Research on applying the combination of number and shape to quadratic function teaching ——A case of comprehensive parabola problem in high school entrance examination
数形结合思想在二次函数教学中的应用研究 ——以一道中考抛物线综合题为例

摘要

数形结合思想是解决二次函数综合问题的核心策略，能够有效联结几何直观与代数推理，显著提升解题效率。本文以上海中考的一道抛物线综合题为切入点，深入探究数形结合思想在二次函数问题中的应用。研究首先拆解原题，该题以抛物线为背景，融合等腰直角三角形的几何性质，考查函数图像与几何图形的综合运用能力；通过坐标设定、方程建立、分类讨论等步骤，将几何特征转化为代数方程，并借助函数图像验证解的合理性，充分体现数形结合的典型应用特征。在此基础上，研究设计了改变几何图形、调整参数范围、引入动态条件等5类变式题目，验证了所提思维模型的普适性。最后，提炼出解决二次函数综合问题的四步通用方法：图形特征提取、代数模型构建、数学运算求解、几何验证优化，并强调数形互译在每一步中的关键作用。本研究不仅为二次函数教学提供理论参考，更通过具体案例展现数形结合思想的实践价值，有助于培养学生的直观想象、逻辑推理等数学核心素养，未来可进一步探索该思想在更高维度数学问题中的应用。

Abstract

The idea of combining number and shape is a core strategy for solving comprehensive problems of quadratic functions. It effectively connects geometric intuition and algebraic reasoning, and greatly improves problem-solving efficiency. Taking a comprehensive parabola problem from the Shanghai High School Entrance Examination as a starting point, this paper deeply explores the application of the number-shape combination idea in quadratic function problems. First, the study deconstructs the original problem, which is set against the background of parabola and integrates the geometric properties of isosceles right triangles to assess students' ability to use function graphs and geometric figures comprehensively. Through coordinate setting, equation establishment and classified discussion, geometric features are transformed into algebraic equations, and the rationality of solutions is verified by function graphs, fully reflecting the typical application characteristics of number-shape combination. On this basis, five types of variant problems are designed, including changing geometric figures, adjusting parameter ranges and introducing dynamic conditions, so as to verify the universality of the proposed thinking model. Finally, a four-step general method is refined: extracting graphic features, constructing algebraic models, solving through mathematical calculation, and optimizing by geometric verification, with emphasis on the key role of mutual translation between number and shape in each step. This study not only provides theoretical references for quadratic function teaching, but also demonstrates the practical value of the number-shape combination idea through concrete cases, which helps to cultivate students' core mathematical competencies such as intuitive imagination and logical reasoning. Further exploration can be made on the application of this idea in higher-dimensional mathematical problems in the future.