{"id":4652,"date":"2025-02-22T15:13:10","date_gmt":"2025-02-22T15:13:10","guid":{"rendered":"https:\/\/flatarmal.is\/?p=4652"},"modified":"2025-02-26T14:50:07","modified_gmt":"2025-02-26T14:50:07","slug":"talnamismunur-og-tugakerfi","status":"publish","type":"post","link":"https:\/\/flatarmal.is\/?p=4652","title":{"rendered":"Talnamismunur og tugakerfi"},"content":{"rendered":"\n

Ing\u00f3lfur G\u00edslason.<\/strong><\/p>\n\n\n\n

St\u00e6r\u00f0fr\u00e6\u00f0ikennslub\u00e6kur eru fullar af \u00e6fingaverkefnum. En nemendur \u00feurfa meira en a\u00f0 gera \u00e6fingar \u2013 \u00feeir \u00feurfa l\u00edka a\u00f0 gl\u00edma vi\u00f0 krefjandi st\u00e6r\u00f0fr\u00e6\u00f0iverkefni. Ein lei\u00f0 til a\u00f0 \u00fatb\u00faa g\u00f3\u00f0 verkefni er a\u00f0 taka venjuleg verkefni \u00far kennslub\u00f3kum og v\u00edkka \u00feau \u00fat \u00feannig a\u00f0 \u00feau s\u00e9u ekki einungis \u00e6fingar heldur kalli l\u00edka \u00e1 alh\u00e6fingar og \u00fear me\u00f0 \u00e1 st\u00e6r\u00f0fr\u00e6\u00f0ilega hugsun. \u00deegar \u00e9g leita\u00f0i a\u00f0 verkefnum um talnaritun \u00ed tugakerfinu \u00ed Stiku 1a<\/em><\/a> fann \u00e9g eftirfarandi verkefni \u00e1 bla\u00f0s\u00ed\u00f0u 13.<\/p>\n\n\n\n

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Mynd 1: d\u00e6mi 1.29 \u00e1 bls. 13 \u00ed Stiku 1a Nemendab\u00f3k<\/em><\/p>\n<\/div>\n\n\n\n

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Mynd 2: Stika 1a Nemendab\u00f3k<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

\u00cd verkefninu er l\u00edklega \u00e6tlast til \u00feess a\u00f0 nemendur finni \u00fat \u00far \u00fev\u00ed a\u00f0 bera saman t\u00f6lur eins og 456, 546, 645 og svo framvegis, og \u00e1tti sig a\u00f0 lokum \u00e1 \u00fev\u00ed a\u00f0 st\u00e6rsta talan s\u00e9 654 en minnsta talan s\u00e9 456. \u00de\u00e1 reikna \u00feau mismuninn sem er 654 \u2013 456 = 198. Tilgangur verkefnisins vir\u00f0ist \u00fev\u00ed vera a\u00f0 auka skilning nemenda \u00e1 tugakerfinu<\/em>. S\u00e9r \u00ed lagi, a\u00f0 gildi hvers t\u00f6lustafs er \u00f3l\u00edkt eftir \u00fev\u00ed hvar hann stendur<\/strong> (sj\u00e1 bl\u00e1a rammann undir verkefninu \u00e1 mynd 1, um t\u00f6luna 325.) \u00deegar \u00e9g las \u00feetta verkefni vakti \u00fea\u00f0 athygli m\u00edna a\u00f0 t\u00f6lurnar standa \u00ed r\u00f6\u00f0, hli\u00f0 vi\u00f0 hli\u00f0 \u00e1 talnal\u00ednunni: 4, 5, 6. M\u00e9r datt \u00ed hug hvort \u00fea\u00f0 hef\u00f0i einhverja s\u00e9rstaka \u00fe\u00fd\u00f0ingu og hvort \u00fea\u00f0 g\u00e6ti veri\u00f0 a\u00f0 a\u00f0rar \u00ferj\u00e1r t\u00f6lur \u00ed r\u00f6\u00f0 g\u00e6fu einhverjar samb\u00e6rilegar \u00fatkomur. \u00c9g sting upp \u00e1 \u00fev\u00ed a\u00f0 lesendur pr\u00f3fi \u00fea\u00f0. Og \u00fe\u00e1 fannst m\u00e9r eins og \u00ed verkefninu hlyti a\u00f0 felast upplagt t\u00e6kif\u00e6ri fyrir unga nemendur a\u00f0 sko\u00f0a regluleika, setja fram og pr\u00f3fa tilg\u00e1tur, alh\u00e6fa og r\u00f6ksty\u00f0ja. H\u00e9r \u00ed framhaldinu ver\u00f0ur \u00fev\u00ed l\u00fdst hvernig h\u00e6gt er a\u00f0 opna og \u00fatv\u00edkka verkefni\u00f0 enn frekar og gera \u00far \u00fev\u00ed ver\u00f0ugra verkefni, nokku\u00f0 dj\u00fapa st\u00e6r\u00f0fr\u00e6\u00f0ilega ranns\u00f3kn.<\/p>\n\n\n\n

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Eiginleikar<\/p>\n\n\n\n

\u00c9g byrja ferli\u00f0 \u00e1 \u00fev\u00ed a\u00f0 skr\u00e1: hverjir eru eiginleikar<\/em> verkefnisins? \u00deetta ger\u00f0i \u00e9g l\u00edka \u00feegar \u00e9g \u00fer\u00f3a\u00f0i verkefni\u00f0 V\u00e9lmenni \u00e1 talnal\u00ednu<\/a> \u00ed Flatarm\u00e1lum (2023). N\u00fa sem \u00fe\u00e1 eru \u00f3teljandi margir m\u00f6guleikar \u00e1 \u00fev\u00ed a\u00f0 nefna st\u00e6r\u00f0fr\u00e6\u00f0ilega eiginleika verkefnisins. Vi\u00f0 sko\u00f0um b\u00e6\u00f0i st\u00e6r\u00f0fr\u00e6\u00f0ilegu fyrirb\u00e6rin og venslin \u00ed verkefninu og \u00ed \u00e6tlu\u00f0u svari. M\u00f6rg st\u00e6r\u00f0fr\u00e6\u00f0iverkefni m\u00e1 t\u00falka \u00feannig a\u00f0 \u00fea\u00f0 s\u00e9 veri\u00f0 a\u00f0 leita a\u00f0 einhverjum st\u00e6r\u00f0fr\u00e6\u00f0ilegum hlut sem uppfyllir einhver st\u00e6r\u00f0fr\u00e6\u00f0ileg skilyr\u00f0i. Til d\u00e6mis, ef nemandinn \u00e1 a\u00f0 leggja saman 5 + 7, \u00fe\u00e1 er veri\u00f0 a\u00f0 leita a\u00f0 framsetningu t\u00f6lu \u00ed tugakerfinu sem er j\u00f6fn 5 + 7, 12. \u00cd verkefninu h\u00e9r er nemandinn be\u00f0inn (a\u00f0 lokum) a\u00f0 finna mismun tveggja talna sem uppfylla \u00e1kve\u00f0in skilyr\u00f0i. Greinum n\u00fa nokkra eiginleika verkefnisins me\u00f0 \u00fea\u00f0 fyrir augum a\u00f0 sko\u00f0a listann \u00e1 eftir og spyrja: hva\u00f0 ef ekki? <\/strong>Hva\u00f0 ef einhver eiginleiki er ekki svona? Hva\u00f0 ef vi\u00f0 breytum einum e\u00f0a fleiri eiginleikum?<\/p>\n\n\n\n

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  • Gefnar eru t\u00f6lur<\/strong><\/li>\n\n\n\n
  • T\u00f6lurnar eru 4<\/strong>, 5<\/strong> og 6<\/strong><\/li>\n\n\n\n
  • T\u00f6lurnar eru \u00ferj\u00e1r<\/strong> talsins<\/li>\n\n\n\n
  • T\u00f6lurnar eru heilar<\/strong><\/li>\n\n\n\n
  • T\u00f6lurnar eru j\u00e1kv\u00e6\u00f0ar<\/strong><\/li>\n\n\n\n
  • T\u00f6lurnar eru n\u00e1grannat\u00f6lur<\/strong> (\u00fe\u00e6r eru hli\u00f0 vi\u00f0 hli\u00f0 \u00e1 talnal\u00ednunni sem heilar t\u00f6lur)<\/li>\n\n\n\n
  • T\u00f6lurnar eru skrifa\u00f0ar me\u00f0 einum t\u00f6lustaf<\/strong><\/li>\n\n\n\n
  • Vi\u00f0 eigum a\u00f0 finna st\u00e6rstu og minnstu t\u00f6luna<\/strong> sem h\u00e6gt er a\u00f0 mynda (me\u00f0 \u00fev\u00ed a\u00f0 skrifa t\u00f6lu \u00ed tugakerfinu)<\/li>\n\n\n\n
  • Vi\u00f0 eigum a\u00f0 reikna mismun<\/strong><\/li>\n<\/ul>\n<\/div>\n<\/div>\n\n\n\n

    H\u00e6gt er a\u00f0 halda \u00e1fram og finna fleiri eiginleika. En h\u00e9r eru nokkrar hugmyndir um hvert m\u00e1 fara me\u00f0 verkefni\u00f0 me\u00f0 \u00fev\u00ed a\u00f0 spyrja: hva\u00f0 ef (ekki)?<\/strong><\/p>\n\n\n\n

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    • Hva\u00f0 ef t\u00f6lurnar v\u00e6ru ekki 4, 5 og 6, heldur 7, 8 og 9 e\u00f0a a\u00f0rar \u00ferj\u00e1r n\u00e1grannat\u00f6lur?<\/li>\n\n\n\n
    • Hva\u00f0 ef t\u00f6lurnar v\u00e6ru ekki \u00ferj\u00e1r talsins heldur tv\u00e6r? E\u00f0a fj\u00f3rar? E\u00f0a fleiri?<\/li>\n\n\n\n
    • Hva\u00f0 ef t\u00f6lurnar v\u00e6ru ekki n\u00e1grannat\u00f6lur heldur v\u00e6ri h\u00e6gt a\u00f0 ra\u00f0a \u00feeim me\u00f0 mismuninn 2 \u00e1 milli? (Eins og 1, 3, og 5, e\u00f0a 2, 4, og 6). E\u00f0a 3 \u00e1 milli?<\/li>\n\n\n\n
    • Hva\u00f0 ef t\u00f6lurnar v\u00e6ru ekki heilar heldur brot? Hva\u00f0 ef einhver tala v\u00e6ri neikv\u00e6\u00f0?<\/li>\n\n\n\n
    • Hva\u00f0 ef t\u00f6lurnar f\u00e6ru \u00fat fyrir tuginn (ekki lengur einn t\u00f6lustafur)?<\/li>\n\n\n\n
    • Hva\u00f0 ef vi\u00f0 reiknu\u00f0um summu en ekki mismun? Margfeldi? Me\u00f0altal?<\/li>\n<\/ul>\n<\/div>\n<\/div>\n\n\n\n

      H\u00e9r m\u00e6tti l\u00edka halda \u00e1fram. \u00cd framhaldinu sko\u00f0um vi\u00f0 a\u00f0allega fyrsta punktinn: hva\u00f0 ef vi\u00f0 f\u00e1um ekki \u00feessar tilteknu t\u00f6lur, heldur a\u00f0rar t\u00f6lur sem eru \u00e1 einhvern h\u00e1tt svipa\u00f0ar? Me\u00f0 \u00f6\u00f0rum or\u00f0um \u00fe\u00e1 pr\u00f3fum vi\u00f0 a\u00f0 nota a\u00f0rar t\u00f6lur me\u00f0 sem flesta s\u00f6mu eiginleika.<\/p>\n\n\n\n

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      Tilg\u00e1tur og alh\u00e6fingar<\/p>\n\n\n\n

      Verkefni\u00f0 \u00ed Stiku 1a skortir eitt mikilv\u00e6gt atri\u00f0i til a\u00f0 geta talist st\u00e6r\u00f0fr\u00e6\u00f0ilegt verkefni \u00ed raun og veru: t\u00e6kif\u00e6ri til \u00feess a\u00f0 alh\u00e6fa<\/strong>, a\u00f0 finna almenna reglu, fullyr\u00f0ingu sem er alltaf s\u00f6nn \u00feegar einhver tiltekin skilyr\u00f0i (eiginleikar) eru til sta\u00f0ar. Til \u00feess a\u00f0 sko\u00f0a hvort einhverjir m\u00f6guleikar eru til alh\u00e6fingar er upplagt a\u00f0 pr\u00f3fa a\u00f0 breyta d\u00e6minu l\u00edtillega. Til d\u00e6mis m\u00e1 reyna a\u00f0 halda sem flestum almennum eiginleikum \u00feess. Ef vi\u00f0 pr\u00f3fum a\u00f0rar t\u00f6lur sem uppfylla l\u00edka flesta (e\u00f0a alla) almenna eiginleika og f\u00e1um s\u00f6mu ni\u00f0urst\u00f6\u00f0u og me\u00f0 upphaflegu t\u00f6lunum \u00fe\u00e1 erum vi\u00f0 komin me\u00f0 efni \u00ed alh\u00e6fingu<\/strong>. \u00deess vegna pr\u00f3fum vi\u00f0 a\u00f0rar \u00ferj\u00e1r t\u00f6lur eins og 7, 8, og 9. \u00cd lj\u00f3s kemur a\u00f0 \u00fear er sami mismunur, 198. \u00deetta \u00e6tti a\u00f0 vekja upp spurninguna: er \u00feetta alltaf svona? Og, j\u00fa, pr\u00f3fun \u00e1 fleiri \u00feremur t\u00f6lum sem eru \u00ed svona r\u00f6\u00f0 gefur alltaf 198. \u00deess vegna setjum vi\u00f0 fram tilg\u00e1tu: mismunur st\u00e6rstu og minnstu \u00feriggja stafa t\u00f6lu sem myndu\u00f0 er \u00far \u00feremur n\u00e1grannat\u00f6lum er 198. <\/strong>\u00deessi tilg\u00e1ta l\u00fdsir alh\u00e6fingu. Fyrir nemendur \u00ed grunnsk\u00f3la m\u00e6tti \u00fev\u00ed b\u00e6ta eftirfarandi spurningu vi\u00f0 verkefni\u00f0:<\/p>\n\n\n\n

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