在平面幾何證明題中使用輔助線的策略

The Strategies in Proving Plane Geometry Problems with Auxiliary Lines

翁聖閔(Sheng-Min Weng)

指導教授 ： 李金城 ； 何淳雪

中原大學/理學院/應用數學研究所/碩士(2014年)

https://doi.org/10.6840/cycu201400876

摘要

摘 要
在中學課程內眾多數學單元之中，「證明」是有效訓練演繹推理能力的方法，但根據許多國中數學老師的教學經驗來看，證明是現今數學教學當中最急需加強的部分。使用輔助線是在平面幾何證明題中最常見的解題方法之一，本研究目的即是針對不同類型的平面幾何證明題給予最適當的輔助線來幫助完成解題，同時也希望能夠讓中學生重拾
對數學證明題的重視，進而幫助中學生能達到於九年一貫數學領域課程的「幾何」主題上的目標：「能以三角形和圓的性質為題材來學習推理」。
本研究所討論的題目適合目前就讀國中九年級以上之中等教育學生練習。研究主題為：試著要以輔助線來幫助幾何證明題時，有何策略？然本研究僅針對平面上的幾何圖形證明題提供解答策略與輔助線技巧，故無法提供所有類型的證明題作參考。
綜觀本研究所討論之題目，可以得知在作平面幾何證明題使用輔助線時，運用到的技巧不外乎從已知條件、圖形本身或其延伸的性質還有欲用之定理來加輔助線或是輔助圓，經過推理論證，使題目最終結論能夠被證明。然而在作這些幾何證明題時，先備知識亦是非常重要的一環。
證明題的解題方式絕對不是只有唯一，解題經驗越豐富，先備知識越完整，則能夠發現更多不同的證明方法。教師也應多用心去了解學生的學習情況，紮實地針對各個學生打好基礎，如此一來學生之間更能無程度差距的互相討論題目，並且讓多數學生能夠參與學習。

關鍵字

幾何證明題

並列摘要

Abstract
In all material covered in high school mathematical class, “proof question” has been
recognized as a highly efficient mean in logic reasoning training. Unfortunately from
practical teaching experience, such training seems to be the Achilles’ tendon in modern
mathematical education. Guide line is the commonest solution in graphic proof question.
The purpose of this study focuses on giving appropriate guide lines in soling graphics
proof questions, while regains the importance of proof question, helping high school
students meet the requirement of new “nine years mathematic education architecture”:
learning logic reasoning from graphics.
Topics covered in this study are designed for students of nine grades or above. The
question becomes: what to do with guide lines, in solving graphics proof questions? This
study may not provide reference for all type of proof questions, but for graphics proof
questions offers guide line-based solving strategies.
By walking through all the material covered in this study, a known fact is that
solving graphics proof question is nothing more than using guide line, properties of
graphical objects themselves or their extended ones. Salted with graphic theories and
logic reasoning, a claim in graphics proof question can be determined, when all the
prerequisite knowledge is well prepared.
Obviously one proof question could be solved in various approaches. With further
experience and richer prerequisite knowledge, better solution could be found in time.
Teachers are expected to understand students’ learning progress and help them build
fundamentals in individual base. So most of students should be able to get involved and
discuss among piers.

並列關鍵字

Proving Plane Geometry Problems

參考文獻

教育部，民92，《國民中小學九年一貫課程綱要數學學習領域》，

詹玉貞，民89，〈波利亞的解題步驟對國中數學資優生學習幾何證明成效之研究〉

李宜芬，民91，〈國三學生突破因附圖造成之論證障礙的學習歷程之研究〉，

2014/06/14。

International Handbook of Mathematics Education,