A multi-solution exploration of geometric calculation problems from the perspective of new definitions and its teaching reflections: A case study based on a senior high school entrance examination problem on “Anti-right Triangles”
定义新视角下几何计算题的多元解法探究与教学思考 ——以一道“反直角三角形”中考题为例

摘要

本文以一道涉及新定义“反直角三角形”的中考题为例，通过几何构造、代数计算、勾股定理及三角函数等多个维度，探究其不同解法。文章旨在透过解法的多样性，挖掘其背后的数学思想（分类讨论、方程思想、转化思想），并结合波利亚解题理论，探讨在核心素养背景下，如何通过一题多解教学提升学生的思维品质和解题能力。

Abstract

This paper takes a senior high school entrance examination problem involving a new definition of “anti-right triangle” as an example, and explores its multiple solutions through various approaches including geometric construction, algebraic computation, the Pythagorean theorem, and trigonometric functions. The aim of this paper is to uncover the underlying mathematical thinking (classification discussion, equation thinking, and transformation thinking) through the diversity of solutions. Furthermore, in conjunction with Pólya’ s problem-solving theory, this paper discusses how to enhance students' thinking quality and problem-solving abilities through multiple-solution instruction in the context of core competencies.