{"id":270,"date":"2023-01-06T15:37:46","date_gmt":"2023-01-06T15:37:46","guid":{"rendered":"http:\/\/flatarmal.is\/?p=270"},"modified":"2025-08-02T07:13:51","modified_gmt":"2025-08-02T07:13:51","slug":"um-verkefnid-velmenni-a-talnalinu","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=270","title":{"rendered":"Um verkefni\u00f0 V\u00e9lmenni \u00e1 talnal\u00ednu"},"content":{"rendered":"<p><strong>Ing\u00f3lfur<\/strong><em><strong> G\u00edslason\u00a0<\/strong><\/em><\/p>\n<p>Verkefni\u00f0 <em>V\u00e9lmenni \u00e1 talnal\u00ednu<\/em> er a\u00f0 finna \u00ed myndbandinu <a href=\"https:\/\/classroom.thenational.academy\/lessons\/negative-numbers-in-context-68t66c?step=1\">Negative numbers<\/a> in context.<\/p>\n<p>Tilgangur verkefnisins \u00ed myndbandinu er a\u00f0 auka skilning nemenda \u00e1 neikv\u00e6\u00f0um t\u00f6lum. Neikv\u00e6\u00f0um t\u00f6lum er gefinn s\u00e1 tilgangur a\u00f0 vera hluti af st\u00e6r\u00f0fr\u00e6\u00f0ilegu l\u00edkani af sta\u00f0setningu og f\u00e6rslu \u00ed tv\u00e6r mismunandi \u00e1ttir.<\/p>\n<p>Sko\u00f0um fyrstu \u00fatf\u00e6rsluna \u00e1 mynd 1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"891\" height=\"464\" class=\"alignnone wp-image-90 size-full\" src=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd1.png\" alt=\"\" wi<span style=\"font-size:0px; color:#ff0000;\" srcset=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd1.png 891w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd1-300x156.png 300w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd1-768x400.png 768w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd1-850x443.png 850w\" sizes=\"auto, (max-width: 891px) 100vw, 891px\" \/>\ufeff<a href=\"https:\/\/www.lambdapy.com\/\">\u7280\u5229\u58eb<\/a><br \/>\n<\/span>dth=&#8220;891&#8243; height=&#8220;464&#8243; \/><\/p>\n<p><em>Mynd 1.<\/em><\/p>\n<p>Gefa m\u00e1 tv\u00e6r skipanir, A og B. A f\u00e6rir v\u00e9lmenni\u00f0 \u00ed austur (til h\u00e6gri) um 5 og B f\u00e6rir<br \/>\nv\u00e9lmenni\u00f0 \u00ed vestur (til vinstri) um 3. Nemendur eru be\u00f0nir a\u00f0 senda v\u00e9lmenni\u00f0 \u00e1 nokkrar<br \/>\ntilteknar t\u00f6lur fr\u00e1 byrjunarreitnum, 0. \u00cd sj\u00e1lfu s\u00e9r er verkefni\u00f0 nokku\u00f0 opi\u00f0, \u00fev\u00ed \u00fea\u00f0 er h\u00e6gt<br \/>\na\u00f0 fara \u00f3l\u00edkar lei\u00f0ir vi\u00f0 a\u00f0 leysa \u00fea\u00f0 og finna m\u00e1 margar mismunandi lausnir. Til d\u00e6mis m\u00e1<br \/>\nkomast \u00e1 4 me\u00f0 ABAB (5-3+5-3) og AABB (5+5-3-3) og AABBBAABBBAABBBAABBB (og \u00f3tal<br \/>\nfleiri lei\u00f0um). \u00dea\u00f0 v\u00e6ri kannski ekki vitlaust a\u00f0 skj\u00f3ta inn spurningu um \u00feetta: <em>Skiptir r\u00f6\u00f0<\/em><br \/>\n<em>skipana einhverju m\u00e1li?<\/em> Einnig m\u00e1 velta fyrir s\u00e9r hversu margar \u00f3l\u00edkar lei\u00f0ir s\u00e9u f\u00e6rar til a\u00f0<br \/>\nkomast \u00e1 tilteknar t\u00f6lur e\u00f0a hva\u00f0a t\u00f6lur vi\u00f0 getum komist \u00e1 ef vi\u00f0 megum bara nota \u00ferj\u00fa A<br \/>\nog tv\u00f6 B. H\u00e9r \u00ed framhaldinu ver\u00f0ur \u00fev\u00ed l\u00fdst hvernig h\u00e6gt er a\u00f0 opna og \u00fatv\u00edkka verkefni\u00f0 enn<br \/>\nfrekar og gera \u00far \u00fev\u00ed ver\u00f0ugra verkefni, nokku\u00f0 dj\u00fapa st\u00e6r\u00f0fr\u00e6\u00f0ilega ranns\u00f3kn.<\/p>\n<h3><strong>Eiginleikar<\/strong><\/h3>\n<p>Vi\u00f0 byrjum \u00e1 a\u00f0 skr\u00e1: hverjir eru<em> eiginleikar<\/em> verkefnisins? H\u00e9r eru alltaf \u00f3teljandi margir<br \/>\nm\u00f6guleikar og \u00fea\u00f0 er enginn einn r\u00e9ttur listi. Tilgangurinn me\u00f0 sl\u00edkum lista er a\u00f0 vi\u00f0 getum sko\u00f0a\u00f0 hann s\u00ed\u00f0ar og spurt <strong>hva\u00f0 ef ekki?<\/strong> Hva\u00f0 ef einhver eiginleiki er ekki svona? Hva\u00f0 ef<br \/>\nvi\u00f0 breytum einum e\u00f0a fleiri eiginleikum?<\/p>\n<ul>\n<li>Vi\u00f0 fer\u00f0umst eftir <strong>talnal\u00ednu<\/strong>,<\/li>\n<li>vi\u00f0 f\u00e1um <strong>tvo m\u00f6guleika<\/strong> \u00e1 a\u00f0 f\u00e6ra v\u00e9lmenni\u00f0,<\/li>\n<li>vi\u00f0 megum fara <strong>5<\/strong> til h\u00e6gri og <strong>3<\/strong> til vinstri,<\/li>\n<li>t\u00f6lurnar 5 og 3 eru b\u00e1\u00f0ar <strong>oddat\u00f6lur<\/strong>,<\/li>\n<li>t\u00f6lurnar 5 og 3 eru b\u00e1\u00f0ar <strong>frumt\u00f6lur<\/strong>,<\/li>\n<li>t\u00f6lurnar 5 og 3 hafa engan <strong>sameiginlegan frum\u00fe\u00e1tt<\/strong>, me\u00f0 \u00f6\u00f0rum or\u00f0um er engin tala<br \/>\nfyrir utan 1 sem gengur upp \u00ed b\u00e1\u00f0ar t\u00f6lurnar.<\/li>\n<\/ul>\n<p>\u00dea\u00f0 er h\u00e6gt a\u00f0 halda \u00e1fram og finna fleiri eiginleika. En h\u00e9r eru nokkrar hugmyndir um \u00fea\u00f0<br \/>\nhvert m\u00e1 fara me\u00f0 verkefni\u00f0:<\/p>\n<ul>\n<li>Hva\u00f0 ef vi\u00f0 fer\u00f0umst ekki \u00e1 talnal\u00ednu heldur \u00ed hnitakerfi, \u00ed tv\u00edv\u00eddd? Megum vi\u00f0 \u00fe\u00e1 fara<br \/>\n\u00ed 4 \u00e1ttir (e\u00f0a fleiri), nor\u00f0ur, austur, su\u00f0ur, vestur. Hva\u00f0 ef vi\u00f0 h\u00f6fum \u00fea\u00f0 \u00ed \u00fer\u00edv\u00eddd?<br \/>\nHva\u00f0 ef vi\u00f0 fer\u00f0umst \u00e1 k\u00faluyfirbor\u00f0i?<\/li>\n<li>Hva\u00f0 ef vi\u00f0 myndum bara f\u00e1 einn m\u00f6guleika \u00e1 f\u00e6rslu? E\u00f0a \u00ferj\u00e1? E\u00f0a fleiri?<\/li>\n<li>Hva\u00f0 ef vi\u00f0 myndum mega hoppa me\u00f0 margf\u00f6ldun e\u00f0a deilingu e\u00f0a \u00e1 einhvern<br \/>\nannan h\u00e1tt?<\/li>\n<li>Hva\u00f0 ef b\u00e1\u00f0ar t\u00f6lurnar v\u00e6ru sl\u00e9ttar t\u00f6lur?<\/li>\n<li>Hva\u00f0 ef t\u00f6lurnar hafa sameiginlegan frum\u00fe\u00e1tt?<\/li>\n<li>Hva\u00f0 ef nemendur f\u00e1 a\u00f0 velja t\u00f6lurnar?<\/li>\n<\/ul>\n<p>H\u00e9r m\u00e6tti l\u00edka halda \u00e1fram.<\/p>\n<h3><strong>Tilg\u00e1tur og alh\u00e6fingar<\/strong><\/h3>\n<p>Upprunalega verkefni\u00f0 um v\u00e9lmenni\u00f0, sem kynnt var \u00ed inngangi \u00feessarar greinar, skortir<br \/>\neinn mikilv\u00e6gan eiginleika til a\u00f0 geta talist st\u00e6r\u00f0fr\u00e6\u00f0ilegt verkefni \u00ed raun og veru: t\u00e6kif\u00e6ri<br \/>\ntil \u00feess a\u00f0 <strong>alh\u00e6fa<\/strong>. A\u00f0 alh\u00e6fa \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i \u00fe\u00fd\u00f0ir a\u00f0 finna einhverja almenna reglu, eitthva\u00f0<br \/>\nsem er alltaf satt, e\u00f0a alltaf satt \u00feegar einhver tiltekin skilyr\u00f0i eru til sta\u00f0ar. \u00deess vegna er<br \/>\n\u00e1hugavert a\u00f0 \u00ed myndbandinu sem fylgir verkefninu fylgir ein spurning \u00ed vi\u00f0b\u00f3t: <em>Geti\u00f0 \u00fei\u00f0<\/em><br \/>\n<em>fundi\u00f0 heila t\u00f6lu sem v\u00e9lmenni\u00f0 kemst ekki til?<\/em> \u00dev\u00ed mi\u00f0ur vantar oft samb\u00e6rilega spurningu \u00ed<br \/>\nverkefni \u00ed kennslub\u00f3kum, spurningu sem kallar \u00e1 alh\u00e6fingu og st\u00e6r\u00f0fr\u00e6\u00f0ilega hugsun. Ef \u00fei\u00f0<br \/>\nhafi\u00f0 lagt ykkur fram um a\u00f0 gl\u00edma vi\u00f0 verkefni\u00f0 \u00fe\u00e1 \u00e6ttu\u00f0 \u00fei\u00f0 a\u00f0 geta sett fram <strong>tilg\u00e1tu<\/strong>:<br \/>\nV\u00e9lmenni\u00f0 kemst hvert sem er.<\/p>\n<h3><strong>R\u00f6ksemdir og sannanir<\/strong><\/h3>\n<p>\u00dea\u00f0 er engin heil tala til sem v\u00e9lmenni\u00f0 kemst ekki \u00e1. \u00deetta m\u00e1<strong> r\u00f6ksty\u00f0ja<\/strong> \u00e1 marga vegu.<br \/>\nLausnir sem hafa komi\u00f0 fram \u00feegar verkefni\u00f0 hefur veri\u00f0 sett fyrir kennaranema ganga<br \/>\nmargar \u00fat \u00e1 a\u00f0 finna nokkrar t\u00f6lur \u00ed r\u00f6\u00f0 sem h\u00e6gt er a\u00f0 komast \u00e1, og vinna \u00fat fr\u00e1 \u00fev\u00ed. Til<br \/>\nd\u00e6mis er h\u00e6gt a\u00f0 sj\u00e1 a\u00f0:<\/p>\n<ul>\n<li>1 = AABBB (5 + 5 &#8211; 3 &#8211; 3 &#8211; 3 = 10 &#8211; 9 = 1)<\/li>\n<li>2 = AB<\/li>\n<li>3 = AAABBBB<\/li>\n<li>4 = ABBA<\/li>\n<li>5 = A<\/li>\n<\/ul>\n<p>N\u00fa h\u00f6fum vi\u00f0 fimm t\u00f6lur \u00ed r\u00f6\u00f0 og \u00fe\u00e1 er h\u00e6gt a\u00f0 f\u00e1 n\u00e6stu fimm t\u00f6lur me\u00f0 \u00fev\u00ed a\u00f0 b\u00e6ta 5 vi\u00f0<br \/>\nhverja \u00feeirra (me\u00f0 \u00fev\u00ed a\u00f0 b\u00e6ta vi\u00f0 A). Sem d\u00e6mi er 6 = 5 + 1 = AAABBB (einu A b\u00e6tt framan<br \/>\nvi\u00f0 lausnina fyrir 1). \u00deannig getum vi\u00f0 f\u00e6rt okkur \u00e1fram upp eftir talnal\u00ednunni: Vi\u00f0 h\u00f6fum<br \/>\nt\u00f6lurnar 1, 2, 3, 4 og 5 og me\u00f0 \u00fev\u00ed a\u00f0 nota A (b\u00e6ta vi\u00f0 5) \u00fat fr\u00e1 hverri \u00feeirra f\u00e1st n\u00e6stu fimm<br \/>\nt\u00f6lur, 6, 7, 8, 9 og 10, og \u00feannig m\u00e1 halda \u00e1fram koll af kolli.<\/p>\n<p>Vi\u00f0 n\u00e1nari umhugsun og sko\u00f0un getum vi\u00f0 einfalda\u00f0 \u00feessi r\u00f6k. Fyrst vi\u00f0 getum komist \u00ed 1, \u00fe\u00e1<br \/>\ngetum vi\u00f0 komist \u00e1 hva\u00f0a j\u00e1kv\u00e6\u00f0u t\u00f6lu sem er. Vi\u00f0 getum endurteki\u00f0 1 (AABBB) eins oft og<br \/>\n\u00fearf til komast \u00feanga\u00f0 sem vi\u00f0 viljum. \u00de\u00e1 f\u00e1st stundum \u00f3\u00fearflegar langar lei\u00f0ir, en \u00fe\u00e6r virka!<br \/>\n\u00c1 sama h\u00e1tt vitum vi\u00f0 a\u00f0 -1 = ABB, og \u00fe\u00e1 er h\u00e6gt a\u00f0 endurtaka ABB eins oft og \u00fearf til a\u00f0 f\u00e1<br \/>\nhva\u00f0a neikv\u00e6\u00f0u t\u00f6lu sem er. Til a\u00f0 komast \u00ed t\u00f6luna 456 notum vi\u00f0 AABBB 456 sinnum \u00ed r\u00f6\u00f0.<\/p>\n<p>Me\u00f0 \u00fev\u00ed a\u00f0 nota algebrulegan rith\u00e1tt getum vi\u00f0 sagt \u00feetta svona: \u00dea\u00f0 a\u00f0 komast \u00e1 j\u00e1kv\u00e6\u00f0u<br \/>\nt\u00f6luna <em>n<\/em>, \u00fe\u00fd\u00f0ir a\u00f0 finna t\u00f6lur <em>x<\/em> og <em>y<\/em> \u00feannig a\u00f0 <em>n<\/em> = 5<em>x<\/em> &#8211; 3<em>y<\/em>. En vi\u00f0 vitum a\u00f0<\/p>\n<p>1 = 5 \u2219 2 &#8211; 3 \u2219 3\u00a0og \u00fe\u00e1 er <em>n<\/em> = <em>n<\/em>(5 \u2219 2 &#8211; 3 \u2219 3) = 2<em>n<\/em> \u2219 5 &#8211; 3<em>n<\/em> \u2219 3. \u00deetta segir okkur \u00fe\u00e1 til<br \/>\nd\u00e6mis a\u00f0 til \u00feess a\u00f0 komast \u00ed t\u00f6luna 456 getum vi\u00f0 nota\u00f0 2 \u2219 456 = 912 sinnum A, og 3 \u2219 456 = 1368 sinnum B.<\/p>\n<p>Myndr\u00e6n lei\u00f0 til a\u00f0 sj\u00e1 \u00feetta:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-91 size-full\" src=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd2.png\" alt=\"\" width=\"947\" height=\"391\" srcset=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd2.png 947w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd2-300x124.png 300w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd2-768x317.png 768w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd2-850x351.png 850w\" sizes=\"auto, (max-width: 947px) 100vw, 947px\" \/><\/p>\n<p><em>Mynd 2.<\/em><\/p>\n<p>\u00c1 mynd 2 s\u00e9st hvernig h\u00e6gt er a\u00f0 komast \u00e1 t\u00f6luna 1 me\u00f0 lei\u00f0inni AABBB. \u00dea\u00f0 m\u00e1 gjarnan sp\u00e1 \u00ed \u00fea\u00f0 hvort r\u00f6\u00f0in \u00e1 skipununum (A og B) skipti m\u00e1li. \u00c1 mynd 3 sj\u00e1um vi\u00f0 almenna tilfelli\u00f0, almenn r\u00f6k fyrir \u00fev\u00ed a\u00f0 \u00fea\u00f0 megi alltaf komast \u00ed t\u00f6lu sem er einum h\u00e6rri en talan sem vi\u00f0 h\u00f6fum n\u00fa \u00feegar komist \u00e1. \u00c9g nota h\u00e9r b\u00f3kstafinn <em>n<\/em> til \u00feess a\u00f0 t\u00e1kna hva\u00f0a t\u00f6lu sem er. S\u00fa tala sem er einum h\u00e6rri en <em>n<\/em> er \u00fe\u00e1 talan <em>n<\/em> + 1 og svo koll af kolli.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-92 size-full\" src=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd3.png\" alt=\"\" width=\"883\" height=\"297\" srcset=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd3.png 883w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd3-300x101.png 300w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd3-768x258.png 768w, https:\/\/flatarmal.is\/wp-content\/uploads\/2022\/12\/mynd3-850x286.png 850w\" sizes=\"auto, (max-width: 883px) 100vw, 883px\" \/><\/p>\n<p><em>Mynd 3.<\/em><\/p>\n<p>\u00cd fr\u00e6\u00f0ilegri st\u00e6r\u00f0fr\u00e6\u00f0i eru r\u00f6ksemdarf\u00e6rslur \u00e1 bor\u00f0 vi\u00f0 \u00feessa nefndar <em>sannanir me\u00f0 \u00ferepun<\/em><br \/>\ne\u00f0a <em>\u00ferepasannanir<\/em>. \u00cd \u00feeirri \u00fatg\u00e1fu sem h\u00e9r er um a\u00f0 r\u00e6\u00f0a \u00fe\u00e1 er s\u00fdnt fram \u00e1 tvo hluti:<\/p>\n<ul>\n<li>a\u00f0 reglan gildi um t\u00f6luna 1 (h\u00e6gt er a\u00f0 komast \u00e1 t\u00f6luna 1),<\/li>\n<li>a\u00f0 ef reglan gildir um einhverja t\u00f6lu \u00fe\u00e1 gildir h\u00fan l\u00edka um n\u00e6stu t\u00f6lu \u00fear \u00e1 eftir (ef h\u00e6gt er a\u00f0 komast \u00e1 einhverja t\u00f6lu \u00fe\u00e1 er h\u00e6gt a\u00f0 komast \u00e1 n\u00e6stu t\u00f6lu \u00e1 eftir \u00ed talnar\u00f6\u00f0inni).<\/li>\n<\/ul>\n<p>Ef \u00feetta tvennt er satt, \u00fe\u00e1 gildir reglan um allar heilar j\u00e1kv\u00e6\u00f0ar t\u00f6lur. \u00cd okkar tilfelli er reglan<br \/>\ns\u00fa a\u00f0 \u201ev\u00e9lmenni\u00f0 kemst \u00ed t\u00f6luna\u201c.<\/p>\n<h3><strong>Og hva\u00f0? Af hverju a\u00f0 fjalla um \u00feetta?<\/strong><\/h3>\n<p>Umfj\u00f6llunin er sett fram h\u00e9r til a\u00f0 s\u00fdna fram \u00e1 a\u00f0 verkefni sem h\u00e6fir grunnsk\u00f3lanemum getur haft mikla \u00fe\u00fd\u00f0ingu og bo\u00f0i\u00f0 upp \u00e1 dj\u00fapa st\u00e6r\u00f0fr\u00e6\u00f0ilega hugsun. Ungt f\u00f3lk er fullf\u00e6rt um a\u00f0 finna \u00feau r\u00f6k sem kynnt hafa veri\u00f0 h\u00e9r a\u00f0 ofan ef \u00feau f\u00e1 hvatningu og \u00f6grandi spurningar fr\u00e1 kennara. \u00deau geta sett r\u00f6kin fram og sannf\u00e6rst af \u00feeim \u00fe\u00f3 a\u00f0 \u00feau \u00feurfi ekki a\u00f0 setja \u00feau formlega fram e\u00f0a a\u00f0 l\u00e6ra formlega um \u00ferepasannanir. \u00derepasannanir eru annars venjulega ekki kynntar f\u00f3lki fyrr en \u00e1 s\u00ed\u00f0asta \u00e1ri \u00ed framhaldssk\u00f3la og \u00fe\u00e1 yfirleitt einungis \u00feeim sem eru \u00e1 n\u00e1tt\u00faruv\u00edsindabraut e\u00f0a annarri braut sem inniheldur mikla st\u00e6r\u00f0fr\u00e6\u00f0i. \u00de\u00e6r eru svo vi\u00f0fangsefni \u00ed h\u00e1sk\u00f3lan\u00e1mi \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i og eru mikilv\u00e6gar \u00ed t\u00f6lvunarfr\u00e6\u00f0i.<\/p>\n<h3><strong>Hver er st\u00e6r\u00f0fr\u00e6\u00f0ilegur kjarni verkefnisins?<\/strong><\/h3>\n<p>Greiningin \u00e1 verkefninu lei\u00f0ir \u00ed lj\u00f3s a\u00f0 ef h\u00e6gt er a\u00f0 komast \u00e1 t\u00f6luna 1, \u00fe\u00e1 er h\u00e6gt a\u00f0 komast hvert sem er \u00e1 j\u00e1kv\u00e6\u00f0a hluta talnal\u00ednunnar (og eins, ef h\u00e6gt er a\u00f0 komast \u00ed -1, \u00fe\u00e1 er h\u00e6gt a\u00f0 komast hvert sem er \u00e1 neikv\u00e6\u00f0a hluta talnal\u00ednunnar). En hva\u00f0a eiginleikar talnanna 5 og -3 gera \u00feetta m\u00f6gulegt? Ein lei\u00f0 til a\u00f0 opna verkefni\u00f0 er a\u00f0 stinga upp \u00e1 a\u00f0 pr\u00f3fa a\u00f0rar t\u00f6lur, anna\u00f0hvort frj\u00e1lst e\u00f0a a\u00f0 gefa tilteknar t\u00f6lur. \u00deegar verkefni\u00f0 var sett fyrir kennaranema haustin 2021 og 2022 voru gefnar t\u00f6lurnar 15 og -18, \u00fea\u00f0 er a\u00f0 segja a\u00f0 skipanir v\u00e9lmennis ur\u00f0u A: 15 skref \u00ed austur og B: 18 skref \u00ed vestur. \u00c1st\u00e6\u00f0an er s\u00fa a\u00f0 \u00feessar t\u00f6lur hafa sameiginlegan \u00fe\u00e1tt, 3. Hva\u00f0a \u00e1hrif skyldi \u00fea\u00f0 hafa? J\u00fa, \u00feetta veldur \u00fev\u00ed a\u00f0 einungis er h\u00e6gt a\u00f0 komast \u00ed t\u00f6lur sem eru margfeldi af 3. \u00dea\u00f0 er a\u00f0 segja, allar skrefast\u00e6r\u00f0ir ver\u00f0a \u00ed \u00ferisvar sinnum t\u00f6flunni og ef v\u00e9lmenni\u00f0 byrjar \u00ed 0 \u00fe\u00e1 kemst \u00fea\u00f0 ekki \u00fat fyrir t\u00f6lurnar \u00ed \u00feeirri t\u00f6flu. Vi\u00f0 \u00feessa \u00fatv\u00edkkun var <strong>eiginleiki talnanna<\/strong> 5 og -3 kanna\u00f0ur og honum<strong> breytt<\/strong>: Hva\u00f0 ef t\u00f6lurnar hafa sameiginlegan frum\u00fe\u00e1tt? Og verkefni\u00f0 sett fram me\u00f0 tveimur tilteknum t\u00f6lum me\u00f0 n\u00fdja eiginleikanum. \u00cd n\u00e6stu spurningu \u00fear \u00e1 eftir var nemendum sagt a\u00f0 velja sj\u00e1lfir skrefast\u00e6r\u00f0ir og a\u00f0 <strong>setja fram tilg\u00e1tu<\/strong> (sem er l\u00edka <strong>alh\u00e6fing<\/strong>) um \u00fea\u00f0 hva\u00f0a eiginleika \u00ferepast\u00e6r\u00f0irnar \u00feurfa a\u00f0 hafa til \u00feess a\u00f0 v\u00e9lmenni\u00f0 komist hvert sem er. \u00dear me\u00f0 hefur verkefni\u00f0 veri\u00f0 opna\u00f0 mj\u00f6g miki\u00f0: Nemendur velja sj\u00e1lfir t\u00f6lur til a\u00f0 pr\u00f3fa og \u00feeir f\u00e1st vi\u00f0 a\u00f0 alh\u00e6fa og r\u00f6ksty\u00f0ja \u00ed samhengi sem er st\u00e6r\u00f0fr\u00e6\u00f0ilega dj\u00fapt og mikilv\u00e6gt.<\/p>\n<p>\u00cd raun og veru n\u00e1lgast nemendur me\u00f0 \u00feessu fr\u00e6ga reglu sem nefnist <a href=\"https:\/\/en.wikipedia.org\/wiki\/B%C3%A9zout%27s_identity\">jafna Bezout<\/a>. \u00dea\u00f0 er s\u00fa sta\u00f0reynd a\u00f0 ef st\u00e6rsti sameiginlegi \u00fe\u00e1ttur tveggja j\u00e1kv\u00e6\u00f0ra heilla talna <em>a<\/em> og <em>b<\/em> er talan <em>d<\/em>, \u00fe\u00e1 er h\u00e6gt a\u00f0 finna heilar t\u00f6lur<em> x<\/em> og <em>y<\/em> \u00feannig a\u00f0<em> d<\/em> = <em>ax<\/em> + <em>by<\/em>. En, l\u00edkt og ekki \u00fearf nau\u00f0synlega a\u00f0 r\u00e6\u00f0a \u00ferepasannanir \u00fe\u00e1 er heldur alls ekki nau\u00f0synlegt a\u00f0 r\u00e6\u00f0a j\u00f6fnu Bezout sem sl\u00edka e\u00f0a kynna hana sem reglu til a\u00f0 l\u00e6ra.<\/p>\n<p>\u00deegar \u00e9g \u00fatv\u00edkka\u00f0i verkefni\u00f0 valdi \u00e9g a\u00f0 taka alh\u00e6finguna sem bent var \u00e1 \u00ed verkefninu (Er h\u00e6gt a\u00f0 komast hvert sem er?) og gera hana a\u00f0 n\u00e1nara sko\u00f0unarefni me\u00f0 \u00fev\u00ed a\u00f0 kanna hva\u00f0 gerist ef vi\u00f0 breytum eiginleikum talnanna tveggja sem gefnar voru sem skrefast\u00e6r\u00f0. \u00dea\u00f0 v\u00e6ri l\u00edka h\u00e6gt a\u00f0 fara \u00ed a\u00f0rar \u00e1ttir eins og kemur fram \u00ed \u201ehva\u00f0 ef\u201c listanum.<\/p>\n<p>Eitt d\u00e6mi um verkefni er eftirfarandi:<\/p>\n<p>V\u00e9lmennin Robbi og T\u00f3ta eru st\u00f6dd \u00ed upphafspunkti hnitakerfis (0,0). H\u00e6gt er a\u00f0 gefa fj\u00f3rar<br \/>\nskipanir fyrir hvert v\u00e9lmenni, samkv\u00e6mt eftirfarandi t\u00f6flu:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-346 \" src=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2023\/01\/Mynd2-4.png\" alt=\"\" width=\"423\" height=\"155\" srcset=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2023\/01\/Mynd2-4.png 440w, https:\/\/flatarmal.is\/wp-content\/uploads\/2023\/01\/Mynd2-4-300x110.png 300w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Spurningar:<\/p>\n<ol>\n<li>T\u00f3ta f\u00e6r\u00f0i sig samkv\u00e6mt skipununum ABCD og enda\u00f0i \u00fear me\u00f0 \u00ed punktinum (2, 3).<br \/>\nKemst Robbi \u00feanga\u00f0? Hvernig e\u00f0a hvers vegna ekki?<\/li>\n<li>Robbi f\u00e6r\u00f0i sig samkv\u00e6mt skipunum s\u00ednum ABCD og enda\u00f0i \u00fear me\u00f0 \u00ed punktinum (-1,3).<br \/>\nKemst T\u00f3ta \u00feanga\u00f0? Hvernig e\u00f0a hvers vegna ekki?<\/li>\n<p><span style=\"font-size:0px; color:#ff0000;\">\ufeff<a href=\"https:\/\/www.csshjxc.com\/\">\u58ef\u967d\u85e5<\/a><br \/>\n<\/span> \t<\/p>\n<li>L\u00fdstu mismunandi m\u00f6guleikum Robba og T\u00f3tu til a\u00f0 komast um \u00ed hnitakerfinu. Hvert<br \/>\nkemst Robbi? Hvert kemst T\u00f3ta?<\/li>\n<li>N\u00fa \u00e1 a\u00f0 hanna skipanir fyrir tv\u00f6 v\u00e9lmenni me\u00f0 sama h\u00e6tti (segja hve m\u00f6rg skref \u00ed hverja<br \/>\n\u00e1tt \u00feau mega fara). \u00deau byrja b\u00e6\u00f0i \u00e1 (0,0) en mega svo aldrei m\u00e6tast \u00e1 neinum \u00f6\u00f0rum sta\u00f0.<\/li>\n<\/ol>\n<p>Finni\u00f0 lei\u00f0ir til a\u00f0 gera \u00feetta.<\/p>\n<p>H\u00e9r m\u00e1 halda \u00e1fram me\u00f0 fleiri spurningar sem sn\u00faast um \u00fea\u00f0 hvers konar mengi af skipunum virka til a\u00f0 komast hvert sem er og hver virka ekki.<\/p>\n<h3><strong>Samantekt: A\u00f0 opna og \u00fatv\u00edkka st\u00e6r\u00f0fr\u00e6\u00f0iverkefni<\/strong><\/h3>\n<p>H\u00e9r hefur veri\u00f0 reynt a\u00f0 gefa inns\u00fdn \u00ed \u00fea\u00f0 hvernig vinna m\u00e1 me\u00f0 gefi\u00f0 st\u00e6r\u00f0fr\u00e6\u00f0iverkefni<br \/>\nog gera \u00far \u00fev\u00ed ver\u00f0ugt verkefni sem reynir \u00e1 st\u00e6r\u00f0fr\u00e6\u00f0ilega hugsun. \u00cd ver\u00f0ugu st\u00e6r\u00f0fr\u00e6\u00f0iverkefni felst a\u00f0 gera tilg\u00e1tur sem fela \u00ed s\u00e9r alh\u00e6fingar og a\u00f0 r\u00f6ksty\u00f0ja \u00fe\u00e6r.<\/p>\n<p>Ein lei\u00f0 til \u00feess a\u00f0 opna og \u00fatv\u00edkka verkefni er a\u00f0:<\/p>\n<ol>\n<li>Gera lista yfir eiginleika verkefnis og \u00feeirra st\u00e6r\u00f0fr\u00e6\u00f0ilegu hluta sem eru \u00ed \u00fev\u00ed.<\/li>\n<li>Fara yfir listann og breyta eiginleikunum: Hva\u00f0 ef (ekki) &#8230;?<\/li>\n<li>Sko\u00f0a \u00fatkomur gegnum alh\u00e6fingar: Er \u00feetta alltaf svona? Hva\u00f0a eiginleikum m\u00e1<br \/>\nbreyta \u00feannig a\u00f0 \u00fea\u00f0 gildi samt alltaf \u00fea\u00f0 sama? Er h\u00e6gt a\u00f0 beina athygli nemenda<br \/>\na\u00f0 \u00feessum hlutum me\u00f0 vel v\u00f6ldum t\u00f6lum, formum e\u00f0a fyrirb\u00e6rum?<\/li>\n<li>Athuga a\u00f0 meginatri\u00f0i\u00f0 s\u00e9 r\u00f6ksemdir og tengsl frekar en eing\u00f6ngu reikningar.<\/li>\n<\/ol>\n<p>\u00deannig er h\u00e6gt a\u00f0 n\u00fdta fyrirliggjandi kennslub\u00e6kur og verkefni til \u00feess a\u00f0 stu\u00f0la a\u00f0 meiri<br \/>\nst\u00e6r\u00f0fr\u00e6\u00f0ihugsun.<\/p>\n<p><em>Ing\u00f3lfur G\u00edslason er a\u00f0junkt vi\u00f0 Menntav\u00edsindasvi\u00f0 H\u00e1sk\u00f3la \u00cdslands<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ing\u00f3lfur G\u00edslason\u00a0 Verkefni\u00f0 V\u00e9lmenni \u00e1 talnal\u00ednu er a\u00f0 finna \u00ed myndbandinu Negative numbers in context. Tilgangur verkefnisins \u00ed myndbandinu er a\u00f0 auka skilning nemenda \u00e1 neikv\u00e6\u00f0um t\u00f6lum. Neikv\u00e6\u00f0um t\u00f6lum er gefinn s\u00e1 tilgangur a\u00f0 vera hluti af st\u00e6r\u00f0fr\u00e6\u00f0ilegu l\u00edkani af sta\u00f0setningu og f\u00e6rslu \u00ed tv\u00e6r mismunandi \u00e1ttir. Sko\u00f0um fyrstu \u00fatf\u00e6rsluna \u00e1 mynd 1. Mynd 1&#8230;.<\/p>\n","protected":false},"author":1,"featured_media":330,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-270","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/270","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=270"}],"version-history":[{"count":21,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/270\/revisions"}],"predecessor-version":[{"id":5368,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/270\/revisions\/5368"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/330"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=270"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=270"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=270"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}