{"id":6676,"date":"2026-06-17T00:33:53","date_gmt":"2026-06-17T00:33:53","guid":{"rendered":"https:\/\/flatarmal.is\/?p=6676"},"modified":"2026-06-18T12:57:20","modified_gmt":"2026-06-18T12:57:20","slug":"rannsoknir-fyrir-staerdfraedinga-a-ollum-aldri","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=6676","title":{"rendered":"Ranns\u00f3knir fyrir st\u00e6r\u00f0fr\u00e6\u00f0inga \u00e1 \u00f6llum aldri"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>Ing\u00f3lfur G\u00edslason<\/em>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color wp-elements-1\" style=\"color:#2ebbcb\"><strong>Opin mi\u00f0ja<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Vi\u00f0 byrjum \u00e1 st\u00e6r\u00f0fr\u00e6\u00f0ispurningu sem \u00e9g setti inn \u00e1 Facebook-h\u00f3pinn St\u00e6r\u00f0fr\u00e6\u00f0ikennarinn \u00ed febr\u00faar 2019.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"452\" src=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.08.34-1024x452.png\" alt=\"\" class=\"wp-image-6678\" style=\"aspect-ratio:2.2656195053334898;width:492px;height:auto\" srcset=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.08.34-1024x452.png 1024w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.08.34-300x132.png 300w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.08.34-768x339.png 768w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.08.34-1536x678.png 1536w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.08.34-850x375.png 850w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.08.34.png 1866w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<div style=\"height:30px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/www.openmiddle.com\/equivalent-statements\/\">Verkefni\u00f0<\/a> fann \u00e9g \u00e1 vefs\u00ed\u00f0unni <a href=\"https:\/\/www.openmiddle.com\/\">Open Middle<\/a> \u00fear sem \u00fea\u00f0 er sagt henta nemendum \u00ed fyrsta bekk \u00ed Bandar\u00edkjunum (sem \u00e6tti a\u00f0 vera mj\u00f6g samb\u00e6rilegur okkar fyrsta bekk). \u00deegar og ef verkefni\u00f0 er lagt fyrir nemendur \u00ed fyrsta bekk er fyrst h\u00e6gt a\u00f0 bi\u00f0ja \u00fe\u00e1 um a\u00f0 nota t\u00f6lustafina fr\u00e1 1 til 9, \u00ed mesta lagi einu sinni hvern, og ra\u00f0a \u00feeim \u00feannig \u00ed reitina a\u00f0 jafnan ver\u00f0i s\u00f6nn. \u00dea\u00f0 g\u00e6ti \u00feurft a\u00f0 undirb\u00faa nemendur me\u00f0 \u00fev\u00ed a\u00f0 l\u00e1ta \u00fe\u00e1 meta hvort einhverjar j\u00f6fnur eins og 3 + 2 = 4 + 1 s\u00e9u sannar, \u00fea\u00f0 er a\u00f0 segja a\u00f0 beina athygli \u00feeirra a\u00f0 \u00fev\u00ed a\u00f0 j\u00f6fnur s\u00e9u anna\u00f0hvort sannar e\u00f0a \u00f3sannar.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Tilgangurinn me\u00f0 verkefninu samkv\u00e6mt vefs\u00ed\u00f0unni er a\u00f0 nemendur auki skilning sinn \u00e1 jafna\u00f0armerkinu: a\u00f0 \u00fea\u00f0 s\u00e9u jafnar st\u00e6r\u00f0ir s\u00edn hvoru megin vi\u00f0 merki\u00f0 og a\u00f0 j\u00f6fnur s\u00e9u anna\u00f0 hvort sannar e\u00f0a \u00f3sannar og a\u00f0 nemendur l\u00e6ri a\u00f0 \u00e1kvar\u00f0a hvort j\u00f6fnur me\u00f0 samlagningu og fr\u00e1dr\u00e6tti s\u00e9u sannar e\u00f0a \u00f3sannar. Eitt af \u00fev\u00ed sem vefst fyrir nemendum a\u00f0 \u00e1tta sig \u00e1 er a\u00f0 jafna\u00f0armerki\u00f0 er merki um j\u00f6fnu\u00f0 en ekki bo\u00f0 um a\u00f0 \u201ereikna \u00fat\u201c. \u00deetta er eitt h\u00e6fnivi\u00f0mi\u00f0i\u00f0 \u00ed sameiginlegu kjarnavi\u00f0mi\u00f0unum sem gefin eru \u00fat \u00ed Bandar\u00edkjunum (Common Core Standards <a href=\"https:\/\/www.thecorestandards.org\/Math\/Content\/1\/OA\/D\/7\/\">Math.Content.1.OA.D.7<\/a>). \u00cd verkefninu m\u00e1 l\u00edka sj\u00e1 kr\u00f6fu um algebrulega hugsun \u00fev\u00ed nemendur ver\u00f0a a\u00f0 \u00edmynda s\u00e9r t\u00f3m s\u00e6ti \u00fear sem geta komi\u00f0 \u00f3l\u00edkar t\u00f6lur. \u00c1 algebrum\u00e1li m\u00e1 \u00fev\u00ed au\u00f0vita\u00f0 segja a\u00f0 vi\u00f0 s\u00e9um a\u00f0 leita lausna \u00e1 j\u00f6fnunni a = b + c = d + e + f \u00fear sem allir b\u00f3kstafir t\u00e1kni \u00f3l\u00edkar t\u00f6lur \u00far mengi heilla talna fr\u00e1 1 upp \u00ed 9. En \u00fea\u00f0 \u00fearf ekki a\u00f0 segja b\u00f6rnum \u00ed fyrsta bekk fr\u00e1 \u00fev\u00ed!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verkefni\u00f0 er af tegundinni <em>Opin mi\u00f0ja<\/em> (e. open middle) sem \u00fe\u00fd\u00f0ir:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Byrjunin er \u201eloku\u00f0\u201c vegna \u00feess a\u00f0 \u00fea\u00f0 er ekkert frelsi til a\u00f0 velja hva\u00f0a spurningum er svara\u00f0.<\/li>\n\n\n\n<li>Svari\u00f0 er l\u00edka \u201eloka\u00f0\u201c vegna \u00feess a\u00f0 \u00fea\u00f0 er einfaldlega r\u00e9tt e\u00f0a rangt.<\/li>\n\n\n\n<li>Mi\u00f0jan er \u201eopin\u201c vegna \u00feess a\u00f0 \u00fea\u00f0 er h\u00e6gt a\u00f0 velja mismunandi m\u00f6guleika.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Verkefni\u00f0 b\u00fd\u00f0ur upp \u00e1 meira en a\u00f0 f\u00e1 nemendur til a\u00f0 reikna og finna r\u00e9tt sv\u00f6r. Tilgangurinn me\u00f0 opinni mi\u00f0ju er a\u00f0 sameina b\u00e6\u00f0i reikni\u00e6fingar og st\u00e6r\u00f0fr\u00e6\u00f0ilega hugsun. St\u00e6r\u00f0fr\u00e6\u00f0ilega hugsun skilgreini \u00e9g h\u00e9r sem hugsun sem er \u00ed samhengi og r\u00f6\u00f0, e\u00f0a n\u00e1nar tilteki\u00f0, hugsun sem er ke\u00f0ja af r\u00f6kr\u00e9ttum \u00e1lyktunum \u00fat fr\u00e1 st\u00e6r\u00f0fr\u00e6\u00f0ilegum eiginleikum hugtaka (\u00feessi skilgreining er bygg\u00f0 \u00e1 skilgreiningu Hung-Hsi Wu \u00ed greininni <a href=\"https:\/\/doi.org\/10.7916\/jmetc.v3i1.729\">Assessment for the Common Core Mathematics Standards<\/a>). Nemandi getur til d\u00e6mis hugsa\u00f0 um verkefni\u00f0 svona:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Greinilega geta t\u00f6lurnar 1, 2, og 3 ekki veri\u00f0 \u00ed s\u00e6tinu lengst til vinstri af \u00fev\u00ed a\u00f0 \u00fe\u00e6r eru of litlar og ef vi\u00f0 hugsum um minnstu m\u00f6gulegu \u00feriggja talna summuna 1 + 2 + 3 = 6, sj\u00e1um vi\u00f0 a\u00f0 minnsta hugsanlega talan \u00ed s\u00e6tinu lengst til vinstri er 7. \u00dea\u00f0 \u00fe\u00fd\u00f0ir a\u00f0 einu m\u00f6guleikarnir \u00e1 t\u00f6lu \u00ed vinstra s\u00e6tinu eru t\u00f6lurnar 7, 8 e\u00f0a 9. \u00dea\u00f0 er l\u00edka lj\u00f3st a\u00f0 h\u00e6sta m\u00f6gulega talan \u00ed \u00feriggja talna summunni er 6 \u00fev\u00ed l\u00e6gstu t\u00f6lurnar \u00e1 m\u00f3ti eru 1 og 2 og 1 + 2 + 6 = 9. Pr\u00f3fum a\u00f0 f\u00e1 \u00fat 9 fyrir st\u00f6ku t\u00f6luna. \u00c9g hugsa m\u00e9r a\u00f0 pr\u00f3fa a\u00f0 nota 1 + 8 fyrir summu tveggja talna og \u00fe\u00e1 vantar mig \u00ferj\u00e1r t\u00f6lur til a\u00f0 f\u00e1 \u00fat 9, sem \u00e9g finn \u00fat me\u00f0 sm\u00e1 pr\u00f3fun a\u00f0 geti veri\u00f0 2 + 3 + 4 (\u00e9g nota talnaskyn mitt til a\u00f0 athuga \u00feessar t\u00f6lur \u00fev\u00ed \u00fe\u00e6r mega ekki vera of st\u00f3rar). Svo ein lausnin er 9 = 1 + 8 = 2 + 3 + 4.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">N\u00e6sta spurning er \u00fe\u00e1 hvort \u00feetta s\u00e9 eini m\u00f6guleikinn \u00e1 lausn og s\u00ed\u00f0an hve margir eru m\u00f6guleikarnir? H\u00e9r er t\u00e6kif\u00e6ri fyrir nemendur a\u00f0 \u00e1tta sig \u00e1 gagnsemi \u00feess a\u00f0 skr\u00e1 m\u00f6guleikana skipulega. Ni\u00f0ursta\u00f0an g\u00e6ti veri\u00f0 tafla \u00ed \u00feessum anda:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"625\" src=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.23.54-1024x625.png\" alt=\"\" class=\"wp-image-6680\" style=\"aspect-ratio:1.6384249664756796;width:469px;height:auto\" srcset=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.23.54-1024x625.png 1024w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.23.54-300x183.png 300w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.23.54-768x469.png 768w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.23.54-1536x937.png 1536w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.23.54-850x519.png 850w, https:\/\/flatarmal.is\/wp-content\/uploads\/2026\/06\/Screenshot-2026-06-16-at-18.23.54.png 1544w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<div style=\"height:29px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Taki\u00f0 eftir \u00fev\u00ed a\u00f0 h\u00e9r er skipulag \u00e1 skr\u00e1setningunni. Til d\u00e6mis eru t\u00f6lur skr\u00e1\u00f0ar \u00ed vaxandi r\u00f6\u00f0 \u00ed hverri summu og \u00ed d\u00e1lki tv\u00f6 breytist fyrri li\u00f0urinn \u00e1 skipulegan h\u00e1tt. \u00deetta hj\u00e1lpar okkur a\u00f0 sj\u00e1 hven\u00e6r vi\u00f0 h\u00f6fum fundi\u00f0 alla m\u00f6guleika. Vi\u00f0 t\u00f6kum svo eftir \u00fev\u00ed a\u00f0 8 er minnsta m\u00f6gulega summan og \u00fe\u00e1 var a\u00f0eins einn m\u00f6guleiki \u00e1 a\u00f0 setja saman j\u00f6fnur \u00e1 forminu sem vi\u00f0 vildum, en svo voru \u00fer\u00edr m\u00f6guleikar ef summan er 9, og \u00fe\u00e1 hugsum vi\u00f0 sem st\u00e6r\u00f0fr\u00e6\u00f0ingar: Hva\u00f0 ef vi\u00f0 leyfum summur upp \u00ed 10? E\u00f0a 11 e\u00f0a h\u00e6rri t\u00f6lur? Einnig er h\u00e6gt a\u00f0 velta fyrir s\u00e9r atri\u00f0um eins og hvort vi\u00f0 eigum a\u00f0 telja m\u00f6guleika \u00f3l\u00edka ef r\u00f6\u00f0 li\u00f0a er \u00f3l\u00edk (\u00fe\u00e1 v\u00e6ri til d\u00e6mis 1 + 8 talin \u00f3l\u00edk 8 + 1), og \u00fe\u00e1 f\u00e1st au\u00f0vita\u00f0 a\u00f0rar ni\u00f0urst\u00f6\u00f0ur um fj\u00f6lda m\u00f6guleika \u00e1 endanum &#8211; sem er \u00e1hugaver\u00f0 og krefjandi spurning fyrir nemendur \u00ed fyrsta bekk.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color wp-elements-2\" style=\"color:#2ebbcb\"><strong>Reiknihugsun \u00ed st\u00e6r\u00f0fr\u00e6\u00f0iranns\u00f3kn<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u00deegar \u00e9g setti verkefni\u00f0 inn \u00ed Facebook-h\u00f3pinn lei\u00f0 ekki langur t\u00edmi \u00fear til einn \u00fe\u00e1tttakandi haf\u00f0i beitt reiknihugsun sinni til a\u00f0 skrifa forrit til a\u00f0 finna lausnirnar. Forriti\u00f0 hans var skrifa\u00f0 \u00ed Python forritunarm\u00e1linu en \u00e9g \u00edmynda m\u00e9r a\u00f0 hann hafi hugsa\u00f0 sem svo, gr\u00f3flega:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. \u00c9g l\u00e6t t\u00f6lvuna b\u00faa til alla m\u00f6guleika \u00e1 a\u00f0 setja inn t\u00f6lur \u00e1 bilinu 1 &#8211; 9 \u00ed sex h\u00f3lf (breytist\u00e6r\u00f0ir).<br><br>2. Fyrir hvern einasta m\u00f6guleika l\u00e6t \u00e9g t\u00f6lvuna athuga:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>hvort j\u00f6fnurnar s\u00e9u sannar (fyrsta talan j\u00f6fn summu n\u00e6stu tveggja og j\u00f6fn summu n\u00e6stu \u00feriggja \u00fear \u00e1 eftir)&nbsp;<\/li>\n\n\n\n<li>hvort li\u00f0irnir \u00ed summunum s\u00e9u \u00ed vaxandi r\u00f6\u00f0 (til \u00feess a\u00f0 vi\u00f0 teljum hvern m\u00f6guleika bara einu sinni en ekki oft, vi\u00f0 viljum a\u00f0 1 + 8 s\u00e9 talinn en ekki 8 + 1).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Ef vi\u00f0 f\u00e6rum okkur skrefi n\u00e6r \u00fev\u00ed a\u00f0 b\u00faa til forrit fyrir t\u00f6lvu getum vi\u00f0 sett saman eftirfarandi reiknirit (alg\u00f3ritma):&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. \u00c9g l\u00e6t t\u00f6lvuna pr\u00f3fa alla m\u00f6guleika \u00e1 a\u00f0 setja inn t\u00f6lur \u00e1 bilinu 1 &#8211; 9 \u00ed sex breytist\u00e6r\u00f0ir sem h\u00e9r ver\u00f0a nefndar p[0], p[1], &#8230; , p[5]. (\u00dea\u00f0 er einfaldara fyrir t\u00f6lvu a\u00f0 vinna me\u00f0 breytist\u00e6r\u00f0ir \u00ed n\u00fameru\u00f0um lista heldur en a\u00f0 nota b\u00f3kstafina a, b, &#8230; , f.)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2. Fyrir hverja ni\u00f0urr\u00f6\u00f0un \u00e1 t\u00f6lum athuga \u00e9g hvort eftirfarandi skilyr\u00f0i eru uppfyllt:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>p[0] = p[1] + p[2] og p[0] = p[3] + p[4] + p[5] (athuga hvort jafnan er s\u00f6nn)<\/li>\n\n\n\n<li>p[1] &lt; p[2] og p[3] &lt; p[4] og p[4] &lt; p[5] (athuga hvort r\u00f6\u00f0in \u00e1 li\u00f0unum s\u00e9 vaxandi)<\/li>\n\n\n\n<li>ef ein tiltekin ni\u00f0urr\u00f6\u00f0un \u00e1 t\u00f6lum uppfyllir b\u00e6\u00f0i skilyr\u00f0in \u00fe\u00e1 hef \u00e9g fundi\u00f0 lausn<\/li>\n\n\n\n<li>skr\u00e1i lausnina.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">H\u00e9r m\u00e1 svo sj\u00e1 forriti\u00f0 sj\u00e1lft, af Facebook spjallinu:<\/p>\n\n\n\n<p class=\"superb-addons-has-padding-top-desktop superb-addons-has-padding-right-desktop superb-addons-has-padding-bottom-desktop superb-addons-has-padding-left-desktop wp-block-paragraph\" style=\"--spbadd-padding-top:0; --spbadd-padding-right:var(--wp--preset--spacing--50); --spbadd-padding-bottom:0; --spbadd-padding-left:var(--wp--preset--spacing--50);\">from itertools import permutations<\/p>\n\n\n\n<p class=\"superb-addons-has-padding-right-desktop superb-addons-has-padding-left-desktop wp-block-paragraph\" style=\"--spbadd-padding-right:var(--wp--preset--spacing--50); --spbadd-padding-left:var(--wp--preset--spacing--50);\">perms = permutations(range(1,10), 6)<\/p>\n\n\n\n<p class=\"superb-addons-has-padding-right-desktop superb-addons-has-padding-left-desktop wp-block-paragraph\" style=\"--spbadd-padding-right:var(--wp--preset--spacing--50); --spbadd-padding-left:var(--wp--preset--spacing--50);\">for p in perms:<\/p>\n\n\n\n<p class=\"superb-addons-has-padding-right-desktop superb-addons-has-padding-left-desktop wp-block-paragraph\" style=\"--spbadd-padding-right:var(--wp--preset--spacing--80); --spbadd-padding-left:var(--wp--preset--spacing--80);\">if (p[0] == p[1] + p[2]) and (p[1] + p[2] == p[3]+p[4]+p[5]) and (p[1] &lt; p[2]) and (p[3] &lt; p[4]) and (p[4] &lt; p[5]):<\/p>\n\n\n\n<p class=\"superb-addons-has-padding-right-desktop superb-addons-has-padding-left-desktop wp-block-paragraph\" style=\"--spbadd-padding-right:var(--wp--preset--spacing--80); --spbadd-padding-left:var(--wp--preset--spacing--80);\">print(&#8216;Lausn: &#8216;+ str(p[0]) + &#8216;=&#8217; +str(p[1])+ &#8216;+&#8217; +str(p[2])+ &#8216;=&#8217; +str(p[3])+ &#8216;+&#8217; +str(p[4])+ &#8216;+&#8217; +str(p[5]))<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Til sk\u00fdringar m\u00e1 nefna a\u00f0 h\u00e9r hefur forritarinn n\u00fdtt s\u00e9r fall (sem heitir permutations) til a\u00f0 skila lista af \u00f6llum umr\u00f6\u00f0unum sex staka \u00far menginu sem inniheldur t\u00f6lurnar 1, &#8230; , 9. Um \u00f6nnur atri\u00f0i vi\u00f0 Python forritun og \u00fea\u00f0 hvernig permutation falli\u00f0 virkar ver\u00f0a lesendur a\u00f0 kynna s\u00e9r \u00e1 eigin sp\u00fdtur ef \u00feeir hafa \u00e1huga. R\u00e9tt er a\u00f0 taka fram a\u00f0 h\u00e6gt er a\u00f0 b\u00faa til \u00f6nnur reiknirit sem fara a\u00f0rar lei\u00f0ir vi\u00f0 a\u00f0 framlei\u00f0a m\u00f6guleikana.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color wp-elements-3\" style=\"color:#2ebbcb\"><strong>N\u00fdjar ni\u00f0urst\u00f6\u00f0ur \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Verkefni\u00f0 sem h\u00e9r var fyrst kynnt getur sta\u00f0i\u00f0 eitt og s\u00e9r sem l\u00edtil st\u00e6r\u00f0fr\u00e6\u00f0iranns\u00f3kn fyrir unga nemendur. Eldri nemendur geta einnig fengist vi\u00f0 \u00fea\u00f0 me\u00f0 \u00fev\u00ed a\u00f0 beita reiknihugsun og t\u00f6lvuforritun. En \u00fea\u00f0 m\u00e1 l\u00edka halda \u00e1fram og \u00fatv\u00edkka spurninguna (eins og nefnt var \u00ed fyrsta hluta greinarinnar). Vi\u00f0 getum velt fyrir okkur hva\u00f0 gerist ef vi\u00f0 breytum einhverjum eiginleikum verkefnisins. \u00c9g hef \u00e1\u00f0ur l\u00fdst \u00feessu ferli vi\u00f0 a\u00f0 \u00fatv\u00edkka spurningar \u00ed greinunum <a href=\"https:\/\/flatarmal.is\/?p=270\">Um verkefni\u00f0 V\u00e9lmenni \u00e1 talnal\u00ednu<\/a> og <a href=\"https:\/\/flatarmal.is\/?p=4652\">Talnamismunur og tugakerfi<\/a>. Ein \u00fatv\u00edkku\u00f0 spurning er: hva\u00f0 ef vi\u00f0 takm\u00f6rkum okkur ekki vi\u00f0 t\u00f6lustafina 1 \u2013 9 heldur spyrjum almennt: <em>hve margar lausnir eru \u00e1 j\u00f6fnunni fyrir t\u00f6lur \u00e1 bilinu fr\u00e1 1 upp \u00ed hva\u00f0a t\u00f6lu sem er<\/em>. Ef vi\u00f0 k\u00f6llum \u00fe\u00e1 t\u00f6lu <em>n<\/em>, \u00fe\u00e1 var s\u00fdnt \u00ed fyrsta hluta greinarinnar a\u00f0 fyrir <em>n <\/em>= 8 er einn svarm\u00f6guleiki og fyrir <em>n <\/em>= 9 eru fj\u00f3rir svarm\u00f6guleikar, og fyrir <em>n <\/em>&lt; 8 er enginn svarm\u00f6guleiki. \u00dea\u00f0 \u00e6tti a\u00f0 vera h\u00e6gt a\u00f0 sty\u00f0ja vi\u00f0&nbsp; nemendur \u00ed efstu bekkjum grunnsk\u00f3la sem hafa fengist vi\u00f0 reiknihugsun og forritun a\u00f0 kanna \u00feetta m\u00e1l me\u00f0 forritun.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Verkefni sem vi\u00f0 vinnum me\u00f0 nemendum \u00ed fyrsta bekk eru sjaldan \u00f3leystar spurningar \u00ed st\u00e6r\u00f0fr\u00e6\u00f0i, spurningar sem enginn hefur \u00e1\u00f0ur svara\u00f0. En \u00fatv\u00edkka\u00f0a spurningin okkar er einmitt sl\u00edk spurning! Eftir vi\u00f0br\u00f6g\u00f0in \u00e1 Facebook setti \u00e9g nefnilega saman l\u00edti\u00f0 forrit til \u00feess a\u00f0 kanna h\u00e6rri t\u00f6lur (\u00fea\u00f0 er a\u00f0 segja a\u00f0 leyfa ekki bara t\u00f6lurnar 1 \u2013 9 heldur fr\u00e1 1 upp \u00ed h\u00e6rri t\u00f6lur en 9). \u00cd lj\u00f3s kom a\u00f0 runan byrjar svona (fr\u00e1 og me\u00f0 fj\u00f6lda m\u00f6guleika a\u00f0 skrifa 8 sem summu af \u00feessu tagi): 1, 4, 10, 20, 36, 60, 93, 141, 200. \u00de\u00e1 spur\u00f0i \u00e9g mig l\u00edka: skyldi einhver hafa sko\u00f0a\u00f0 \u00feetta \u00e1\u00f0ur? \u00c9g leit \u00fev\u00ed inn \u00e1 s\u00ed\u00f0u \u00e1 netinu sem heldur utan um \u00feekktar heilt\u00f6lurunur, <a href=\"https:\/\/oeis.org\/\">The On-Line Encyclopedia of Integer Sequences (OEIS)<\/a>. \u00dear var rununa ekki a\u00f0 finna! Svo \u00e9g sendi hana inn sem till\u00f6gu a\u00f0 n\u00fdrri runu og eftir nokkrar athugasemdir f\u00e9kkst h\u00fan sam\u00feykkt. H\u00fan er n\u00fana <a href=\"https:\/\/oeis.org\/A362717\">runa n\u00famer 362717<\/a> \u00ed runusafninu, og er l\u00fdst svona:<\/p>\n\n\n\n<p class=\"superb-addons-has-padding-right-desktop superb-addons-has-padding-left-desktop wp-block-paragraph\" style=\"--spbadd-padding-right:var(--wp--preset--spacing--50); --spbadd-padding-left:var(--wp--preset--spacing--50);\">Fj\u00f6ldi lei\u00f0a til a\u00f0 skrifa <em>a<\/em> + <em>b<\/em> + <em>c<\/em> = <em>d<\/em> + <em>e<\/em> = <em>f<\/em>  me\u00f0 {<em>a<\/em>, <em>b<\/em>, <em>c<\/em>, <em>d<\/em>, <em>e<\/em>, <em>f<\/em>} sem hlutmengi \u00ed [<em>n<\/em>] af st\u00e6r\u00f0 6 og <em>a<\/em> &lt; <em>b<\/em> &lt; <em>c<\/em> og <em>d<\/em> &lt; <em>e<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u00cd \u00feessari skilgreiningu \u00fe\u00fd\u00f0ir [<em>n<\/em>] mengi heilu talnanna fr\u00e1 0 upp \u00ed <em>n<\/em>. \u00c1 s\u00ed\u00f0unni m\u00e1 einnig sj\u00e1 Python-forrit sem framlei\u00f0ir rununa sj\u00e1lfa, en h\u00fan byrjar \u00e1 a\u00f0 telja fr\u00e1 og me\u00f0 <em>n<\/em> = 0, svo runan byrjar \u00e1 \u00e1tta n\u00fallum en s\u00ed\u00f0an taka vi\u00f0 1, 4, 10, 20, og svo framvegis.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color wp-elements-4\" style=\"color:#2ebbcb\"><strong>Opin mi\u00f0ja: einfalt d\u00e6mi me\u00f0 m\u00f6guleikum til \u00fatv\u00edkkana<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Vi\u00f0 h\u00f6fum h\u00e9r liti\u00f0 \u00e1 eitt st\u00e6r\u00f0fr\u00e6\u00f0iverkefni sem sameinar b\u00e6\u00f0i reikning og ranns\u00f3kn. Reikni\u00e6fingin er falin inni \u00ed st\u00e6rri ranns\u00f3kn, \u00fear sem nemendur \u00feurfa a\u00f0 reikna, pr\u00f3fa og skr\u00e1, en reikningurinn \u00fej\u00f3nar st\u00e6rra markmi\u00f0i. Fyrir unga nemendur er krefjandi a\u00f0 reikna \u00fat margar summur, a\u00f0 finna fleiri og fleiri \u00f3l\u00edka m\u00f6guleika til a\u00f0 pr\u00f3fa, a\u00f0 skr\u00e1 m\u00f6guleikana skipulega og a\u00f0 \u00fatsk\u00fdra hvernig \u00feeir vita a\u00f0 \u00feeir hafi fundi\u00f0 allar lausnir. Fyrir eldri e\u00f0a lengra komna nemendur m\u00e1 beina athyglinni a\u00f0 skipulegri talningu og reiknihugsun, a\u00f0 l\u00e1ta t\u00f6lvu finna og telja m\u00f6guleikana. \u00dea\u00f0 vildi svo til a\u00f0 \u00ed \u00feessu verkefni var h\u00e6gt a\u00f0 b\u00e6ta talnarunu vi\u00f0 safn \u00feekktra talnaruna sem s\u00fdnir a\u00f0 stundum er h\u00e6gt a\u00f0 skapa alveg n\u00fdja st\u00e6r\u00f0fr\u00e6\u00f0i me\u00f0 \u00fev\u00ed a\u00f0 sko\u00f0a eiginleika verkefna og breyta \u00feeim.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-media-text is-stacked-on-mobile\" style=\"grid-template-columns:38% auto\"><figure class=\"wp-block-media-text__media\"><img loading=\"lazy\" decoding=\"async\" width=\"510\" height=\"380\" src=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2025\/02\/Ingolfur.jpg\" alt=\"\" class=\"wp-image-4707 size-full\" srcset=\"https:\/\/flatarmal.is\/wp-content\/uploads\/2025\/02\/Ingolfur.jpg 510w, https:\/\/flatarmal.is\/wp-content\/uploads\/2025\/02\/Ingolfur-300x224.jpg 300w\" sizes=\"auto, (max-width: 510px) 100vw, 510px\" \/><\/figure><div class=\"wp-block-media-text__content\">\n<p class=\"wp-block-paragraph\">I<strong><strong>ng\u00f3lfur G\u00edslason<\/strong>, lektor \u00ed st\u00e6r\u00f0fr\u00e6\u00f0imenntun vi\u00f0 Menntav\u00edsindasvi\u00f0 H\u00e1sk\u00f3la \u00cdslands.<\/strong><\/p>\n<\/div><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ing\u00f3lfur G\u00edslason. Opin mi\u00f0ja Vi\u00f0 byrjum \u00e1 st\u00e6r\u00f0fr\u00e6\u00f0ispurningu sem \u00e9g setti inn \u00e1 Facebook-h\u00f3pinn St\u00e6r\u00f0fr\u00e6\u00f0ikennarinn \u00ed febr\u00faar 2019. Verkefni\u00f0 fann \u00e9g \u00e1 vefs\u00ed\u00f0unni Open Middle \u00fear sem \u00fea\u00f0 er sagt henta nemendum \u00ed fyrsta bekk \u00ed Bandar\u00edkjunum (sem \u00e6tti a\u00f0 vera mj\u00f6g samb\u00e6rilegur okkar fyrsta bekk). \u00deegar og ef verkefni\u00f0 er lagt fyrir nemendur \u00ed&#8230;<\/p>\n","protected":false},"author":1,"featured_media":6725,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[61,220,215,214,216,156,221,218,21,96,52,23,219,217],"class_list":["post-6676","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","tag-ad-opna-og-utvikka-verkefni","tag-heiltolurunur","tag-jafnadarmerki","tag-opin-midja","tag-reikniaefing","tag-reiknihugsun","tag-runa","tag-skipulogd-skraning","tag-staerdfraedi","tag-staerdfraedileg-hugsun","tag-staerdfraedinam","tag-staerdfraedirannsokn","tag-talnaruna","tag-tolvuforritun"],"_links":{"self":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/6676","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=6676"}],"version-history":[{"count":37,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/6676\/revisions"}],"predecessor-version":[{"id":6751,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/posts\/6676\/revisions\/6751"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=\/wp\/v2\/media\/6725"}],"wp:attachment":[{"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6676"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6676"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/flatarmal.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6676"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}