【People's Education Press】High School Mathematics Compulsory Volume 2 (Version A)

📚 Content Summary

This textbook covers five major chapters: plane vectors and their applications, complex numbers, introductory solid geometry, statistics, and probability. It aims to cultivate core mathematical competencies—logical reasoning, mathematical modeling, and data analysis—by integrating "numbers" with "shapes."

Master core mathematical thinking and explore the deep integration of vectors, complex numbers, and geometry.

Author: Zhang Jianyue, Li Zenghu

Acknowledgments: This textbook has been reviewed and approved by the Expert Committee of the National Textbook Committee (2019); related funding and copyright belong to People's Education Press.

🎯 Learning Objectives

1. Understand the physical background of vectors, grasp fundamental concepts such as vectors, equal vectors, and collinear vectors, along with their geometric representations.
2. Master vector addition, subtraction, scalar multiplication, and their geometric meanings, and apply coordinate-based linear operations proficiently.
3. Understand the definition, properties, and concept of projection vectors for the dot product of vectors, and comprehend its physical significance (e.g., work).
4. Understand the algebraic representation of complex numbers z = a+bi, conjugate complex numbers, and the concept of modulus.
5. Master the algebraic operations of complex numbers—addition, subtraction, multiplication, and division—and use conjugate complex numbers to handle division.
6. Be able to transform complex number operations into vector operations in the complex plane (parallelogram rule, triangle rule), and apply the geometric meaning of modulus to solve distance and locus problems.
7. Accurately identify structural characteristics of prisms, pyramids, frustums, and solids of revolution, and recognize simple composite solids.
8. Master surface area and volume formulas for cylinders, cones, frustums, and spheres, and understand the role of Zu Geng's Principle in deriving volumes.
9. Master the theorems for determining and properties of spatial parallelism and perpendicularity, and apply axiomatic methods for rigorous geometric proofs.
10. Students can distinguish between and implement simple random sampling and stratified random sampling, and select appropriate data collection methods based on context.

🔹 Lesson 1: Plane Vectors and Their Applications

Overview: This lesson covers the foundational theory of plane vectors and their extensive applications in geometry and trigonometry. Students will begin with physical contexts (such as displacement and force) to understand the geometric representation and linear operations (addition, subtraction, scalar multiplication) of vectors, then progress to mastering the dot product, projections, and coordinate representations of vectors. Finally, using the fundamental theorem of plane vectors, vector tools are introduced into triangle studies, enabling deeper exploration of the Law of Sines and Law of Cosines, and the solution of practical measurement and geometric proof problems.

Learning Outcomes:

• Understand the physical background of vectors, master basic concepts such as vectors, equal vectors, and collinear vectors, along with their geometric representations.
• Proficiently perform vector addition, subtraction, scalar multiplication, and understand their geometric meanings; apply coordinate-based linear operations.
• Grasp the definition, properties, and concept of projection vectors for the dot product of vectors, and comprehend its physical significance (e.g., work).

🔹 Lesson 2: Algebraic and Geometric Properties of Complex Numbers

Overview: This lesson aims to help high school students deeply understand complex numbers from both algebraic and geometric perspectives. It covers fundamental concepts of complex numbers, the one-to-one correspondence between complex numbers and points or vectors in the complex plane, and the algebraic rules for arithmetic operations, along with their geometric interpretations involving translation, scaling, and rotation. Finally, the lesson introduces the trigonometric form of complex numbers, achieving unification between algebraic and trigonometric forms.

Learning Outcomes:

• Understand the algebraic representation z = a+bi, conjugate complex numbers, and the concept of modulus.
• Master the algebraic operations of complex numbers—addition, subtraction, multiplication, and division—and use conjugate complex numbers to handle division.
• Be able to convert complex number operations into vector operations in the complex plane (parallelogram rule, triangle rule), and apply the geometric meaning of modulus to solve distance and locus problems.

🔹 Lesson 3: Introduction to Solid Geometry

Overview: This unit guides students from intuitive perception to rational analysis, systematically studying the structural characteristics of three-dimensional geometric solids. Topics include definitions, intuitive drawing techniques, surface area and volume calculations (including Zu Geng’s Principle) for basic solids (prisms, pyramids, frustums, spheres), logical determination and properties of parallelism and perpendicularity among points, lines, and planes in space, culminating in the axiomatic framework of Euclidean geometry.

Learning Outcomes:

• Accurately identify structural features of prisms, pyramids, frustums, and solids of revolution, and recognize simple composite solids.
• Master surface area and volume formulas for cylinders, cones, frustums, and spheres, and understand the role of Zu Geng’s Principle in volume derivation.
• Master the theorems for determining and properties of spatial parallelism and perpendicularity, and apply axiomatic methods for rigorous geometric proofs.

🔹 Lesson 4: Foundations of Statistics and Data Analysis

Overview: This unit guides students through the core process of statistics: from data collection (random sampling, various data acquisition methods) to data organization (frequency distribution tables and histograms, selection of statistical charts), to data analysis and inference (estimating populations from samples, percentiles). Through the case study of “Company Employees’ Obesity Status,” students will learn to apply statistical theory to real-world complexity and understand the probabilistic nature of statistical conclusions.

Learning Outcomes:

• Students can distinguish between and implement simple random sampling and stratified random sampling, and choose appropriate data collection methods based on context.
• Students can accurately create and interpret frequency distribution histograms, and use percentiles to assess data positions within a population.
• Students can select optimal statistical graphs according to analytical goals, and write statistical case reports based on data analysis.

🔹 Lesson 5: Probability Theory and Random Simulation

Overview: This instructional design covers the foundational core concepts of probability theory, starting with the definition of random experiments, introducing sample spaces and set-theoretic descriptions of events. By exploring logical relationships and operations among events, students will master methods for calculating probabilities in classical probability models (equally likely outcomes), and ultimately transition from theoretical derivation to numerical experimentation via the stability of frequency, culminating in random simulation and Monte Carlo methods—completing a full cognitive loop from theory to practice.

Learning Outcomes:

• Correctly define random experiments, sample spaces, and various types of random events, and describe relationships between events (inclusion, union, intersection, mutually exclusive, complementary) using set language.
• Understand the characteristics of classical probability models, and compute probabilities of simple random events using counting methods.
• Comprehend the relationship between frequency and probability, master the basic ideas of random simulation, and apply Monte Carlo methods to solve complex real-world probability problems.