【People's Education Press】Junior High School Mathematics Grade 9 Part 1
This textbook is the standard mathematics curriculum for the first semester of ninth grade in junior high school, covering five core modules: quadratic equations, quadratic functions, rotation, circles, and an introduction to probability. It aims to develop students' logical thinking and mathematical modeling skills.
Course Overview
📚 Content Summary
This textbook is the standard mathematics curriculum for the first semester of Grade 9 in junior high school, covering five core modules: quadratic equations, quadratic functions, rotation, circles, and an introduction to probability. It aims to develop students’ logical thinking and mathematical modeling abilities.
Master the core logic of middle school mathematics and embark on an advanced exploration journey into functions and geometry.
Author: Lin Qun
Acknowledgments: Reviewed and approved by the Ministry of Education (2013), Second Prize for National Outstanding Textbooks at the First National Textbook Construction Award
🎯 Learning Objectives
- Identify quadratic equations, proficiently convert equations into standard form, and accurately determine the coefficients of the quadratic term, linear term, and constant term.
- Understand the meaning of roots of an equation and verify roots by substitution.
- Formulate quadratic equations based on real-world contexts such as geometric area or quantitative relationships, and master solving simple quadratic equations using the definition of square roots (foundation of completing the square).
- Understand and master the graphical characteristics of quadratic functions y=ax^2 and y=a(x-h)^2+k (opening direction, axis of symmetry, vertex).
- Skillfully apply the translation rule “add to the left, subtract from the right; add to the top, subtract from the bottom” to transform function expressions.
- Determine the expression of a quadratic function using the method of undetermined coefficients and explain the relationship between the x-axis intercepts of a parabola and the roots of the corresponding equation.
- Understand and master the properties of rotation, and the concepts of central symmetry and centrally symmetric figures; identify and locate centers of symmetry.
- Derive and memorize the coordinate characteristics of points symmetric about the origin, and perform basic rotational transformations of figures in the Cartesian coordinate system.
- Apply knowledge of rotation and symmetry to pattern design, and understand the mathematical principles behind rotationally symmetric figures in real life.
- Understand and master positional relationships: determine the position of a point relative to a circle or a line relative to a circle using numerical relationships (d and r).
Lessons 共 5 课时 · 预计 15.0h
Lessons
Lesson
本课程介绍了如何将生活情境转化为一元二次方程模型，重点讲解了方程的一般形式 $ax^2+bx+c=0$ ($a \neq 0$) 及其系数识别。通过学习方程的根的概念与验证方法，学生能够掌握将复杂代数式规范化并求解的逻辑思维。
本课程深入探讨了二次函数 $y=ax^2$ 的几何性质、平移规律及其与一元二次方程的内在联系。通过学习参数 $a$ 对开口与宽窄的影响、顶点式 $y=a(x-h)^2+k$ 的平移变换，以及利用待定系数法和判别式分析函数与 $x$ 轴的交点，学生将掌握数形结合的建模方法，并能运用二次函数解决实际生活中的最值问题。
本节课系统介绍了旋转变换的几何定义、核心要素及性质，重点探讨了中心对称图形的识别准则及其在坐标系中的代数规律。通过数形结合的方法，学生将掌握如何利用坐标变换（x, y）→（-x, -y）快速处理旋转问题，并理解旋转对称在几何优化与工业设计中的实际应用。
本课程系统讲解了圆的几何性质与度量计算，重点探讨了直线与圆、圆与圆的位置关系判定，以及切线的判定与性质定理。通过切线长定理与三角形内切圆的应用，学生将掌握如何利用数形结合与面积法解决复杂的几何综合问题。
本节课介绍了概率论的基础知识，通过区分必然事件、不可能事件与随机事件，学习了如何利用频率的稳定性在大量试验中估计概率。学生将掌握概率的定义及其在 $[0, 1]$ 区间的取值规律，并学会通过几何模型和统计数据分析解决实际生活中的可能性问题。