{"id":118,"date":"2013-12-12T21:24:00","date_gmt":"2013-12-12T20:24:00","guid":{"rendered":"http:\/\/public.kvcso.cz\/?p=118"},"modified":"2020-03-28T21:29:49","modified_gmt":"2020-03-28T20:29:49","slug":"vyuziti-programu-geogebra-pri-vyuce-linearnich-funkci-na-zs","status":"publish","type":"post","link":"https:\/\/ucitel.kvcso.cz\/?p=118","title":{"rendered":"Vyu\u017eit\u00ed programu GEOGEBRA p\u0159i v\u00fduce line\u00e1rn\u00edch funkc\u00ed na Z\u0160"},"content":{"rendered":"\n<p>N\u011bkolik n\u00e1m\u011bt\u016f pro vyu\u017eit\u00ed programu GEOGEBRA p\u0159i v\u00fduce algebry. Vzhledem k dostupnosti tohoto v\u00fdborn\u00e9ho n\u00e1stroje, jeho voln\u00e9 \u0161i\u0159itelnosti a mo\u017en\u00e9m p\u0159\u00edstupu p\u0159es po\u010d\u00edta\u010d i tablet je k dispozici aplikace s mnoha mo\u017enostmi vyu\u017eit\u00ed.<\/p>\n\n\n\n<p>P\u0159i zav\u00e1d\u011bn\u00ed pojmu line\u00e1rn\u00ed funkce v 9.\u00a0ro\u010dn\u00edku nar\u00e1\u017e\u00edme \u010dasto na probl\u00e9m pochopen\u00ed &#8222;z\u00e1vislosti jedn\u00e9 prom\u011bnn\u00e9 na druh\u00e9&#8220;. P\u0159esto\u017ee \u017e\u00e1ci b\u011b\u017en\u011b pou\u017e\u00edvaj\u00ed vzorce v geometrii, ve fyzice a v chemii, pokud jim chceme z\u00e1vislost zav\u00e9st jako matematick\u00fd pojem, p\u0159ipad\u00e1 jim, \u017ee mluv\u00edme o n\u011b\u010dem \u00fapln\u011b teoretick\u00e9m.<\/p>\n\n\n\n<p>Za\u010dneme t\u00edm, \u017ee sezn\u00e1m\u00edme \u017e\u00e1ky s obecn\u00fdm p\u0159edpisem&nbsp;<em>y=ax+b<\/em>. Vysv\u011btl\u00edme parametry&nbsp;<em>a<\/em>&nbsp;a&nbsp;<em>b<\/em>&nbsp;a nau\u010d\u00edme je sestrojit graf line\u00e1rn\u00ed funkce. M\u016f\u017eeme vych\u00e1zet z jejich zku\u0161enost\u00ed s grafem p\u0159\u00edm\u00e9 \u00fam\u011brnosti a nav\u00e1zat na znalosti o n\u011bm. P\u0159i dal\u0161\u00edm zkoum\u00e1n\u00ed line\u00e1rn\u00ed funkce m\u016f\u017eeme vyu\u017e\u00edt GeoGebru, kter\u00e1 \u017e\u00e1k\u016fm umo\u017en\u00ed vytv\u00e1\u0159et vlastn\u00ed grafy funkc\u00ed, ani\u017e by tyto r\u00fdsovali do se\u0161it\u016f. Pr\u00e1ce s GeoGebrou slou\u017e\u00ed ke zkoum\u00e1n\u00ed vlastnost\u00ed funkc\u00ed a hled\u00e1n\u00ed souvislost\u00ed mezi parametry&nbsp;<em>a<\/em>&nbsp;a&nbsp;<em>b<\/em>&nbsp;a tvarem grafu funkce.<\/p>\n\n\n\n<p>Pokud \u017e\u00e1ky nau\u010d\u00edme z\u00e1kladn\u00ed ovl\u00e1d\u00e1n\u00ed GeoGebry, je mo\u017en\u00e9 je nechat s funkcemi experimentovat. V podstat\u011b sta\u010d\u00ed sezn\u00e1mit je s mo\u017enost\u00ed zad\u00e1vat funkce pomoc\u00ed vstupn\u00edho pole. D\u016fle\u017eitou pom\u016fckou je i zapnut\u00e9 algebraick\u00e9 okno, ve kter\u00e9m se objevuj\u00ed p\u0159edpisy jednotliv\u00fdch objekt\u016f v n\u00e1kresn\u011b.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/geogebra\/geogebra1.jpg\" alt=\"Geogebra1\" \/><\/figure><\/div>\n\n\n\n<p>Dal\u0161\u00edm pomocn\u00edkem je pro n\u011b ur\u010dit\u011b zm\u011bna barvy grafu funkce ve vlastnostech objekt\u016f. P\u0159\u00edpadn\u011b popis objektu. Z\u00e1pis funkce v algebraick\u00e9m okn\u011b m\u00e1 stejnou barvu jako graf funkce, co\u017e je v\u00fdborn\u00e1 pom\u016fcka pro orientaci v objektech.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/geogebra\/geogebra23.jpg\" alt=\"Geogebra23\" \/><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Prvn\u00ed \u00fakol zn\u00ed:<\/strong><\/h4>\n\n\n\n<p><em>Nar\u00fdsuj grafy funkc\u00ed y=2x; y=x; y=3x; y=0,5x; y=0,2x. D\u016fle\u017eit\u00e1 je funkce y=x, tu p\u0159ebarvi na \u010dervenou. Funkce, kter\u00e9 maj\u00ed graf &#8222;nad n\u00ed&#8220;, zobraz mod\u0159e, funkce, kter\u00e9 maj\u00ed graf &#8222;pod n\u00ed&#8220;, zobraz zelen\u011b.<\/em><\/p>\n\n\n\n<p>V\u00fdsledn\u00e1 pr\u00e1ce m\u016f\u017ee vypadat t\u0159eba takto:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/geogebra\/geogebra4.jpg\" alt=\"Geogebra 4\" \/><\/figure><\/div>\n\n\n\n<p>Pokud tuto pr\u00e1ci zvl\u00e1dnou, m\u016f\u017eeme pokl\u00e1dat dopl\u0148uj\u00edc\u00ed ot\u00e1zky.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>Co je spole\u010dn\u00e9 pro modr\u00e9\/zelen\u00e9 funkce?<\/em><\/li><li><em>Zkus vymyslet p\u0159edpis dal\u0161\u00ed funkce, kter\u00e1 by mohla b\u00fdt modr\u00e1\/zelen\u00e1?<\/em><\/li><li><em>Na \u010dem z\u00e1vis\u00ed strmost funkce?<\/em><\/li><\/ul>\n\n\n\n<p>D\u016fle\u017eit\u00e9 je ale, \u017ee \u017e\u00e1ci sami objev\u00ed z\u00e1vislost naklon\u011bn\u00ed grafu na parametru&nbsp;<em>a<\/em>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Dal\u0161\u00ed \u00fakol je:<\/strong><\/h4>\n\n\n\n<p><em>Nar\u00fdsuj grafy funkc\u00ed y=2x; y=-2x; y=-0,5x; y=0,5x; y=-x;y=x. Funkce se z\u00e1porn\u00fdm line\u00e1rn\u00edm \u010dlenem p\u0159ebarvi zelen\u011b, funkce s kladn\u00fdm line\u00e1rn\u00edm \u010dlenem zobraz mod\u0159e.<\/em><\/p>\n\n\n\n<p>Pr\u00e1ce \u017e\u00e1k\u016f m\u016f\u017ee vypadat t\u0159eba takto:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/geogebra\/geogebra5.jpg\" alt=\"Geogebra 5\" \/><\/figure><\/div>\n\n\n\n<p>N\u00e1sleduj\u00ed ot\u00e1zky typu:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>Co spojuje modr\u00e9\/zelen\u00e9 funkce?<\/em><\/li><li><em>Zkus vymyslet p\u0159edpis pro dal\u0161\u00ed modr\u00e9\/zelen\u00e9 funkce.<\/em><\/li><\/ul>\n\n\n\n<p>Z\u00e1v\u011brem m\u016f\u017eeme se \u017e\u00e1ky pojmenovat funkce jako rostouc\u00ed a klesaj\u00edc\u00ed a vysv\u011btlit si z\u00e1vislost t\u00e9to vlastnosti u line\u00e1rn\u00ed funkce na parametru&nbsp;<em>a<\/em>.<\/p>\n\n\n\n<p>V tuto chv\u00edli je \u010das za\u010d\u00edt se zab\u00fdvat parametrem&nbsp;<em>b<\/em>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Za\u010dneme jednoduch\u00fdm \u00fakolem:<\/strong><\/h4>\n\n\n\n<p><em>Sestroj grafy funkc\u00ed y=2x; y=2x+1; y=2x-1; y=2x+2; y=2x-2,5. V\u0161imni si, jak\u00fdm bodem na ose y jednotliv\u00e9 grafy proch\u00e1z\u00ed. Body na n\u00e1kresn\u011b vyzna\u010d.<\/em><\/p>\n\n\n\n<p>Pr\u00e1ce \u017e\u00e1k\u016f vypad\u00e1 takto:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/geogebra\/geogebra6.jpg\" alt=\"Geogebra 6\" \/><\/figure><\/div>\n\n\n\n<p>P\u0159i rozboru vypracovan\u00e9ho \u00fakolu se je\u0161t\u011b m\u016f\u017eeme vr\u00e1tit k line\u00e1rn\u00edmu \u010dlenu&nbsp;<em>a<\/em>. P\u0159\u00edpadn\u00e9 na\u0161e ot\u00e1zky mohou zn\u00edt:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>Co je zaj\u00edmav\u00e9 na grafech v\u0161ech funkc\u00ed?<\/em><\/li><li><em>\u010c\u00edm je zp\u016fsobena jejich rovnob\u011b\u017enost?<\/em><\/li><li><em>Um\u00ed\u0161 navrhnout dal\u0161\u00ed funkci, kter\u00e1 bude m\u00edt s dan\u00fdmi funkcemi rovnob\u011b\u017en\u00fd graf?<\/em><\/li><\/ul>\n\n\n\n<p>T\u00edm upozorn\u00edme \u017e\u00e1ky op\u011bt na vztah mezi&nbsp;<em>a<\/em>&nbsp;a &#8222;strmost\u00ed&#8220; funkce.<\/p>\n\n\n\n<p>Ot\u00e1zky t\u00fdkaj\u00edc\u00ed se absolutn\u00edho \u010dlenu mohou b\u00fdt:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>Co maj\u00ed spole\u010dn\u00e9 pr\u016fse\u010d\u00edky graf\u016f jednotliv\u00fdch funkc\u00ed s p\u0159edpisy t\u011bchto funkc\u00ed?<\/em><\/li><li><em>Pro\u010d v\u0161echny pr\u016fse\u010d\u00edky maj\u00ed x-ovou sou\u0159adnici nulovou?<\/em><\/li><li><em>Vyvo\u010f z obecn\u00e9 rovnice y=ax+b p\u0159edpis pro pr\u016fse\u010d\u00edk s osou y.<\/em><\/li><\/ul>\n\n\n\n<p>Z\u00e1v\u011brem je ur\u010den\u00ed pr\u016fse\u010d\u00edku line\u00e1rn\u00ed funkce s osou y pro libovoln\u00fd p\u0159edpis, ani\u017e by bylo t\u0159eba graf konstruovat.<\/p>\n\n\n\n<p>Trochu t\u011b\u017e\u0161\u00ed je pro \u017e\u00e1ky ur\u010dit pr\u016fse\u010d\u00edk s osou&nbsp;<em>x<\/em>, ale i to je mo\u017en\u00e9 zvl\u00e1dnout. Obdobn\u011b zad\u00e1me &#8222;\u0161ikovn\u00e9&#8220; funkce, a \u017e\u00e1ci hledaj\u00ed pr\u016fse\u010d\u00edk s osou<em>&nbsp;x<\/em>.<\/p>\n\n\n\n<p><em>Sestroj grafy funkc\u00ed y=0,5x-1; y=-2x-5; y=-x+4; y=3x-1,5. Ozna\u010d pr\u016fse\u010d\u00edky graf\u016f s osou x.<\/em><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">V\u00fdsledn\u00e1 pr\u00e1ce:<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/geogebra\/geogebra7.jpg\" alt=\"Geogebra 7\" \/><\/figure><\/div>\n\n\n\n<p>A pt\u00e1me se d\u00e1l:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>Co maj\u00ed spole\u010dn\u00e9ho v\u0161echny pr\u016fse\u010d\u00edky?<\/em><\/li><li><em>Pro\u010d v\u0161echny pr\u016fse\u010d\u00edky maj\u00ed y-ovou sou\u0159adnici nulovou?<\/em><\/li><li><em>Vyvo\u010f z obecn\u00e9 rovnice y=ax+b p\u0159edpis pro pr\u016fse\u010d\u00edk s osou x.<\/em><\/li><\/ul>\n\n\n\n<p>V tuto chv\u00edli u\u017e jsou \u017e\u00e1ci schopni vy\u0161et\u0159it pr\u016fb\u011bh line\u00e1rn\u00ed funkce &#8211; monot\u00f3nnost a pr\u016fse\u010d\u00edky s osami. M\u011bli by b\u00fdt schopni navrhnout p\u0159edpis funkce, kter\u00e1 je rovnob\u011b\u017en\u00e1 nebo kter\u00e1 m\u00e1 stejn\u00fd pr\u016fse\u010d\u00edk s osou&nbsp;<em>y<\/em>.<\/p>\n\n\n\n<p>Dal\u0161\u00edm zaj\u00edmav\u00fdm \u00fakolem pro \u017e\u00e1ky m\u016f\u017ee b\u00fdt i \u0159e\u0161en\u00ed soustavy dvou line\u00e1rn\u00edch rovnic. GeoGebra um\u00ed samoz\u0159ejm\u011b vy\u010d\u00edslit jakoukoli rovnici o dvou nezn\u00e1m\u00fdch.<\/p>\n\n\n\n<p><em>Vy\u0159e\u0161 soustavu rovnic x-2y=0; (x+3)\/2=(1-y)\/4. Ob\u011b rovnice zadej do vstupn\u00edho pole a najdi v n\u00e1kresn\u011b \u0159e\u0161en\u00ed.<\/em><\/p>\n\n\n\n<p>Jednoduch\u00e9 \u0159e\u0161en\u00ed m\u016f\u017ee vypadat t\u0159eba takto:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/geogebra\/geogebra8.jpg\" alt=\"Geogebra 8\" \/><\/figure><\/div>\n\n\n\n<p>N\u00e1sleduj\u00ed ot\u00e1zky:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>V co se zm\u011bnily ob\u011b rovnice? Co je tedy vlastn\u011b rovnice o dvou nezn\u00e1m\u00fdch?<\/em><\/li><li><em>Co je \u0159e\u0161en\u00edm soustavy dvou rovnic?<\/em><\/li><li><em>Jak bychom poznali soustavu, kter\u00e1 nem\u00e1 \u017e\u00e1dn\u00e9 \u0159e\u0161en\u00ed?<\/em><\/li><li><em>Jak bychom poznali soustavu, kter\u00e1 m\u00e1 nekone\u010dn\u011b mnoho \u0159e\u0161en\u00ed?<\/em><\/li><\/ul>\n\n\n\n<p>Toto je jen p\u00e1r n\u00e1m\u011bt\u016f na vyu\u017eit\u00ed programu GeoGebra p\u0159i v\u00fduce algebry na z\u00e1kladn\u00ed \u0161kole. Jej\u00ed v\u00fdhodou je dostupnost a velmi jednoduch\u00e9 ovl\u00e1d\u00e1n\u00ed. V podstat\u011b na jej\u00ed vyu\u017eit\u00ed sta\u010d\u00ed pouze p\u0159ipojen\u00ed k internetu a maxim\u00e1ln\u011b desetiminutov\u00e1 uk\u00e1zka z\u00e1kladn\u00edho ovl\u00e1d\u00e1n\u00ed. V\u011b\u0159\u00edm, \u017ee \u017e\u00e1ky tato forma zaujme a \u017ee pochop\u00ed pojem funkce jako ned\u00edln\u00e9 sou\u010d\u00e1sti matematiky.<\/p>\n\n\n\n<p><strong>Autorka: Marta Pincov\u00e1<\/strong> <\/p>\n","protected":false},"excerpt":{"rendered":"<p>N\u011bkolik n\u00e1m\u011bt\u016f pro vyu\u017eit\u00ed programu GEOGEBRA p\u0159i v\u00fduce algebry. Vzhledem k dostupnosti tohoto v\u00fdborn\u00e9ho n\u00e1stroje, jeho voln\u00e9 \u0161i\u0159itelnosti a mo\u017en\u00e9m p\u0159\u00edstupu p\u0159es po\u010d\u00edta\u010d i tablet je k dispozici aplikace s mnoha mo\u017enostmi vyu\u017eit\u00ed. P\u0159i zav\u00e1d\u011bn\u00ed pojmu line\u00e1rn\u00ed funkce v 9.\u00a0ro\u010dn\u00edku nar\u00e1\u017e\u00edme \u010dasto na probl\u00e9m pochopen\u00ed &#8222;z\u00e1vislosti jedn\u00e9 prom\u011bnn\u00e9 na druh\u00e9&#8220;. P\u0159esto\u017ee \u017e\u00e1ci b\u011b\u017en\u011b pou\u017e\u00edvaj\u00ed vzorce&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,7],"tags":[37,15,34],"class_list":["post-118","post","type-post","status-publish","format-standard","hentry","category-sekce-ict","category-sekce-matematiky-a-informatiky","tag-geogebra","tag-ict","tag-matematika"],"_links":{"self":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts\/118","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=118"}],"version-history":[{"count":1,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts\/118\/revisions"}],"predecessor-version":[{"id":119,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts\/118\/revisions\/119"}],"wp:attachment":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}