{"created":"2025-05-30T07:41:16.323984+00:00","id":2000122,"links":{},"metadata":{"_buckets":{"deposit":"06413692-dd9a-4c68-9f6e-04131d06e674"},"_deposit":{"created_by":8,"id":"2000122","owner":"8","owners":[8],"pid":{"revision_id":0,"type":"depid","value":"2000122"},"status":"published"},"_oai":{"id":"oai:socu.repo.nii.ac.jp:02000122","sets":["1623632832836:1735019262973"]},"author_link":[],"item_30001_bibliographic_information17":{"attribute_name":"bibliographic_information","attribute_value_mlt":[{"bibliographicIssueDates":{"bibliographicIssueDate":"2025-03-31","bibliographicIssueDateType":"Issued"},"bibliographicNumberOfPages":"6","bibliographicPageEnd":"98","bibliographicPageStart":"93","bibliographicVolumeNumber":"8","bibliographic_titles":[{"bibliographic_title":"山陽小野田市立山口東京理科大学紀要","bibliographic_titleLang":"ja"},{"bibliographic_title":"Bulletin of Sanyo-Onoda City University","bibliographic_titleLang":"en"}]}]},"item_30001_creator2":{"attribute_name":"著者","attribute_type":"creator","attribute_value_mlt":[{"creatorAffiliations":[{"affiliationNames":[{"affiliationName":"山陽小野田市立山口東京理科大学","affiliationNameLang":"ja"}]}],"creatorNames":[{"creatorName":"田中,俊光","creatorNameLang":"ja"},{"creatorName":"Toshimitsu,TANAKA","creatorNameLang":"en"}]}]},"item_30001_description8":{"attribute_name":"内容記述","attribute_value_mlt":[{"subitem_description":"３次元の図形「多面体」の点の数をＴ、辺の数をＨ、面の数をＭと表すと、Ｔ－Ｈ＋Ｍ＝２が成り立つ。これがオイラーの多面体定理である。この定理は、多面体のＴ、Ｈ、Ｍを数えることで比較的容易に見つけることができる。２次元の図形「多角形」では、Ｔ－Ｈ＋Ｍ＝１が成り立つ。これをオイラーの多角形定理と呼ぼう。多角形のＴ、Ｈ、Ｍの関係については、図形が単純すぎてＴ＝Ｈ、Ｍ＝１に目をうばわれ、Ｔ－Ｈ＋Ｍ＝１を見つけるのが難しい。そこで、図形を多少複雑にするために「仕切り線」を入れてＴ＝Ｈ、Ｍ＝１を意図的にくずすことで見つけることができる。ここで、２次元のオイラーの多角形定理Ｔ－Ｈ＋Ｍ＝１と３次元のオイラーの多面体定理Ｔ－Ｈ＋Ｍ＝２を比較すると、４次元の図形「多胞体」ではＴ－Ｈ＋Ｍ＝３になりそうだと予測できる。\n　ところが、２次元の図形「多角形」に「仕切り線」を入れたように、３次元の図形「多面体」に「仕切り面」を入れてＴ、Ｈ、Ｍを数えると、Ｔ－Ｈ＋Ｍ＝２が成り立たなくなる。真のオイラーの多面体定理は、胞の数をHoとしてＴ－Ｈ＋Ｍ－Ho＝１なのである。\nここで再び、２次元のオイラーの多角形定理Ｔ－Ｈ＋Ｍ＝１と３次元のオイラーの多面体定理Ｔ－Ｈ＋Ｍ－Ho＝１を比較すると、４次元の図形「多胞体」では、胞で囲まれた４次元の小部屋の数を？と表すとＴ－Ｈ＋Ｍ－Ho＋？＝１になりそうだと予測できる。図形の構成要素を次元の順に並べているので、＋と－が交互に出てきて、次元が１つ上がるたびに左辺の項が１つずつ増えているのである。\n　本稿では、研究仮説を「生徒の思考に沿った授業づくりレベルで各次元のオイラーの図形定理を考察することで、次元を超えた図形概念を育てるための手だてを一般化できる」とする。中学２年の授業「次元を超える－図形の点・辺・面…の数－」（２時間計画）をつくる過程をもとに、「３次元のオイラーの多面体定理を見つける」→「『仕切り線』を入れて２次元のオイラーの多角形定理を見つける」→「『仕切り面』を入れて考察し、３次元のオイラーの多面体定理を修正する」→「４次元・５次元のオイラーの図形定理を予測する」という授業の流れと、「次元の順に整理する」「比較しやすいように学習プリントを工夫する」の２つをベースに「２次元の図形と３次元の図形で同様のことを行わせる」→「規則をもとに４次元・５次元の図形をイメージさせる」→「一度次元を下げてデータを増やす」という手だてについて述べていく。","subitem_description_language":"ja","subitem_description_type":"Abstract"}]},"item_30001_file22":{"attribute_name":"ファイル情報","attribute_type":"file","attribute_value_mlt":[{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2025-03-31"}],"filename":"SU10008000012.pdf","filesize":[{"value":"2.1 MB"}],"format":"application/pdf","url":{"url":"https://socu.repo.nii.ac.jp/record/2000122/files/SU10008000012.pdf"},"version_id":"c949be50-5b0e-439a-b21e-850b792bf0b9"}]},"item_30001_language10":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"jpn"}]},"item_30001_publisher9":{"attribute_name":"出版者","attribute_value_mlt":[{"subitem_publisher":"山陽小野田市立山口東京理科大学","subitem_publisher_language":"ja"},{"subitem_publisher":"Sanyo-Onoda City University","subitem_publisher_language":"en"}]},"item_30001_resource_type11":{"attribute_name":"item_30001_resource_type11","attribute_value_mlt":[{"resourcetype":"departmental bulletin paper","resourceuri":"http://purl.org/coar/resource_type/c_6501"}]},"item_30001_subject7":{"attribute_name":"主題","attribute_value_mlt":[{"subitem_subject":"mathematics education","subitem_subject_language":"en","subitem_subject_scheme":"Other"},{"subitem_subject":"dimension","subitem_subject_language":"en","subitem_subject_scheme":"Other"},{"subitem_subject":"Euler’s polyhedron theorem","subitem_subject_language":"en","subitem_subject_scheme":"Other"},{"subitem_subject":"partition line","subitem_subject_language":"en","subitem_subject_scheme":"Other"},{"subitem_subject":"partition surface","subitem_subject_language":"en","subitem_subject_scheme":"Other"},{"subitem_subject":"数学教育","subitem_subject_language":"ja","subitem_subject_scheme":"Other"},{"subitem_subject":"次元","subitem_subject_language":"ja","subitem_subject_scheme":"Other"},{"subitem_subject":"オイラーの多面体定理","subitem_subject_language":"ja","subitem_subject_scheme":"Other"},{"subitem_subject":"仕切り線","subitem_subject_language":"ja","subitem_subject_scheme":"Other"},{"subitem_subject":"仕切り面","subitem_subject_language":"ja","subitem_subject_scheme":"Other"}]},"item_30001_title0":{"attribute_name":"タイトル","attribute_value_mlt":[{"subitem_title":"次元の考え方をもとにした図形概念を育てる数学教育 －オイラーの多面体定理の発展１－","subitem_title_language":"ja"},{"subitem_title":"Mathematics Education That Nurtures Geometric Concepts  Based on The Concept of Dimensions －Development of Euler’s Polyhedron Theorem １－","subitem_title_language":"en"}]},"item_title":"次元の考え方をもとにした図形概念を育てる数学教育 －オイラーの多面体定理の発展１－","item_type_id":"40001","owner":"8","path":["1735019262973"],"pubdate":{"attribute_name":"PubDate","attribute_value":"2025-03-31"},"publish_date":"2025-03-31","publish_status":"0","recid":"2000122","relation_version_is_last":true,"title":["次元の考え方をもとにした図形概念を育てる数学教育 －オイラーの多面体定理の発展１－"],"weko_creator_id":"8","weko_shared_id":-1},"updated":"2025-05-30T07:50:25.159278+00:00"}