📚 Year 10 Edexcel Maths: Core Knowledge Points Review | Year 10 Edexcel 数学：核心知识点梳理
Year 10 is a pivotal year in the Edexcel GCSE Mathematics course. Students consolidate foundational skills and begin to tackle more complex, interconnected topics across Number, Algebra, Geometry, Statistics and Probability. A solid grasp of these core concepts is essential for success in Year 11 and the final examinations. This article provides a structured review of the key knowledge points covered in Year 10, with clear explanations and relevant examples.
十年级是 Edexcel GCSE 数学课程的关键年。学生既要巩固基础技能，又要开始处理更复杂、相互关联的课题，涵盖数字、代数、几何、统计和概率等。扎实掌握这些核心概念是十一年级和最终考试取得成功的基石。本文梳理了十年级的核心知识点，配以清晰的解释和相关例子。
1. Number: Types, Operations and Applications | 数字：类型、运算与应用
Students must be confident working with integers, fractions, decimals and percentages. They should perform calculations with directed numbers, apply the order of operations (BIDMAS/BODMAS) and use written methods for multiplication and division.
学生需要熟练处理整数、分数、小数和百分比。他们应能进行有向数计算，应用运算顺序（括号、指数、乘除、加减），并使用笔算方法进行乘除运算。
Work with standard form (scientific notation) is also essential: express large and small numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Calculations with standard form require adding/subtracting indices when multiplying/dividing, and adjusting to ensure a remains within the required range.
掌握标准形式（科学记数法）也很关键：将大小数字表示为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。用标准形式进行计算时，乘法/除法需对指数进行加减，并调整使 a 保持在要求范围内。
In addition, students encounter surds and their simplification. They learn to manipulate expressions involving √, simplify surds such as √48 = 4√3, rationalise denominators like 5/√2, and use exact values in geometric problems.
此外，学生会接触到根式及其化简。他们学习处理含有 √ 的式子，化简根式如 √48 = 4√3，进行分母有理化如 5/√2，并在几何题中使用精确值。
2. Algebra: Manipulating Expressions | 代数：表达式变换
Algebraic fluency is a cornerstone of Year 10. Students expand brackets, including double brackets and expressions of the form (x + a)(x + b), leading to quadratic expressions. Factorisation includes taking out a common factor, factorising quadratics of the form x² + bx + c, and recognising the difference of two squares: a² − b² = (a + b)(a − b).
代数熟练度是十年级的基石。学生展开括号，包括双括号以及形如 (x + a)(x + b) 的式子，从而得到二次表达式。因式分解包括提取公因式、分解形如 x² + bx + c 的二次式，并识别平方差：a² − b² = (a + b)(a − b)。
They also simplify algebraic fractions by factorising numerators and denominators, change the subject of a formula, and work with indices laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.
他们也通过因式分解分子分母来化简代数分式，变更公式中的主项，以及运用指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。
3. Algebra: Solving Equations and Inequalities | 代数：解方程与不等式
Linear equations with unknowns on both sides, including those with fractions and brackets, are solved using inverse operations. Pupils learn to set up equations from word problems. The solution to an equation is the value that makes the statement true.
含有未知数在两侧的线性方程，包括含分数和括号的方程，通过逆运算求解。学生学会根据文字题列出方程。方程的解就是使等式成立的那个值。
Quadratic equations are solved by factorising, using the null factor law: if (x − p)(x − q) = 0, then x = p or x = q. For non‑factorisable quadratics, they may explore the quadratic formula:
二次方程通过因式分解求解，利用零因子律：若 (x − p)(x − q) = 0，则 x = p 或 x = q。对于不能因式分解的二次方程，他们可能探索二次公式：
x = (-b ± √(b² − 4ac)) / 2a
Simultaneous equations – both linear and one linear with one quadratic – are solved graphically and algebraically (elimination and substitution). Inequalities are represented on number lines and solved algebraically, remembering to reverse the inequality sign when multiplying or dividing by a negative.
联立方程——两个线性方程以及一个线性一个二次的情况——通过图像法和代数法（消元法和代入法）求解。不等式在数轴上表示并代数求解，切记当乘以或除以负数时要反转不等号。
4. Graphs: Straight Lines and Quadratic Curves | 图像：直线与二次曲线
Year 10 extends graphing skills to include plotting and interpreting straight‑line graphs in the form y = mx + c, where m is the gradient and c is the y‑intercept. Parallel and perpendicular lines are explored: parallel lines have equal gradients; the product of gradients of perpendicular lines is −1.
十年级将绘图技能扩展到绘制和解释直线图像，形式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。探讨平行线和垂直线：平行线斜率相等；垂直线斜率的乘积为 −1。
Quadratic functions y = x² + bx + c are plotted, identifying roots, y‑intercept and the turning point (minimum or maximum). Students also sketch simple cubic and reciprocal graphs, recognising their typical shapes. Real‑life graphs such as distance–time and speed–time are interpreted: gradient represents speed/acceleration, and area under a speed–time graph gives distance.
绘制二次函数 y = x² + bx + c 的图像，并识别根、y轴截距以及转折点（最小值或最大值）。学生也会勾勒简单的三次函数和反比例函数图像，辨认其典型形状。解释实际生活中的图像，如距离–时间图和速度–时间图：斜率代表速度/加速度，速度–时间图下的面积代表距离。
5. Ratio, Proportion and Percentages | 比、比例与百分比
Ratios are simplified and used to divide quantities into given parts. Students apply the unitary method and proportional reasoning to solve problems involving best buys, recipes and scale drawings. Direct proportion (y = kx) and inverse proportion (y = k/x) are modelled algebraically and graphically.
简化比，并用它按给定部分分配数量。学生运用归一法和比例推理解决涉及最划算购买、食谱和比例尺图的问题。正比例（y = kx）和反比例（y = k/x）以代数和图像方式建模。
Percentages are extended to repeated percentage change: compound interest and decay (e.g. depreciation). The multiplier method is used, applying a decimal multiplier raised to the power of the number of periods. Reverse percentages and original value problems are also covered.
百分比拓展到重复百分比变化：复利和衰减（如折旧）。使用乘数法，将小数乘数按周期数乘方。反推百分比和求原值问题也有涉及。
6. Geometry: Angles, Polygons and Parallel Lines | 几何：角、多边形与平行线
Angle facts are consolidated: angles on a straight line sum to 180°, around a point sum to 360°, vertically opposite angles are equal. For parallel lines, corresponding, alternate and co‑interior (allied) angles are used to deduce missing values.
巩固角的基本事实：平角为 180°，周角为 360°，对顶角相等。对于平行线，利用同位角、内错角和同旁内角推导缺失的角。
Interior and exterior angles of polygons are studied: sum of exterior angles of any convex polygon is 360°; sum of interior angles = (n − 2) × 180°. Regular polygons provide equal interior and exterior angles, calculated using these rules.
学习多边形的内角和外角：任何凸多边形的外角和为 360°；内角和 = (n − 2) × 180°。正多边形具有相等的内角和外角，可根据这些规则计算。
7. Geometry: Perimeter, Area and Volume | 几何：周长、面积与体积
Formulae for area and circumference of a circle (A = πr², C = 2πr or πd) are used along with sector area and arc length as fractions of a circle’s area/circumference. Composite shapes combining rectangles, triangles, circles and trapezia require students to break down figures and apply known area formulas.
运用圆的面积和周长公式（A = πr²，C = 2πr 或 πd），以及扇形面积和弧长（作为圆面积/周长的分数）。由矩形、三角形、圆和梯形组成的复合图形要求学生分解图形并应用已知面积公式。
For 3D shapes, volume of prisms is calculated as area of cross‑section × length. Surface area of cuboids, cylinders and prisms requires summing the areas of all faces. Cones, spheres and pyramids appear later, but Year 10 may introduce volume of a pyramid as ⅓ × base area × height.
对于三维图形，棱柱的体积计算为横截面积 × 长。长方体、圆柱和棱柱的表面积需要将所有面的面积相加。圆锥、球体、棱锥稍后出现，但十年级可能引入棱锥体积公式：⅓ × 底面积 × 高。
8. Transformations and Vectors | 变换与向量
Transformations include reflection (mirror line), rotation (centre, angle, direction), translation (vector) and enlargement (centre and scale factor, including fractional and negative scale factors). Describing a transformation fully requires all relevant details.
变换包括反射（镜像线）、旋转（中心、角度、方向）、平移（向量）和放大（中心与比例因子，包括分数和负比例因子）。完整描述一个变换需要给出所有相关细节。
Vectors are introduced as a column or component form, representing translations. Students add, subtract and multiply vectors by scalars, calculate magnitude using Pythagoras’ theorem, and find resultant vectors geometrically. Simple geometric proofs using vectors are also explored.
向量以列向量或分量形式引入，表示平移。学生进行向量的加法、减法、数乘，用勾股定理计算模长，并在几何上求合向量。也探索用向量进行简单几何证明。
9. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
Pythagoras’ theorem states that in a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It is used to find missing lengths and to check whether a triangle is right‑angled.
勾股定理指出，在直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²。它用于求未知边长以及检验三角形是否为直角三角形。
Trigonometric ratios (sin, cos, tan) are defined in right‑angled triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Students apply SOHCAHTOA to find unknown sides and angles. Exact values for 0°, 30°, 45°, 60° and 90° should be memorised.
三角比（正弦、余弦、正切）在直角三角形中定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。学生运用 SOHCAHTOA 求未知边和角。应熟记 0°、30°、45°、60° 和 90° 的精确值。
10. Probability | 概率
Probability is measured on a scale from 0 to 1, and the sum of probabilities for all mutually exclusive outcomes is 1. Students construct sample spaces and use them to calculate theoretical probabilities. Relative frequency is used to estimate probabilities from experiments.
概率以 0 到 1 之间的数值度量，所有互斥结果的概率之和为 1。学生构建样本空间并用其计算理论概率。相对频率用于根据实验估计概率。
Tree diagrams are essential for independent and dependent events: branch probabilities are multiplied along the path, and outcomes are added where appropriate. Conditional probability (P(A|B) = P(A and B)/P(B)) is introduced, often modelled with two‑way tables or Venn diagrams.
树状图对独立和相依事件至关重要：沿路径将分支概率相乘，并适当将结果相加。条件概率（P(A|B) = P(A 且 B)/P(B)）被引入，常借助双向表或维恩图来建模。
11. Statistics | 统计
Data handling includes collecting, organising and representing data in bar charts, pie charts, frequency polygons, scatter graphs and cumulative frequency diagrams. Students interpret and construct these diagrams, identifying trends and correlations.
数据处理包括收集、整理数据并用条形图、饼图、频率多边形、散点图和累积频率图表示。学生解读并绘制这些图表，识别趋势和相关性。
Averages and measures of spread are central: mean, median, mode and range for discrete and grouped data. From a frequency table, the mean is Σ(fx)/Σf, and the modal class interval is identified. Quartiles and interquartile range are found from cumulative frequency curves or by formula. Box plots provide a visual summary of the five‑number summary.
平均数和离散量数是核心：离散数据和分组数据的平均数、中位数、众数和极差。对于频数表，平均数 = Σ(fx)/Σf，并找出众数所在组区间。四分位数和四分位距可从累积频率曲线或公式求得。箱线图提供了五数概括的可视化摘要。
Scatter graphs are used to assess correlation (positive, negative, none) and draw a line of best fit to make predictions, understanding interpolation and extrapolation limits.
散点图用于评估相关性（正相关、负相关、无相关），并绘制最佳拟合线进行预测，理解内插和外推的局限性。
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