・ Ryokuyou High School * 3Kanazawa Technical College The purpose of this study was to develop a lesson which connects the three representational modes of the geometric definition , the figure , and the equation of an ellipse through mathematical activities . In order to encourage learner autonomy , we implemented an experimental class based on the constructive approach with internal and external transla ・ tions of the three representational modes , i . e。 1inguistic , illustrative , and

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... representation . We used the graphing softWare GRAPES to support stUdents ' unskillfU1 mathematical aCtivities. The following results were obtained : ( 1 ) Although students felt that the experimental class was difficult , they tackled mathematical activities positively and enthusiastically throughout the learning process of the constructive approach . Approximately 70 percent of students were able to understand the equations of an ellipse. ( 2 ) As a result of internal and external translations of the three representational modes dependent on students ' abilities , students were able to construct and acquire the connection between the geometric definition , the figure , and the equation of an ellipse with their teacher and other students . ( 3)W 合concluded that using the GRAPES method in this type of experimental class is a useful approach to confirm the 丘 gure and the graph of the equation of an ellipse visually and intuit { ve1 ¥