{"id":283,"date":"2020-10-10T14:38:45","date_gmt":"2020-10-10T12:38:45","guid":{"rendered":"http:\/\/eduhm.fr\/?page_id=283"},"modified":"2020-10-11T08:33:11","modified_gmt":"2020-10-11T06:33:11","slug":"3e-calcul-litteral","status":"publish","type":"page","link":"http:\/\/eduhm.fr\/?page_id=283","title":{"rendered":"3e calcul litt\u00e9ral"},"content":{"rendered":"\n<p>Cette s\u00e9ance porte \u00e0 la fois sur le calcul litt\u00e9ral et sur un travail de m\u00e9thode (choix de la bonne approche pour aborder une question math\u00e9matique). Le support choisi est constitu\u00e9 d\u2019extraits de Diophante, Vi\u00e8te et Fermat.<\/p>\n\n\n\n<p>La fiche \u00e9l\u00e8ve :\u00a0<a href=\"http:\/\/eduhm.fr\/wp-content\/uploads\/2020\/10\/3_litteral_fiche_eleve.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">3_litteral_fiche_eleve<\/a><\/p>\n\n\n\n<p><em>Niveau : troisi\u00e8me \/ Ann\u00e9e 2011<\/em><\/p>\n\n\n\n<p><strong>Extrait 1 : La dimension historique au service des \u00e9l\u00e8ves<\/strong><\/p>\n\n\n\n<p>Les premi\u00e8res minutes de la s\u00e9ance (coup\u00e9es au montage) sont consacr\u00e9es \u00e0 des rappels sur les identit\u00e9s remarquables. Plusieurs \u00e9l\u00e8ves vont successivement au tableau pour y noter les r\u00e9sultats vus les ann\u00e9es pr\u00e9c\u00e9dentes.<\/p>\n\n\n\n<p>Au d\u00e9but de l\u2019activit\u00e9, l\u2019enseignant \u00e9voque rapidement les noms des auteurs qui seront rencontr\u00e9s dans l\u2019heure, mais il insiste surtout sur l\u2019objectif m\u00e9thodologique de la s\u00e9ance.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-790\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-790\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>\n\n\n<p><strong>Extrait 2 : Approche par essais-erreurs sur le probl\u00e8me de Diophante<\/strong><\/p>\n\n\n\n<p>La fiche donn\u00e9e aux \u00e9l\u00e8ves comporte quelques phrases permettant de situer historiquement l\u2019extrait de Diophante propos\u00e9. Ces phrases sont lues \u00e0 voix haute par un \u00e9l\u00e8ve.<\/p>\n\n\n\n<p>L\u2019histoire vient l\u00e9gitimer le probl\u00e8me propos\u00e9 aux \u00e9l\u00e8ves qui s\u2019engagent ensuite dans une phase de recherche et une premi\u00e8re m\u00e9thode, par essais-erreurs, est propos\u00e9e.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-132\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-132\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>\n\n\n<p><strong>Extrait 3 : Deuxi\u00e8me m\u00e9thode<\/strong><\/p>\n\n\n\n<p>Dans cet extrait une \u00e9l\u00e8ve propose une approche par le calcul litt\u00e9ral. Sur cet exemple, il n\u2019est pas possible \u00e0 des \u00e9l\u00e8ves de troisi\u00e8me de mener le calcul jusqu\u2019au bout, mais la m\u00e9thode est quand m\u00eame retenue par le professeur.<\/p>\n\n\n\n<p>Comme dans l\u2019extrait pr\u00e9c\u00e9dent, le travail est focalis\u00e9 sur les math\u00e9matiques et les aspects historiques sont provisoirement mis de c\u00f4t\u00e9.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-208\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-208\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>\n\n\n<p><strong>Extrait 4 : Bilan d\u2019\u00e9tape et premi\u00e8res difficult\u00e9s<\/strong><\/p>\n\n\n\n<p>L\u2019enseignant fait une rapide synth\u00e8se des deux m\u00e9thodes employ\u00e9es dans la premi\u00e8re partie de l\u2019activit\u00e9 et engage ensuite la deuxi\u00e8me partie du travail sur des extraits de texte de Vi\u00e8te.<\/p>\n\n\n\n<p>La lecture de ce texte, \u00e9crit en langage naturel, va entrainer des difficult\u00e9s de compr\u00e9hension.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-92\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-92\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>\n\n\n<p><strong>Extrait 5 : Texte historique \u2013 travail math\u00e9matique<\/strong><\/p>\n\n\n\n<p>Dans cet extrait, le professeur rebondit sur une difficult\u00e9 (attendue) des \u00e9l\u00e8ves dans la confusion entre les expressions \u00ab&nbsp;somme des carr\u00e9s&nbsp;\u00bb et \u00ab&nbsp;carr\u00e9 de la somme&nbsp;\u00bb.<\/p>\n\n\n\n<p>La lecture du texte historique donne l\u2019occasion d\u2019un travail sur le langage math\u00e9matique, d\u2019une mise en regard des expressions de la langue naturelle et du langage symbolique.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-695\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-695\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>\n\n\n<p><strong>Extrait 6 : Calcul num\u00e9rique et calcul litt\u00e9ral<\/strong><\/p>\n\n\n\n<p>Au tableau, un \u00e9l\u00e8ve pr\u00e9sente le d\u00e9but de la v\u00e9rification de l\u2019\u00e9galit\u00e9 sur un exemple. Il parvient \u00e0 exprimer en chiffres et en symboles le premier membre de l\u2019\u00e9galit\u00e9 mais peine \u00e0 traduire l\u2019expression \u00ab&nbsp;carr\u00e9 de la diff\u00e9rence&nbsp;\u00bb. Des \u00e9changes avec la classe permettent de terminer ce travail qui est l\u2019occasion de revenir sur les priorit\u00e9s entre les op\u00e9rations.<\/p>\n\n\n\n<p>D\u2019un travail sur la terminologie, l\u2019enseignant passe aux calculs num\u00e9riques, puis revient au calcul litt\u00e9ral comme outil de preuve. Comme pour le probl\u00e8me de Diophante, deux aspects sont en jeu : le choix d\u2019une m\u00e9thode et l\u2019interpr\u00e9tation du texte.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-112\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-112\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>\n\n\n<p><strong>Extrait 7 : Fin de la partie math\u00e9matique<\/strong><\/p>\n\n\n\n<p>Comme au d\u00e9but de la s\u00e9ance, la r\u00e9solution par le calcul litt\u00e9ral est mise en parall\u00e8le de calculs sur des valeurs num\u00e9riques.<\/p>\n\n\n\n<p>On note la pr\u00e9sence de formulations st\u00e9r\u00e9otyp\u00e9es comme \u00ab&nbsp;soustraire, c\u2019est ajouter l\u2019oppos\u00e9&nbsp;\u00bb et \u00ab&nbsp;un exemple ne prouve rien&nbsp;\u00bb qui fonctionnent comme des rep\u00e8res pour les \u00e9l\u00e8ves.<\/p>\n\n\n\n<p>L\u2019extrait se termine par la mise en valeur du calcul litt\u00e9ral comme m\u00e9thode pour \u00ab&nbsp;pouvoir g\u00e9n\u00e9raliser&nbsp;\u00bb.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-453\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-453\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>\n\n\n<p><strong>Extrait 8 : Et sur le plan historique ?<\/strong><\/p>\n\n\n\n<p>A l\u2019issue de la s\u00e9ance, le professeur r\u00e9investit la dimension historique de l\u2019activit\u00e9 en engageant une discussion \u00e9pist\u00e9mologique sur l\u2019\u00e9volution des math\u00e9matiques. L\u2019enseignant semble attendre une mise en regard des diff\u00e9rents types de langage utilis\u00e9s en math\u00e9matiques (naturel \/ symbolique).<\/p>\n\n\n\n<p>Les \u00e9l\u00e8ves r\u00e9affirment l\u2019int\u00e9r\u00eat du calcul litt\u00e9ral comme outil de preuve g\u00e9n\u00e9rale, mais ne parviennent&nbsp; pas \u00e0 se situer dans le d\u00e9bat initi\u00e9 par l\u2019enseignant. Le professeur propose alors sa propre vision du \u00ab&nbsp;progr\u00e8s&nbsp;\u00bb math\u00e9matique. L\u2019\u00e9criture symbolique serait plus \u00ab&nbsp;simple&nbsp;\u00bb, plus \u00ab&nbsp;lisible&nbsp;\u00bb que l\u2019\u00e9criture en langage naturelle, \u00ab&nbsp;\u00e0 condition d\u2019avoir le code&nbsp;\u00bb.<\/p>\n\n\n<div class=\"passster-form\" id=\"ps-689\">  \n  <form class=\"password-form\" method=\"post\" autocomplete=\"off\" action-xhr=\"http:\/\/eduhm.fr\/?page_id=283\" target=\"_top\">\n    <h4>Protected Area<\/h4>\n    <p>This content is password-protected. Please verify with a password to unlock the content.<\/p>\n    <fieldset>\n      <span class=\"ps-loader\"><img src=\"http:\/\/eduhm.fr\/wp-content\/plugins\/content-protector\/assets\/public\/ps-loader.svg\"\/><\/span>\n      <input placeholder=\"Enter your password..\" type=\"password\" tabindex=\"1\" name=\"passster_password\" id=\"passster_password\" class=\"passster-password\" autocomplete=\"off\" data-protection-type=\"password\" data-list=\"[PASSSTER_LIST]\" data-area=\"[PASSSTER_AREA]\" data-protection=\"[PASSSTER_PROTECTION]\">\n      \n      <button name=\"submit\" type=\"submit\" class=\"passster-submit\" data-psid=\"ps-689\"  data-submit=\"...Checking Password\">Saisissez le mot de passe pour acc\u00e9der \u00e0 la vid\u00e9o<\/button>\n       <div class=\"passster-error\"><\/div>\n    <\/fieldset>\n  <\/form>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Cette s\u00e9ance porte \u00e0 la fois sur le calcul litt\u00e9ral et sur un travail de m\u00e9thode (choix de la bonne approche pour aborder une question math\u00e9matique). Le support choisi est constitu\u00e9 d\u2019extraits de Diophante, Vi\u00e8te et Fermat. La fiche \u00e9l\u00e8ve :\u00a03_litteral_fiche_eleve Niveau : troisi\u00e8me \/ Ann\u00e9e 2011 Extrait 1 : La dimension historique au service &hellip; <\/p>\n<p class=\"link-more\"><a href=\"http:\/\/eduhm.fr\/?page_id=283\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;3e calcul litt\u00e9ral&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/eduhm.fr\/index.php?rest_route=\/wp\/v2\/pages\/283"}],"collection":[{"href":"http:\/\/eduhm.fr\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/eduhm.fr\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/eduhm.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/eduhm.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=283"}],"version-history":[{"count":3,"href":"http:\/\/eduhm.fr\/index.php?rest_route=\/wp\/v2\/pages\/283\/revisions"}],"predecessor-version":[{"id":384,"href":"http:\/\/eduhm.fr\/index.php?rest_route=\/wp\/v2\/pages\/283\/revisions\/384"}],"wp:attachment":[{"href":"http:\/\/eduhm.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}