 28 This is the date chosen by Philippe Sanmarco, (ex-PS, most municipal), elected to that position last spring, to convene a board of directors of the corporation (SEM), in order to confirm his resignation. The reproach
elected system of governance between the presidency and the direction of the Week and a strategy problem with a policy that does not correspond to the resolution passed municipal February 1, 2008 which allowed the agreement between Convention citizen movement created by Philippe Sanmarco and Gaudin-Muselier ticket for municipal elections.
At the request of the Mayor of Marseille Jean-Claude Gaudin (UMP), first met on April 20, the outgoing president had suspended its decision. But, after a new interview on May 4, Philippe Sanmarco has confirmed. "I have requested a ten-point, those that seem necessary for the exercise of the President." Requests which could not respond favorably Jean-Claude Gaudin.
"Political agreements with Philippe Sanmarco remain unchanged and will continue to be director of the company," said Mayor of Marseille. In addition, I will attend this meeting which will be elected the new president. "The council holds as MPM, 25% of the capital, reward the candidacy of Dominique Vlasto, Deputy Tourism and MEP.
Sanmarco As for Philip, he states that it will vote on May 28 "for a president who accepts his ten proposals." While ensuring that it stays in the local majority.
The Ten Philippe Sanmarco not to resign
"The foundations of a minimum acceptable position for any president are: 1 - The company accounts must be validated by an outside agency. The last control is the regional chamber of accounts and over 10 years ago. No organization can afford to rely on word of its CEO so long without external control. It is a decision to be taken immediately.
2 - This control should be first on the whole account. But also on activities, one by one. The PRI is in the forefront of balances to ask for one can not deploy a new political center without having a record of 10 years.
3 - Regarding the sequence of operations in the city center, the deliberation of the municipal council dated February 1, 2008 must be strictly respected. Now not only is this not the case, but there trends reverse. Worse, having failed to impose the automatic renewal of its concession, the CEO is now dragging things or makes them more and more impossible to resolve, to the detriment of the interests of the city. In order to see the end of 2009 we are in a deadlock and has emerged as the savior after putting in jeopardy the public interest. This issue has become emblematic and urgent: it is the reason why the CEO should be immediately withdrawn in favor of a Deputy CEO reporting to the President and Council, in conjunction direct the General Secretariat of the Town.
4 - Regarding the other major business areas, the long-standing requirements of the CRC must be strictly observed, "the dealer must provide annually to the community, a plan of the discounted cash transaction showing the schedule of revenue and expenditure and a note of economic conditions on the physical and financial completion of the transaction during the past year compared to original estimates and forecasts for the coming year ... . Therefore the operations of the center city (PRI, zac de la Bourse), but also that of Zac Capelette that du Rouet, that of Saint Martha, those of Vallon Régny and Jarre is an emergency organized meetings within the spirit and letter of the remarks of the CRC, which is indeed the aspiration of all elected officials concerned, especially as these documents are not limited to developments soothing as is unfortunately the present.
5 - For the future, the CEO must be divested of any communication activity, exclusively reserved to the President. The monthly newsletter will be suspended in order that its purpose and its contents are reviewed in depth.
6 - The CEO's personal situation should be clarified immediately when he refused to disclose his employment by threatening consequences of a possible early departure which nobody knows the conditions and surrounded by a total opacity.
7 - The CEO must agree to confirm that he did not hide behind hypothetical decisions of the Mayor's office and improperly utilized. Moreover we must integrate the fact that there are other shareholders that the city of Marseille and that all must be respected. In particular, it is time to realize that MPM is a separate community which also depends on the future of Marseille Development.
8 - A question asked by the CEO a director must be a written response. This document must be given to the directors who made the request, unless the CEO is happy to read it and keep it for himself.
9 - A scientific committee composed of independent persons will be created with the board of directors will solicit his opinion on topics of their choice. More generally the use of external expertise will be developed.
10 - To carry out its mission, the chairman may be assisted by one or two assistants of his choice ".

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Marseille 2013: Culture as an ultimate low blow

article found on: http://www.millebabords.org / by Lei Chapacans

MARSEILLE, carelessness
There are cities that seem to resist against all odds, despite the insults and contempt, with multiple attacks throughout the centuries have tried to make them like any other. Few are those who, despite the insistence of the blows were not processed urban-setting, center and periphery sterile dormitory. Marseille or at least what is left of them. Mocked and maligned in France until the mid-90s, she knew not to deny what is left of life. Overturning as clever as doubtful all the flaws that he had joined, the city lazy, dirty, vulgar and cosmopolitan has become fashionable lately. Semantic processing of words experienced and conceptualized in terms dishes were allowed to market valuation. Marseille, a city hospital, cosmopolitan, just not violent enough to be annoying, officially became a city in which to live. The spectrum of Haussmann, Attila the straight line is more haunted places. Rue de la République, real imperial bloodletting, whose implementation had not fulfilled wishes to return to a town near the center of Marseilles timid bourgeoisie, is presented in a new European bourgeoisie, as coveted as fantasy, as all modern eldorado opportunities for investment and housing, offices which, moreover, remain empty. Euromed tirelessly to commence, with the arrogance of the builders on the deserts, a restructuring of the European territory in regional clusters for competitiveness, operates around the new "neighborhood" of Joliette business with great blows of expropriation and expulsion. This project to construct an "area of peace, security and shared prosperity" is embodied in the destruction of the last enclaves of the popular center for deportation at the periphery and building poles for alternative virtual population acquired mobility or speculation. One of the last major European cities with a popular center bend Does the spine facing annihilation forms of collusion in order to substitute their shapeless conglomerates, separated by deserts and highways? In the absence of resistance size, sabotage of parking meters, actions against evictions, swarming rebellious lives in this administration and bankruptcies in the Lords are all cries of a city that refuses to be called only by deception that city .
flourishing culture and gentrification
When the sounds of jackhammers and the gigantism of cardboard no longer seem sufficient arise so little sneaky plans for a vital culture. Banners and brochures flower to celebrate the victory of Marseille-Provence as European Capital of Culture in 2013. The former Orangina-Schweppes CEO, Jacques Pfister, president of the event and the Chamber of Commerce and Industry, celebrates the marriage of culture and enterprise. To this patron of modern times, man of culture and great fighter against the demonization of the company, one role of the artist is to trace the financers economic moment they live. The event would have no issues but would participate in a politics of civilization. One of the many goals would be to reassure foreign companies that install on reception conditions and lives of their frames, families and employees in a city where culture must be well secured [ 1]. Others, like Rudy Ricciotti, architect of the future MuCEM, who still live together between the poor would not be an aesthetic project or a political project, complain that there is not enough rich people in Marseille and hope that the event could bring a little corn, a few beautiful women convertible car [ 2]. The consensus among policy makers and economic and the temporary absence of dissenting voices among the entertainers would it be due to the financial attractiveness of the famous 98 million? Citizens dubious say leaders should be content with a mobilization strategy that would become an actor of the event. This project, whose principal axes [ 3] have been thoroughly defined the attributes important to a culture with a European international show. The crumbs of a local folklore multiculturalist doubtful based on what was a city of Marseille in Marseille mode tweak the brand. Desertions and initiatives to fight in the field of art, a struggle which will manifest itself by a radical questioning of the role of guard dog on a lead financial authority and active participant in the collapse recent popular enclaves attributed to the artist, will pose challenges to those projects where culture and peace are forever linked. We can not say "No" and sabotage by all means the negation of what remains of life from below. Because if the love we begin by saying positive things, we should fall into the error of claiming, which is a mistake because when the love making these positive claims, he helps himself to convert these things, idea of themselves manipulated by trade, and the State Capital. Lei
Chapacans
Readings for an offensive Cities and social resistance, "Agone No. 38/39, 2008 Bendy Glu, Culture & Propaganda" "Lille 2004" European Capital of Culture in Agone, No. 34, 2005 Bruno Le Dantec, The Nameless City in Marseilles in the mouths of those who murdered, Red Dog, 2007 Website "The party is over" - http://lafeteestfinie.free.fr/
Published in No. 21 Offensive
OLS Chapacans c / o Thousand Babord Consolat 61 street, 13001 Marseille chapacans@riseup.net http://offensive.samizdat.net/ http://atheles.org/offensive/
[ 1] Remarks at the end of the radio program: Marseille for its economic policy makers http://www.grenouille888.org/dyn/spip.php? rubrique450
[ 2] Remarks Everything happens during the show! (21'30) On November 7, 2008 on France Culture http://www.tv-radio.com/ondemand/france_culture/TOUTARRIVE/TOUTARRIVE20081107.ram
[ 3] We will return later on the merits of this project.

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Lab: volume calculation (3rd)

By correcting the homework of my third on the geometry in space, I realized they did not figure, they imagined any section by a plane making practical calculations.
calculations were correct but difficult to read without the figure.
To force them to ask these questions, I proposed a practical session.
I arranged the tables different everyday objects.
The orders were to calculate the internal volume.
They had at their disposal rules, compasses, calipers and if they asked, bits of string. The work was expected
conducting a poster presenting their approach.
Lacking figure drawn in the statement, they were obliged to make a name for the points.
That day, my classroom was a bit like a trade fair at all.



The junction box for electrical outlets.
Two hemispheres + cylinder
Besides the use of calipers, no problem.



The roll of Scotch double-sided
difference of two cylinders
Using calipers.
No difficulties, but to show that the volumes do not always add.

jar of jam.
a right prism + 1 cylinder

The main difficulty was to calculate the area of the base, which is a hexagon.
Using calipers.

lampshades and funnels
truncated cone and truncated pyramid.
The calculation is more complicated and more ambitious. Indeed, we must find the position of the truncated apex.
Using the Pythagorean theorem, the theorem of Thales.
resolution of an equation.


Some students used several methods to locate the approximate position of the top: using multiple rules, pieces of string (it had to be cut to length, because impossible to remove with your finger on the right mark.)


Some students came to a very fine resolution of the problem of lampshades, others were content to make an approximation.
Anyway, they all made plans.