Stage de Pratique Accompagnée, M2 Agrégation
Rapport de Stage
Marguerite Flammarion & Elio Joseph 13 décembre 2016
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Stage de Pratique Accompagnée
CRCG
Nous remercions en premier lieu l’Université de Paris Saclay et le département de mathématiques d’Orsay qui, grâce à la formation qu’ils nous proposent, ont donné lieu à ce stage de pratique accompagnée qui se révéla, comme nous le verrons par la suite, enrichissant humainement et professionnellement. Nous sommes reconnaissants à Madame Françoise Issard-Roch qui nous af fecta fecta au lycée lycée René Cassin d’Arpajon d’Arpajon et qui géra géra nos retar retard d avec avec le sourire. sourire. Nous remer remercions Arpajon, Arpajon, ville vil le agréa agréable ble tant par son charme charme que par son calme. alme. Comment Comment ne pas appré apprécier la qualité qualité environnem environnemental entalee de la ville d’Arp d’Arpaa jon ? Les espaces espaces verts, les pelouses, pelouses, les parterr parterres es fleuris, soigneusement entretenus entretenus par les jardiniers de la ville, incitent à la promenade. Un effort constant est fourni par la commune pour embellir et préserver le cadre de vie. Parmi les récentes réalisations, l’ouverture du parc de la Prairie - espace de détente et de loisirs - permet la poursuite des cheminements et liaisons douces, dans le cadre de la promenade de l’Orge. Nous ne pouvons que remercier madame Marie-Christine Lévi qui nous enseigna l’enseignement et nous apprit à apprendre. Nous ne sommes que ses élèves mais, grâce à elle, nous nous sommes élevés. Toute notre gratitude va à Madame Florence Lefebvre , femme agréable tant par son autorité que par sa bonne humeur. Pédagogue née, nous la remercions pour ses conseils avisés, son partage généreux de sa grande expérience et sa bienveillante supervision. Nous rendons grâce à tous nos professeurs passés, présents et futurs pour nous avoir insufflé la passion de la mathématique et de son enseignement et présenté la muse de la divine inspiration. On le sait maintenant : ils sont toujours présents. Mais nous ne pouvons cesser de nous répandre en remerciements tant leur générosité gratuite et leur constante amitié est hors du commun des mortels. Ils ont relu nos sujets, nos corrections, ont partagé leurs idées, leur expérience, leurs échecs, leur réussite. Nous les bénissons par ordre alphabétique : Julien Brice, Sandrine Gauthier, Florian Granger, Laurène Latieule-Cogné, Coralie Lemonnier. Ils mériteraient un paragraphe à eux tous seuls : Gédé Gédéon on Chevallier, Chevallier, le paladin aladin au grand grand coeur présent présent aux moments moments difficiles ; Cyril Falcon, la tête chercheuse qui nous cultiva de sa science et de ses dimensions. Nous remer remercions le lycée lycée René Cassin qui nous accepta accepta en stage stage ; monsieur monsieur le proviseur Bertrand Bertrand Villain Vil lain pour sa confiance confiance ; mesdames Marie-Françoise Marie-Françoise Gire et Elisabeth Elisabeth Pezon-Sigal pour leur gentil lesse et leur patience patience ; madame Aurélie Aurélie Maupas qui nous permit permit de nous sustente sustenterr aux jours de disette disette ; ainsi que tout le personnel personnel
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Stage de Pratique Accompagnée
CRCG
évidemment évidemment les profess professeurs eurs de mathémat mathématiques iques du lycé lycée René Cassin. Cassin. Nos pensées ensées vont particulièrement à monsieur Jean-François Alliot, madame Séverine Perrache, monsieur Cyril Vargas et madame Sandrine Bozec, qui nous enrichirent notre expérience en nous permettant d’assister à un de leurs cours. Nous exprimons exprimons notre notre sympathi sympathiee à mademois mademoiselle elle Marie Marie Escot Escot et monsieur monsieur Emmanuel Emmanuel Magnin, Magnin, dit "Manu", "Manu", qui nous supportè supportèrrent sans relâche relâche et égayèr égayèrent ent nos journées de leurs sourires. Enfin, ces remerciements seraient vains s’ils ne mettaient une fois de plus à l’honneur celle qui fut la véritable actrice de cette aventure. Parfois discrète mais toujours présente par sa poigne de fer et son rire joyeux, elle fut la meilleure des tutrices et restera à jamais dans nos coeurs :
Florence Lefebvre.
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Stage de Pratique Accompagnée
CRCG
Instruisez-le d’exemple et rendez-le parfait, Expliquant à ses yeux vos leçons par l’effet. Corneille, Le Cid, Acte I, Scène 3.
Introduction Laissez-moi vous conter une histoire. Autrefois, la riche tribu Ayayonga vivait dans un riche pays couvert de champs fertiles et de prés prés verdoyants. verdoyants. Les Les Ayayonga s’étaient rendus célébre célébress non pas pas par le bétail gras qu’ils élevaient, ni par les nombreuses variétés de blé qu’ils cultivaient, mais par le sel gemme qu’ils trouvaient en abondance dans leurs terres. Casser des blocs de sel n’était pas difficile. C’était plus facile, en tout cas, que cultiver le blé et élever élever le bétail étail ! Av Aveec le sel, sel, on pouvai ouvaitt tout tout achete acheterr : le blé et le bétail, étail, de magnimagni fiques tissus, des bijoux et des armes. Ainsi, les Ayayonga abandonnaient abandonnaient peu à peu le travail, négligeant leurs terres et leurs bêtes, entiérement occupés à l’extraction du sel qui était la source de leurs richesses. Le puissant dieu Mguri-mgori considérait avec avec mé méccontenteme ontentement nt l’attitud l’attitudee des Ayayonga Ayayonga.. Il décida, décida, en fin de compte compte,, de les punir pour leur paresse. Mais auparavant, il voulait leur accorder une chance. Ainsi, un vieillard nommé Bakomba s’adressa au peuple ayayonga : "Ecoutez-moi, hommes ayayonga ayayonga ! Je suis assez vieux, hommes, hommes, pour me permettr ermettree de vous parler parler comme comme le puissant Mguri-mgori me l’a ordonné. Le puissant dieu est en colère contre vous, car vous négligez votre bétail et ne cultivez plus votre terre. Vous ne faites qu’extraire le sel et amasser amasser les richesses richesses,, oubliant oubliant vos dieux, dieux, y compris ompris le grand grand Mguri-mgori Mguri-mgori !" Bakom Bakombba n’eut ’eut pas le temps temps de finir. finir. Les Ayayo Ayayonga nga se jetèr jetèrent ent sur lui et l’ac l’accusèrent de convoitise et d’ambition. Ils décidèrent de s’en débarrasser. Ils attachèrent le pauvre vieillard à un arbre et le lapidèrent avec des blocs de sel. Avant de mourir, Bakomba s’écria : "Le puissant Mguri-mgori vous regarde et son oeil affligé détruira votre votre arro arrogante gante richesse richesse !" A peine eut-il eut-il pronon prononccé ces paroles aroles que le ciel se couvrit couvrit de lourds nuages noirs qui commencèrent à déverser des trombes d’eau sur la terre. Dans son affliction, Mguri-mgori inonda de ses larmes le pays des Ayayonga, faisant fondre fondre leur richesse, le sel. Lorsque Lorsque le soleil brilla à nouveau nouveau dans le ciel, il ne restait restait rien de la tribu ayayonga, de ses champs et pâturages, de ses mines de sel. A leur place s’étendaient à perte de vue des marais salés, dans lesquels aucune végétation ne pousse, aucun poisson, aucun crocodile ne vit. On appelle ces marais Topé-tope, "Larmes salées ".
Ce conte nous enseigne que la paresse et la négligence sont rarement récompensées. C’est donc avec les conseils du sage Bakomba en tête que nous avons effectué notre stage de pratique encadrée, au cours de notre seconde année de Master Agrégation Agrégation Second dégré – Parcours Parcours Mathématiques. Mathématiques.
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Stage de Pratique Accompagnée
CRCG
Sujet du Stage age Stag Stagee de pratique acc accomp ompagné gnée Lieu du Sta Stage Lycée René Cassin, à Arpajo ajon (91) Dates Dates du stag stagee Les lundis lundis du 10 10 octobr octobree au 12 décem décembre bre 2016 Duré Duréee du stag stagee 32 heur heures es d’ob d’obse serv rvat atio ionn et de prat pratiq ique ue Classes obs observées De la secon econdde à la terminal nale Clas Classe se ense enseig igné néee Termi ermina nale le Scie Scienntifiq tifique ue numér uméroo 4 Tutrice du stage Madame Florence Lefebvre Parlons un peu du lycée qui nous a accueilli pour notre stage. Le lycée René Cassin n’a pas de biographie sur le site de l’Académie de Versailles. En revanche, nous pouvons donner une idée de son niveau : 1526 sur les 2277 lycées en 2016 (ce classement évoluera possiblement en 2017), mais aussi 25 sur les 42 lycées du département la même année. Enfin, l’Express lui accorde la note de 14,4/20 en méthodologie, thodologie, soit 18/25 en écriture fractionnaire.
Figure
vide.
1 – La salle des profs,
2 – Le lycée René Cassin Figure
Donnons maintenant une biographie de René Cassin. L’image à venir est disponible à l’adresse http://img.over-blog-kiwi. com/0/53/84/84/201305/phpIZYEu6.
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Stage de Pratique Accompagnée
Figure 3
– R.C.
CRCG
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Stage de Pratique Accompagnée
CRCG
le reste de sa vie. Il prit alors une décision héroïque : celle de rendre la guerre horsla-loi. Dans la seconde partie de sa vie, René Cassin eut une influence déterminante au sein de la France libre, puis en droit international public. Il passa l’arme à droite (il était gaucher) le 20 février 1976 à Paris. Finalement, ce que René Cassin laissa à la postérité fut son fils : Marc. En observation nous avons observé plusieurs classes : une terminale scientifique spécialité mathématiques, une première scientifique, une seconde générale, une terminale scientifique spécialité ISN et enfin une terminale scientifique normale. Nous n’avons pas observer de première STMG suite à un imprévu de dernière minute. En enseignement nous avons pris en charge la classe numéro quatre de terminale scientifique pour leur enseigner les nombres complexes. Nous avons observé un nombre d’heure certain, nous avons enseigné environ seize heures. Le contact avec notre tutrice fut aisé. Nous l’avons appelée par voie cellulaire afin de convenir d’un rendez-vous, qui fut fixé au premier lundi de notre stage. Nous pénétrâmes dans la salle des professeurs et là, nous la vîmes. Nous balbutiâmes un "euh... bonjour... nous cherchons madame Lefebvre ". "C’est MOI," nous réponditelle d’une voix profonde. Nous fîmes connaissance. connaissance. Tout Tout de suite, elle nous plut. "Je ne considère pas être une prof modèle affirmait-elle ; pourtant, à nos yeux, elle modèle " nous affirmait-elle l’était. Ainsi, nous passâmes deux mois incroyables dans cette aventure obscurément lumineuse.
Florence Flore, anse Le fait vraiment
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Stage de Pratique Accompagnée
4 4 S S T T s s b b o o é S p s e r S è T 1 s s b b o o é p s N S S I T é n s p o i t S a T v r s e s b b o o e c i r ) t u h t 2 ( c e 4 v a S n T e s i t b e r o t n e 1 2
s e x e l p m o c s e r b m o n s e l s u o t r u s e m m a r g o r p e l t u o t e d e r t i p a h c e l t u o t e d s r u o c e l t u o t e d n o i t a r a p
h 1 h g r g 1 a r i o M a l E D M D T D T T e c i r t u P é t p s c A e S e v d n T a 2 n s s e b i b o t o e r t n e n e i t e r h 1 o t n i e g l r a E t s e M r g r u s r o a u c M o c s r u o c h o g r 1 i l a o E i l s M s E r r u u s o o r c c u o S E c N I E O R R I T T
e e s o u i e l d n i r E 2 o l s D b g T O n fi n e i t e r t n e
n e i t p e a r j t n e
e l h b 1 t a s e g r r l a s u a n M r n i A o D v T e d o i l e E l s b r a u t s e o l c r u s a n O r n N i A o O v G e I D R T 6 7 8
CRCG
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Stage de Pratique Accompagnée
1.2 1.2
CRCG
Phase Phase d’ d’obs obser erv vation ation
Nous avons observé différentes classes, allant de la seconde générale et technologique nologique à la terminale terminale scientifique. scientifique. Plus précisément, nous avons observé quatre heures 1 de cours en une classe de terminale scientifique, cette classe-même que nous allions enseigner par la suite. Nous avons également observé trois heures de cours d’option de spécialité mathématique en classe terminale scientifique. Ces sept heures d’observation se firent pendant les cours de notre tutrice, madame Florence Lefebvre. Mais ce n’est pas tout, et c’est loin d’être fini, nous avons également observé une heure de cours d’option de spécialité I.S.N., en classe de terminale scientifique, avec l’aimable autorisation du talentueux monsieur Alliot Jean-François. Ultérieurement, nous avons observé une heure de cours en classe de première scientifique, en compagnie de nos camarades de classe sous la tutelle de madame Perrache Séverine. Subséquemment, nous avons observé une heure de cours d’A.P. (lire "Aide Personnali Personnalisée"), sée"), en classe de seconde générale et technologiq technologique, ue, plus exactement exactement un 2 tiers de celle-ci , grâce au chaleureux accueil de l’aimable monsieur Vargas Cyril. Enfin, nous avons observé une heure de cours en classe de seconde générale et technologique, cette fois-ci en classe entière, grâce à la touchante hospitalité de madame Bozec Sandrine. Sandrine. Il était initialement prévu que nous observassions une heure de cours en classe de première première S.T.M.G., S.T.M.G., cependant, pour des raisons indépendantes indépendantes de notre volonté, volonté, l’accès à cet événement nous fut refusé. De même, nous ne pûmes assister à aucun C.D.C. C.D.C. ("Conseil ("Conseil De Classe"). Classe"). Tout ceci nous permit de faire de multiples observations instructives, selon les axes décrits ci-après. 1.2.1 1.2.1
Le choi choix x des des axes axes d’ob d’obse serv rvati ation on
Nous avons, conformément aux instructions en vigueur, effectué chacun une observation ciblée selon deux axes préétablis à l’avance. Nous avions préparé des fiches préalablement remplies pour être plus efficaces dans notre observation au moment crucial. Nous appuyons notre étude sur les quelques séances d’observation effectuées. Plutôt que d’insérer des images des feuill complétées, complétées, parfois avec un soin certain,
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Stage de Pratique Accompagnée
CRCG
• La discipline : J’ai choisi cet axe d’observation car la pratique d’une bonne disci-
pline est un sujet qui me tient particulièrement à cœur. Même si les professeurs se plaisent à dire qu’ils ne sont pas là pour faire la police, cela fait en pratique pleinement partie de leurs charges, souvent contre leur gré. Ainsi, une bonne pratique de la discipline permet une classe tenue, et une classe tenue permet un enseignement enseignement soutenu. J’ai donc un intérêt profond pour comprendre les mécanismes derrière cette science : les stratégies qui fonctionnent, ainsi que celles qui ne fonctionnent pas. Cet axe est celui qui est le point de départ à une classe attentive et intéressée, tous les autres axes découlent donc de celui-ci. • Les différents types d’activité : J’ai choisi cet axe d’observation car je pense que l’alternance entre différents types d’activités est une clef permettant de mieux prendre ne compte la diversité des élèves et ainsi de mieux attirer leur attention. J’ai également choisi d’étudier l’activité réelle des élèves en fonctions des différentes activités car c’est un sujet qui me tient particulièrement à cœur. En effet, c’est une des vocations du professeur de réussir à intéresser ses élèves en tout temps, tout lieu. Il lui incombe de réussir à faire participer tout le monde, même les élèves les plus en difficulté. Ainsi, une plus grande implication de la part des élèves élèves permet p ermet un enseignement enseignement plus soutenu soutenu et mieux accueilli. J’ai donc un intérêt profond pour comprendre les mécanismes derrière la didactique des savoirs savoirs disciplinaires disciplinaires au lycée : les stratégies stratégies qui fonctionnent, fonctionnent, ainsi que celles qui ne fonctionnent pas. J’ai notamment lu avec attention le livre Parler pour que les élèves écoutent, écouter pour que les élèves parlent , de Adele Faber et Elaine Mazlish, qui traite de ce thème. Cet axe est la pierre d’angle d’une classe attentive et intéressée. Pourquoi nous posons nous la question de pourquoi nous avons choisi ces axes d’observ d’observation ? L’importance de la réflexion post-évènement est indéniable, c’est en psychanalysant ce qui a été fait, pourquoi cela a été fait et comment cela aurait pu être fait que nous pouvons tirer un profit maximum de notre expérience. Perpendiculairement à la réflexion communicationnelle, les conjectures poussent les analystes à déployer leurs les atouts de stratégie. En effet, afin de maîtriser l’expertise référentielle, chaque entité doit désenclaver les déboires organisatifs. Ainsi, il est important de se poser la question de la cause de ces axes, et non d’accepter seulement un choix qui pourrait sembler aléatoire ou même chaotique.
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Stage de Pratique Accompagnée
CRCG
Débu Débutt du cour courss Mili Milieu eu du cours cours Fin Fin du cours cours éléments déclencheurs déclencheurs de ces moments d’inattent d’inattention ion ? • Quels ont été les éléments 1. 2. 3. 4.
5. 6. 7. 8.
• Le professeur applique les punitions suivantes : ◦ Séparation des élèves ◦ Exercices supplémentaires ◦ Interrogation supplémentaire ◦ Correspondance avec les parents ◦ Exclusion ◦ Détention ◦ Autre moyens ont été utilisés par le professeur pour récupérer l’attenti l’attention on ? • Quels moyens 1. 2.
5. 6.
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Stage de Pratique Accompagnée
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Stage de Pratique Accompagnée
CRCG
journée peut être résumée dans le graphique suivant. suivant.
Figure 1.4
– Attention de l’élève en fonction de l’heure de la journée.
Ainsi, le matin est plus propice à un cours magistral et l’après-midi à un travail dirigé. C’est donc ce que nous avons essayé de faire en pratique une fois la phase d’observation terminée. • Ce qui est ressorti du deuxième point d’observation est assez étrange. En effet, les moments d’inattention ce sont plus fréquemment produits dans l’ordre : en milieu de cours, en début de cours et enfin en fin de cours. A quoi cela est-il dû? Il est possible, voire probable, que comme la plage du "milieu" de cours est temporellement plus large que celle du début, c’est ainsi fort naturel que ce soit concentré le plus grand nombre de moments d’inattention. En proportionnalité moment d’inattention sur temps de la plage, nous ne sommes pas
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Stage de Pratique Accompagnée
CRCG
◦ Correction des exercices évènement sortant sortant de l’ordinaire l’ordinaire arrive arrive ◦ Dès qu’un petit évènement ◦ Effacement trop rapide du tableau Nous pouvons ainsi en conclure que chaque petite chose est propice à l’inattention. • Listons les différentes punitions qu’applique le professeur, toujours avec leur fréquence d’occurrence. d’occurrence.
◦ Correspondance avec les parents ◦ Exclusion (menace uniquement, si carnet non signé ou exercices non faits) Nous remarquons que le nombre de punition est donc très faible. Les élèves doivent être très sages? • Listons les différents moyens qu’emploie le professeur pour récupérer l’attention de ses élèves, encore avec leur fréquence d’occurrence.
◦ Demande explicite de silence ◦ Tape dans ces mains (plutôt occurrent) ◦ Chuuuuut (vraiment très occurrent) ◦ Réprimande nominatives (très occurrent) ◦ Ironie, sarcasme (super occurrent) ◦ Voix qui monte ◦ Faire une blague • Enfin, pour conclure ce beau travail, croisons le type de réaction du professeur
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Stage de Pratique Accompagnée 1.2.3 1.2.3
CRCG
Axe Axe : les différe différen nts types types d’ac d’activ tivité itéss
............................ Date : ............................ Activité
............................ Heure : ............................
............................ Classe : ............................
Temps Nombre Comm Commen entt les les élèv élèves es Oral/Écrit d’élèves participent participent-ils -ils ? participant
Recherche d’exercices Correction d’exercices Intervention Intervention d’élève Manipulation Debriefing contôle Interrogation Travail en groupe Autre Les différentes activités chronologiquement :
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Stage de Pratique Accompagnée
CRCG
Voici mes fiches d’observation (résumées).
TS Spécialité Date Horaire Classe Effectif Enseignant 10/1 10/100 13h3 13h300 - 15h2 15h200 TS spéc spécia iali lité té 19 élèv élèves es Mme Mme Lefe Lefebbvre vre Les différentes activités chronologiquement :
◦ Debriefing contrôle (10 min) : La correction est polycopiée, les élèves posent des
questions questions à l’oral. l’oral. ◦ Correction des exercices (35 min) : Les élèves passent au tableau et répondent aux questions. Ils donnent donnent d’autres pistes à l’oral. l’oral. L’enseignan L’enseignante te passe au tableau tableau également. ◦ Recherche d’exercices (10 min) : Tous les élèves participent à l’écrit. ◦ Pause (2 min) ◦ Recherche et correction d’exercices (25 min) : Les élèves parlent, recherchent et
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Stage de Pratique Accompagnée
CRCG
◦ Debriefing devoir maison (10 min) ◦ Correction d’exercices (15 min) : Les élèves se lèvent, prennent le stylo et corrigent
au tableau. ◦ Cours (30 min) : Les élèves travaillent à l’oral sur des exemples simples puis généraux. Ils traitent beaucoup d’exemples. Ils découpent et collent un cours polycopié mais ils écrivent écrivent eux-mêmes les exemples. exemples. ◦ Pause (15 min) ◦ Recherche d’exercices (15 min) : Les élèves recherchent des exercices type. Ils les traitent à l’écrit, dans le cours. ◦ Cours (15 min) ◦ Recherche d’exercices (5 min) ◦ Cours (5 min) ◦ Recherche d’exercices (15 min)
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Stage de Pratique Accompagnée
CRCG
2nde -AP Date Horaire Classe Effectif En E nseignant 21/1 21/111 12h3 12h300 - 13h2 13h255 2nde 2nde AP 11 élèv élèves es M. Varga argass Les différentes activités chronologiquement : ◦ Accueil, distribution des feuilles (5 min) ◦ TP algorithmique (10 min) : La feuille de cours/TD est polycopiée. Les élèves travaillent au brouillon. L’enseignant passe dans les rangs et corrige au tableau. Les élèves répondent et posent des questions. ◦ Rappels de cours (5 min) ◦ TP algorithmique (10 min) ◦ Correction de l’exercice (15 min) implémentent l’algorithme sur leur calculatrice, ◦ Manipulation (10 min) : Les élèves implémentent à l’aide du support polycopié.
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Stage de Pratique Accompagnée
CRCG
• Déroulement de la séquence et place de la séance : La séquence se déroule
en quatre séances de cours consacrées comme ceci : algèbre, équations, géométrie, trigonométri trigonométriee ; suivies d’une séance de devoir sur table, puis de quelques quelques séances d’annales; le tout agrémenté d’un devoir maison. La séance étudiée / analysée plus précisément ici est : la première séance de cours, autrement dit celle portant sur l’algèbre. • Objectifs de la séance : Les objectifs de la séance sont les suivants :
développer leurs premières premières connaissances, connaissances, essentielles essentielles pour la ◦ Aider les élèves à développer suite de la séquence, sur les concepts fondamentaux des nombres complexes, notamment via une approche historique de l’intérêt pratique des nombres complexes, anisi qu’une vision très algébrique de C. ◦ Développer les capacités de calculs dans le R-espace vectoriel C en insérant dans le cours plusieurs exemples et exercices corrigés, notamment sur les calculs de double distributivité utilisant le fait que i2 = −1. ◦ Insister sur les démonstrations en accord avec la pédagogie de la filière scientifique, par exemple en insistant sur l’importance de l’unicité de l’écriture sous
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Stage de Pratique Accompagnée
CRCG
ne représen représente te rien rien de concret concret pour eux ? L’appr L’approche oche historiq historique ue que nous avons avons prévue est censée prévenir cette difficulté. • Déroulement prévu (les différentes étapes et leurs durées ( sic), éventuellement une organisation particulière de la classe, du tableau, supports, TICE..... (sic )) : Les différentes étapes prévues sont les suivantes. Première prise de contact avec les élèves. Nous ferons l’appel pour essayer de
retenir leurs prénoms. Cours de type exposé présentant une approche historique de la notion de nombre complexe. Cours formel d’introduction aux nombres complexes, agrémenté d’exemple et autres exercices et tous genres. Nous ne prévoyons pas d’organisation particulière de la classe, les élèves ayant déjà été disciplinés par notre tutrice depuis le début de l’année. Nous utiliserons le tableau blanc et ses feutres qui vont avec. Nous nous appli-
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Stage de Pratique Accompagnée
CRCG
Bien évidemment, nous porterons une attention toute particulière à la rigueur mathématique. Par exemple, nous démontrerons rigoureusement les théorèmes et les propriétés qui en découlent lorsque leur démonstration est accessible à un élève de terminale scientifique. Cette séance étant la première de la séquence, nous comptons sur la fraîcheur mentale, la curiosité et l’absence de lacunes desdits élèves sur ce sujet nouveau pour que leur activité soit réelle et leur intérêt certain. Nous veillerons sans relâche à leur constante implication dans le cours à travers une participation active de
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Stage de Pratique Accompagnée
CRCG
trigonométri trigonométriee ; suivies d’une séance de devoir sur table, puis de quelques quelques séances d’annales; le tout agrémenté d’un devoir maison. La séance étudiée / analysée plus précisément ici est : la troisième séance de cours, autrement dit celle portant sur la géométrie. • Objectifs de la séance : Les objectifs de la séance sont les suivants :
◦ Aider les élèves à acquérir des connaissances essentielles sur les concepts fon-
damentaux de représentation géométrique des nombres complexes, notamment b onne compréhensio et appréhension appréhension du plan d’Argand-Cauch d’Argand-Cauch via une bonne
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Stage de Pratique Accompagnée
CRCG
Accueil chaleureux des élèves et ramassage des copies de devoirs maison. Correction des exercices que nous avions donnés à finir. La correction sera
effectuée par des élèves volontaires. Cours sur la représentation géométrique d’un nombre complexe et introduction de nouveau nouveau vocabulaire. vocabulaire. Cours sur le module d’un nombre complexe. Nous ne prévoyons pas d’organisation particulière de la classe, nos élèves étant
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Stage de Pratique Accompagnée
CRCG
Afin d’obtenir une activité réelle optimale des élèves pendant ce travail complexe, nous veillerons à leur offrir du temps pour chercher les exemples et autres exercices que nous traiterons traiterons dans le cours. Nous aurons à cœur d’interroger d’interroger régulièrement régulièrement nos élèves sur tel ou tel point du cours, ainsi que de circuler dans les rangs pour surveiller l’avancement des troupes et répondre à d’éventuelles questions. • Évalutation ou travaux de rédaction prévus : Il n’y aura pas d’évaluation pendant cette séance. En revanche, nous évaluerons les acquis de cette séance au cours de la séance prochaine. Ils nous rendront le devoir maison qu’ils auront
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Stage de Pratique Accompagnée
CRCG
Plus forts. Le sujet du devoir maison est en annexe. Nous avons choisi de commencer par un petit exercice burlesque, mais qui n’en n’était pas moins pédagogique. Les élèves devaient simplifier ¯ ¯ ¯ ¯ ¯ ¯ z z¯
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Stage de Pratique Accompagnée
CRCG
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Stage de Pratique Accompagnée
CRCG
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Stage de Pratique Accompagnée
CRCG
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Stage de Pratique Accompagnée
CRCG
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TS4
Nombres complexes et géométrie Exercice Exercice 1.
On considèr considèree quatre quatre points points A,B,C,D d’affixes respectifs respectifs z A = −2, z B = 1 + i, z C − 1 − 2i et z D = 2 − i. C = −1 1. Faire un dessin. dessin. Remar emarque : De manièr manièree génér générale ale,, que ce soit soit demand demandéé ou non, non, il est toujours toujours extrêmement recommandé de faire un dessin lorsqu’on a à résoudre un problème de géométrie. 2. Montrer Montrer que que ABCD est un parallélogramme. 3. Que le triang triangle le ODC est rectangle isocèle. 4. Détermin Déterminer er l’affixe de H centre de ABCD .
5. Détermin Déterminer er l’affixe de C symétrique de C par rapport à D . Exercice Exercice 2.
Déterminer l’ensemble des points M d’affixe z du plan tels que 1. |z | = 4, 2. arg(z ) = − 3.
π
4
z + 1 ∈ iR. z − 1
[2π ],
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TS4
Nombres complexes et géométrie
Exercice Exercice 1.
On considère quatre points A,B,C,D d’affixes A,B,C,D d’affixes respectifs z A = −2, z B = 1 + i, z C = −1 − 2i et z et z D = 2 − i. 1. Faire un dessin. dessin. Remarque : De manière générale, que ce soit demandé ou non, il est toujours extrêmement recommandé de faire un dessin lorsqu’on a à résoudre un problème de géométrie.
2. Montrer Montrer que que ABCD est ABCD est un parallélogramme. −→ = z B − zA = 1 + i − (−2) = 3 + i, et z −−→ = z D − zC = 2 − i + 1 + 2i On a z AB et z CD 2i = 3 + i, donc les affixes sont égaux, donc les vecteurs sont égaux, donc on a un parallélogramme. 3. Que le triang triangle le O ODC DC est est rectangle isocèle. (2−i)(− )(−1+2i) −zO On calcule z calcule z = zzD = −21−−i2i = (2− = 5 C −zO Avec les propriétés du module, |z | z | =
zD −zO z −z C
O
=
5i i . 5 = i. |zD −zO | |zC −zO |
=
OD OC
.
Or | Or |zz | = |i| = 1, donc O donc OD D = O = OC C , donc le triangle est isocèle en O. O . −−→ −−→ De plus on a OC d’affixe OC d’affixe 2 − i et OD d’affixe −1 − 2i, donc on a les coordonnées. On −−→ −−→ calcule le p.s. OC .OD = OD = −2 + 2 = 0. 0. 4. Détermine Déterminerr l’affixe de H de H centre centre de ABCD. ABCD . H H est le milieu des diagonales, donc H H milieu de [AC ], donc zH = 1 3 2 (−2 − 1 − 2i) = − 2 − i.
1 2 (zA
+ z C ) =
5. Détermine Déterminerr l’affixe de C de C symétrique de C par C par rapport à D. D . ), soit 2(2 On a D milieu de [C [ C C ], donc z donc z C vérifie z vérifie z D = 21 (zC + zC soit 2(2 − i) = −1 − 2i + zC soit z soit z C = 5.
Exercice Exercice 2.
Déterminer l’ensemble des points M d’affixe M d’affixe z z du plan tels que 1. |z | = 4 |z | = 4 ⇐⇒ OM = 4, l’ensemble de points cherché est donc le cercle de centre O et de rayon 4 rayon 4.. 2. arg(z arg(z ) = − π4 [2π [2π ]
−−→ arg(z arg(z ) = − π4 [2π [2π ] ⇐⇒ (u, u,OM ) OM ) = −π/ − π/4[2 4[2π π ], l’ensemble de points cherché est donc la demi-droite ouverte blabla... z + 1 3. ∈ iR z−1 z ¯+1 ¯ ¯ Posons Z := zz +1 = 1, zz+1 −1 . On a Z ∈ iR ⇐⇒ Z = −Z , or Z = z¯−1 . On a si z −1 = z ¯+1 ¯ − z + z¯ − 1 = −z z¯ − z + z¯ + 1 ⇐⇒ 2z z¯ = 2 ⇐⇒ |z |2 = 1 ⇐⇒ |z | = 1. z ¯−1 ⇐⇒ z z L’ensemble cherché est donc le cercle de centre O et de rayon 1 (privé de 1). 1 ).
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Nombres complexes et forme trigonométrique Exercice 1.
Déterminer l’ensemble des nombres complexes 1.
| | = 4,
:
z
2. arg(z ) = 3.
z vérifiant
z + + z
1 1
− ∈
π
−4
[2π],
iR.
Exercice 2.
Mettre sous forme trigonométrique les complexes suivants : 1 + i , 3 Mettre sous forme algébrique les complexes suivants : z =
√ 2(cos( 5
π
6
) + i sin(
z =
5π )). 6
−4
i
π
et i. π
π
π
π
π
3(cos( ) + i sin( ), z = cos( ) + i sin( ) et 3 3 2 2
Exercice 3.
1. Mettre Mettre sous forme trigonométri trigonométrique que (1 + i )8 . 2. Détermin Déterminer er tous les entiers entiers
n tel
que (1 + i ) est réel. n
Nombres complexes et forme trigonométrique Exercice 1.
Déterminer l’ensemble des nombres complexes 1.
| | = 4,
:
z
2. arg(z ) = 3.
z vérifiant
z + + z
1 1
− ∈
π
−4
[2π],
iR.
Exercice 2.
Mettre sous forme trigonométrique les complexes suivants : 1 + i , 3 Mettre sous forme algébrique les complexes suivants : z =
√ 2(cos( 5
π
) + i sin(
5π
)).
z =
π
−4
i
et i. π
3(cos( ) + i sin( ), z = cos( ) + i sin( ) et 3 3 2 2
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TS4
Sujet Bac Antilles-Guyanne 2013 On considère la suite (z ) à termes complexes définie par entier naturel n, par
z 0
n
z n+1
=
= 1 + i + i et, pour tout
z n + z n
3
| |.
+ ib , où a est la partie réelle de Pour tout entier naturel n, on pose : z = a + i et b est la partie imaginaire de z . Le but de cet exercice est d’étudier la convergence des suites (a ) et (b ). n
n
n
n
n
z n
n
n
n
Partie Partie A
1. Donner
a0
et
b0 .
2. Calcul Calculer er z 1, puis en déduire que
√
1+ 2 = et 3
a1
b1
1 = . 3
3. On considère l’algorithme l’algorithme suivant suivant : Variables :
A et e t B des
nombres réels K et N des nombres entiers Initialisation : Affecter à A la valeur 1 Affecter à B la valeur 1 Traitement : Entrer la valeur de N Pour K variant de 1 à N A + A2 + B 2 Affecter à A la valeur 3 B Affecter à B la valeur 3 FinPour Afficher A
√
(a) On exécute cet algorithme algorithme en saisissant saisissant N = 2. Recopier et compléter le tableau ci-dessous contenant l’état des variables au cours de l’exécution de l’algorithme (on arrondira les valeurs calculées à 1 à 100 4 près). −
K
A
B
1 2 (b) Pour Pour un nombre nombre N donné, à quoi correspond la valeur affichée par l’algorithme rithme par rapport à la situation situation étudiée dans cet exercice exercice ? Partie Partie B
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TS4 ( b ) ? En déduire l’expression de b en fonction 2. Quelle est est la nature nature de la suite suite ( de n, et déterminer la limite de (b ). n
n
n
3. (a) On rappelle que pour p our tous nombres nombres complexes z et
+ z | |z | + |z | |z +
z
:
(inégalité triangulaire).
Montrer que pour tout entier naturel n,
|z | 2 |3z | . n
+1
n
(b) Pour Pour tout entier naturel naturel n, on pose u = z . Montrer par récurrence que, pour tout entier naturel n,
| |
n
n
2 √ n
un
3
2.
En déduire que la suite ( suite (u ) converge vers une limite que l’on déterminera. n
(c) Montre Montrerr que, que, pour tout entier entier nature naturell n, a u . En déduire que la suite (a ) converge vers une limite que l’on déterminera.
| | n
n
n
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Figure 1.6
– Une perle trouvée dans une interrogation surprise
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