【People's Education Press】Junior High School Mathematics Grade 8, Lower Term

📚 Content Summary

This course covers the core content of the eighth-grade lower secondary mathematics curriculum, focusing on quadratic radicals, the Pythagorean theorem, parallelograms, linear functions, and data analysis. Through theoretical exploration and mathematical activities, it aims to develop students’ logical reasoning and problem-solving abilities.

Deepen mathematical thinking, master the core mysteries of algebra and geometry.

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

Acknowledgments: This book is developed according to the "Compulsory Education Mathematics Curriculum Standards (2011 Edition)" issued by the Ministry of Education.

🎯 Learning Objectives

1. Understand and apply the multiplication rule for quadratic radicals (\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}) and division rule (\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}) for computation and simplification.
2. Identify and simplify radicands into "simplest quadratic radicals," mastering two core criteria for simplification.
3. Master the addition and subtraction rules for quadratic radicals, able to combine like radicals by analogy with combining like terms in polynomial operations.
4. Understand and master multiple proof methods of the Pythagorean theorem (e.g., Zhao Shuang’s string diagram), and apply the theorem to represent irrational numbers on the number line.
5. Understand the concepts of original and converse propositions, and be able to prove and apply the converse of the Pythagorean theorem to determine right triangles.
6. Gain preliminary understanding of the derivation and application of Heron-Qin Jiushao’s formula, and have basic awareness of mathematical history such as Fermat’s Last Theorem.
7. Understand and master the properties (sides, angles, diagonals) and determination theorems of parallelograms, rhombuses, and squares.
8. Grasp the concept of distance between two parallel lines and its application in geometric proofs.
9. Master the properties of the midline of a triangle and apply them to solve problems involving position and length relationships of segments.
10. Master the graphical method: Accurately draw function graphs using the plotting method (listing, plotting points, connecting lines), and extract information from the graph.

🔹 Lesson 1: Quadratic Radicals

Overview: This lesson centers on the four arithmetic operations of quadratic radicals and their simplification standards. Students will explore and discover the multiplication and division rules of quadratic radicals, understand and master the definition of “simplest quadratic radicals,” and learn to perform addition and subtraction operations by analogy with combining like terms in polynomials, thus building a complete system of quadratic radical operations.

Learning Outcomes:

• Understand and apply the multiplication rule for quadratic radicals (\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}) and division rule (\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}) for calculation and simplification.
• Identify and convert radicands into “simplest quadratic radicals,” mastering the two core criteria for simplification.
• Master the addition and subtraction rules for quadratic radicals, able to combine radicals with identical radicands by analogy with combining like terms in polynomial operations.

🔹 Lesson 2: The Pythagorean Theorem

Overview: This course delves deeply into the theoretical framework of the Pythagorean theorem, starting from various geometric proof methods (such as the string diagram and trapezoid method), extending to logical judgment via its converse. It covers how to geometrically construct irrational numbers on the number line, introduces the application of Heron-Qin Jiushao’s formula in calculating triangle areas, and ultimately expands students' mathematical horizons through Fermat’s Last Theorem, establishing connections from fundamental geometry to cutting-edge number theory.

Learning Outcomes:

• Understand and master multiple proof methods of the Pythagorean theorem (e.g., Zhao Shuang’s string diagram), and apply the theorem to represent irrational numbers on the number line.
• Master the concepts of original and converse propositions, and be able to prove and apply the converse of the Pythagorean theorem to identify right triangles.
• Gain preliminary understanding of the derivation and application of Heron-Qin Jiushao’s formula, and have basic awareness of mathematical history such as Fermat’s Last Theorem.

🔹 Lesson 3: Parallelograms

Overview: This lesson aims to explore the core categories of quadrilaterals, focusing on the definitions, properties, and determination methods of parallelograms, rhombuses, and squares. Through logical reasoning and geometric proofs, students will grasp the evolutionary relationships among shapes and learn about the important geometric property of the triangle midline, as well as the definition of distance between parallel lines.

Learning Outcomes:

• Understand and master the properties (sides, angles, diagonals) and determination theorems of parallelograms, rhombuses, and squares.
• Master the concept of distance between two parallel lines and its application in geometric proofs.
• Master the properties of the midline of a triangle and apply them to solve problems involving segment position and length relationships.

🔹 Lesson 4: Linear Functions

Overview: This unit covers a deep exploration from graphical representation of functions to specific function models (proportional functions and linear functions). The course begins by establishing the concept of "shape" through the plotting method, then thoroughly studies the definitions, graphical characteristics, and algebraic properties of linear functions and their special form—proportional functions. Finally, by exploring the intrinsic connections between linear functions, equations, and inequalities, it applies mathematical modeling to real-life decision-making scenarios such as “choosing options.”

Learning Outcomes:

• Master the graphical method: Accurately draw function graphs using the plotting method (listing, plotting points, connecting lines), and extract information from the graph.
• Understand the essence of linear functions: Master the definitions, graphical features (meaning of k and b), and monotonicity of proportional and linear functions.
• Comprehensive application ability: Construct linear function models to solve real-world problems, and use functional perspectives to analyze solutions to systems of equations and inequalities.

🔹 Lesson 5: Data Analysis

Overview: This instructional design covers core statistical concepts for describing data distribution. It first uses measures such as mean (including weighted mean), median, and mode to characterize the “central tendency” of data; then employs variance to measure the “variability” of data; finally emphasizing the core idea of statistics—that sample characteristics are used to estimate population characteristics. These tools collectively form the foundation for quantitative data analysis.

Learning Outcomes:

• Understand the difference between arithmetic mean and weighted mean, and compute weighted means based on the practical meaning of weights.
• Accurately identify or calculate the median and mode of a data set, and interpret their statistical significance in specific contexts.
• Master the formula for calculating variance, and understand the relationship between variance and the degree of data fluctuation.