📚 Year 10 OCR Mathematics: Core Topic Breakdown | Year 10 OCR 数学：核心知识点梳理
In Year 10 of the OCR GCSE Mathematics course, students consolidate and extend their understanding of fundamental mathematical concepts. This article provides a structured breakdown of the core topics, helping you to identify key areas, build confidence, and prepare effectively for assessments. From number skills and algebra to geometry, statistics and probability, mastering these foundations is essential for success in Year 11 and final exams.
在OCR GCSE数学课程的十年级阶段，学生需要巩固并拓展对基础数学概念的理解。本文系统梳理了核心知识点，帮助你抓住重点、建立信心，并有效准备各种评估。从数字运算和代数，到几何、统计与概率，扎实掌握这些基础对于十一年级学习和最终考试的成功至关重要。
1. Number: Fractions, Decimals and Percentages | 数字：分数、小数与百分数
You must be able to fluently convert between fractions, decimals and percentages. For example, to convert a fraction to a decimal, divide the numerator by the denominator; to convert a percentage to a fraction, write it over 100 and simplify.
你必须熟练掌握分数、小数和百分数之间的转换。例如，将分数转换为小数，用分子除以分母；将百分数转换为分数，除以100并化简。
Adding and subtracting fractions require a common denominator. Multiply numerators and denominators when multiplying fractions, and use ‘keep, change, flip’ for division.
分数加减需要通分。分数相乘时分子乘分子、分母乘分母，除法使用‘保持不变、变除为乘、翻转除数’法则。
Find a percentage of an amount by using a multiplier (e.g., 15% increase → ×1.15). For reverse percentages, divide by the original multiplier to find the amount before a change.
使用乘数（如15%增长→×1.15）求某数的百分比。逆向百分数问题中，用原乘数去除以求得变化前的量。
2. Surds and Indices | 根式与指数
Simplify surds by expressing the number under the root as a product of a square number and another integer, e.g., √50 = √(25×2) = 5√2.
化简根式，将根号下的数分解为一个平方数与另一个整数的乘积，例如 √50 = √(25×2) = 5√2。
Rationalise denominators by multiplying the numerator and denominator by the conjugate surd, e.g., 1/(√a + b) → multiply by (√a − b)/(√a − b).
分母有理化，将分子分母同乘共轭根式，如 1/(√a + b) → 乘以 (√a − b)/(√a − b)。
Apply the laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.
运用指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。
3. Algebraic Expressions and Expanding Brackets | 代数表达式与去括号
Simplify expressions by collecting like terms, e.g., 3x + 2y − x + 5y = 2x + 7y.
合并同类项化简表达式，例如 3x + 2y − x + 5y = 2x + 7y。
Expand single brackets by multiplying each term inside by the term outside: a(b + c) = ab + ac. For double brackets, use FOIL or the grid method: (x + a)(x + b) = x² + (a+b)x + ab.
单项括号去括号：用外面的项乘以括号内的每一项：a(b + c) = ab + ac。双括号展开用FOIL或表格法：(x + a)(x + b) = x² + (a+b)x + ab。
4. Factorising and Solving Quadratic Equations | 因式分解与解二次方程
Factorise quadratics of the form x² + bx + c by finding two numbers that multiply to c and add to b. For example, x² + 5x + 6 = (x + 2)(x + 3).
对形如 x² + bx + c 的二次式进行因式分解，找出两个乘积为c、和为b的数。例如，x² + 5x + 6 = (x + 2)(x + 3)。
Solve quadratic equations by factorising, then setting each bracket equal to zero. For x² − 4x = 0, factorise to x(x − 4) = 0 → solutions x = 0 or x = 4.
通过因式分解解二次方程，然后令每个因式为零。例如 x² − 4x = 0，分解为 x(x − 4) = 0 → 解为 x = 0 或 x = 4。
Also use the quadratic formula when factorisation is not straightforward: x = [−b ± √(b² − 4ac)] / (2a).
当因式分解不易时，可用二次公式：x = [−b ± √(b² − 4ac)] / (2a)。
5. Linear Equations and Inequalities | 线性方程与不等式
Solve linear equations with unknowns on both sides by balancing, e.g., 2x + 3 = x − 5 → x = −8.
通过等式平衡解含未知数两边的线性方程，如 2x + 3 = x − 5 → x = −8。
Solve linear inequalities similarly, but remember to reverse the inequality sign when multiplying or dividing by a negative number. Represent solutions on a number line.
解线性不等式方法类似，但注意乘以或除以负数时，不等号方向要改变。解集表示在数轴上。
6. Sequences: nth Term | 数列：第n项
Find the nth term of a linear (arithmetic) sequence: if the term-to-term difference is d, the expression is dn + (a − d), where a is the first term.
求线性（等差）数列的第n项：若公差为d，则表达式为 dn + (a − d)，a为首项。
Recognise quadratic sequences where the second difference is constant. The nth term is of the form an² + bn + c; find a as half the second difference.
辨别二次序列，其二阶差分为常数。第n项形式为 an² + bn + c；a等于二阶差分的一半。
7. Ratio, Proportion and Direct/Inverse Proportion | 比、比例与正反比例
Simplify ratios by dividing by common factors, and divide a quantity in a given ratio. Use the unitary method for proportion problems.
通过除以公因数化简比，并按给定比例分配数量。用归一法解决比例问题。
Direct proportion: y ∝ x → y = kx. Inverse proportion: y ∝ 1/x → y = k/x. Use given values to find constant k.
正比关系：y ∝ x → y = kx。反比关系：y ∝ 1/x → y = k/x。利用已知值求常数k。
8. Geometry: Angles and Polygons | 几何：角度与多边形
Angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°. Vertically opposite angles are equal.
角度基本性质：直线上角和为180°，绕一点周角360°。对顶角相等。
Interior and exterior angles of polygons: sum of exterior angles = 360°. Interior angle = 180° − exterior angle. Sum of interior angles = (n − 2) × 180°.
多边形的内角与外角：外角和为360°。内角 = 180° − 外角。内角和 = (n − 2) × 180°。
9. Perimeter, Area and Volume | 周长、面积与体积
Recall and use formulas: area of triangle = ½ × base × height, area of trapezium = ½(a+b)h, area of circle = πr², circumference = 2πr or πd.
熟记并运用公式：三角形面积 = ½ × 底 × 高，梯形面积 = ½(a+b)h，圆面积 = πr²，圆周长 = 2πr 或 πd。
Volume of prisms = area of cross-section × length. Volume of pyramid = ⅓ × base area × height. Surface area is the sum of area of faces.
棱柱体积 = 横截面积 × 长。棱锥体积 = ⅓ × 底面积 × 高。表面积为各个面的面积之和。
10. Pythagoras’ Theorem and Basic Trigonometry | 勾股定理与基础三角学
Pythagoras’ theorem: in a right-angled triangle, a² + b² = c², where c is the hypotenuse. Use to find missing sides.
勾股定理：在直角三角形中，a² + b² = c²，c为斜边。用于求未知边长。
Trigonometric ratios: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. Use SOH CAH TOA to remember. Apply to find angles and sides.
三角比：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。用 SOH CAH TOA 记忆。用于求角度和边长。
11. Statistics: Averages and Data Representation | 统计：平均数与数据表示
Calculate mean, median, mode and range from a list or frequency table. Mean = sum of values ÷ number of values; median is the middle value when ordered.
从列表或频数表计算平均数、中位数、众数和极差。平均数 = 数值总和 ÷ 数据个数；中位数为排序后中间的值。
Construct and interpret bar charts, pie charts, and scatter graphs. Line of best fit on scatter graphs shows correlation.
构建并解读条形图、饼图和散点图。散点图上的最佳拟合线展示相关性。
12. Probability: Simple and Combined Events | 概率：简单事件与复合事件
Probability of an event = number of favourable outcomes / total number of outcomes. Probabilities always between 0 and 1.
事件概率 = 有利结果数 / 总结果数。概率值始终介于0和1之间。
For combined events, use sample space diagrams, two-way tables, and tree diagrams. Remember to multiply along branches for ‘and’ and add for ‘or’ when events are mutually exclusive.
复合事件使用样本空间图、双向表和树形图。记住：对于‘且’分支相乘，对于互斥事件‘或’时相加。
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