📚 Year 10 WJEC Mathematics: Core Concepts Guide | WJEC 数学 Year 10 核心概念梳理
This guide walks you through the essential topics of the Year 10 WJEC Mathematics course. Each section breaks down key ideas, helping you to revise systematically and tackle both calculator and non-calculator papers with confidence.
本文梳理 WJEC 数学 Year 10 课程的核心知识点，分节讲解重要概念，帮助你系统复习，在计算器与非计算器试卷中都能从容应对。
1. Number Basics | 数的基本概念
Whole numbers (0, 1, 2, 3, …) and integers (including negatives like -1, -2) are the building blocks. You need to be fluent in addition, subtraction, multiplication and division with positive and negative numbers, using the correct order of operations (BIDMAS/BODMAS).
正整数 (0, 1, 2, 3…) 和整数 (含 -1, -2 等负数) 是基础。要能熟练进行正负数的加减乘除，并遵循正确的运算顺序 (括号、指数、乘除、加减)。
Prime numbers have exactly two distinct factors: 1 and the number itself (2, 3, 5, 7, 11, …). A factor is a whole number that divides another number exactly, while a multiple is the result of multiplying a number by an integer. The highest common factor (HCF) and lowest common multiple (LCM) are often used in fraction arithmetic and problem-solving.
质数只有两个不同的正因数：1 和它自身 (如 2, 3, 5, 7, 11…)。因数是能整除某数的整数，倍数则是某数乘以整数得到的积。最大公因数 (HCF) 和最小公倍数 (LCM) 常用于分数运算和应用题。
Place value determines the magnitude of a digit within a number. Rounding to decimal places or significant figures helps to give sensible estimates. Always check whether the question asks for rounding or an exact answer.
位值决定数字中每个数码的量级。将数字四舍五入到指定的小数位或有效数字可以得到合理的估算值。务必看清题目要求的是近似值还是精确答案。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
A fraction represents a part of a whole. You must be able to find equivalent fractions, simplify fractions, and convert between mixed numbers and improper fractions. When adding or subtracting fractions, first rewrite them with a common denominator.
分数表示整体的一部分。要掌握等值分数、约分以及带分数与假分数的互化。在加减分数时，需先通分化为同分母再计算。
Decimals can be added, subtracted, multiplied and divided; careful alignment of the decimal point is essential. To convert a fraction to a decimal, divide the numerator by the denominator. Recurring decimals can be expressed using dot notation or converted back to fractions using algebraic methods.
小数可进行加、减、乘、除运算，小数点对齐至关重要。分数化小数只需用分子除以分母。循环小数可用点记号表示，也可通过代数方法还原为分数。
Percentages mean ‘out of 100’. To find a percentage of an amount, multiply by the percentage and divide by 100. Percentage increase or decrease problems should be approached by identifying the multiplier (e.g. 1.15 for a 15% increase). Reverse percentages require dividing by the multiplier.
百分数表示 ‘百分之几’。求一个数的百分之几，先乘以百分数再除以 100。百分增减问题可先确定乘数 (如增长 15% 时乘数为 1.15)，反向百分数则要用除法反推原值。
3. Indices and Standard Form | 指数与标准形式
Indices (powers) show repeated multiplication. The key rules are: am × an = am+n, am ÷ an = am-n, (am)n = amn. Also remember that a0 = 1 and a-n = 1/an.
指数 (幂) 表示连乘。基本法则：am × an = am+n，am ÷ an = am-n，(am)n = amn。同时记住 a0 = 1，a-n = 1/an。
Square roots (√) and cube roots (∛) are the inverse of squaring and cubing. For example, √64 = 8 because 8² = 64. You should be able to estimate roots and recognise perfect squares and cubes.
平方根 (√) 和立方根 (∛) 分别是平方与立方的逆运算。例如 √64 = 8 因为 8² = 64。应能估算根的值并识别常见的完全平方数与完全立方数。
Standard form is used for very large or very small numbers, written as a × 10n where 1 ≤ a < 10 and n is an integer. e.g. 34 000 = 3.4 × 10⁴ and 0.0052 = 5.2 × 10⁻³. Ensure you can convert between ordinary numbers and standard form, and carry out calculations with standard form numbers.
标准形式用于表示极大或极小的数，写作 a × 10n，其中 1 ≤ a < 10，n 为整数。例如 34 000 = 3.4 × 10⁴，0.0052 = 5.2 × 10⁻³。务必熟练普通数与标准形式的互化，并能用标准形式进行四则运算。
4. Algebra: Expressions | 代数：表达式
Algebraic terms consist of numbers, letters and powers. Like terms (same variable and power) can be collected by adding or subtracting their coefficients. For example, 3x + 5x = 8x, but 3x + 5y cannot be simplified further.
代数项由数、字母和幂组成。同类项 (变量和指数相同) 可通过系数加减合并，如 3x + 5x = 8x，但 3x + 5y 不能进一步化简。
Expanding brackets means multiplying each term inside the bracket by the term outside. For single brackets: a(b + c) = ab + ac. For double brackets, use FOIL: (x + a)(x + b) = x² + (a+b)x + ab. Watch out for negative signs and always check you can factorise back.
展开括号即将括号外部的项与内部每一项相乘。单项式乘括号：a(b + c) = ab + ac。两项式相乘可用首外内尾法：(x + a)(x + b) = x² + (a+b)x + ab。注意负号，并要能反过来进行因式分解。
Factorising reverses expansion. Look for the highest common factor (HCF) of the terms; for quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b. Always check by expanding your factorised brackets.
因式分解是展开的逆运算。先提取各项的最大公因数 (HCF)；对于 x² + bx + c 形式的二次式，需找到两个乘积为 c、和为 b 的数。完成因式分解后，务必展开验证。
5. Algebra: Equations and Inequalities | 代数：方程与不等式
Solving linear equations means finding the value of the variable that makes the equation true. Use inverse operations to isolate the unknown: if 2x + 3 = 11, subtract 3, then divide by 2 to get x = 4. Always check your answer by substitution.
解线性方程即求出使等式成立的变量值。通过逆运算逐步分离未知数：若 2x + 3 = 11，先减 3 再除以 2 得 x = 4。解出后务必代入原方程检验。
Equations may have brackets or variables on both sides. Expand first, then gather like terms to one side. For fractions, multiply every term by the common denominator to clear denominators.
方程可能含有括号或两边都有变量。先展开括号，再将同类项移到一边。若含分数，可乘以公分母消去分母。
Inequalities are similar to equations but use <, >, ≤, ≥. Remember that multiplying or dividing by a negative number reverses the inequality sign. Solutions are often displayed on number lines using open or closed circles.
不等式的解法类似方程，但使用 <, >, ≤, ≥ 符号。注意乘或除以负数时，不等号方向要反转。解集常用数轴表示，空心圈表示不含，实心圈表示包含端值。
6. Sequences | 数列
A sequence is an ordered list of numbers following a rule. Arithmetic sequences (linear) have a constant difference between terms. For example, in 5, 8, 11, 14,… the term-to-term rule is ‘add 3’.
数列是按特定规律排列的一串数。等差数列 (线性数列) 相邻两项的差恒定，例如 5, 8, 11, 14,… 相邻差为 +3。
The nth term of an arithmetic sequence is given by an = a₁ + (n – 1)d, where a₁ is the first term and d is the common difference. In Year 10 this is usually written as nth term = dn + (a₁ – d), e.g. 3n + 2. You can use the nth term to find any term or to check if a number belongs to the sequence.
等差数列的第 n 项公式为 aₙ = a₁ + (n – 1)d，其中 a₁ 为首项，d 为公差。在 Year 10 往往写成形如 dn + (a₁ – d)，如 3n + 2。利用通项公式可求任意项或判断某数是否在数列中。
Other sequences include geometric (multiplying by a constant ratio) and special sequences like square numbers (1, 4, 9, 16, …). Recognise simple patterns and be able to continue sequences and describe the rule in words.
其他数列包括等比数列 (每次乘以固定比值) 以及特例，如平方数 (1, 4, 9, 16, …)。要能识别简单模式、续写数列并用语言描述规律。
7. Coordinates and Straight Line Graphs | 坐标与直线图像
Coordinates are written as (x, y) on a Cartesian plane. The midpoint of two points (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2). To find the length of a horizontal or vertical segment, simply subtract the differing coordinates.
坐标在笛卡尔平面上表示为 (x, y)。两点 (x₁, y₁) 和 (x₂, y₂) 的中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。水平或竖直线段的长度只需将对应坐标相减即可。
Straight line graphs have an equation of the form y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). To find the gradient between two points, use m = change in y / change in x.
直线图像的方程为 y = mx + c，其中 m 为斜率 (坡度)，c 为 y 轴截距 (直线与 y 轴交点的纵坐标)。任意两点间的斜率可由 m = y 的变化量 / x 的变化量 求得。
To plot a straight line, create a table of values by substituting x into the equation. You can also identify m and c to draw the line quickly. Parallel lines have the same gradient; perpendicular lines have gradients that multiply to -1.
绘制直线时，可代入 x 值列表求 y 再描点连线。也可通过识别 m 和 c 快速作图。平行线斜率相等；垂直线斜率乘积为 -1。
8. Ratio, Proportion and Rates | 比、比例与速率
Ratios compare quantities and are most useful when simplified to the form a : b with whole numbers and no common factor. To divide a quantity in a given ratio, add the parts together, then multiply the total by each part’s fraction.
比用于比较数量，通常化为最简整数比 a : b (无公因数)。按比例分配总量时，将份数相加，然后用总量乘以各份所占分数即可。
Direct proportion means two quantities increase at the same rate: y = kx. Inverse proportion means as one increases, the other decreases: y = k/x. Word problems often involve recipes, exchange rates, or scaling lengths and areas.
正比关系指两个量以相同速率变化：y = kx；反比关系中一个量增加则另一个减少：y = k/x。应用题常涉及食谱缩放、汇率换算或长度与面积的缩放。
Rates link different measures, such as speed = distance/time, or density = mass/volume. Use formula triangles or rearrange formulas to solve for the unknown quantity. Pay attention to units and convert where necessary (e.g. minutes to hours).
速率联系不同量纲，如 速度 = 路程/时间、密度 = 质量/体积。可使用公式三角或移项法求未知量。务必注意单位，必要时要换算 (如分钟化为小时)。
9. Angles and Shapes | 角与形状
Basic angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, and vertically opposite angles are equal. In a triangle, angles sum to 180°; in a quadrilateral they sum to 360°.
基本角度事实：直线上的邻角之和为 180°，绕一点一周的角度之和为 360°，对顶角相等。三角形内角和为 180°，四边形内角和为 360°。
When a transversal crosses parallel lines, alternate angles are equal, corresponding angles are equal, and interior (co‑interior) angles sum to 180°. Use these properties to find missing angles in diagrams involving parallel lines.
平行线被横线所截时，内错角相等，同位角相等，同旁内角之和为 180°。利用这些性质可求出含有平行线的图形中的未知角。
Polygons are named by the number of sides. The sum of interior angles of an n-sided polygon is (n – 2) × 180°. The sum of exterior angles (one at each vertex) is always 360°. For regular polygons, each interior angle = (n – 2) × 180° / n.
多边形按边数命名。n 边形的内角和为 (n – 2) × 180°。外角和恒为 360° (每个顶点取一个外角)。对于正多边形，每个内角 = (n – 2) × 180° / n。
10. Area, Perimeter and Volume | 面积、周长与体积
Perimeter is the distance around a 2D shape; simply add all side lengths. For a circle, the circumference (perimeter) is C = 2πr or C = πd, where r is the radius and d is the diameter.
周长是二维图形边界的总长，将各边长相加即可。圆的周长 (圆周) 为 C = 2πr 或 C = πd，其中 r 为半径，d 为直径。
Area formulas: rectangle A = l × w, triangle A = ½ × base × height, parallelogram A = base × height, trapezium A = ½(a + b)h. For a circle, A = πr². Remember to use perpendicular height, not slant edges. For composite shapes, split into simpler parts and sum the areas.
面积公式：矩形 A = 长×宽，三角形 A = ½×底×高，平行四边形 A = 底×高，梯形 A = ½(a + b)h。圆的面积 A = πr²。注意高必须是垂直高，而非斜边。对于组合图形，可分割为简单图形后求和。
Volume of a prism = area of cross-section × length. Common prisms include cuboids (V = l × w × h) and cylinders (V = πr²h). Capacity is often measured in litres; 1 litre = 1000 cm³. Learn to work with surface area as well.
棱柱的体积 = 横截面积 × 长度。常见棱柱包括长方体 (V = 长×宽×高) 和圆柱 (V = πr²h)。容量常用升表示，1 升 = 1000 cm³。同时要掌握表面积的计算。
11. Pythagoras’ Theorem and Trigonometry | 毕达哥拉斯定理与三角学
Pythagoras’ theorem applies to right‑angled triangles: a² + b² = c², where c is the hypotenuse (longest side opposite the right angle). It can be used to find a missing side length or to check if a triangle is right‑angled.
毕达哥拉斯定理适用于直角三角形：a² + b² = c²，其中 c 是斜边 (直角对边，最长边)。该定理可用于求未知边长或验证三角形是否为直角三角形。
Trigonometry relates angles and side lengths in right‑angled triangles. The three ratios are: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use the appropriate ratio depending on which two sides are known or required.
三角学联系直角三角形的角与边长。三个比值分别为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。根据已知或所求的两边选择合适比值。
To find a missing angle, use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹ on your calculator. Always ensure your calculator is in degree mode. Labelling the sides (hypotenuse, opposite, adjacent) correctly relative to the given angle is the first critical step.
求未知角时，需使用计算器上的反三角函数 sin⁻¹、cos⁻¹、tan⁻¹，并确保计算器处于角度模式。正确标记相对于已知角的斜边、对边和邻边是解题的关键第一步。
12. Probability and Statistics | 概率与统计
Probability values range from 0 (impossible) to 1 (certain). The probability of an event happening plus the probability of it not happening equals 1: P(A’) = 1 – P(A). For equally likely outcomes, P(event) = number of favourable outcomes / total number of outcomes.
概率值介于 0 (不可能) 到 1 (必然) 之间。事件发生与不发生的概率之和为 1：P(A’) = 1 – P(A)。对于等可能结果，P(事件) = 有利结果数 / 全部可能结果数。
Sample space diagrams and two-way tables help list all possible outcomes for combined events. Probability trees are used for successive events; multiply along branches for combined probability and add probabilities for mutually exclusive branches.
样本空间图和双向表可帮助列出组合事件的所有可能结果。概率树用于连续事件，沿分支相乘得到联合概率，并将互斥分支的概率相加。
Statistics involve collecting, representing and interpreting data. Measures of central tendency: mean (sum of values ÷ number of values), median (middle value), mode (most frequent). Range measures spread: range = highest – lowest. Be able to read bar charts, pictograms, pie charts and line graphs.
统计学涉及数据的收集、呈现与解读。集中趋势量数：平均数 (总和÷数据个数)、中位数 (居中值)、众数 (出现次数最多的值)。极差衡量离散程度：极差 = 最大值 – 最小值。要能读懂条形图、象形图、饼图和折线图。
For grouped frequency tables, you can estimate the mean by using midpoints of intervals. Clearly label axes, scales and titles when drawing statistical diagrams. Outliers can affect the mean, so the median is sometimes more representative.
对于分组频数表，可用组中值估算平均数。绘制统计图表时，要清晰标注轴、刻度和标题。极端值
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