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

📚 Content Summary

This textbook is the second volume of the elective compulsory mathematics for ordinary high schools (A edition), strictly aligned with the 2017 revised mathematics curriculum standards. It covers sequences (arithmetic and geometric sequences, mathematical induction) and the derivatives of single-variable functions along with their applications. Through rich real-world contexts, it guides students to understand mathematical concepts and cultivate core mathematical competencies.

Explore the mysteries of sequences and lay the foundation of calculus—develop rigorous mathematical logic and modeling skills.

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

Acknowledgments: This textbook has been reviewed and approved by the National Textbook Committee Expert Review Panel (2019)

🎯 Learning Objectives

1. Master core definitions: Identify and describe the definition of a sequence, general term formula, recursive formula, and sum of the first n terms.
2. Strengthen logical reasoning: Skillfully apply the two fundamental steps of mathematical induction to prove propositions related to arithmetic and geometric sequences.
3. Cross-disciplinary application: Use sequence models to solve real-world problems such as Halley’s Comet return cycles, basic reproduction number R_0, and radioactive decay.
4. Apply analogical reasoning to explore relationships between arithmetic and geometric sequences, and understand the operational rules of the "Collatz Conjecture."
5. Deeply comprehend the limit-based definition of derivatives, and distinguish between average rate of change and instantaneous rate of change.
6. Grasp the geometric meaning of derivatives, internalize the mathematical idea of "approximating curves with straight lines," and proficiently use basic elementary function derivative formulas for calculations.

🔹 Lesson 1: Sequences

Overview: This lesson covers foundational concepts of sequences, in-depth study of arithmetic sequences, logical proofs using mathematical induction, and an initial introduction to the concept of derivatives. Students will transition from analyzing discrete numerical patterns (sequences) to continuous rate-of-change analysis (derivatives), understanding mathematics’ central role in describing natural phenomena such as comet returns, radioactive decay, and disease transmission.

Learning Outcomes:

• Master core definitions: Identify and describe the definition of a sequence, general term formula, recursive formula, and sum of the first n terms.
• Strengthen logical reasoning: Skillfully apply the two fundamental steps of mathematical induction to prove propositions related to arithmetic and geometric sequences.
• Cross-disciplinary application: Use sequence models to solve real-world problems such as Halley’s Comet return cycles, basic reproduction number R_0, and radioactive decay.

🔹 Lesson 2: Derivatives of Single-Variable Functions and Their Applications

Overview: This instructional design transitions from a comprehensive review of sequences—including the famous Collatz Conjecture and analogical reasoning—to the core concept of derivatives. The focus lies in understanding the shift from "average rate of change" to "instantaneous rate of change," defining derivatives via limits, and mastering their significance in geometry (slope of tangent lines) and physics (instantaneous velocity and acceleration). The lesson concludes with practical application of derivative computations for basic elementary functions and intuitive understanding of derivative function graphs.

Learning Outcomes:

• Apply analogical reasoning to explore relationships between arithmetic and geometric sequences, and understand the operational rules of the "Collatz Conjecture."
• Deeply comprehend the limit-based definition of derivatives, and distinguish between average rate of change and instantaneous rate of change.
• Grasp the geometric meaning of derivatives, internalize the mathematical idea of "approximating curves with straight lines," and proficiently use basic elementary function derivative formulas for calculations.