📚 Year 10 Eduqas Mathematics: Core Topics Overview | 10年级 Eduqas 数学：核心知识点梳理
This guide covers the essential mathematical knowledge expected in Year 10 of the Eduqas GCSE course. From foundational number skills to geometry, algebra, probability and statistics, each section highlights key concepts and common techniques that will support your progress throughout the year. Use it as a checklist for revision, making sure you can apply each idea confidently to problem‑solving and exam‑style questions.
本指南涵盖了 Eduqas GCSE 课程 10 年级阶段必备的数学核心知识。从基础数的运算到几何、代数、概率与统计，每个小节都突出了关键概念和常用技巧，帮助你在整个学年中稳步前进。你可以把它当作一份复习清单，确保自己能够自信地将每个知识点应用于解决问题和应对考试题型。
1. Number Basics and Operations | 数的基本运算
Place value underpins all work with integers and decimals. It allows you to compare, order and round numbers to a given power of 10, decimal place or significant figure. Rounding follows the rule: if the next digit is 5 or more, round up.
位值是所有整数和小数运算的基础。利用位值可以对数字进行比较、排序，并按要求精确到 10 的幂、小数位数或有效数字进行四舍五入。四舍五入的规则是：如果下一位数字大于或等于 5，则向上进位。
Operations with directed numbers (positive and negative) rely on consistent sign rules. To add a negative number, subtract its absolute value. When multiplying or dividing, two signs that are the same give a positive result, while two different signs give a negative result.
带符号数（正数和负数）的运算需要遵循一致的符号规则。加上一个负数相当于减去它的绝对值。乘除运算中，同号得正，异号得负。
The order of operations is given by BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction). Always work from left to right within the same level of priority, and perform multiplications or divisions before additions or subtractions.
运算顺序遵循 BIDMAS 法则：括号、指数、除法与乘法、加法与减法。在同一优先级内按从左到右的顺序进行计算，并且乘除运算要优先于加减运算。
Factors, multiples, highest common factor (HCF) and lowest common multiple (LCM) are tools for simplifying fractions and solving numerical problems. A prime number has exactly two distinct factors. Expressing a number as a product of prime factors helps to find HCF and LCM efficiently.
因数、倍数、最大公因数（HCF）和最小公倍数（LCM）是化简分数和解决数值问题的工具。质数恰好有两个不同的因数。将数字写成质因数乘积的形式，可以高效地求出 HCF 和 LCM。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
0 to get a percentage. Common equivalents such as 1/3 ≈ 0.333..., 1/8 = 0.125 and 1/20 = 5% should be known by heart.
你必须能够在分数、小数和百分数之间流畅地转换。分数 a/b 等价于 a ÷ b；将小数乘以 100 就得到百分数。一些常见的等值关系，如 1/3 ≈ 0.333…，1/8 = 0.125 和 1/20 = 5%，应当牢记在心。
When adding or subtracting fractions, first write them with a common denominator. For multiplication, multiply the numerators together and the denominators together. To divide by a fraction, multiply by its reciprocal.
分数相加减时，先化为同分母。乘法是将分子相乘、分母相乘。除以一个分数相当于乘以它的倒数。
Finding a percentage of an amount and working with percentage change, including repeated percentages such as compound interest, require a clear understanding of multipliers. For an increase of 20%, multiply by 1.2; for a decrease, multiply by 0.8. The reverse percentage problem can be solved by dividing by the multiplier.
求一个量的百分之几、计算百分数增减（包括像复利这样的连续百分数变化）都需要理解“乘数”的概念。增加 20% 时，乘以 1.2；减少时乘以 0.8。逆向求原值的问题可以通过除以乘数来解决。
mple, let x = 0.3 recurring, then 10x = 3.3 recurring, subtract to get 9x = 3, so x = 1/3.
循环小数可通过代数方法转化为分数。例如，设 x = 0.3 循环，则 10x = 3.3 循环，两式相减得 9x = 3，从而 x = 1/3。
3. Indices and Standard Form | 指数与标准形式
The laws of indices simplify expressions involving powers. The key rules are:
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 a⁻ⁿ = 1/aⁿ
指数法则可简化含幂的表达式。核心规则如下：
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 a⁻ⁿ = 1/aⁿ
uantities, such as the speed of light (3.0 × 10⁸ m/s) or the mass of a proton (1.67 × 10⁻²⁷ kg). You need to convert between ordinary numbers and standard form and to perform calculations with them.
标准形式（科学记数法）把一个数写作 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。这种形式对于非常大或非常小的量特别有用，比如光速（3.0 × 10⁸ m/s）或质子的质量（1.67 × 10⁻²⁷ kg）。你需要能够在普通数字和标准形式之间进行转换，并能进行标准形式下的计算。
Fractional indices represent roots: a½ = √a, a⅓ = ∛a, and in general a1/q = ᵠ√a. Combining this with the power rule gives ap/q = (ᵠ√a)ᵖ.
分数指数代表方根：a½ = √a，a⅓ = ∛a，一般来说 a1/q = ᵠ√a。与幂规则结合可以得到 ap/q = (ᵠ√a)ᵖ。
4. Algebraic Expressions and Manipulation | 代数式与运算
, etc.) can be collected by adding or subtracting their coefficients. The distributive law a(b + c) = ab + ac is used to expand brackets.
代数运算是指将表达式改写成更简单或更有用的形式。同类项（含有相同变量部分，如 x、x² 等）可以通过加减系数进行合并。分配律 a(b + c) = ab + ac 用于去括号展开。
To expand a product of two binomials, such as (x + 3)(x – ultiply each term in the first bracket by each term in the second bracket and simplify: x² – 2x + 3x – 6 = x² + x – 6. The special pattern (a + b)(a – b) = a² – b² is known as the difference of two squares.
展开两个二项式的乘积（例如 (x + 3)(x – 2)）时，要将第一个括号中的每一项分别乘以第二个括号中的每一项，然后合并：x² – 2x + 3x – 6 = x² + x – 6。特殊模式 (a + b)(a – b) = a² – b² 被称为平方差公式。
ctorised into (x + p)(x + q) where p + q = b and pq = c. For higher tier, more complex quadratics and factorising by grouping are also required.
因式分解是展开的逆过程。开始时应先寻找公因数。形如 x² + bx + c 的二次式通常可以分解为 (x + p)(x + q)，其中满足 p + q = b 且 pq = c。对高阶内容而言，还需要掌握更复杂的二次式以及分组分解法。
Algebraic fractions follow the same rules as numerical fractions. Simplify by first factorising the numerator and denominator, then cancel common factors. Take care with restrictions where the denominator must not be zero.
代数分式的运算规则与数字分式相同。化简时先将分子和分母因式分解，再约去公因子。要注意分母不能为零的限制条件。
5. Solving Equations and Inequalities | 解方程与不等式
Solving linear equations means isolating the unknown variable on one side. Use inverse operations in a logical order: undo addition or subtraction first, then multiplication or division. Equations with brackets or with the unknown on both sides can be solved by expanding and collecting like terms.
解线性方程意味着将未知数隔离在等式的一侧。按合理的顺序运用逆运算：先消除加减，再处理乘除。含有括号或未知数出现在两边的方程，可通过展开和合并同类项求解。
Forming equations from words or diagrams is a vital skill. Read the problem carefully, define the variable clearly, translate the information into an equation and then solve it. Always check that your answer makes sense in context.
根据文字或图表建立方程是一项重要技能。仔细阅读题目，清晰地定义变量，将信息转化为方程，然后求解。最后一定要检查答案在情境中是否合理。
e same way as equations, but with one crucial exception: if you multiply or divide both sides by a negative number, the inequality sign must be reversed. Solutions are often shown on a number line or as intervals.
不等式用符号 <、>、≤ 和 ≥ 表示。它们的解法与方程类似，但有一个关键例外：如果两边同时乘以或除以一个负数，不等号的方向必须反转。解集通常表示在数轴上或写成区间形式。
or must be zero. For example, x² + 5x + 6 = 0 factorises to (x + 2)(x + 3) = 0, giving x = –2 or x = –3. The quadratic formula and completing the square are needed for higher tier.
利用因式分解解二次方程的依据是：如果乘积为零，则至少有一个因子为零。例如，x² + 5x + 6 = 0 分解为 (x + 2)(x + 3) = 0，从而得到 x = –2 或 x = –3。高阶内容还需要掌握二次公式和配方法。
6. Sequences and Graphs | 数列与图像
t difference between terms. The nth term of an arithmetic sequence is given by aₙ = a + (n – n difference. Using this formula, you can generate terms or find a specific term directly.
数列是按照某个规则排列的一串数。等差（线性）数列相邻两项的差为常数。等差数列的第 n 项公式为 aₙ = a + (n – 1)d，其中 a 是首项，d 是公差。利用此公式可生成数列或直接求出任意项。
Other important patterns include geometric sequences (where the ratio between consecutive terms is constant), square numbers, cube numbers and triangular numbers. Recognising these patterns helps in problem‑solving contexts.
其他重要的模式还包括等比数列（相邻项之比为常数）、平方数、立方数和三角形数。识别这些模式有助于在解题过程中快速上手。
onships of the form y = mx + c. Here, m stands for the gradient (steepness) and c is the y‑intercept (the point where the line crosses the y‑ Parallel lines have the same gradient; perpendicular lines have gradients that multiply to –1.
线性图像表示 y = mx + c 形式的关系。其中 m 表示斜率（倾斜程度），c 是 y 截距（直线与 y 轴交点的纵坐标）。平行线斜率相同；垂直线斜率之积为 –1。
You should be able to plot straight‑line graphs, find their equations from given points or from a graph, and interpret real‑life graphs such as distance‑time and velocity‑time graphs. In a distance‑ ntal line indicates the object is stationary. Quadratic graphs (y = x²) produce a smooth U‑shaped parabola; cubic and reciprocal graphs appear at higher tier.
你需要会画直线图像，根据给出的点或图像求直线方程，并能解读现实生活中的图像，如距离 ‑ 时间图和速度 ‑ 时间图。在距离 ‑ 时间图中，斜率代表速度；水平线表示物体静止。二次图像（y = x²）会得到一条光滑的 U 形抛物线；三次图像和反比图像属于高阶内容。
7. Ratio, Proportion and Compound Measures | 比与比例及复合量度
Ratio is used to compare quantities. Simplifying a ratio involves dividing all parts by their common factor. Sharing an amount in a given ratio requires finding the value of one “share” by dividing by the total number of parts, then multiplying by the needed number of shares.
比用来比较数量。化简比时，将比的各项除以它们的公因子。按给定比例分配总量时，需先用总份数除总量得到每一份的值，再分别乘以相应的份数。
oportion, covered at higher tier, follows the rule y = k/x, producing a curved graph called a hyperbola.
正比例意味着两个量以相同速率增减：y = kx。其图像是一条通过原点的直线。反比例属于高阶内容，遵循 y = k/x 的规律，图像为双曲线。
Understanding these as rates allows you to use formula triangles or rearranged equations to find any missing component. Conversions between units are essential, for instance 72 km/h = 72 ÷ 3.6 = 20 m/s.
复合量度将不同的单位组合在一起。主要例子包括速度（= 距离 ÷ 时间）、密度（= 质量 ÷ 体积）和压强（= 力 ÷ 面积）。将这些关系理解为“速率”，你就可以运用公式三角形或变形公式求出任意未知量。单位换算必不可少，例如 72 km/h = 72 ÷ 3.6 = 20 m/s。
Percentage increase and decrease, as well as repeated percentage
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