{"id":595,"date":"2024-03-09T17:12:43","date_gmt":"2024-03-09T16:12:43","guid":{"rendered":"https:\/\/auxmaths.fr\/?p=595"},"modified":"2024-03-09T17:37:38","modified_gmt":"2024-03-09T16:37:38","slug":"nombre-rectangle-et-multiplication","status":"publish","type":"post","link":"https:\/\/auxmaths.fr\/index.php\/2024\/03\/09\/nombre-rectangle-et-multiplication\/","title":{"rendered":"Nombre rectangle et multiplication"},"content":{"rendered":"\n

La conception intuitive qu\u2019ont les \u00e9l\u00e8ves de la multiplication est celle de l\u2019addition r\u00e9p\u00e9t\u00e9e. Bien que n\u00e9cessaire de prime abord, elle se r\u00e9v\u00e8le insuffisante d\u00e8s lors que les probl\u00e8mes abord\u00e9s rel\u00e8vent par exemple du champ des nombres d\u00e9cimaux (sous-entendu non entiers). L\u2019objectif de l\u2019application accessible ici et pr\u00e9sent\u00e9e ci-dessous<\/a> est de d\u00e9passer cette conception par l\u2019introduction d\u2019images mentales du produit de deux nombres. Cet outil param\u00e9trable et vid\u00e9o-projetable s\u2019appuie sur le concept de nombre rectangle. La d\u00e9marche n\u2019est qu\u2019indicative et la notion n\u00e9cessiterait bien entendu d\u2019\u00eatre travaill\u00e9e plus longuement avec les \u00e9l\u00e8ves\u00a0; il ne s\u2019agit ici que de pr\u00e9senter l\u2019outil et sa motivation.<\/p>\n\n\n\n

\n

\u00ab L\u2019apprentissage des structures multiplicatives dont font partie la multiplication, la division et les fractions, tout en relevant d\u2019un socle commun, se r\u00e9percute sur les apprentissages ult\u00e9rieurs comme les raisonnements proportionnels et alg\u00e9briques. \u00bb<\/em><\/p>\nDe la multiplication au fractions\u00a0: r\u00e9concilier intuition et sens math\u00e9matique (synth\u00e8se du CSEN<\/a>)<\/strong><\/cite><\/blockquote>\n\n\n\n

Le produit 4 \u00d7 6 se repr\u00e9sente par un rectangle de largeur 4 et de longueur 6. On ne parle pas ici d\u2019aire (m\u00eame si cette grandeur est intrins\u00e8quement li\u00e9e au concept) mais d\u2019une r\u00e9partition en 4 bandes de 6 unit\u00e9s (repr\u00e9sent\u00e9e chacune par un carr\u00e9).<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Cette repr\u00e9sentation s\u2019appuie sur la conception intuitive de la multiplication et permet de visualiser de fa\u00e7on \u00e9vidente sa commutativit\u00e9, en \u00e9changeant simplement la position des facteurs.<\/p>\n\n\n\n

Cette conception intuitive peut encore \u00eatre utilis\u00e9e pour calculer le produit 4 \u00d7 6,7\u00a0: il s\u2019agit l\u00e0 encore de le repr\u00e9senter par 4 bandes de 6 unit\u00e9s et 7\/10. On d\u00e9nombre alors 24 unit\u00e9s et 28 dixi\u00e8mes qui permet \u00e0 l\u2019\u00e9l\u00e8ve en s\u2019appuyant sur sa connaissance de la fraction d\u00e9cimale d\u2019aboutir au r\u00e9sultat escompt\u00e9.<\/p>\n\n\n

\n
\"\"
(Remarque\u00a0: les captures donnent l\u2019aper\u00e7u final de l\u2019animation\u00a0; le d\u00e9coupage est en r\u00e9alit\u00e9 progressif)<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n

Le produit 6,7 \u00d7 4 est plus d\u00e9licat \u00e0 concevoir. On pourrait certes \u00e9tendre la commutativit\u00e9 de la multiplication sur les entiers ou s\u2019appuyer sur un argument g\u00e9om\u00e9trique pour affirmer qu\u2019il est \u00e9gal au produit 4 \u00d7 6,7\u00a0mais c\u2019est \u00e0 mon sens un raccourci \u00e0 \u00e9viter\u00a0:<\/p>\n\n\n\n

    \n
  • On ne pourra pas avoir la m\u00eame d\u00e9marche quand les deux facteurs seront d\u00e9cimaux.<\/li>\n\n\n\n
  • Dans une situation contextualis\u00e9e, elle s\u2019oppose trop frontalement \u00e0 l\u2019intuition\u00a0: l\u2019achat de 6,7 kg d\u2019un produit \u00e0 4 \u20ac le kg est diff\u00e9rent de l\u2019achat de 4 kg d\u2019un produit \u00e0 6,7 \u20ac le kg, m\u00eame si au final, la somme pay\u00e9e est la m\u00eame.<\/li>\n<\/ul>\n\n\n\n

    \u00a0Aussi, on repr\u00e9sentera dans un premier temps le produit 6,7 \u00d7 4 par 6 bandes de 4 unit\u00e9s et 7\/10 d\u2019une bande suppl\u00e9mentaire de 4 unit\u00e9s.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    On ne manquera pas de faire remarquer que les r\u00e9sultats sont identiques et qu\u2019il s\u2019agit donc effectivement du m\u00eame produit : la multiplication garde sa commutativit\u00e9 quand il s\u2019agit de nombres d\u00e9cimaux.<\/p>\n\n\n\n

    Le produit 6,7 \u00d7 4,3 constitue le dernier cas et se traite de fa\u00e7on identique. Le seul obstacle r\u00e9side dans l\u2019apparition des centi\u00e8mes comme dixi\u00e8mes de dixi\u00e8mes, mais la difficult\u00e9 est lev\u00e9e par la repr\u00e9sentation.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Cette repr\u00e9sentation en rectangle permet d\u2019aborder des situations contextualis\u00e9es et de justifier le recours \u00e0 la multiplication. Un exemple prototypique de prix au kilogramme peut se mod\u00e9liser de fa\u00e7on analogue et exprime visuellement la similarit\u00e9 de la situation.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    \u00c9videmment, cette mod\u00e9lisation n\u2019est que transitoire et on devra s\u2019en d\u00e9tacher une fois l\u2019automatisation acquise. Celle-ci passe d\u2019abord par l\u2019apprentissage d\u2019un algorithme de calcul et l\u00e0-encore, on peut s\u2019appuyer sur le mod\u00e8le du nombre rectangle.<\/p>\n\n\n\n

    Par exemple, en juxtaposant 100 rectangles repr\u00e9sentant le produit 6,7 \u00d7 4,3 on fait appara\u00eetre le rectangle repr\u00e9sentant 67 \u00d7 43 dont le calcul est connu des \u00e9l\u00e8ves.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Ces repr\u00e9sentations figur\u00e9es permettent de soutenir la compr\u00e9hension des structures math\u00e9matiques multiplicatives et rendent visibles des propri\u00e9t\u00e9s pertinentes telles que la commutativit\u00e9 de la multiplication, la distributivit\u00e9 et la relation inverse entre multiplication et division\u2026 mais cela est une autre histoire.<\/p>\n","protected":false},"excerpt":{"rendered":"

    La conception intuitive qu\u2019ont les \u00e9l\u00e8ves de la multiplication est celle de l\u2019addition r\u00e9p\u00e9t\u00e9e. Bien que n\u00e9cessaire de prime abord, elle se r\u00e9v\u00e8le insuffisante d\u00e8s lors que les probl\u00e8mes abord\u00e9s rel\u00e8vent par…<\/p>\n","protected":false},"author":1,"featured_media":605,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-595","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-coin-des-profs"],"yoast_head":"\nNombre rectangle et multiplication - Aux Maths ! et c\u00e6tera<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/auxmaths.fr\/index.php\/2024\/03\/09\/nombre-rectangle-et-multiplication\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Nombre rectangle et multiplication - Aux Maths ! et c\u00e6tera\" \/>\n<meta property=\"og:description\" content=\"La conception intuitive qu\u2019ont les \u00e9l\u00e8ves de la multiplication est celle de l\u2019addition r\u00e9p\u00e9t\u00e9e. 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