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Feat: import de la séquence sur Pythagore

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Bertrand Benjamin 2021-09-21 09:16:41 +02:00
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Découverte du théorème de Pythagore avec les 4e de Mouthe
#########################################################
:date: 2021-09-21
:modified: 2021-09-21
:tags: Geométrie, Pythagore
:category: 4e
:authors: Bertrand Benjamin
:summary: Découverte du théorème Pythagore, du carré et de la racine carré.
Étape 1: Mesure de l'hypoténuse
===============================
On dessine 3 triangles rectangles avec 2 longueurs à chaque fois et on demande de calculer la longueur du 3e côté (l'hypoténuse). Au début, la seule méthode possible est de tracer et mesurer. Il faudra s'assurer qu'il y est au moins un triangle qui ne puisse pas être tracé sur le cahier (trop grand ou trop petit). On discutera ensuite la limite cette méthode: l'imprécision.
Quelques valeurs de triplets de Pythagore primitifs (`source <http://villemin.gerard.free.fr/Wwwgvmm/Addition/TripProp.htm>`_):
::
a, b, c
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
20, 21, 29
12, 35, 37
9, 40, 41
28, 45, 53
11, 60, 61
16, 63, 65
33, 56, 65
48, 55, 73
13, 84, 85
36, 77, 85
39, 80, 89
65, 72, 97
20, 99, 101
60, 91, 109
Étape 2: Animation pour induites le théorème
============================================
On présente l'animation `puzzle <./puzzle.ggb>`_ (ou sous `la version mepc <./puzzle_bis.ggb>`_) en leur expliquant que ce découpage a permis aux mathématiciens de **calculer** la longueur manquante.
On leur demande d'exploiter cette découverte pour calculer la longueur pour le triangle 5-12. Après un travail de groupe, si l'idée n'a pas émergée, on peut faire un croquis pour y calculer l'aire des carrés.
Cahier de bord: une note sur l'écriture a*a qui peut être réécrite avec un carré.
Étape 3: Pratique du proto-théorème
===================================
Réutilisation de ce qui a été fait l'étape 2 sur 3 exemples (sans utilisation de la racine carré). Chaque groupe produit un début de rédaction afin de garder une trace pour le cahier de bord.
Cahier de bord: On colle de `puzzle deplié <./B1_Puzzle_Pythagore.pdf>`_, on écrit l'égalité des aires.
Étape 4: cette égalité des aires est-elle vraie pour tous les triangles?
========================================================================
On pose cette question aux élèves. Ils doivent y répondre en illustrant. Cette étape va permettre de continuer à habituer les élèves à ces calculs d'aires.
Cahier de bord: On écrit que ce n'est le cas que pour les triangles rectangles.

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