High School Mathematics (Selected Compulsory Volume 1, A Edition) | People's Education Press

📚 Content Summary

This textbook aligns with the Curriculum Standards for Ordinary High School Mathematics (2017 Edition), focusing on core content including spatial vectors and solid geometry, equations of lines and circles, and equations of conic sections. By integrating numerical and geometric thinking with vector methods, the course aims to enhance students’ core competencies in mathematical modeling, logical reasoning, and intuitive visualization.

Master vector tools and analytic geometry to unlock the door to spatial mathematical thinking.

Author: People's Education Press Curriculum and Textbook Research Institute, Center for Research and Development of Secondary School Mathematics Curriculum and Textbooks

Acknowledgments: Approved by the National Textbook Committee Expert Review Panel (2019)

🎯 Learning Objectives

1. Understand and master the concept of spatial vectors, linear operations, and necessary and sufficient conditions for collinearity and coplanarity.
2. Grasp the fundamental theorem of spatial vectors and be proficient in establishing a spatial rectangular coordinate system for coordinate operations of vectors.
3. Use direction vectors and normal vectors of spatial vectors to determine positional relationships—parallelism and perpendicularity—between lines and planes in space.
4. Master algebraic representations of lines: deeply understand the concepts of inclination angle and slope, and skillfully apply point-slope form, general form, and parametric equations to describe lines.
5. Quantify geometric relationships and distances: master the conditions for parallelism and perpendicularity between two lines, and be proficient in using distance formulas for points, point-to-line, and between parallel lines.
6. Construct mathematical models of circles: determine standard and general equations of circles given specific conditions, and analyze positional relationships among points, lines, and circles, as well as between two circles.
7. Master standard equations: derive standard equations of ellipses, hyperbolas, and parabolas based on given conditions, and conduct case analysis according to focus positions.
8. Analyze geometric properties: skillfully identify and compute vertices, foci, major/minor axes (real/imaginary axes), eccentricity of conic sections, asymptotes of hyperbolas, and directrices of parabolas.
9. Solve positional relationships: learn to use the discriminant method to address problems involving common points between lines and ellipses, and master techniques for computing chord length and finding midpoints' loci.

🔹 Lesson 1: Spatial Vectors and Solid Geometry

Overview: This unit extends the study of plane vectors into three-dimensional space, establishing an algebraic framework for spatial vectors. Through learning linear operations, dot product operations, and coordinate representations of spatial vectors, students will master vector-based methods for analyzing spatial line-plane relationships and metric problems, transforming solid geometry from geometric intuition to algebraic computation.

Learning Outcomes:

• Understand and master the concept of spatial vectors, linear operations, and necessary and sufficient conditions for collinearity and coplanarity.
• Grasp the fundamental theorem of spatial vectors and be able to establish a spatial rectangular coordinate system for vector coordinate operations.
• Use direction vectors and normal vectors of spatial vectors to determine parallelism and perpendicularity between lines and planes in space.

🔹 Lesson 2: Equations of Lines and Circles

Overview: This unit forms the foundational core of high school analytic geometry, emphasizing the use of coordinate methods to convert geometric figures into algebraic equations. By exploring the inclination angle and slope of lines, various equation forms, and standard and general equations of circles, students will master a “three-step approach” for solving geometric problems. The course aims to cultivate integrated numerical-geometric thinking, enabling precise resolution of positional relationships, distances, and loci through algebraic computation.

Learning Outcomes:

• Master algebraic representation of lines: deeply understand the concepts of inclination angle and slope, and skillfully use point-slope form, general form, and parametric equations to describe lines.
• Quantify geometric relationships and distances: master conditions for parallelism and perpendicularity between two lines, and be proficient in using distance formulas between two points, point-to-line, and between parallel lines.
• Construct mathematical models of circles: determine standard and general equations of circles from given conditions, and analyze positional relationships among points, lines, and circles, as well as between two circles.

🔹 Lesson 3: Equations of Conic Sections

Overview: This lesson covers the core content of analytic geometry: ellipses, hyperbolas, and parabolas. Through coordinate methods, the course studies the standard equations and geometric properties of these three conic sections, explores the positional relationships between lines and ellipses, and uses information technology to investigate point trajectories. Finally, it reveals the intrinsic connection between the graph of quadratic functions and parabolas.

Learning Outcomes:

• Master standard equations: derive standard equations of ellipses, hyperbolas, and parabolas based on given conditions, and perform case analysis depending on focus positions.
• Analyze geometric properties: skillfully identify and compute vertices, foci, major/minor axes, eccentricity of conic sections, asymptotes of hyperbolas, and directrices of parabolas.
• Solve positional relationships: learn to use the discriminant method to handle problems involving common points between lines and ellipses, and master methods for computing chord length and finding midpoints’ loci.