Year 10 OCR Maths: Core Topics Overview | Year 10 OCR 数学：核心知识点梳理

📚 Year 10 OCR Maths: Core Topics Overview | Year 10 OCR 数学：核心知识点梳理

Year 10 is a crucial year for the OCR GCSE Mathematics course (J560), where students consolidate foundational skills and begin tackling the more demanding Higher Tier content. This article provides a structured walkthrough of the core topics you will encounter, from number operations to statistics. Understanding these building blocks will set you up for success in both the Foundation and Higher papers, as many concepts interlink across different areas of the syllabus.

对于 OCR GCSE 数学课程（代码 J560）而言，十年级是承上启下的关键一年：学生既要巩固基础技能，又要开始接触更具挑战性的进阶内容。本文系统梳理了从数与运算到统计的核心知识点。无论你参加基础卷还是高阶卷，这些知识模块都相互关联，牢牢掌握它们将为后续学习打下扎实的基础。

1. Number and Arithmetic | 数与算术

You must be confident with the four operations (addition, subtraction, multiplication and division) for integers, fractions and decimals. Key skills include finding prime factors, highest common factor (HCF) and lowest common multiple (LCM) using factor trees, and applying the order of operations (BIDMAS/BODMAS). Negative number calculations often cause errors, so practise questions involving −3 × 5 and 4 − (−7) carefully.

你必须熟练掌握整数、分数和小数的四则运算（加、减、乘、除）。核心技能包括利用因数树求质因数、最大公因数（HCF）和最小公倍数（LCM），并正确运用运算顺序（BIDMAS/BODMAS）。负数运算经常失分，要仔细练习诸如 −3 × 5 和 4 − (−7) 之类的题目。

Upper and lower bounds are introduced in contexts of rounding and measurement. When a length is given as 12 cm to the nearest cm, the lower bound is 11.5 cm and the upper bound is 12.5 cm. This feeds into error intervals and later into bounds for calculations.

在近似值和测量中开始接触上下界的概念。若某长度四舍五入到整厘米记为 12 cm，其下界为 11.5 cm，上界为 12.5 cm。据此可写出误差区间，并进一步处理涉及四则运算的界值问题。

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2. Fractions, Decimals and Percentages | 分数、小数与百分数

Converting fluently between fractions, decimals and percentages is essential. For example, 3/8 = 0.375 = 37.5%. You should be able to find a percentage of an amount, increase or decrease by a percentage, and calculate percentage change. Reverse percentages (finding the original amount before a percentage change) are a common exam challenge: if a price after a 20% reduction is £64, the original price is £64 ÷ 0.8 = £80.

在分数、小数和百分数之间熟练转换是基本要求。例如 3/8 = 0.375 = 37.5%。要能够求一个数的百分之几、按百分比增减以及计算百分比变化。逆向百分数（已知增减后的量求原值）是考试中常见的难点：若降价 20% 后价格为 £64，原价就是 £64 ÷ 0.8 = £80。

Repeated percentage changes, such as compound interest and depreciation, use a multiplier raised to a power. The formula for compound interest is Amount = P × (1 + r/100)ⁿ, where n is the number of years. This overlaps with the topic of growth and decay.

重复百分比变化，比如复利和折旧，要用到乘方运算。复利计算公式为 金额 = P × (1 + r/100)ⁿ，其中 n 代表年数。这与增长与衰减的内容相互关联。

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3. Indices and Standard Form | 指数与标准形式

The laws of indices are vital: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Negative indices mean reciprocals, e.g. 2⁻³ = 1/2³ = 1/8. Fractional indices link to roots: 8¹/³ = ∛8 = 2. OCR expects you to simplify expressions like (3x²)³ = 27x⁶ confidently.

指数法则是重中之重：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。负指数表示倒数，如 2⁻³ = 1/2³ = 1/8。分数指数与根式相通：8¹/³ = ∛8 = 2。OCR 要求能够熟练化简类似 (3x²)³ = 27x⁶ 的表达式。

Standard form is written as A × 10ⁿ where 1 ≤ A < 10. It is used for very large or very small numbers, such as the speed of light 3.0 × 10⁸ m/s. You must be able to convert, order and calculate with standard form without a calculator on the non-calculator papers.

标准形式写为 A × 10ⁿ（1 ≤ A < 10）。它用于表示极大或极小的数，如光速 3.0 × 10⁸ m/s。你应能在非计算器试卷上完成标准形式之间的转换、排序和四则运算。

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4. Algebraic Expressions and Equations | 代数表达式与方程

Simplifying algebraic expressions includes collecting like terms, expanding brackets and factorising. Double bracket expansion: (x + 3)(x − 2) = x² + x − 6. Factorising quadratics in the form x² + bx + c is a key skill; for Higher Tier, you also factorise ax² + bx + c. Solving linear equations like 3x − 7 = 2x + 5 leads to x = 12, and you must check your solution.

代数式的化简包括合并同类项、展开括号和因式分解。双括号展开：(x + 3)(x − 2) = x² + x − 6。形如 x² + bx + c 的二次式因式分解是核心技能；进阶卷还要求处理 ax² + bx + c 的分解。解一元一次方程如 3x − 7 = 2x + 5 得到 x = 12，并需检验答案。

Forming equations from worded problems is frequently tested. For example, “The perimeter of a rectangle is 28 cm; length is 2 cm more than width” leads to 2(w + 2) + 2w = 28, so w = 6 cm. Simultaneous equations are solved by elimination or substitution; Higher students also meet one linear and one quadratic.

根据文字题列出方程是常考题型。例如，“矩形周长为 28 cm，长比宽多 2 cm”可列出 2(w + 2) + 2w = 28，解得 w = 6 cm。联立方程组通过消元法或代入法求解；进阶学生还会遇到一个线性、一个二次的方程组。

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5. Inequalities | 不等式

Solving linear inequalities is similar to equations, but remember to reverse the sign when multiplying or dividing by a negative number. E.g., −2x < 8 ⇒ x > −4. Representing inequalities on a number line uses open and closed circles. You also need to list integer values that satisfy combined inequalities like −3 ≤ 2x + 1 < 5.

解一元一次不等式与解方程类似，但要注意当乘或除以负数时，不等号方向要改变。例如 −2x < 8 ⇒ x > −4。在数轴上表示不等式时需使用空心和实心圆点。还要能列出满足组合不等式（如 −3 ≤ 2x + 1 < 5）的所有整数解。

Quadratic inequalities, such as x² − 4x − 5 > 0, are Higher Tier material. Solve the quadratic equation to find critical values, then test regions or sketch the graph to determine the solution set (x < −1 or x > 5).

二次不等式（例如 x² − 4x − 5 > 0）是进阶内容。先解相应二次方程求出临界值，再通过分段检验或画草图确定解集（x < −1 或 x > 5）。

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6. Sequences | 数列

Recognising and continuing sequences is a Foundation skill; you should spot patterns in linear sequences and simple quadratic sequences. The nth term of an arithmetic sequence is given by a + (n−1)d, where a is the first term and d the common difference. For OCR, you may need to find the nth term of a linear sequence like 7, 12, 17, 22… which is 5n + 2.

识别并续写数列是基础技能；你应能找出线性数列和简单二次数列的变化规律。等差数列的第 n 项公式为 a + (n−1)d，其中 a 为首项，d 为公差。OCR 考试中会要求写出线性数列 7, 12, 17, 22… 的第 n 项，即 5n + 2。

Quadratic sequences (Higher Tier) have a second difference that is constant. For the sequence 3, 10, 21, 36…, the nth term involves n². You derive the rule by halving the second difference to get the coefficient of n², then solving the remaining linear part.

二次数列（进阶）具有恒定的二次差分。对于数列 3, 10, 21, 36…，其第 n 项包含 n²。推导时先将二次差分除以 2 得到 n² 的系数，再解剩余线性部分。

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7. Graphs of Functions | 函数图像

Plotting straight-line graphs from y = mx + c is essential. m represents gradient and c the y-intercept. Parallel lines have the same gradient; perpendicular lines have gradients whose product is −1 (Higher). You should be able to find the midpoint and length of a line segment, as well as the equation of a line through two points.

根据 y = mx + c 绘制直线图像是核心内容。m 表示斜率，c 表示 y 轴截距。平行线斜率相等；垂直线斜率之积为 −1（进阶）。要会求线段的中点坐标和长度，以及根据两点写出直线方程。

Quadratic graphs (y = x²) are parabolas; solving quadratic equations graphically involves finding intersections with the x-axis. Higher students also sketch cubic and reciprocal functions, and use graphs to solve equations like x² − 2x − 3 = 0. Real-life graphs such as distance–time and velocity–time are examined, focusing on interpreting gradients and areas under graphs.

二次函数图像（y = x²）为抛物线；通过图像解二次方程即寻找与 x 轴的交点。进阶学生还要绘制三次函数和反比例函数草图，并利用图像求解诸如 x² − 2x − 3 = 0 的方程。生活情境图（如距离-时间图、速度-时间图）也常出现，侧重斜率与图像下面积的解读。

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8. Ratio, Proportion and Rates | 比、比例与比率

Ratios are used to compare quantities. Simplify ratios to their lowest terms and divide a quantity in a given ratio. For example, divide £50 in the ratio 2:3 gives 2/5 × £50 = £20 and 3/5 × £50 = £30. Direct proportion means y = kx; inverse proportion is y = k/x (Higher). Recognising proportional relationships from tables and graphs is tested.

比用来比较数量。要求能将比化为最简形式并按给定比例分配总量。例如，将 £50 按 2:3 分配，即 2/5 × £50 = £20，3/5 × £50 = £30。正比例表示为 y = kx；反比例表示为 y = k/x（进阶）。考试中会考查从表格和图像中识别比例关系。

Rates of change include speed (distance/time), density (mass/volume) and pressure (force/area). Convert compound units, e.g. from km/h to m/s. Best buy and recipe proportion questions require scaling quantities up or down using unitary methods.

变化率包括速度（距离/时间）、密度（质量/体积）和压强（力/面积）。要能转换复合单位，例如从 km/h 换算到 m/s。最佳购买方案和食谱比例题则需要运用归一法按比例增减量。

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9. Angles and Polygons | 角与多边形

Angle facts: on a straight line (180°), around a point (360°), vertically opposite angles equal. In parallel lines, alternate, corresponding and co-interior angles are used to find missing angles. Triangles: sum of interior angles = 180°. Polygons: sum of interior angles = (n − 2) × 180°, and each exterior angle of a regular polygon = 360°/n.

基本角度知识：平角为 180°，周角为 360°，对顶角相等。在平行线中，利用内错角、同位角和同旁内角求未知角。三角形内角和为 180°。多边形内角和公式为 (n − 2) × 180°；正多边形每个外角 = 360°/n。

Circle theorems (Higher) include: angle at centre is twice angle at circumference, angle in a semicircle is 90°, angles in the same segment are equal, and opposite angles in a cyclic quadrilateral sum to 180°. You will need to give reasons, not just state angle values.

圆定理（进阶）包括：圆心角等于圆周角的两倍、半圆上的圆周角为 90°、同弧上圆周角相等、圆内接四边形对角互补（和为 180°）。解答时不仅要写出角度值，还要陈述理由。

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10. Pythagoras and Trigonometry | 勾股定理与三角比

Pythagoras’ theorem: For a right-angled triangle, a² + b² = c², where c is the hypotenuse. OCR often embeds Pythagoras in 3D problems, requiring you to find the diagonal of a cuboid. Trigonometric ratios (SOHCAHTOA) apply to right-angled triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Know exact values for 0°, 30°, 45°, 60° and 90°.

勾股定理：对于直角三角形，a² + b² = c²（c 为斜边）。OCR 常将勾股定理嵌入到立体几何中，要求计算长方体的空间对角线。三角比（SOHCAHTOA）用于直角三角形：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。需熟记 0°、30°、45°、60° 和 90° 的精确值。

The sine rule and cosine rule (Higher) are used for non-right-angled triangles. Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A. They are applied to find missing sides or angles, and to solve bearing problems.

正弦定理和余弦定理（进阶）用于任意三角形。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。用于求解未知边或角度，也可解决方位角问题。

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11. Probability | 概率

Probability is measured on a scale from 0 to 1. The sum of probabilities of all possible outcomes is 1. You must understand mutually exclusive events and use the OR rule: P(A or B) = P(A) + P(B). Tree diagrams are essential for combined events; remember to multiply along branches for ‘AND’ probabilities and add for ‘OR’.

概率的范围从 0 到 1。所有可能结果的概率之和为 1。要理解互斥事件并使用加法法则：P(A 或 B) = P(A) + P(B)。树状图是处理复合事件的核心工具；沿分支相乘求“且”的概率，再将相关分支相加求“或”的概率。

Conditional probability (Higher) appears when an event affects the probability of another. The notation P(A|B) means ‘probability of A given B’. Tree diagrams for conditional probabilities show changing probabilities on the second branches. OCR also expects you to use Venn diagrams to record sample spaces and calculate probabilities.

条件概率（进阶）出现在一个事件影响另一个事件概率时。符号 P(A|B) 表示“在 B 发生的前提下 A 发生的概率”。条件概率的树状图中，第二层分支的概率会根据条件改变。OCR 还要求使用维恩图表示样本空间并计算概率。

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12. Statistics | 统计

Data handling includes designing data collection sheets, understanding types of data (qualitative/quantitative, discrete/continuous) and sampling methods. Averages: mode, median, mean and range are calculated from lists, frequency tables and grouped frequency tables. The mean from a grouped table uses the midpoint × frequency formula.

数据处理包括设计数据收集表，理解数据类型（定性/定量、离散/连续）和抽样方法。平均数：众数、中位数、平均数和极差需要从列表、频数表和分组频数表中计算。由分组表求平均数时需用组中点值 × 频数公式。

Graphical representations: bar charts, pie charts, scatter graphs and cumulative frequency diagrams (Higher). A line of best fit on a scatter graph illustrates correlation. Cumulative frequency graphs lead to estimating median, quartiles and interquartile range, and box plots are drawn to compare distributions. OCR emphasises interpreting these diagrams critically.

统计图表：条形图、饼图、散点图和累积频数图（进阶）。散点图中的最佳拟合线用于表明相关性。累积频数图可用于估算中位数、四分位数和四分位距，据此绘制箱线图以比较不同数据分布。OCR 强调要在情境下对图表进行批判性解读。

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