{"id":3378,"date":"2023-12-01T18:19:08","date_gmt":"2023-12-01T17:19:08","guid":{"rendered":"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=3378"},"modified":"2026-04-27T12:45:57","modified_gmt":"2026-04-27T10:45:57","slug":"le-billard","status":"publish","type":"post","link":"https:\/\/math.univ-lyon1.fr\/dream\/?p=3378","title":{"rendered":"Le billard"},"content":{"rendered":"\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/\">Accueil<\/a> \u00bb <a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=3328\">Ressources p\u00e9dagogiques<\/a> \u00bb <a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=77\">Des exemples de SDRP<\/a> \u00bb Le billard<\/p>\r\n<h1>L&rsquo;\u00e9nonc\u00e9 de la situation<\/h1>\r\n<p>On consid\u00e8re un billard de forme rectangulaire qui est quadrill\u00e9 de fa\u00e7on r\u00e9guli\u00e8re (c&rsquo;est-\u00e0-dire qu&rsquo;il a un nombre entier de lignes et un nombre entier de colonnes).<br \/>Aux 4 sommets du billard il y a une ouverture qui permet d&rsquo;envoyer un rayon lumineux le long des diagonales du quadrillage.<br \/>Le rayon lumineux \u00ab rebondit \u00bb sur les c\u00f4t\u00e9s du rectangle et ne peut sortir du billard que s&rsquo;il arrive sur un des 4 sommets<\/p>\r\n<p><iframe loading=\"lazy\" title=\"Le Probl\u00e8me du Billard\" src=\"https:\/\/tube-sciences-technologies.apps.education.fr\/videos\/embed\/b56a73c4-900e-4375-a06d-0d9759cc5aae\" width=\"560\" height=\"315\" frameborder=\"0\" sandbox=\"allow-same-origin allow-scripts allow-popups allow-forms\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p><strong>Question<\/strong><\/p>\r\n<p>Existe-t-il un moyen de d\u00e9terminer \u00e0 l&rsquo;avance le nombre de carreaux travers\u00e9s par le rayon lumineux dans le billard en fonction de ses dimensions?<\/p>\r\n<p><strong>Un exemple<\/strong><\/p>\r\n<p><img decoding=\"async\" src=\"https:\/\/clarolineconnect.univ-lyon1.fr\/file\/resource\/media\/2483675\" alt=\"Image_Billard.png\" \/><\/p>\r\n<h1>Les math\u00e9matiques en jeu<\/h1>\r\n<ul>\r\n<li>Notions de multiples et de diviseurs<\/li>\r\n<li>Notion de nombres premiers et de nombres premiers entre eux<\/li>\r\n<li>Agrandir ou r\u00e9duire une figure en utilisant la conservation des angles et la proportionnalit\u00e9 entre les longueurs de la figure initiale et celles de la figure \u00e0 obtenir<\/li>\r\n<li>Calculer le PGCD et le PPCM de deux nombres entiers<\/li>\r\n<li>Calcul litt\u00e9ral lors des conjectures et des preuves.<\/li>\r\n<\/ul>\r\n<h4>Analyse math\u00e9matique<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/LeBillard_Analyse_Maths.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Analyse didactique<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/LeBillard_Analyse_didactique.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Exemples de mise en \u0153uvre<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/LeBillard_Mise_en_oeuvre.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h4>Pour aller plus loin<\/h4>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/LeBillard_Pour_Aller_Plus_Loin.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charger le document <\/a><\/p>\r\n<h2>Retours d&rsquo;exp\u00e9riences<\/h2>\r\n<p>Des retours d&rsquo;exp\u00e9riences partag\u00e9s par les enseignant.e.s du groupe d&rsquo;exp\u00e9rimentation qui ont mis en place le probl\u00e8me du Billard\u00a0 dans le cadre d&rsquo;un enseignement fond\u00e9 sur les probl\u00e8mes<\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Existe-t-il un moyen de d\u00e9terminer \u00e0 l\u2019avance le nombre de carreaux travers\u00e9s par le rayon lumineux dans le billard en fonction de ses dimensions?<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[106],"tags":[],"niveau":[113,114],"type_situation":[115],"theme_maths":[118],"probleme_associe":[130],"type-expe":[],"class_list":{"0":"post-3378","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-sdrp","7":"niveau-cycle-4","8":"niveau-lycee","9":"type_situation-sdrp","10":"theme_maths-nombres-et-calculs","11":"probleme_associe-le-billard","12":"czr-hentry"},"rttpg_featured_image_url":null,"rttpg_author":{"display_name":"St\u00e9phanie CROQUELOIS","author_link":"https:\/\/math.univ-lyon1.fr\/dream\/?author=3"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?cat=106\" rel=\"category\">SDRP<\/a>","rttpg_excerpt":"Existe-t-il un moyen de d\u00e9terminer \u00e0 l\u2019avance le nombre de carreaux travers\u00e9s par le rayon lumineux dans le billard en fonction de ses dimensions?","_links":{"self":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3378","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3378"}],"version-history":[{"count":10,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3378\/revisions"}],"predecessor-version":[{"id":3762,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3378\/revisions\/3762"}],"wp:attachment":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3378"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3378"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3378"},{"taxonomy":"niveau","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fniveau&post=3378"},{"taxonomy":"type_situation","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype_situation&post=3378"},{"taxonomy":"theme_maths","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftheme_maths&post=3378"},{"taxonomy":"probleme_associe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fprobleme_associe&post=3378"},{"taxonomy":"type-expe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype-expe&post=3378"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}