📚 Year 10 Eduqas Maths: Core Knowledge Overview | 10年级Eduqas数学：核心知识点梳理
Year 10 marks a crucial stage in the Eduqas GCSE Mathematics course, where you consolidate the foundations built in earlier years and begin to tackle the deeper, interconnected concepts that will feature in your final examinations. This overview brings together the core knowledge you will encounter across Number, Algebra, Geometry, Statistics, and Probability. A clear grasp of these topics will help you approach problem-solving with confidence and make steady progress throughout the year.
10年级是Eduqas GCSE数学课程中至关重要的一年。在这一年，你将巩固此前打下的基础，并开始学习更深层、相互关联的概念，这些概念将出现在最终考试中。本文梳理了你在数、代数、几何、统计与概率等领域将会接触到的核心知识点。清晰掌握这些主题，能让你自信应对问题解决，并在全年稳步前进。
1. Numbers and the Number System | 数字与数系
The number system underpins almost every mathematical topic. You need to be comfortable working with natural numbers (1, 2, 3, …), integers (…, −2, −1, 0, 1, 2, …), rational numbers (numbers that can be written as a fraction, such as ½ or 0.75), and irrational numbers (numbers that cannot be expressed as a simple fraction, like √2 and π). Recognising prime numbers, which have exactly two distinct factors, is essential for writing prime factor decompositions and finding Highest Common Factors (HCF) and Lowest Common Multiples (LCM).
数系是几乎所有数学主题的基础。你需要熟练处理自然数（1, 2, 3, …）、整数（…, −2, −1, 0, 1, 2, …）、有理数（可以写成分数的数，如½或0.75）和无理数（不能表示为简单分数的数，如√2和π）。认识质数（只有两个不同因数的数）对于分解质因数、求最大公因数（HCF）和最小公倍数（LCM）至关重要。
egative indices give reciprocals, so a⁻ⁿ = 1/aⁿ. Standard form is used to write very large or very small numbers efficiently as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
理解指数记法是另一关键领域：对于任意非零数a，aᵐ × aⁿ = aᵐ⁺ⁿ 且 aᵐ ÷ aⁿ = aᵐ⁻ⁿ。零指数规则表明 a⁰ = 1，负指数则得出倒数，即 a⁻ⁿ = 1/aⁿ。标准形式用于高效书写极大或极小的数，形式为 A × 10ⁿ，其中 1 ≤ A < 10 且 n 为整数。
2. Fractions, Decimals, and Percentages | 分数、小数与百分数
minator; to convert a decimal to a percentage, multiply by 100. You must also be able to order these forms and apply them in real-life contexts such as calculating percentage increase, decrease, or finding the original amount after a percentage change. Working with recurring decimals and being able to prove their equivalent fractions (for example, showing 0.3̇ = ⅓) deepens your number sense.
你需要熟练地在分数、小数和百分数之间进行转换。要将分数化为小数，用分子除以分母；要将小数化为百分数，则乘以100。你还必须能够对这些形式的数进行排序，并将其应用于实际情境，例如计算百分数的增减，或求百分数变化后的原值。处理循环小数并证明其对应分数（例如，证明 0.3̇ = ⅓）能深化你的数感。
When adding or subtracting fractions, it is vital to find a common denominator; for multiplication, multiply numerators together and denominators together; for division, multiply by the reciprocal. In the context of percentages, a solid understanding of multipliers — such as using 1.15 for a 15% increase or 0.85 for a 15% decrease — simplifies compound interest and depreciation calculations.
进行分数的加减运算时，必须找到公分母；乘法则将分子相乘、分母相乘；除法则是乘以其倒数。在百分数的运算中，牢固掌握乘数 —— 如用1.15表示增加15%，或用0.85表示减少15% —— 能简化复利和折旧的计算。
3. Ratio and Proportion | 比与比例
Ratio is a way to compare quantities, often written in the form a : b. You should be able to simplify ratios by dividing by common factors and to share an amount into a given ratio by finding the total number of parts. Proportion goes further to describe how one quantity changes in relation to another. Direct proportion means that as one quantity doubles, the other doubles as well; inverse proportion means that as one doubles, the other halves. Key problems involve best buys, recipe scaling, and exchange rates.
比是比较数量的一种方式，通常写作 a : b 的形式。你应该能够通过除以公因数化简比，并通过总份数将某个量按给定比分配。比例则进一步描述一个量如何随另一个量变化。正比例意味着一个量翻倍时另一个也翻倍；反比例则意味着一个量翻倍时另一个减半。关键问题包括最佳购买、配方缩放和汇率换算。
You will meet the unitary method: finding the value of a single unit first, then scaling up or down as needed. Understanding the difference between ratio and proportion is essential, especially when interpreting graphs: direct proportion is represented by a straight line through the origin, whereas inverse proportion gives a curve that never touches the axes.
你会用到归一法：先求出单个单位的值，再根据需要放大或缩小。理解比与比例的区别至关重要，尤其是在解读图像时：正比例表示为一条通过原点的直线，而反比例则给出永不相交坐标轴的曲线。
4. Algebraic Expressions and Equations | 代数表达式与方程
+ ac, and factorising expressions back into brackets, for example identifying the highest common factor. At Year 10, you will extend this to factorising quadratic expressions of the form x² + bx + c into (x + p)(x + q), where p + q = b and pq = c. For Higher tier students, factorising quadratics with a coefficient of x² greater than 1 is also expected.
代数运用字母代替未知数或变量来总结算术规律。你需要熟练掌握去括号，如 a(b + c) = ab + ac，以及将表达式因式分解回括号形式，比如找出最高公因式。在10年级，你将进一步将形如 x² + bx + c 的二次表达式因式分解为 (x + p)(x + q)，其中 p + q = b 且 pq = c。对于更高层级的考生，还需掌握 x² 系数大于1的二次式的因式分解。
Solving linear equations involves isolating the unknown on one side using inverse operations, remembering that whatever you do to one side you must do to the other. For linear inequalities, the same steps apply, with the critical rule that multiplying or dividing by a negative number reverses the inequality sign. You will also form and solve equations from word problems, turning written descriptions into algebraic models.
解线性方程需通过逆运算将未知数孤立在方程的一边，并牢记对方程一边所做的一切也必须对另一边做。对于线性不等式，步骤相同，但关键法则是在乘以或除以负数时不等号方向反转。你还将学习如何从文字题中建立并求解方程，将文字描述转化为代数模型。
5. Sequences | 数列
A sequence is an ordered list of numbers following a rule. The most common sequences at this stage are arithmetic (linear) sequences, where the difference between consecutive terms is constant. You must be able to find the nth term of an arithmetic sequence in the form an + b, where a is the common difference. Once you have the nth term, you can check whether a given number is in the sequence, find any term, or describe the sequence's behaviour as n grows.
数列是按规则排列的一列数。现阶段最常见的数列是等差（线性）数列，其中相邻两项的差恒定。你必须能够求出等差数列第n项的表达式 an + b，其中 a 是公差。一旦得到第n项公式，你便可以判断某个给定数是否在数列中、求出任意项或描述当 n 增大时数列的趋势。
Other sequences you may explore include geometric sequences (where each term is multiplied by a constant ratio), Fibonacci-style sequences, and quadratic sequences where the second differences are constant. For quadratic sequences, the nth term takes the form an² + bn + c, and using a table of differences helps determine the coefficients. Recognising special sequences such as triangle numbers or square numbers also forms part of the wider curriculum.
你可能会接触的其他数列包括等比数列（每一项乘以一个常数比）、斐波那契式数列以及二次差为常数的二次数列。对于二次数列，第n项公式为 an² + bn + c，借助差分表有助于确定系数。识别三角形数、平方数等特殊数列也是更广泛课程的一部分。
6. Graphs and Functions | 图形与函数
= mx + c, where m is the gradient and c is the y-intercept. Knowing how to find the gradient between two points and how to write the equation of a line that is parallel or perpendicular to a given line is fundamental. Quadratic graphs y = x² and y = ax² + bx + c produce smooth parabolas, and you need to identify their roots, turning points, and axes of symmetry.
图形能直观地表示变量之间的关系。你必须能够根据 y = mx + c 形式的方程绘制直线图形，其中 m 为斜率，c 为y轴截距。懂得如何求两点间的斜率以及如何写出与给定直线平行或垂直的直线方程是基础技能。二次函数图形 y = x² 和 y = ax² + bx + c 形成平滑的抛物线，你需要识别其根、顶点和对称轴。
f(x) = k, and finding simple composite and inverse functions. Recognising the effect of transformations — translations along the axes, reflections, and stretches — tions builds a deeper understanding of how algebraic changes affect the geometric picture.
函数描述了从输入到输出的映射，常写作 f(x)。在10年级，你将学习如何对给定 x 值求函数值、解 f(x) = k，以及求简单的复合函数和反函数。识别图形变换 —— 沿坐标轴的平移、对称和拉伸 —— 对函数图像的影响，能让你更深入地理解代数变化如何影响几何图像。
7. Geometry: Angles and Polygons | 几何：角与多边形
), vertically opposite angles (equal), and angles within parallel lines: corresponding, alternate, and co-interior (allied) angles. These rules allow you to calculate unknown angles and prove relationships in diagrams. For polygons, the sum of interior angles of an n-sided polygon is (n − 2) × 180°, and each exterior angle of a regular polygon is 360°/n.
角的性质是几何推理的基本组成部分。你需要掌握以下规则：一点周围的角度和为360°，直线上的角度和为180°，对顶角相等，以及平行线中的同位角、内错角和同旁内角。这些规则能帮助你计算未知角度并证明图形中的关系。对于多边形，n 边形的内角和为 (n − 2) × 180°，正多边形的每个外角为 360°/n。
Properties of triangles and quadrilaterals are also essential. Knowing the sum of angles in a triangle is 180°, and recognising scalene, equilateral, and right-angled triangles helps in problem solving. For quadrilaterals, you must know the characteristic angle and side properties of squares, rectangles, parallelograms, rhombuses, trapeziums, and kites. Circle geometry may be introduced in Year 10, focusing on the language of radius, diameter, chord, tangent, and the angle in a semicircle being 90°.
三角形和四边形的性质同样不可或缺。知道三角形内角和为180°，并能识别不等边三角形、等边三角形和直角三角形，有助于解决问题。对于四边形，你必须掌握正方形、长方形、平行四边形、菱形、梯形和风筝形的典型角度与边长性质。在10年级可能会引入圆几何，重点关注半径、直径、弦、切线的术语以及半圆上的圆周角为90°等概念。
8. Perimeter, Area, and Volume | 周长、面积与体积
by C = πd or C = 2πr. Area formulas for common shapes include: rectangle (length × width), triangle (½ × base × height), parallelogram (base × vertical height), trapezium (½(a + b)h), and circle (πr²). You must also be able to find the area of shaded regions by subtracting one shape's area from another.
计算直线型和组合图形的周长需要将所有边长相加。对于圆，周长公式为 C = πd 或 C = 2πr。常见图形的面积公式包括：长方形（长×宽）、三角形（½×底×高）、平行四边形（底×垂直高）、梯形（½(a+b)h）和圆（πr²）。你还必须能够通过减除一个图形的面积来求阴影部分的面积。
s, for which volume = πr²h and surface area = 2πrh + 2πr². Understanding how to work with compound shapes made up of cuboids, triangular prisms, and cylinders is a key skill. At the Higher tier, you may also work with spheres, cones, and pyramids, using the curved surface area and volume formulas provided in the exam.
体积与表面积将这些概念扩展到三维空间。对于棱柱，体积 = 横截面面积 × 长度。你会接触到圆柱，其体积 = πr²h，表面积 = 2πrh + 2πr²。理解如何处理由长方体、三棱柱和圆柱组合而成的复合形体是一项关键技能。在更高层级，你可能还会处理球、圆锥和棱锥，并使用考试提供的曲面面积和体积公式。
9. Pythagoras’ Theorem and Trigonometry | 毕达哥拉斯定理与三角函数
In right-angled triangles, Pythagoras’ theorem links the three sides:
a² + b² = c²
where c is the hypotenuse, the longest side opposite the right angle. This relationship allows you to find a missing side length and to determine whether a triangle is right-angled.在直角三角形中，毕达哥拉斯定理将三边联系起来：
a² + b² = c²
其中 c 为斜边，即直角对边中最长的一边。利用这一关系，你可以求出缺失的边长并判断某个三角形是否为直角三角形。Trigonometry extends this by relating angles to side ratios. The three primary ratios are sine, cosine and tangent:
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
三角函数则进一步将角度与边长比关联起来。三个基本比为正弦、余弦和正切：
sin θ = 对边/斜边, cos θ = 邻边/斜边, tan θ = 对边/邻边
You must be able to choose the correct ratio, set up an equation, and solve for a missing side or angle (using inverse trigonometric functions). Trigonometry is also applied to solve problems involving angles of elevation and depression, bearings, and later, in three-dimensional contexts.
你必须能够选择正确的三角比，建立方程，并求解缺失的边长或角度（借助反三角函数）。三角函数也用于解决涉及仰角、俯角和方位角的问题，后续还将拓展至三维空间。
10. Statistics | 统计
Statistics is about collecting, representing, and interpreting data. Averages — mean, median, and mode — summarise a dataset, while the range describes its spread. You need to be able to calculate these for both raw data and grouped frequency tables, where you will estimate the mean using midpoints of class intervals. Cumulative frequency diagrams and box plots are used to present grouped data and identify quartiles and the interquartile range.
统计关乎数据的收集、呈现与解读。平均数 —— 平均、中位数和众数 —— 用来概括数据集，而极差则描述其离散程度。你需要能够为原始数据和分组频数表计算这些指标，在分组数据中需借助组中值估算平均数。累积频数图和箱形图用于展示分组数据并识别四分位数和四分位距。
You will construct and interpret a variety of charts: bar charts, pie charts, line graphs, and scatter graphs. For scatter graphs, you should be able to describe correlation (positive, negative, or none) and, where appropriate, draw a line of best fit to make predictions. Understanding that correlation does not imply causation is a vital statistical insight that features in many exam questions.
你将绘制并解读多种图表：条形图、饼图、折线图和散点图。对于散点图，你应能描述相关关系（正相关、负相关或无相关），并在适当时画出最佳拟合线以进行预测。理解相关关系并不意味着因果关系是一项重要的统计洞察，在许多考题中都会涉及。
11. Probability | 概率
Probability measures the chance that an event will happen, expressed as a number between 0 (impossible) and 1 (certain). For equally likely outcomes, the probability of an event is given by (number of favourable outcomes) / (total number of outcomes). You must understand and use terms like sample space, mutually exclusive events, and exhaustive outcomes. The probability that an event does not happen is 1 minus the probability that it does.
概率衡量事件发生的可能性，用0（不可能）到1（必然）之间的数字表示。对于等可能结果，事件的概率由（有利结果数）/（总结果数）给出。你必须理解并运用样本空间、互斥事件和穷举结果等术语。某个事件不发生的概率等于1减去它发生的概率。
Tree diagrams are the go-to tool for displaying the outcomes of two or more successive events. You need to label branches with probabilities, remembering that probabilities on branches from the same point add up to 1. To find the probability of a sequence of independent events, multiply along the branches; for events that can happen in more than one way, add the probabilities of the relevant paths. Venn diagrams offer another way to visualise sets and their intersections, both for probability and for understanding data sets.
树状图是展示两个或多个相继事件结果的首选工具。你需要用概率标记分支，并牢记从同一点射出的分支概率之和为1。求一系列独立事件发生的概率，要沿分支相乘；对于可以多种方式发生的事件，则将相关路径的概率相加。韦恩图提供了另一种可视化集合及其交集的方法，既可用于概率计算，也可用于理解数据集。
f an event is affected by the occurrence of another event. This is formally written as P(A| is often tackled with modified tree diagrams or Venn diagrams, and you use the formula P(A|B) = P(A ∩ ). A clear grasp of these core concepts will set you up for success in Year 11 and beyond.
随着学习的深入，你将接触到条件概率，即某事件的概率受另一事件发生的影响。条件概率正式写作 P(A|B)，常借助修改后的树状图或韦恩图来解答，使用的公式为 P(A|B) = P(A ∩ B) / P(B)。清晰掌握这些核心概念将为你11年级及以后的学习打下成功的基础。
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