【People's Education Press】Junior High School Mathematics Grade 8 Part 1

📚 Content Summary

This textbook is part of the compulsory education curriculum, designed for eighth-grade students in their first semester. The content covers the properties of triangles, criteria and properties of congruent triangles, axisymmetric figures, operations with polynomials including multiplication and factorization, and fundamental operations and equations involving rational expressions. The goal is to develop students' geometric logical reasoning abilities and algebraic computational skills.

Explore the beauty of geometric proofs, master the core of algebraic computation.

Author: Lin Qun

Acknowledgments: Approved by the Ministry of Education 2013, Second Prize in the National Excellent Textbook Award, First National Textbook Construction Award

🎯 Learning Objectives

1. Identification and Construction: Accurately identify and draw altitudes, medians, and angle bisectors of triangles; understand the concept of centroid.
2. Conceptual Analysis: Understand the stability of triangles and the instability of quadrilaterals, and explain their real-world applications.
3. Computation and Reasoning: Master the use of the triangle angle sum theorem and exterior angle properties for angle calculations; know the formulas for interior and exterior angles of polygons and solve practical polygon problems.
4. Master Concepts and Properties: Accurately identify corresponding vertices, sides, and angles of congruent triangles, and apply “corresponding sides are equal, corresponding angles are equal” in computations and proofs.
5. Master Criteria: Skillfully apply SSS and SAS to determine triangle congruence, and write formal proof processes.
6. Understand Logical Limits: Through exploration, recognize that SSA (side-side-angle) cannot be used as a criterion for congruence, fostering rigorous geometric logic.
7. Master Axis Symmetry Concept: Identify axisymmetric figures and accurately draw the symmetric image of a given figure about a specified line.
8. Deepen Understanding of Isosceles/Equilateral Triangles: Master the properties “equal sides imply equal angles” and “three lines coincide,” and use “equal angles imply equal sides” to identify isosceles triangles.
9. Computation and Proof: Apply the 30°-angle property of right triangles for length calculations, and understand the inequality relationship between larger sides and larger angles in triangles.
10. Proficiently Apply Exponent Rules: Accurately perform calculations using the rules for multiplying powers with the same base, power of a power, and power of a product.

🔹 Lesson 1: Basic Properties of Triangles and Exploration of Polygons

Overview: This lesson covers the core content of Chapter Eleven in junior high school mathematics. Starting from basic segments in triangles (altitudes, medians, angle bisectors) and their stability, it explores the interior and exterior angle properties of triangles, ultimately extending to the formulas for interior and exterior angles of polygons and the tiling of the plane using regular polygons.

Learning Outcomes:

• Identification and Construction: Accurately identify and draw altitudes, medians, and angle bisectors of triangles; understand the concept of centroid.
• Conceptual Analysis: Understand the stability of triangles and the instability of quadrilaterals, and explain their real-world applications.
• Computation and Reasoning: Skillfully apply the triangle angle sum theorem and exterior angle properties for angle calculations; master the formulas for interior and exterior angles of polygons and solve real-world polygon problems.

🔹 Lesson 2: Criteria and Applications of Congruent Triangles

Overview: This course aims to help second-year junior high students grasp the core concepts, properties, and foundational criteria (SSS and SAS) for congruent triangles. By observing geometric transformations such as translation, reflection, and rotation, students will learn to identify corresponding elements and use triangle congruence properties to solve problems involving lengths and angles. The lesson also explores the derivation and application of the SSS and SAS congruence theorems.

Learning Outcomes:

• Master Concepts and Properties: Accurately identify corresponding vertices, sides, and angles of congruent triangles, and use “corresponding sides are equal, corresponding angles are equal” in computations and proofs.
• Master Criteria: Skillfully apply SSS and SAS to determine triangle congruence and write formal proof procedures.
• Understand Logical Limits: Through exploration, understand why SSA (side-side-angle) cannot serve as a criterion for congruence, cultivating rigorous geometric logic.

🔹 Lesson 3: Axis Symmetry Transformations and Properties of Isosceles Triangles

Overview: This lesson covers the core theory of axis symmetry transformations and their in-depth application in isosceles triangles. Students will begin with the symmetry of figures, gradually mastering the properties and criteria for isosceles and equilateral triangles, and extending to the special properties of 30° angles in right triangles and the inequality relationships among sides and angles within triangles.

Learning Outcomes:

• Master Axis Symmetry Concept: Identify axisymmetric figures and accurately draw the symmetric image of a given figure about a specified line.
• Deepen Understanding of Isosceles/Equilateral Triangles: Skillfully apply the properties “equal sides imply equal angles” and “three lines coincide,” and use “equal angles imply equal sides” to identify isosceles triangles.
• Computation and Proof: Use the 30°-angle property of right triangles for length calculations, and understand the inequality relationship where larger sides correspond to larger angles in triangles.

🔹 Lesson 4: Polynomial Multiplication and Factoring Techniques

Overview: This lesson provides a complete knowledge system covering exponent operations, polynomial multiplication formulas, and their reverse process—factoring. Students will master the basic rules for multiplying powers with the same base, powers of powers, and powers of products. They will then learn to use difference of squares and perfect square formulas for rapid polynomial multiplication, and finally gain mastery over core techniques for factoring polynomials.

Learning Outcomes:

• Proficiently apply the rules for multiplying powers with the same base, powers of powers, and powers of products for accurate calculations.
• Flexibly use the difference of squares and perfect square formulas for polynomial multiplication and simplification.
• Understand the basic concept of factoring and skillfully apply common factor extraction and formula methods (difference of squares, perfect square) to fully factor polynomials.

🔹 Lesson 5: Rational Expression Operations and Solving Rational Equations

Overview: This lesson covers the core knowledge system of rational expressions in junior high school mathematics, starting from the basic concepts and conditions for existence of rational expressions, delving into equivalent transformations such as simplification and common denominator operations. Building on this foundation, students will master operation rules for rational expressions, learn to represent small positive numbers using scientific notation, and ultimately focus on solving rational equations.

Learning Outcomes:

• Understand the concept of rational expressions, master conditions for meaningful expressions, and perform simplification, common denominator operations, and reduction to simplest form.
• Perform mixed operations including multiplication, division, exponentiation, and addition/subtraction of rational expressions proficiently, and use scientific notation to represent positive numbers less than 1.
• Grasp the fundamental approach to solving rational equations (transformation thinking), skillfully eliminate denominators to solve equations, and verify solutions.