Year 10 AQA Mathematics: Core Knowledge Essentials | Year 10 AQA 数学：核心知识点梳理

📚 Year 10 AQA Mathematics: Core Knowledge Essentials | Year 10 AQA 数学：核心知识点梳理

Year 10 marks the stage where AQA GCSE Mathematics really begins to build depth. This article walks you through the essential knowledge areas you need to master — from number operations and algebra to geometry, probability and statistics. Use it as your roadmap to stay on top of the syllabus and build confidence for the exams ahead.

Year 10 是 AQA GCSE 数学真正开始深入的关键阶段。本文带你逐一梳理必须掌握的核心知识领域——从数字运算、代数到几何、概率与统计。把它当作你的学习路线图，稳稳跟上课纲、建立考前信心。

1. Number Types and Operations | 数字类型与运算

You need to be fully comfortable with different number families: natural numbers, integers, decimals, fractions, and negatives. The order of operations (often remembered as BIDMAS or BODMAS – Brackets, Indices, Division/Multiplication, Addition/Subtraction) is tested repeatedly.

你必须对各种数字类型了如指掌：自然数、整数、小数、分数和负数。运算顺序（常记作 BIDMAS 或 BODMAS——括号、指数、除/乘、加/减）是反复考查的重点。

• BIDMAS priority: Always solve brackets first, then powers, then division or multiplication (left to right), and finally addition or subtraction (left to right).
• BIDMAS 优先级：先算括号，再算乘方，然后从左到右进行乘除运算，最后从左到右进行加减运算。

Rounding and estimation are powerful check tools. You should know how to round to a given number of decimal places or significant figures, and how to use rounded numbers to estimate an answer before calculating precisely.

取整与估值是强有力的验证工具。你需要掌握如何按要求保留指定的小数位数或有效数字，并在精确计算前利用取整后的数值估计答案。

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

Interchanging between fractions, decimals and percentages must become second nature. Key conversions like 1/8 = 0.125 = 12.5% should be memorised. You will also be expected to perform operations with fractions, including mixed numbers, without a calculator in the non‑calculator paper.

分数、小数和百分比之间的转换必须成为本能。需熟记 1/8 = 0.125 = 12.5% 等关键等值关系。你还需要能在非计算器考卷中完成分数（包括带分数）的四则运算。

• Adding/Subtracting fractions: find a common denominator first.
• 分数加减法：先通分，找到公分母。
• Multiplying: multiply numerators, multiply denominators; simplify if possible.
• 乘法：分子乘分子，分母乘分母；能化简则化简。
• Dividing: flip the second fraction (multiply by the reciprocal).
• 除法：颠倒第二个分数（乘以倒数）。

Percentage increase and decrease are modelled by multipliers. An increase of 15% uses the multiplier 1.15; a decrease of 15% uses 0.85. Reverse percentages are solved by dividing by the original multiplier.

百分比增减可通过乘数建模。增加 15% 用乘数 1.15；减少 15% 用乘数 0.85。逆向百分比问题可除以原乘数求解。

3. Powers and Roots | 幂与根号

Index laws govern how powers combine. For any positive base a: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m–n, (a^m)ⁿ = a^mn. You must also understand negative and fractional indices: a^–1 = 1/a, a^1/2 = √a.

指数律支配着幂的运算方式。对任意正数 a：a^m × aⁿ = a^m+n，a^m ÷ aⁿ = a^m–n，(a^m)ⁿ = a^mn。你还需要理解负指数和分数指数：a^–1 = 1/a，a^1/2 = √a。

Standard form expresses very large or very small numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. You should be able to order numbers in standard form, perform calculations with them, and convert between standard form and ordinary numbers confidently.

标准形将极大或极小数字表示为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。你应能对标准形数字排序、进行计算，并在标准形与普通记数法间自如转换。

4. Algebraic Expressions | 代数表达式

Algebraic fluency starts with understanding terms, coefficients, and like terms. A term like 5x² has coefficient 5; only terms with identical variable parts can be combined.

代数熟练度始于理解项、系数和同类项。像 5x² 这样的项系数为 5；只有变量部分完全相同的项才能合并。

Expanding brackets uses the distributive law: a(b + c) = ab + ac. For double brackets, every term in the first bracket must multiply every term in the second: (x + a)(x + b) = x² + (a + b)x + ab.

去括号运用分配律：a(b + c) = ab + ac。对于双括号，第一个括号中的每一项都要乘第二个括号中的每一项：(x + a)(x + b) = x² + (a + b)x + ab。

Factorising is the reverse process. Look for the highest common factor (HCF) first, then recognise quadratics of the form x² + bx + c that factorise into (x + p)(x + q), where p+q = b and pq = c.

因式分解是相反的过程。先提取最大公因子，再识别形如 x² + bx + c 的二次式，分解为 (x + p)(x + q)，其中 p+q = b，pq = c。

Substitution means replacing variables with given numbers. Stay disciplined with brackets when substituting to protect negative signs and operations.

代入法是用给定数值替换变量。代值时务必保留括号，以守护负号和运算的正确性。

5. Equations and Inequalities | 方程与不等式

Solving linear equations requires doing the same operation to both sides. If x appears on both sides, collect x‑terms on one side and constants on the other. Always check your solution by substituting back into the original equation.

解线性方程必须对等号两边执行相同操作。若 x 在两边都出现，将含 x 项移到一边，常数移到另一边。始终把解代回原方程检验。

Inequalities are solved similarly, but remember the golden rule: if you multiply or divide by a negative number, you must reverse the inequality sign. The solution can be shown on a number line with open circles for < or > and closed circles for ≤ or ≥.

解不等式方法类似，但切记黄金法则：乘或除以一个负数时，必须反转不等号。解集可用数轴表示，用空心圆表示 < 或 >，实心圆表示 ≤ 或 ≥。

If −2x ≤ 6, then x ≥ −3

Quadratic equations often appear in factorised form. Set the expression equal to zero and use the fact that if ab = 0, then a = 0 or b = 0. Always write both possible solutions.

二次方程常以分解形式出现。令表达式等于零，利用若 ab = 0 则 a = 0 或 b = 0 的性质。记得写出两个可能的解。

6. Sequences | 数列

A sequence is a list of numbers following a rule. The term‑to‑term rule tells you how to go from one term to the next; the position‑to‑term rule (nth term) links the term directly to its position n. For linear sequences, the nth term is of the form an + b, where a is the common difference.

数列是按某一规则排列的一组数。项间规则告诉你如何从一项得到下一项；位置规则（第 n 项公式）则直接将项与其位置 n 关联。线性数列的第 n 项为 an + b 形式，其中 a 是公差。

To find the nth term of a linear sequence, subtract consecutive terms to find a (the common difference), then work out the b value that would give the first term when n = 1. Generate terms from an nth term formula by substituting n = 1, 2, 3, …

求线性数列第 n 项时，用相邻项减差得 a（公差），再推出使 n=1 时得到首项的 b 值。由第 n 项公式生成各项时，依次代入 n = 1, 2, 3 … 即可。

Special sequences, such as square numbers (n²) and cube numbers (n³), should be recognised. Context‑based sequences, like patterns of matchsticks, often require you to find a linear nth term that matches the visual growth.

平方数数列 (n²)、立方数数列 (n³) 等特殊数列必须能识别。基于情境的数列（如火柴棍图案）通常需要你找出与图案增长规律吻合的线性第 n 项。

7. Coordinates and Straight Line Graphs | 坐标与直线图

Coordinates are written as (x, y) and plotted on the Cartesian plane. The x‑axis is horizontal; the y‑axis is vertical. You must be able to find the midpoint of a segment and the distance between two points using the grid, and later with Pythagoras’ theorem.

坐标写作 (x, y)，绘制在笛卡尔平面上。x 轴为水平方向，y 轴为垂直方向。你必须能利用坐标格求线段的中点以及两点间的距离；后续会与勾股定理结合。

The equation of a straight line is y = mx + c, where m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis). Gradient is found by rise ÷ run between two points. Parallel lines have the same gradient.

直线方程为 y = mx + c，其中 m 是斜率（坡度），c 是 y 轴截距（直线与 y 轴的交点）。斜率由两点间的垂直增量除以水平增量得出。平行线的斜率相同。

You should be able to plot a line from its equation by creating a table of values, or by marking the y‑intercept and using the gradient to find a second point. Recognising graphs from their equations is a key exam skill.

你需要能通过列表取值绘制给定方程的直线图，也能通过标出 y 轴截距并利用斜率找到第二个点来画线。根据方程识别对应图形是重要的考试技能。

8. Ratio, Proportion and Rates of Change | 比率、比例与变化率

Ratios compare parts of a whole and are usually written in their simplest form by dividing by common factors. When sharing a quantity in a given ratio, find the total number of parts, divide the quantity by that total to find the value of one part, then scale accordingly.

比率用于比较整体中的各部分，通常用除以公因数的方式化为最简形式。按给定比率分配数量时，先求总份数，用总量除以总份数得到每一份的值，再进行缩放。

Direct proportion means that two quantities increase at the same rate: y = kx. You can solve proportion problems using the unitary method or a multiplier. Best‑buy problems ask you to compare unit prices (price per gram, per litre, etc.) to determine the best value.

正比例意味着两个量以相同速率增加：y = kx。解比例题可使用归一法或乘数法。最优购置问题要求你比较单位价格（每克、每升的价格等）以确定最划算的选择。

Compound measures blend two units together, such as speed (km/h), density (g/cm³) and pressure. Use the formula triangles or rearrange equations like distance = speed × time. Always ensure units match before substituting.

复合度量将两个单位结合，如速度 (km/h)、密度 (g/cm³) 和压强。使用公式三角形或改写方程式，如 distance = speed × time。代值前务必确保单位一致。

9. Geometry Foundations: Angles, Area and Volume | 几何基础：角度、面积与体积

Angle facts form the backbone of geometric reasoning. You must know:

角度关系是几何推理的支柱。你必须掌握：

Angles on a straight line sum to 180°	直线上的角之和等于 180°
Angles around a point sum to 360°	一点周围的角之和为 360°
Vertically opposite angles are equal	对顶角相等
Angles in a triangle sum to 180°	三角形内角和为 180°
Angles in a quadrilateral sum to 360°	四边形内角和为 360°

Area of common shapes: triangle A = ½ × base × height; parallelogram A = base × height; trapezium A = ½(a + b) × height. For circles, circumference C = 2πr and area A = πr². You must leave answers in terms of π unless told otherwise.

常见图形面积：三角形面积 = ½ × 底 × 高；平行四边形面积 = 底 × 高；梯形面积 = ½(上底+下底) × 高。圆的周长 C = 2πr，面积 A = πr²。除非题目另有要求，答案保留 π 形式。

Volume of a prism is found by multiplying the area of its cross‑section by its length. For a cylinder (a circular prism), volume = πr²h. Surface area involves computing the total area of all faces.

棱柱的体积等于横截面积乘以长度。圆柱体（一种圆形棱柱）体积 = πr²h。表面积需计算所有面的总面积。

10. Pythagoras’ Theorem and Basic Trigonometry | 勾股定理与基础三角学

In any right‑angled triangle, Pythagoras’ theorem links the legs (a, b) and the hypotenuse (c, the longest side opposite the right angle): a² + b² = c². You can use this relationship to find a missing side, or to check if a triangle is right‑angled.

在任何直角三角形中，勾股定理将两条直角边 (a, b) 与斜边 (c，直角的对边，为最长边) 联系起来：a² + b² = c²。你可以利用这一关系求未知边长，或判定三角形是否为直角三角形。

Pythagoras: side² + side² = hypotenuse²

Basic trigonometry introduces three ratios for a right‑angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Label the sides relative to the angle you are working with before applying ‘SOH CAH TOA’.

基础三角学引入直角三角形中的三种比值：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。使用 ‘SOH CAH TOA’ 前，先根据所考虑的角度标注各边。

When finding an angle, use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹. Always express the final angle to the required accuracy and check that the triangle is indeed right‑angled.

求角度时，使用反三角函数 sin⁻¹、cos⁻¹、tan⁻¹。最终角度按题目要求保留精度，并确认三角形确实是直角三角形。

11. Probability | 概率

Probability is measured on a scale from 0 (impossible) to 1 (certain). The probability of an event = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.

概率的度量范围从 0（不可能）到 1（必然）。若所有结果等可能，事件的概率 = 有利结果的数量 / 所有可能结果的总数。

Mutually exclusive events cannot happen at the same time; adding their probabilities gives the probability that one or the other occurs. The sum of probabilities of all possible mutually exclusive outcomes is 1.

互斥事件不能同时发生；将它们的概率相加即得到其中任一事件发生的概率。所有可能的互斥结果概率之和为 1。

Tree diagrams are used for sequences of independent events. Multiply along branches for ‘AND’ probabilities; add across branches for ‘OR’ probabilities. Always check that probabilities on branches from the same point sum to 1.

树形图用于表示一系列独立事件。沿着分支相乘可求 ‘AND’ 概率；跨分支相加可得 ‘OR’ 概率。务必检查从同一点发出的各分支概率之和为 1。

Experimental probability (relative frequency) may differ from theoretical probability in a small number of trials, but tends towards the theoretical value with more trials.

实验概率（相对频率）在少量试验中可能与理论概率有差异，但随着试验次数增多，会趋近理论值。

12. Statistics: Averages and Charts | 统计：平均数与图表

The three main averages are mean (sum of values ÷ number of values), median (middle value when ordered) and mode (most frequent). The range (maximum – minimum) measures spread. Know when each measure is most useful; the mean is sensitive to outliers.

三种主要平均数为：均值（数值之和÷数值个数）、中位数（排序后居中的值）和众数（出现频率最高的值）。极差（最大值−最小值）衡量离散度。需知何时用哪种度量；均值对极端值较敏感。

Frequency tables summarise data; modal class is the group with the highest frequency. From a frequency table you can estimate the mean using midpoints.

频数表可汇总数据；众数所在组是频数最高的组。利用频数表的中点值可估算均值。

Statistical charts: bar charts for discrete/comparative data, pie charts for proportions, scatter graphs for relationship between two quantitative variables. Correlation can be positive, negative or none; a line of best fit can be drawn to make predictions (interpolation, not extrapolation).

统计图表：条形图用于离散或比较类数据，饼图展示比例，散点图展示两个定量变量间的关系。相关性可为正、负或无；可画一条最佳拟合线进行预测（内推法，不可外推）。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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