Mastering Conceptual Intension and Extension to Skillfully Solve New Definition Problems in the Gaokao Mathematics (Part 1)工程配资
高考数学新定义问题已成为区分考生能力的重要题型。通过准确把握概念的内涵与外延，考生能够有效破解这类问题，提高解题效率和高考成绩。
New definition problems in the Gaokao Mathematics have become a key question type for distinguishing students' abilities. By accurately grasping the intension (essential attributes) and extension (scope of application) of concepts, students can effectively deconstruct such problems, thereby improving their solution efficiency and overall test performance.
展开剩余94%概念内涵是指概念的本质属性，而外延则是概念的应用范围和边界，两者共同构成了数学概念的完整认知框架。
The intension of a concept refers to its essential attributes, while the extension refers to its applicable range and boundaries. Together, they form a complete cognitive framework for understanding mathematical concepts.
在面对新定义问题时，考生需要先明确新概念的核心本质，再确定其适用范围，从而构建解题思路。
When confronting a new definition problem, students first need to clarify the core essence (intension) of the new concept, then determine its scope of application (extension), thus constructing a logical path to solve the problem.
概念的含义：通过揭示事物的本质属性而反映事物的思维形式。事物的性质和关系统称为属性，属性分为本质和非本质属性。
Concept Defined: A concept is a form of thinking that reflects the essential attributes of things. Properties and relationships of things are collectively called attributes, which are divided into essential and non-essential attributes.
任何概念都是内涵与外延的统一。准确把握概念，既要弄清概念的内涵，又要分清概念的外延。
Every concept is a unity of intension and extension. To accurately grasp a concept, one must understand both its intension and delineate its extension.
内涵：指概念所反映的事物的本质属性，反映事物“质”的规定性，概念所反映的事物究竟“是什么”。
Intension: Refers to the essential attributes of the things reflected by the concept. It deals with the "qualitative" aspect, answering "what"the thing fundamentally is.
外延：指具有概念所反映的本质属性的事物的范围。说明概念所反映的事物“有哪些”。
Extension: Refers to the range of things that possess the essential attributes reflected by the concept. It specifies "which"things are included in the concept's scope.
一、新定义问题的识别与内涵外延的提取方法
I. Identifying New Definition Problems and Methods for Extracting Intension and Extension
新定义问题通常通过文字描述、符号表示或几何模型引入陌生概念，要求考生在短时间内理解并应用。
New definition problems typically introduce unfamiliar concepts through textual descriptions, symbolic representations, or geometric models, requiring candidates to understand and apply them within a short time frame.
解决这类问题的第一步是准确提取新定义的内涵和外延。
The first step in solving such problems is to accurately extract the intension(essential attributes) and extension(scope of application) of the new definition .
以2024年新高考Ⅰ卷第19题为例，题目引入了“对k的可拆分函数”的新定义：
Taking the 2024 New Gaokao (National College Entrance Examination) Volume I, Question 19 as an example, which introduces the new definition of a "k-separable function":
“如果函数y=f(x)在其定义域内存在实数x₀，使得f(kx₀)=f(k)f(x₀)(k为常数)成立，则称函数y=f(x)为’对k的可拆分函数’。”
"If a function y=f(x) has a real number x0 within its domain such that the equation f(kx0)=f(k)f(x0) holds (where k is a constant), then the function y=f(x) is called a 'k-separable function'."
内涵提取：这里的内涵是“存在实数x₀，使得f(kx₀)=f(k)f(x₀)”这一等式成立，即函数在自变量k倍时具有某种乘积性质。考生需要理解这是函数满足某种特定关系的本质特征。
Extracting the Intension: The intension here is the essential property that "there exists a real number x0 such that the equation f(kx0)=f(k)f(x0) holds." Candidates need to understand that this equation represents the core characteristic the function must satisfy—a specific multiplicative property when the independent variable is scaled by k.
外延确定：外延则涉及该性质的适用范围，如对于不同类型的函数（线性函数、指数函数、对数函数等），这一性质是否成立，以及成立的条件是什么。
Determining the Extension: The extension involves the range of scenarios where this property applies. This includes examining whether this property holds for different types of functions (e.g., linear functions, exponential functions, logarithmic functions) and under what specific conditions it holds . Analyzing the extension helps in understanding the boundaries and limitations of the new concept.
在解题过程中，考生需要将新定义与已学知识进行关联。
In the problem-solving process, candidates need to link the new definition to prior knowledge.
例如，对于f(kx₀)=f(k)f(x₀)，可以联想到指数函数的性质f(kx)=f(k)f(x)，从而推测可能的解题方向。For example, the expression f(kx0)=f(k)f(x0)can be associated with the property of exponential functions, where f(kx)=f(k)f(x), thereby suggesting a potential direction for solving the problem.
通过这种内涵与外延的明确，考生能够将陌生问题转化为熟悉的知识领域，提高解题效率。
By clarifying the intension and extension in this way, candidates can transform unfamiliar problems into familiar domains of knowledge, thereby enhancing their problem-solving efficiency.
二、新定义问题的典型题型与解题策略
II. Typical Types of New Definition Problems and Solution Strategies
新定义问题在高考数学中主要分为三类：概念新定义型、运算新定义型和规则新定义型。每种类型都需要不同的内涵与外延提取策略。
New definition problems in the Gaokao mathematics exam are primarily categorized into three types: concept-based, operation-based, and rule-based new definitions. Each type requires distinct strategies for extracting their intension and extension .
1.概念新定义型：这类问题通过重新定义数学概念，考查考生的理解与应用能力。例如2025年新高考Ⅱ卷第19题引入的”零点存在性定理的逆向应用”，需要考生理解传统定理的本质，并进行逆向思维。
Concept-based New Definitions
These problems redefine mathematical concepts to test students' comprehension and application skills. For example, Question 19 in the 2025 New Gaokao Volume II introduced the "reverse application of the Zero-Point Existence Theorem," requiring students to understand the essence of the traditional theorem and engage in reverse thinking .
2.运算新定义型：这类问题通过定义新的运算规则，考查考生的逻辑推理能力。例如2021年全国新高考Ⅱ卷中的“ω(n)=a₀+a₁+…+a_k”，要求考生理解二进制表示下数字各位数的和，并应用这一新运算解决相关问题 。
Operation-based New Definitions
These problems define new operational rules to assess students' logical reasoning abilities. An instance is the definition "ω(n)=a₀+a₁+…+a_k" from the 2021 National New Gaokao Volume II, which required candidates to understand the sum of the digits of a number in its binary representation and apply this new operation to solve related problems .
3.规则新定义型：这类问题通过设定新的规则或条件，考查考生的转化能力。例如2020年浙江卷第10题中定义的集合S与T的关系，需要考生理解“对于任意的x,y∈S，若x≠y，则xy∈T”和”对于任意的x,y∈T，若x<y，则y/x∈S”的规则本质。
Rule-based New Definitions
These problems establish new rules or conditions to evaluate students' ability to transform and apply knowledge. For example, Question 10 in the 2020 Zhejiang Volume defined a relationship between sets S and T, demanding students grasp the essence of the rules: "for any x, y ∈ S, if x ≠ y, then xy ∈ T" and "for any x, y ∈ T, if x < y, then y/x ∈ S" .
针对这些题型，解题策略主要包括：
To tackle these types of problems, the main solution strategies include:
１．性质优先分析法：先判断新函数的对称性、单调性、奇偶性等性质，利用性质缩小研究范围。例如在分析“双重切点”问题时，先确定曲线的对称性，再进行具体计算。
Priority Analysis of Properties: Begin by determining properties of the new entity, such as symmetry, monotonicity, or parity (odd/even nature), and use these properties to narrow the scope of investigation. For example, when analyzing a "double point of tangency" problem, first identify the symmetry of the curve before proceeding with specific calculations .
２．构造示例法：通过构造具体的例子来帮助理解和解决问题 。例如在处理“对k的可拆分函数”问题时，可以先构造一些简单的函数（如线性函数、指数函数）进行验证，再推广到一般情况。
Method of Constructing Examples: Aid understanding and problem-solving by constructing specific examples. For instance, when dealing with the "k-separable function" problem, one can first construct and verify simple functions (such as linear or exponential functions) before generalizing to the broader case .
３．化归与建模：将新定义问题转化为熟悉的问题进行处理 。例如将“点到几何图形的距离”转化为几何体体积计算问题，需要将抽象定义转化为具体的几何模型。
Reduction and Modeling: Transform the new definition problem into a familiar problem for processing. For example, converting the "distance from a point to a geometric figure" into a problem of calculating the volume of a geometric solid requires translating the abstract definition into a concrete geometric model.
三、内涵与外延明确对解题效率的提升
III. The Enhancement of Problem-Solving Efficiency through Clarifying Intension and Extension
准确把握概念的内涵与外延，能够显著提升解题效率。
Accurately grasping the intension and extension of concepts can significantly enhance problem-solving efficiency.
内涵明确使考生能够抓住问题本质，避免在次要细节上浪费时间；外延明确则帮助考生确定解题范围，避免无效尝试。
Clear intension allows candidates to grasp the essence of a problem, avoiding wasted time on secondary details; clear extension helps candidates define the scope of the solution, preventing ineffective attempts.
以2023年高考数学全国乙卷第16题为例，题目引入了“点到几何图形的距离”的新定义：
Taking Question 16 from the 2023 Gaokao Mathematics National Volume B as an example, the problem introduces a new definition of "distance from a point to a geometric figure":
“定义‘点到几何图形的距离’为这个点到几何图形上各点距离中的最小值。现有边长为2的正方形ABCD，则到定点A距离为1的点围成的几何体的体积为______；该正方形ABCD区域（包括边界以及内部的点）记为Ω，则到Ω距离等于1的点所围成的几何体的体积为______。”
"Define the 'distance from a point to a geometric figure' as the minimumdistance from that point to all points on the geometric figure. Given a square ABCD with side length 2, the volume of the solid formed by all points whose distance to the fixed point A is 1 is ______; the region of square ABCD (including its boundary and interior points) is denoted as Ω, and the volume of the solid formed by all points whose distance to Ω equals 1 is ______."
内涵分析：这里的内涵是“点到几何图形的距离是点到图形上各点距离的最小值”，即最短距离的概念。
Intension Analysis: The intension here is that "the distance from a point to a geometric figure is the minimumdistance from the point to all points on the figure" – the concept of the shortest distance.
外延应用：外延则涉及这一概念在不同几何图形上的应用。
Extension Application: The extension involves applying this concept to different geometric figures.
对于第一个空，到定点A距离为1的点围成的几何体是一个半径为1的球，体积为4/3π；对于第二个空，到正方形区域Ω距离等于1的点所围成的几何体是一个正方体、四个半圆柱和四个四分之一球的组合体，体积需要通过几何体的特征进行计算。
For the first blank, the set of points whose distance to the fixed point A is 1 forms a sphere of radius 1, with a volume of 4π/3. For the second blank, the set of points whose distance to the square region Ω equals 1 forms a composite solid: a cube, four half-cylinders, and four quarter-spheres. Its volume requires calculation based on the features of this composite geometric solid.
通过准确把握“点到几何图形的距离”这一新定义的内涵与外延，考生能够快速确定解题方向，避免在复杂的几何关系中迷失。
By accurately grasping the intension and extension of the new definition "distance from a point to a geometric figure," candidates can quickly determine the direction for solving the problem, avoiding getting lost in complex geometric relationships.
内涵明确使考生能够将抽象定义转化为具体的几何模型，外延明确则帮助考生确定几何体的构成，从而高效完成解题。
Clear intension enables candidates to translate the abstract definition into a concrete geometric model, and clear extension helps them identify the components of the geometric solid, thereby solving the problem efficiently.
创新思维探索者作者：宏哥十堰郧中频道 2025.12.22.工程配资
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