國中資優數學之相似形的探討

本研究旨在瞭解學習相似形教材之迷思，以及探究國中數理資優課程適用之相似形教材為何。研究採用教材內容分析法，蒐羅相關相似形試題之後，開始進行整理。本研究經由文獻探討與試題分析後獲致以下結論：一、在國中學習相似形甚至是幾何問題時，巧添合適的輔助線有助於邏輯思考與分析圖形的能力。二、本研究整理出的幾何解題技法如下：（一）線段延伸與裁切角度以製造直角三角形。（二）當原圖形不易分析與解題，則割補成特殊的、簡單的或完整的新圖形，即能建立條件與結論的橋樑。（三）旋轉是利用平面圖形繞定點旋轉某個角度後，因改變位置後重組成新的幾何圖形，使原本無關聯之幾何量建立內在聯繫，因此旋轉變換的關鍵在於找到旋轉中心。（四）圖形是線對稱圖形常作對稱軸當輔助線，若非線對稱圖形，則選擇某一線段作為對稱軸，以建立內部關聯。（五）面積切割與比例線段是運用輔助線將圖形裁切成共高三角形或在分點位置做平行製造相似三角形。（六）線性獨立法是運用代數式去分析圖形線性獨立性與相依性以列出最簡關係式。茲根據以上結論，提出以下幾點建議供教育相關單位及學校參考：1.培養幾何師資專業能力，在教學上注重圖像思考、解題的多樣性與重視實作；2.使用動態幾何軟體模擬，觀察圖形變換，實證所學之幾何概念；3.充實教材內容，並輔以幾何證明強化邏輯推演與教材深度，以達到教材的完整性。

關鍵字

資優教育；課程設計；相似形

並列摘要

The purpose of this research is to understand the myth of learning similar triangles and explore what is the appropriate courses for the mathematical gifted students. This research was based on the analysis of teaching materials, which was begun to sort out after collecting the related questions of similar triangles. After demonstrated literature review and item analysis has come to the conclusion as the followings: Ⅰ)Suitable auxiliary lines helps to logical thinking and analytical graphic capabilities in learning similar triangles and even geometry problem in junior high school . Ⅱ)In this study, the geometry problem-solving techniques are induced to solve difficult problems. Based on those conclusions stated above, has raised the following recommendations as the reference for educational administrative authorities: 1) It is important to teachers to have professional competence about image thinking, problem-solving diversity and implementation.2) Dynamic geometry software is used to simulated and observed the graph transformation to verify the geometric conceptions. 3) There is a need of enriching the teaching materials and increasing the remaining geometry textbooks.