指向类比推理能力培养的探究案例——几何体的表面积与体积的关系探究

An Inquiry Case Aiming at Cultivating Analogical Reasoning Ability—Exploration of the Relationship between Surface Area and Volume of Geometric Solids

摘要: 类比推理是高中数学核心思维能力之一，是学生实现知识迁移、建构数学规律的重要载体。本文以人教版高中数学导数章节拓展内容为依托，以几何体表面积与体积的导数关系为探究主题，设计指向学生类比推理能力培养的探究式案例。以圆与球的公式导数规律为切入点，引导学生发现二维、三维对称几何体中“面积导数为周长、体积导数为表面积”的特殊规律，通过正方形、正方体引发认知冲突，逐步修正猜想。进而依次对正方体、长方体、圆柱体等典型几何体展开探究，通过变量替换、几何直观分析、代数推导验证等方式，揭示几何体表面积与体积导数关系的本质：该规律的成立与否并非取决于几何图形形态，而是由几何体的生长扩张方式与自变量选取方式决定，选取几何体中心到表面的距离为特征自变量时，各类几何体均可实现体积导数与表面积的统一对应关系。整个过程遵循“观察–猜想–验证–修正–归纳”的探究主线，让学生亲历从特例发现、冲突思辨到本质提炼的完整数学探究过程，有效实现二维到三维的思维跃迁，在掌握几何导数核心规律的同时，锤炼类比推理、质疑求证、模型建构的数学核心素养，为高中数学导数几何意义拓展教学与思维能力培养提供实践参考。

Abstract: Analogical reasoning is one of the core thinking abilities in senior high school mathematics. It serves as a vital carrier for students to realize knowledge transfer and construct mathematical rules. Based on the extended content of the Derivatives chapter in the People’s Education Press senior high school mathematics textbook, this paper takes the derivative relationship between the surface area and volume of geometric solids as its research theme, and designs an inquiry‑based teaching case oriented to cultivating students’ analogical reasoning ability. Starting from the derivative rules of formulas for circles and spheres, the teaching guides students to discover the special rule that “the derivative of area equals perimeter, and the derivative of volume equals surface area” for two‑dimensional and three‑dimensional symmetric geometric figures. Cognitive conflicts are triggered through squares and cubes to revise conjectures step by step. Typical geometric solids including cubes, cuboids and cylinders are further explored. By means of variable substitution, geometric intuitive analysis and algebraic derivation and verification, this paper reveals the essence of the derivative relationship between the surface area and volume of geometric solids: whether the rule holds does not depend on the shape of geometric figures, but on the growth‑expansion mode of geometric solids and the selection of independent variables. When the distance from the center of a geometric solid to its surface is selected as the characteristic independent variable, the unified correspondence between the derivative of volume and surface area can be achieved for various geometric solids. Following the inquiry thread of “observation‑conjecture‑verification‑revision‑induction”, the whole teaching process enables students to experience a complete mathematical inquiry from discovering special cases and reflecting on conflicts to extracting essences. It effectively realizes the thinking leap from two dimensions to three dimensions. While mastering the core rules of geometric derivatives, students’ key mathematical competencies such as analogical reasoning, questioning and verification, and model construction are developed. This paper provides practical references for the extended teaching of the geometric meaning of derivatives and the cultivation of thinking abilities in senior high school mathematics.