{"id":2360,"date":"2016-11-29T12:57:01","date_gmt":"2016-11-29T11:57:01","guid":{"rendered":"http:\/\/agora.xtec.cat\/iesm-jmzafra\/?p=2360"},"modified":"2016-12-09T10:34:05","modified_gmt":"2016-12-09T09:34:05","slug":"anem-x-matematiques-geometria-amb-el-geogebra","status":"publish","type":"post","link":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/eso\/eso-3\/anem-x-matematiques-geometria-amb-el-geogebra\/","title":{"rendered":"Anem x + matem\u00e0tiques: geometria amb el GeoGebra"},"content":{"rendered":"<p><a href=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/anemxmates-geogebra.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2377 alignleft\" src=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/anemxmates-geogebra.jpg\" alt=\"anemxmates-geogebra\" width=\"300\" height=\"141\" \/><\/a>Aquest dissabte vaig estar a la quarta sessi\u00f3 de <a href=\"http:\/\/abeam.feemcat.org\/course\/view.php?id=32\" target=\"_blank\">Anem x + matem\u00e0tiques<\/a>, que tractava sobre geometria, i per treballar vam fer servir l\u2019aplicaci\u00f3 digital del GeoGebra, una eina molt \u00fatil per treballar matem\u00e0tiques.<\/p>\n<p>Primer vam treballar amb la figura del triangle; buscant el baricentre, l\u2019ortocentre, l\u2019incentre i el circumcentre.<\/p>\n<ul>\n<li><strong>Baricentre<\/strong>: punt que es troba a la intersecci\u00f3 de les mitjanes, l\u00ednies que uneixen els v\u00e8rtexs i el punt mitj\u00e0 del costat oposat.<\/li>\n<\/ul>\n<p><a href=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/baricentre.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2366 alignnone\" src=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/baricentre.jpg\" alt=\"baricentre\" width=\"237\" height=\"195\" srcset=\"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/baricentre.jpg 547w, https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/baricentre-300x247.jpg 300w\" sizes=\"auto, (max-width: 237px) 100vw, 237px\" \/><\/a><\/p>\n<ul>\n<li><span lang=\"ca-ES\"><strong>Ortocentre<\/strong>: <\/span><span lang=\"ca-ES\">punt on es creuen les tres altures d&#8217;un triangle. <\/span><span lang=\"ca-ES\">Si el triangle \u00e9s obtusangle, l&#8217;ortocentre \u00e9s exterior. Si el triangle \u00e9s acutangle, l&#8217;ortocentre \u00e9s interior. Si el triangle \u00e9s rectangle, l&#8217;ortocentre coincideix amb el v\u00e8rtex.<\/span><\/li>\n<\/ul>\n<p><a href=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/ortocentre.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2367\" src=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/ortocentre.jpg\" alt=\"ortocentre\" width=\"236\" height=\"173\" srcset=\"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/ortocentre.jpg 572w, https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/ortocentre-300x220.jpg 300w\" sizes=\"auto, (max-width: 236px) 100vw, 236px\" \/><\/a><\/p>\n<ul>\n<li><span lang=\"ca-ES\"><strong>Incentre<\/strong>: <\/span><span lang=\"ca-ES\">punt interior on es tallen les bisectrius dels seus angles, <\/span><span lang=\"ca-ES\">i coincideix amb el centre de la circumfer\u00e8ncia inscrita del triangle.<\/span><\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2459 alignnone\" src=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/incentre-1-300x207.jpg\" alt=\"incentre\" width=\"300\" height=\"207\" srcset=\"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/incentre-1-300x207.jpg 300w, https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/incentre-1.jpg 627w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<ul>\n<li><strong>Circumcentre<\/strong>: punt on es creuen les mediatrius d\u2019un triangle. \u00c9s un punt equidistant als v\u00e8rtex, i coincideix amb el centre de la circumfer\u00e8ncia circumscrita del triangle. Si \u00e9s un triangle rectangle, el circumcentre estar\u00e0 sobre un dels costats del triangle (la hipotenusa). Si el triangle \u00e9s obtusangle, el circumcentre estar\u00e0 fora del triangle. Si el triangle \u00e9s acutangle, el circumcentre estar\u00e0 dins el triangle<\/li>\n<\/ul>\n<p><a href=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/Circumcentre.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2369\" src=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/Circumcentre.jpg\" alt=\"circumcentre\" width=\"255\" height=\"192\" srcset=\"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/Circumcentre.jpg 747w, https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/Circumcentre-300x225.jpg 300w, https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/Circumcentre-200x150.jpg 200w\" sizes=\"auto, (max-width: 255px) 100vw, 255px\" \/><\/a><\/p>\n<p>Un cop trobats els quatre punts, vam poder construir la recta d\u2019Euler, una recta que talla el baricentre, l\u2019ortocentre, l\u2019incentre i el circumcentre d\u2019un triangle.<\/p>\n<p><a href=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/Recta-dEuler.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2370\" src=\"http:\/\/agora.xtec.cat\/iesm-jmzafra\/wp-content\/uploads\/usu164\/2016\/11\/Recta-dEuler.jpg\" alt=\"recta-deuler\" width=\"343\" height=\"291\" srcset=\"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/Recta-dEuler.jpg 667w, https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-content\/uploads\/usu164\/2016\/11\/Recta-dEuler-300x255.jpg 300w\" sizes=\"auto, (max-width: 343px) 100vw, 343px\" \/><\/a><\/p>\n<p>Aix\u00ed vam poder estudiar les propietats i caracter\u00edstiques dels centres dels triangles.<\/p>\n<p>Despr\u00e9s de treballar aix\u00f2 i fer una petita pausa, vam fer altres activitats relacionades amb la geometria. Primer, amb el <a href=\"https:\/\/www.geogebra.org\/?lang=es\" target=\"_blank\">GeoGebra<\/a>, hav\u00edem de construir tres circumfer\u00e8ncies tangents entre si. Era una mica dif\u00edcil aconseguir-ho amb el programa, aix\u00ed que vam utilitzar una activitat de la p\u00e0gina <a href=\"http:\/\/procomun.educalab.es\/es\/ode\/view\/1416349610215\">Proyecto Gauss<\/a>.<\/p>\n<p>Despr\u00e9s vam seguir treballant trigonometria, centrant-nos en els triangles auxiliars. Per fer el triangle auxiliar d&#8217;un triangle, has de trobar els punts mitjos dels tres segments que el formen i unir-los. El triangle i el seu triangle auxiliar tenen en com\u00fa que els seus angles oposats s\u00f3n iguals l&#8217;un respecte a l&#8217;altre. Un segment del triangle ser\u00e0 el doble que el seu segment oposat en el seu triangle auxiliar. El triangle auxiliar representa 1\/4 de l&#8217;\u00e0rea total del triangle. El triangle i el triangle auxiliar s\u00f3n semblants.<\/p>\n<p>A continuaci\u00f3 ens vam centrar en els quadril\u00e0ters, i igual que hav\u00edem fet amb els triangles, vam treballar els quadril\u00e0ters auxiliars. Per aconseguir que la figura auxiliar fos un rombe, el quadril\u00e0ter havia de ser un rectangle, i perqu\u00e8 la figura auxiliar fos un rectangle, el quadril\u00e0ter havia de tenir els costats perpendiculars.<\/p>\n<p>Seguint amb els quadril\u00e0ters, vam treballar el centre de massa o centre de gravetat. Primer, divid\u00edem el quadril\u00e0ter en dos triangles, i trob\u00e0vem el centre de massa de cadascun. Despr\u00e9s, un\u00edem els dos centres de masses, i el punt en qu\u00e8 tallaven la diagonal, era el centre de masses del quadril\u00e0ter. Aquest m\u00e8tode pot ser una mica pesat si es fa manualment, per\u00f2 amb el GeoGebra \u00e9s realment molt senzill. Despr\u00e9s, vam estudiar quan el centre de gravetat coincidia amb les diagonals del <em>Paral\u00b7lelogram de Varignon i Wittenbauer<\/em>.<\/p>\n<p>Aix\u00f2 va ser tot el que vam treballar de matem\u00e0tiques, per\u00f2 la classe tamb\u00e9 ens va servir per aprendre molt\u00edssimes possibilitats del GeoGebra que desconeix\u00edem.<\/p>\n<p>Enlla\u00e7 de les activitats treballades: <a href=\"http:\/\/venxmas.fespm.es\/temas\/geometria-dinamica-explorando-los.html?lang=es\">Geometria din\u00e1mica. <em>Explorando los triangulos y sus centros<\/em><\/a><\/p>\n<p>Fins la propera sessi\u00f3!<\/p>\n<p>Duna Galup<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aquest cop la Duna Galup de 3r A ens explica la seva experi\u00e8ncia amb el geogebra dintre l&#8217;activitat Anem x+ Matem\u00e0tiques<\/p>\n","protected":false},"author":34,"featured_media":2377,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center 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