一道高考数学题目的多种解法与思考——以2025年全国新课标II卷第16题为例

Multiple Solutions and Reflections on a College Entrance Examination Mathematics Problem—Taking Question 16 of the 2025 National New Curriculum Standard Volume II as an Example

摘要: 圆锥曲线问题作为历年高考数学试卷中的一个重点与难点，主要考查学生对知识的灵活性、创新性应用。以2025年高考数学新课标二卷第16题为例，从不同思维视角切入来分析解决问题，并归纳思考解法中的共性思维，实现从“一题多解”到“多解归一”，给予一线数学教师习题教学的策略建议：立足单元教学，培养学生对知识的综合应用能力；聚焦解法本质，提升学生的数学思维与解题素养。帮助学生分析解法的逻辑起点与核心思路，实现从“盲目机械解题”到“真正学习”的转变。

Abstract: Conic section problems, as a key and difficult topic in the mathematics papers of the National College Entrance Examination over the years, mainly examine students’ flexible and innovative application of knowledge. Taking Question 16 of the 2025 college entrance exam Mathematics National New Curriculum Standard Volume II as an example, this paper analyzes and solves the problem from different thinking perspectives, and summarizes the common thinking in the solution methods. It realizes the transformation from “multiple solutions to one problem” to “integrating multiple solutions into one core logic”, and provides strategic suggestions for front-line mathematics teachers in exercise teaching: basing on unit teaching to cultivate students’ comprehensive application ability of knowledge; focusing on the essence of solutions to improve students’ mathematical thinking and problem-solving literacy. This helps students analyze the logical starting point and core ideas of solutions, and achieves the transformation from “blind and mechanical problem-solving” to “genuine learning”.