【People's Education Press】High School Mathematics Compulsory Course First Volume (A Edition)

📚 Content Summary

This textbook serves as an introductory resource for high school mathematics, covering core topics such as sets and common logical expressions, quadratic functions, equations, and inequalities, function concepts and properties, exponential and logarithmic functions, and trigonometric functions. It aims to develop students' mathematical core competencies, logical reasoning abilities, and awareness of mathematical modeling.

Open the door to high school mathematics—master core concepts and rigorous logical thinking.

Author: People's Education Press Curriculum & Textbook Research Institute, Center for High School Mathematics Curriculum and Textbook Research & Development

Acknowledgments: This textbook received the National Special Award for Outstanding Textbooks at the First National Textbook Construction Awards. It is developed based on the General Senior High School Mathematics Curriculum Standards (2017 Edition).

🎯 Learning Objectives

1. Accurately determine the definiteness, distinctness, and unordered nature of sets, and proficiently use enumeration and descriptive methods to represent sets.
2. Master the application of Venn diagrams and the formula for calculating the number of elements in a set: $ \text{card}(A \cup B) = \text{card}(A) + \text{card}(B) - \text{card}(A \cap B) $.
3. Understand and distinguish between sufficient conditions, necessary conditions, and necessary-and-sufficient conditions; apply logical language to describe geometric properties and criteria for determination.
4. Grasp the fundamental facts about comparing real numbers, and use inequality properties for algebraic proofs and size comparisons.
5. Understand the geometric background and applicability conditions (positive, fixed, equal) of basic inequalities, and solve simple extremum problems.
6. Master the solution flowchart for quadratic inequalities, comprehend the correspondence among the graph of a quadratic function, roots of equations, and solution sets of inequalities, and solve complex practical application problems.
7. Use set theory and mapping language to define functions, and understand the standard for identifying "identical functions" (same domain and correspondence).
8. Skillfully apply the three representation methods to describe variable relationships, with special emphasis on writing analytical expressions and drawing graphs of piecewise functions.
9. Strictly prove the monotonicity and even/odd nature of functions using definitions, and find extrema within given intervals.
10. Understand the concepts of nth roots and fractional exponents, and master their operational properties.

🔹 Lesson 1: Sets, Logic, and Mathematical Foundations

Overview: This module introduces foundational tools in high school mathematics: set theory and common logical expressions. It begins with basic concepts of sets, element characteristics, and various representation methods (enumeration and description), using Venn diagrams to solve problems involving the count of set elements. It then delves into propositional logic, focusing on definitions of sufficient, necessary, and necessary-and-sufficient conditions, applying them to derive and explore geometric propositions (e.g., parallelogram criteria).

Learning Outcomes:

• Accurately determine the definiteness, distinctness, and unordered nature of sets, and proficiently use enumeration and descriptive methods to represent sets.
• Master the application of Venn diagrams and the formula for calculating the number of elements in a set: $ \text{card}(A \cup B) = \text{card}(A) + \text{card}(B) - \text{card}(A \cap B) $.
• Understand and distinguish between sufficient, necessary, and necessary-and-sufficient conditions, and apply logical language to describe geometric properties and determination theorems.

🔹 Lesson 2: Properties of Inequalities and Solving Quadratic Equations

Overview: This unit covers the core logic and computational methods related to "inequalities" in high school mathematics. Starting from fundamental facts about comparing real number sizes, it systematically reviews inequality properties and their proofs, introduces the crucial "basic inequality" (arithmetic mean–geometric mean), and ultimately applies these to standardized solving of quadratic inequalities and their modeling applications in real life. Through the integrated perspective of functions, equations, and inequalities, it helps students build a complete cognitive framework for quadratic models.

Learning Outcomes:

• Master the fundamental facts about comparing real numbers, and use inequality properties for algebraic proofs and size comparisons.
• Understand the geometric background and applicability conditions (positive, fixed, equal) of basic inequalities, and solve simple extremum problems.
• Master the solution flowchart for quadratic inequalities, comprehend the correspondence among the graph of a quadratic function, equation roots, and solution sets, and solve complex real-world application problems.

🔹 Lesson 3: Function Concepts, Properties, and Fundamental Models

Overview: This unit covers the core foundations of functions in high school mathematics, redefining functions from the perspective of set theory and exploring three representation forms: analytical, graphical, and tabular methods. It deeply investigates fundamental properties of functions—monotonicity, even/odd symmetry, maximum/minimum values—and introduces power function models and the special y = x + 1/x model. Using real-life examples such as personal income tax calculations, it demonstrates the application of piecewise functions in solving complex practical problems, while reviewing the evolution of the function concept from geometric quantities to mappings.

Learning Outcomes:

• Use set theory and mapping language to define functions, and understand the criterion for identifying "identical functions" (same domain and correspondence).
• Skillfully apply the three representation methods to describe relationships between variables, with particular focus on writing analytical expressions and drawing graphs of piecewise functions.
• Prove the monotonicity and even/odd nature of functions rigorously using definitions, and find extrema over given intervals.

🔹 Lesson 4: Deep Dive into Exponential and Logarithmic Functions

Overview: This lesson plan covers a comprehensive theoretical system from radicals and fractional exponents to exponential and logarithmic functions. It emphasizes the intrinsic relationship between exponentiation and logarithms as inverse operations, and illustrates their central role in solving real scientific problems and numerical computations through models such as radioactive decay and the bisection method for root finding.

Learning Outcomes:

• Understand the concepts of nth roots and fractional exponents, and master their operational properties.
• Proficiently grasp the concepts, graphical features, and monotonicity of exponential and logarithmic functions.
• Apply the change-of-base formula for logarithms to perform complex logarithmic operations and simplifications.

🔹 Lesson 5: Mathematical Modeling: Constructing Function Models

Overview: This session focuses on transforming real-world problems into mathematical function models. By collecting, organizing, and analyzing data, students learn to select the most appropriate function model (linear, quadratic, power, exponential, or logarithmic) using data-fitting techniques to address optimization and prediction challenges in real life. The core lies in understanding the modeling cycle: "real problem → mathematical model → mathematical result → real conclusion."

Learning Outcomes:

• Identify and describe the basic steps for building function models to solve real-world problems (understanding the problem, modeling, solving, verifying).
• Learn to use scatter plots to observe data distribution trends and perform preliminary data fitting based on observed patterns.
• Understand the characteristics of different growth-type functions, and choose the optimal function model based on the context and data trend of the real problem.

🔹 Lesson 6: Trigonometric Functions and Their Applications in Identities

Overview: This unit presents a complete logical framework starting from the concept of arbitrary angles, defining trigonometric functions via the unit circle, and then studying their algebraic properties (identity transformations) and analytical properties (graphs and behaviors). Emphasis is placed on mastering the graph transformation rules of functions like y = A\sin(\omega x + \phi) using the "five-point method," and applying trigonometric models to solve real-world periodic phenomena such as tides, Ferris wheels, and waveforms. Finally, Taylor series are introduced to extend the numerical approximation capabilities of trigonometric functions.

Learning Outcomes:

• Master the concepts of arbitrary angles, coterminal angles, and the unit circle definition; skillfully apply identities, induction formulas, sum-difference formulas, and double-angle formulas for trigonometric identity transformations.
• Use the "five-point method" to draw sine, cosine, and tangent curves, analyze and extract periodicity, symmetry, monotonicity, and extrema of functions.
• Build trigonometric models to solve real-life periodic problems, and understand the fundamental idea of using Taylor series for approximating trigonometric functions numerically.