📚 Year 10 CCEA Mathematics: Essential Topics Review | 核心知识点梳理
Year 10 is a pivotal stage in the CCEA mathematics journey, where you consolidate foundation skills and bridge into more advanced GCSE topics. This article provides a bilingual, section-by-section recap of the essential knowledge you need – from number operations and algebra to Pythagoras, trigonometry, statistics and transformations. Each topic is presented with clear explanations in English, followed immediately by the Chinese translation, helping you revise efficiently and deepen understanding.
Year 10 是 CCEA 数学学习的关键阶段，既要夯实基础技能，又要衔接更深层的 GCSE 内容。本文以中英双语形式，按章节梳理你必须掌握的核心知识点 – 从数运算和代数，到勾股定理、三角学、统计与变换。每个主题先给出英文讲解，紧接着配中文翻译，帮助你高效复习、加深理解。
1. Number Skills and Operations | 数与运算技巧
Number skills are the backbone of all mathematical reasoning. Fluency with integers, fractions, decimals and percentages enables you to model real-life situations accurately.
数运算是所有数学推理的支柱。熟练运用整数、分数、小数和百分比能让你精确地模拟生活中的问题。
When multiplying or dividing integers, remember the sign rules: same signs give a positive result, different signs give a negative result. For example, (-8) ÷ (-2) = 4 and (-5) × 3 = -15.
进行整数乘除运算时，牢记符号法则：同号得正，异号得负。例如 (-8) ÷ (-2) = 4，而 (-5) × 3 = -15。
To add or subtract fractions, first write them with a common denominator. For multiplication, simply multiply the numerators and multiply the denominators; for division, multiply by the reciprocal of the divisor.
分数的加减需要先化成同分母。乘法直接将分子相乘、分母相乘；除法则乘以除数的倒数。
Converting between fractions, decimals and percentages is a daily maths skill. For instance, 3/8 = 0.375 = 37.5%. A percentage increase of 20% on £60 means new value = 60 × 1.2 = £72.
分数、小数和百分比之间的互化是日常数学技能。比如 3/8 = 0.375 = 37.5%。在 £60 的基础上增加 20% 意味着新金额 = 60 × 1.2 = £72。
2. Algebraic Manipulation | 代数变换
Algebra is the language of patterns and generalizations. You need to simplify expressions confidently and expand or factorise brackets in order to solve equations later.
代数是描述模式和规律的语言。你需要自信地化简表达式，并能展开或分解括号，为后续解方程打好基础。
Collect like terms by combining terms that have exactly the same variable part. For example, 3x + 5y – 2x + y simplifies to x + 6y.
合并同类项就是把含有完全相同字母的部分相加或相减。例如 3x + 5y – 2x + y 化简为 x + 6y。
Expanding brackets uses the distributive law: a(b + c) = ab + ac. A double bracket product such as (x + 2)(x – 5) expands to x² – 5x + 2x – 10, which then simplifies to x² – 3x – 10.
去括号运用分配律：a(b + c) = ab + ac。双括号相乘如 (x + 2)(x – 5) 展开得 x² – 5x + 2x – 10，再合并为 x² – 3x – 10。
Factorising reverses expansion. Take out the highest common factor: 6x² – 9x = 3x(2x – 3). A quadratic like x² + 7x + 12 factorises to (x + 3)(x + 4) by finding two numbers that multiply to 12 and add to 7.
因式分解是展开的逆运算。先提公因式：6x² – 9x = 3x(2x – 3)。像 x² + 7x + 12 这样的二次式，可分解为 (x + 3)(x + 4)，即寻找两个相乘得 12、相加得 7 的数。
Substitution means replacing letters with given numbers. For instance, when a = -2 and b = 5, the value of 3a + b² is 3(-2) + 25 = 19.
代入法就是将字母替换为给定的数值。例如 a = -2、b = 5 时，3a + b² 的值是 3(-2) + 25 = 19。
3. Solving Equations and Inequalities | 解方程与不等式
Forming and solving equations is at the heart of problem solving. You must be able to isolate the unknown and interpret the solution in context.
列方程和解方程是解决问题的核心。你必须能够求出未知数，并结合实际问题解释解的含义。
For a linear equation like 4x – 7 = 2x + 9, first collect x terms on one side and numbers on the other: 4x – 2x = 9 + 7 → 2x = 16, so x = 8.
对于一元一次方程 4x – 7 = 2x + 9，首先将含 x 的项移到一边，常数移到另一边：4x – 2x = 9 + 7 → 2x = 16，因此 x = 8。
Simple quadratic equations (where one side is zero and the other factorises) are solved by setting each factor to zero. For x² + 5x + 6 = 0, factorise to (x + 2)(x + 3) = 0, giving x = -2 or x = -3.
简单的二次方程（一边为零且另一边可分解）通过令每个因式等于零求解。对于 x² + 5x + 6 = 0，分解为 (x + 2)(x + 3) = 0，得 x = -2 或 x = -3。
Linear inequalities are solved in a similar way to equations, but remember to flip the inequality sign when multiplying or dividing by a negative number. Show the solution on a number line with open or closed circles.
一元一次不等式的解法与方程类似，但当乘以或除以负数时，不等号方向要改变。解集可用空心或实心圆圈在数轴上表示。
4. Sequences and Finding the nth Term | 数列与求第 n 项
Sequences help you recognise patterns and predict future terms. The nth term formula lets you find any term directly without generating all previous terms.
数列帮助你识别规律并预测后续的项。通项公式能让你直接求出数列中的任意项，而无需逐项递推。
An arithmetic sequence has a constant difference d between consecutive terms. The nth term is given by aₙ = a₁ + (n – 1)d. For the sequence 3, 7, 11, 15, …, a₁ = 3, d = 4, so aₙ = 3 + 4(n – 1) = 4n – 1.
等差数列中相邻两项的差 d 恒定。第 n 项公式为 aₙ = a₁ + (n – 1)d。以数列 3, 7, 11, 15 … 为例，a₁ = 3，d = 4，因此 aₙ = 3 + 4(n – 1) = 4n – 1。
If a sequence has a second difference that is constant, it is quadratic. Its nth term takes the form an² + bn + c, but finding this usually requires solving simultaneous equations.
若数列的二次差为常数，则它是二次数列。其通项形式为 an² + bn + c，但推导通常需要解联立方程。
Generating terms simply involves substituting n = 1, 2, 3 … into the nth term expression. For the rule 5n + 2, the first three terms are 7, 12 and 17.
生成数列只需将 n = 1, 2, 3 … 代入通项表达式。对于规则 5n + 2，前三项为 7、12 和 17。
5. Coordinates and Straight Line Graphs | 坐标与直线图像
Graphs provide a visual way of representing relationships between variables. Mastering the straight line graph equips you with a powerful problem-solving tool.
图像是展现变量之间关系的直观方式。掌握直线图像能让你拥有一个强有力的解题工具。
All straight lines (except vertical ones) can be written in the form y = mx + c, where m is the gradient and c is the y-intercept. The gradient is found by m = (change in y)/(change in x).
所有直线（除垂直线外）都可以写成 y = mx + c 的形式，其中 m 代表斜率，c 代表 y 轴截距。斜率通过 m = (y 的变化量)/(x 的变化量) 求得。
To plot a line, choose three x-values, calculate the corresponding y-values and plot the points. If they lie in a straight line, your calculations are correct.
绘制直线时，选取三个 x 值，计算对应的 y 值，描点。若三点共线，说明计算正确。
Horizontal lines have equation y = constant (m = 0); vertical lines have equation x = constant (gradient is undefined). Parallel lines share the same gradient; perpendicular lines have gradients that multiply to -1.
水平线的方程为 y = 常数（m = 0）；垂直线的方程为 x = 常数（斜率无定义）。平行线的斜率相等；垂直线的斜率之积为 -1。
6. Ratio, Proportion and Rates of Change | 比、比例与变化率
Ratio and proportion are used to compare quantities and scale amounts up or down. They appear in recipes, maps, speed problems and many real-world contexts.
比和比例用于比较数量并按比例缩放。它们常见于配方、地图、速度问题以及许多现实情境中。
A ratio compares parts of a whole. The ratio 3 : 5 means for every 3 of one item there are 5 of the other. To share £64 in the ratio 3 : 5, first find the total number of parts (3 + 5 = 8); one part is 64 ÷ 8 = £8, so the shares are £24 and £40.
比用于比较整体中的部分。比 3 : 5 表示每 3 个 A 对应 5 个 B。要将 £64 按 3 : 5 分配，先计算总份数（3 + 5 = 8）；每份为 64 ÷ 8 = £8，因此两部分分别为 £24 和 £40。
Direct proportion means two quantities increase or decrease at the same rate. If 5 pens cost £3.50, then 8 pens cost (3.50/5) × 8 = £5.60. The unitary method works reliably here.
正比例意味着两个量以相同的变化率增加或减少。如果 5 支笔花费 £3.50，那么 8 支笔花费 (3.50/5) × 8 = £5.60。归一法在此非常可靠。
Rates link two different units. Speed = distance/time, density = mass/volume. Always check that units are consistent before calculating, and convert if necessary (e.g. minutes to hours, g to kg).
速率连接两种不同的单位。速度 = 距离/时间，密度 = 质量/体积。计算前务必检查单位是否一致，必要时应进行单位换算（如分钟换为小时，克换为千克）。
7. Perimeter, Area and Volume | 周长、面积与体积
Geometry in two and three dimensions appears frequently in exams and practical tasks. You must recall formulas and apply them to composite shapes.
二维和三维几何在考试和实际任务中经常出现。你需要记住相关公式并将其应用于组合图形。
Perimeter is the distance around a shape. For a rectangle with length l and width w, perimeter = 2l + 2w. The circumference of a circle is C = 2πr or πd.
周长是图形外部轮廓的长度。长为 l、宽为 w 的矩形周长为 2l + 2w。圆的周长为 C = 2πr 或 πd。
Area formulas: rectangle A = lw; triangle A = ½bh; parallelogram A = bh; trapezium A = ½(a + b)h; circle A = πr². For composite shapes, split the shape into standard parts, sum or subtract areas.
面积公式：矩形 A = lw；三角形 A = ½bh；平行四边形 A = bh；梯形 A = ½(a + b)h；圆面积 A = πr²。遇到组合图形时，可将其拆分成标准图形，再对面积进行相加或相减。
Volume of a prism = area of cross-section × length. For a cylinder, this becomes πr²h. Surface area is the total area of all faces; for a cube of side s, SA = 6s².
棱柱的体积 = 横截面的面积 × 长度。对于圆柱，体积为 πr²h。表面积指所有面的面积之和；边长为 s 的立方体表面积为 6s²。
8. Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem applies exclusively to right-angled triangles and links the squares of the three sides. It is one of the most frequently used results in geometry.
勾股定理仅适用于直角三角形，它将三条边的平方联系起来。它是几何中最常用的结论之一。
The theorem states: a² + b² = c², where c is the hypotenuse (the longest side opposite the right angle). To find a missing shorter side, rearrange to a² = c² – b².
定理表示为：a² + b² = c²，其中 c 是斜边（直角对边中最长的边）。若要求一条直角边，移项得 a² = c² – b²。
In a triangle with shorter sides 6 cm and 8 cm, the hypotenuse is √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
若直角三角形的两条直角边分别为 6 cm 和 8 cm，则斜边为 √(6² + 8²) = √(36 + 64) = √100 = 10 cm。
Real-life applications include finding the distance between two points on a coordinate grid, ladder problems, and determining whether a triangle is right-angled by checking if a² + b² equals the square of the longest side.
实际应用包括计算坐标网格上两点间的距离、梯子问题，以及通过验证 a² + b² 是否等于最长边的平方来判断三角形是否为直角三角形。
9. Right-Angled Triangle Trigonometry | 直角三角形三角学
Trigonometry extends Pythagoras by linking angles to side lengths. The three basic trigonometric ratios are essential for solving right-angled triangle problems where an angle or side is unknown.
三角学将勾股定理扩展，把角度与边长联系起来。三个基本的三角函数比是求解直角三角形未知边或未知角的必要工具。
Label the sides relative to a chosen angle θ: Opposite (O), Adjacent (A) and Hypotenuse (H). Then sin θ = O/H, cos θ = A/H, tan θ = O/A. A handy mnemonic is SOH CAH TOA.
对选定的角 θ 标注各边：对边 (O)、邻边 (A) 和斜边 (H)。则有 sin θ = O/H, cos θ = A/H, tan θ = O/A。实用的记忆口诀是 SOH CAH TOA。
To find a missing side, choose the ratio that uses the known side and the side you want. For an angle θ = 30° and adjacent side 10 cm, hypotenuse = 10/cos 30° ≈ 11.55 cm.
求缺失的边长时，选择包含已知边和所求边的三角比。例如，已知 θ = 30°，邻边 10 cm，则斜边 = 10/cos 30° ≈ 11.55 cm。
To find an unknown angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹). If the opposite is 5 and hypotenuse is 8, then θ = sin⁻¹(5/8) ≈ 38.68°.
求未知角时，使用反三角函数 (sin⁻¹, cos⁻¹, tan⁻¹)。假设对边为 5，斜边为 8，则 θ = sin⁻¹(5/8) ≈ 38.68°。
10. Statistics: Averages and Data Representation | 统计：平均数与数据表示
Statistics involves collecting, organising and interpreting data. Averages and charts help you summarise data sets and spot trends.
统计学涉及数据的收集、整理和解读。平均数和图表能帮助你概括数据集并发现趋势。
The three averages are mean (sum of values ÷ number of values), median (middle value when ordered) and mode (most frequent value). The range measures spread: range = highest value − lowest value.
三大平均数是均值（总和 ÷ 个数）、中位数（排序后位于中间的值）和众数（出现次数最多的值）。极差衡量数据的分散程度：极差 = 最大值 − 最小值。
Bar charts display frequency for discrete categories. Pie charts show proportions of a whole. Scatter graphs reveal correlation between two variables; a line of best fit can be drawn to estimate values.
条形图展示离散类别的频数。饼图展示各部分占整体的比例。散点图揭示两个变量之间的相关关系；添加一条最佳拟合线可以估算数值。
For grouped data, the mean can be estimated using midpoints: mean ≈ Σ
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