【People's Education Press】Junior High School Mathematics Grade 9 Part 2

📚 Content Summary

This is the final mathematics textbook for middle school, primarily covering four core modules: inverse proportion functions, similarity, trigonometric functions of acute angles, and projections and views. The course emphasizes understanding mathematical concepts through observation, exploration, and logical reasoning, and applying mathematical models to describe patterns, perform measurements, and solve spatial cognition problems in real-life contexts.

Explore the mysteries of functions and geometric logic—the concluding chapter in building core mathematical competencies for middle school.

Author: Lin Qun

Acknowledgments: This book was co-edited by Tian Zaijin, Xue Bin, and Li Haidong, with Zhang Jianyue as chief editor, and published by People's Education Press.

🎯 Learning Objectives

1. Understand concepts and expressions: Recognize inverse proportion functions, master the three forms y = \frac{k}{x}, y = kx^{-1}, and xy = k, and clearly identify the constraints k \neq 0 and x \neq 0.
2. Master graphs and properties: Accurately describe the quadrant distribution and monotonicity of inverse proportion function graphs, and understand the symmetry of hyperbolas.
3. Apply the method of undetermined coefficients: Use the method of undetermined coefficients to determine the expression of an inverse proportion function based on given point coordinates or graph information.
4. Understand and determine: Accurately define similar figures, identify the center of similarity by connecting corresponding points, and calculate the similarity ratio using side length ratios.
5. Coordinate operations: Master the coordinate transformation rules when the origin is the center of similarity (multiples of k or -k), and perform scaling operations (enlargement and reduction) of figures in the coordinate plane.
6. Comprehensive application: Skillfully apply the properties of similar polygons (equal corresponding angles, proportional side lengths) and triangle similarity criteria to solve problems involving height measurement, area calculation, and complex geometric proofs.
7. Master definitions and special values: Accurately understand and memorize the definitions of sine, cosine, and tangent, and fluently recall the trigonometric values for 30°, 45°, and 60° angles.
8. Develop problem-solving skills: Proficiently use scientific calculators to compute values, and apply the Pythagorean theorem and trigonometric relationships to solve right triangles.
9. Practical application ability: Identify and apply concepts such as elevation angle, depression angle, slope, and azimuth to solve real-world problems including height measurement, navigation avoidance of reefs, and engineering slope construction.
10. Understand the definitions of projection, parallel projection, central projection, and orthographic projection, and master the properties of points, lines, and planes under orthographic projection.

🔹 Lesson 1: Properties and Comprehensive Application Practice of Inverse Proportion Functions

Overview: This lesson aims to guide students in abstracting mathematical models of inverse proportion functions from real-world problems (such as relationships between speed, pressure, and area), deeply exploring the characteristics of the expression y = \frac{k}{x} (k \neq 0), the shape of its graph (hyperbola), and how its properties change with the coefficient k. Through the method of undetermined coefficients and the integration of algebra and geometry, students will learn how to determine the expression and analyze the combined graphical relationship between inverse proportion and linear functions within the same coordinate system.

Learning Outcomes:

• Understand concepts and expressions: Recognize inverse proportion functions, master the three forms y = \frac{k}{x}, y = kx^{-1}, and xy = k, and clearly identify the constraints k \neq 0 and x \neq 0.
• Master graphs and properties: Accurately describe the quadrant distribution and monotonicity of inverse proportion function graphs, and understand the symmetry of hyperbolas.
• Apply the method of undetermined coefficients: Determine the expression of an inverse proportion function using the method of undetermined coefficients based on given point coordinates or graph information.

🔹 Lesson 2: Practice in Similarity Determination and Homothetic Transformations

Overview: This lesson focuses on homothetic transformations of figures and comprehensive applications of similar shapes. Students will learn how to define and locate the center of homothety and the similarity ratio, master the coordinate calculation rules for homothetic transformations in the Cartesian coordinate system, and deeply understand the similarities and differences among translation, rotation, reflection, and homothety. Through practical cases such as The Sea Island Mathematical Manual, students will enhance their ability to apply similarity theorems to solve real-world problems involving height measurement, navigation, and industrial processing.

Learning Outcomes:

• Understand and determine: Accurately define similar figures, identify the center of similarity by connecting corresponding points, and calculate the similarity ratio using side length ratios.
• Coordinate operations: Master the coordinate transformation rules when the origin is the center of homothety (multiples of k or -k), and perform scaling operations (enlargement and reduction) of figures in the coordinate plane.
• Comprehensive application: Skillfully apply the properties of similar polygons (equal corresponding angles, proportional side lengths) and triangle similarity criteria to solve problems involving height measurement, area calculation, and complex geometric proofs.

🔹 Lesson 3: Trigonometric Functions of Acute Angles and Applications in Right Triangle Measurement

Overview: This instructional design covers core ninth-grade mathematics content: the definition of trigonometric functions of acute angles and their wide-ranging applications in solving right triangles. From abstract ratio definitions to memorizing values for special angles, and then using calculators to handle complex numerical computations, the curriculum culminates in applying mathematical models to real-world scenarios such as navigation, construction, hydraulics, and field surveying. The focus lies in cultivating students’ ability to transform real-world problems into mathematical models based on right triangles.

Learning Outcomes:

• Master definitions and special values: Accurately understand and memorize the definitions of sine, cosine, and tangent, and fluently recall the trigonometric values for 30°, 45°, and 60° angles.
• Develop problem-solving skills: Proficiently use scientific calculators to compute values, and apply the Pythagorean theorem and trigonometric relationships to solve right triangles.
• Practical application ability: Identify and apply concepts such as elevation angle, depression angle, slope, and azimuth to solve real-world problems including height measurement, navigation avoidance of reefs, and engineering slope construction.

🔹 Lesson 4: Projection Rules, Reconstruction from Views, and 3D Modeling

Overview: This course systematically introduces the fundamental theory of projection, with a focus on the properties of orthographic projection and its central role in the "three-view" system. Students will learn to draw and interpret three views following the principles of "length alignment, height levelness, width equality," and reconstruct 3D solids from three views to perform engineering calculations of surface area and volume. Finally, through project-based learning, students will transform 2D drawings into 3D models, achieving a leap from theory to practice.

Learning Outcomes:

• Understand the definitions of projection, parallel projection, central projection, and orthographic projection, and master the properties of points, lines, and planes under orthographic projection.
• Master the formation principles, positional layout, and projection relationships (length alignment, height levelness, width equality) of the three views (front, left, top).
• Reconstruct 3D solids from three views, and use unfoldings to calculate surface area and volume of complex geometric solids (e.g., tents, sealed cans).