지오지브라와 함께하는 이차곡선 교수·학습

양성현

경희대학교문리과대학수학과
경희대학교교육대학원교육학과(수학교육)
성균관대학교교과교육학과수학교육학전공교육학박사

약력
경희고등학교교사
현한국교육과정평가원연구위원

논문학회학회지게재년도
GeoGebra를활용한역동적인시각적표상에기반한
이차곡선지도방안
대한수학교육학회학교수학2011
GeoGebra의구성단계기능을활용한고등학교
수학영재문제해결과정의창의성평가사례연구
한국영재학회영재교육연구2014
GeoGebra활용문항출제연수를통한
평가문항제작사례연구
한국수학교육학회수학교육논문집2015
GeoGebra를활용한각등변형습유의일반화한국수학사학회한국수학사학회지2016
GeoGebra를활용한연립이차방정식교수학습방안연구영남수학회영남수학회지2021
GeoGebra의네비게이션바와구성단계기능을활용한
과정중심평가가능성탐색
대한수학교육학회수학교육학연구2023

수상경력
제3회교육정보화연구대회(국무총리상표창)
-GeoGebra를활용한기하와대수의연결성에관한연구
제11회교실수업개선실천사례연구발표대회(서울시교육감표창)
-GeoGebra를활용한수학에대한자신감찾기프로젝트

Ⅰ.GeoGebra활용관련기초사항
1.GeoGebra초기화면설정·························································································
2.GeoGebra의대상에대한설정사항··········································································2
3.‘구성단계네비게이션바’와‘구성단계’기능··················································5
4.도구상자를이용한작도와명령어를이용한작도···················································7

Ⅱ.원뿔의단면과이심률
1.원뿔의단면················································································································11
2.포물선:원뿔을자르는평면과원뿔의밑면이이루는각이모선과원뿔의밑면이이루는각과같을때···················································································15
3.타원:원뿔을자르는평면과원뿔의밑면이이루는각이모선과원뿔의밑면이이루는각보다작을때···············································································23
4.쌍곡선:원뿔을자르는평면과원뿔의밑면이이루는각이모선과원뿔의밑면이이루는각보다클때···················································································30
5.원뿔곡선의이심률·····································································································38

Ⅲ.포물선의자취와성질
1.자취를이용한포물선의작도··················································································43
2.GeoGebra를활용하여포물선의성질확인하기···················································55

Ⅳ.타원의자취와성질
1.자취를이용한타원의작도······················································································95
2.GeoGebra를활용하여타원의성질확인하기·····················································121

Ⅴ.쌍곡선의자취와성질
1.자취를이용한쌍곡선의작도················································································157
2.GeoGebra를활용하여쌍곡선의성질확인하기·················································173

이차곡선은원뿔을절단해서얻을수있는곡선으로매우흥미롭고다양한성질들을내포하고있으며,고등학교선택중심교육과정의진로선택과목인〈기하〉에서다루어지고있다.2015개정수학과교육과정에서는이차곡선에대한실용성및유용성과가치를중요시하도록하였으며,이차곡선이원뿔의절단을통하여얻을수있는곡선임을인식하게하여기하적대상과대수적대상의수학적연결성을강조하여지도하도록하고있다.2022개정수학과교육과정에서도이차곡선과관련하여내용체계의핵심아이디어를통하여‘원뿔을절단하여만든곡선을방정식으로표현하는것은기하적성질을탐구하는데유용한방법이다.’라고명시하고있다.그러나교과서의구성은이차곡선이원뿔을절단해서얻을수있는곡선임을이해하기보다는원뿔을자른단면이‘포물선처럼,타원처럼,쌍곡선처럼’보이는곡선의형태를인지하는수준에그치고있다.물론원뿔을자르는평면과원뿔의밑면이이루는각에따라포물선,타원,쌍곡선이되는원리를이심률과함께세부적으로다루고,이차곡선의다양한성질을엄밀한증명과함께도입하는것또한학생들의학습부담이가중되기때문에용이치않을수있다.
본고는이차곡선에대하여두가지측면으로나누어구성하였다.첫째는원뿔의단면이이차곡선이되는원리를GeoGebra를이용하여구현하는방법과그원리를수록하였다.둘째는GeoGebra의자취기능을이용하여포물선,타원,쌍곡선을여러가지방법으로작도하고각각의성질을다양하게수록하였다.이차곡선의자취관련내용은상당부분교과서에서다루고있는내용에기반하고있으나,이차곡선의성질관련내용은교육과정을고려하여구성한내용은아니다.그러나본고에수록된내용은다양한방법으로활용이가능할수있다.학교상황에따라교사가재구성한다면수학동아리활동또는영재교육에서충분히활용할수있을것이다.학생들의수학에대한흥미와학습동기유발은수학학습에중요한요소이다.GeoGebra를활용하여기하적이고시각적인교수･학습방안을병렬적으로첨가하여지도하는것은학생들의시각적표상을증진시키고대수와기하의연결성을확고히해줌으로써대수와기하의다면적이해를촉진할수있다.GeoGebra를활용하여현실세계를시뮬레이션하는활동은학생들에게수학의실용성을증대시켜수학외적연결성을깨닫게할수있으며대수적으로국한되어있는지도의영역에기하적인요소를가미함으로써수학내적연결성을증대시켜학생들의다면적이해를촉진함과더불어학생스스로연결적사고를할수있게도와줄수있다.본고는단순히GeoGebra를이용하여독자에게이차곡선의자취와성질을시각적으로보여주는데그치지않고제작된도구를교수･학습에서활용할수있는방안을제시하고자하였다.
GeoGebra의‘2차원기하창’에서‘슬라이더’와‘자취’관련도구와명령어를활용하면이차곡선의자취와성질을간단하면서도역동적으로표현할수있다.필자가각각의이차곡선의성질에대하여간략하게제시한〈증명〉이최선의방법이아닐수있으며,GeoGebra를이용하여이차곡선의다양한성질을작도하는방법또한필자가작성한방법이외에도다양한방법이존재할것이다.교재에서다루어지는모든내용은교사들의접근성이용이하도록‘한글2022’와‘GeoGebraClassic5’만을사용하여제작하였다.또한교사와학생이함께만들어감으로써시각적효과를넘어조작적효과를증진시킬수있도록모든자료의제작방법을구성단계별로세부적으로제시하였다.예비교사와현직교사가본고를통하여수학학습소프트웨어를활용한이차곡선교수･학습의다양성을인식할수있는계기가되길바라며교수･학습상황에서작은보탬이될수있기를바라는마음이다.단한분의선생님이라도필자가본고에담고자하는마음을이해해주시고,그로인하여단한명의학생이라도기하의아름다움을느낄수있게된다면행복할것이다.