📚 Year 10 WJEC Mathematics: Core Topic Review | WJEC 十年级数学核心知识点梳理
Year 10 is a pivotal stage in the WJEC GCSE Mathematics course, where the foundational knowledge acquired in earlier years is consolidated and extended. This review outlines the essential topics every student should master, from number skills and algebra to geometry, statistics, and the first steps in trigonometry. A secure grasp of these core areas builds confidence for the examination and provides a solid platform for the more advanced work in Year 11.
十年级是 WJEC 普通中等教育证书数学课程的关键阶段，此前所学的基础知识在这一年中得到巩固与拓展。本文梳理了每位学生都应掌握的核心主题，涵盖数字技能、代数、几何、统计以及三角函数初步。牢固掌握这些核心内容不仅能增强应考信心，也为十一年级更高阶的学习奠定坚实基础。
1. Number Skills and Place Value | 数字技能与位值
Place value is the understanding that a digit’s worth depends on its position within a number. In the number 75.36, the 7 represents 7 tens, the 5 is 5 units, the 3 is three tenths and the 6 is six hundredths.
位值是指一个数字的值取决于它在数中的位置。在数字 75.36 中，7 代表 7 个十，5 代表 5 个一，3 代表十分之三，6 代表百分之六。
Ordering integers, decimals and negative numbers correctly relies on placing them on a number line. Negative numbers further left are smaller; for example, −8 < −3 < 0 < 2.5.
正确排序整数、小数和负数需要将它们放在数轴上。越靠左的负数越小，例如 −8 < −3 < 0 < 2.5。
The four operations (addition, subtraction, multiplication, division) must be applied following the BIDMAS order: Brackets, Indices, Division & Multiplication (left to right), Addition & Subtraction (left to right).
四则运算（加、减、乘、除）必须遵循 BIDMAS 顺序：先括号，再指数，然后乘除（从左到右），最后加减（从左到右）。
A prime number has exactly two distinct factors, 1 and itself. The highest common factor (HCF) and lowest common multiple (LCM) of two numbers can be found by listing factors and multiples or by using prime factor trees.
质数恰好有两个不同的因数：1 和它本身。两个数的最大公因数 (HCF) 和最小公倍数 (LCM) 可以通过列举因数与倍数，或使用质因数树来求出。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
Converting between fractions, decimals and percentages is a core skill. To change a fraction to a decimal, divide the numerator by the denominator; for example ¾ = 3 ÷ 4 = 0.75. To express as a percentage, multiply the decimal by 100%, giving 75%.
在分数、小数和百分数之间进行转换是一项核心技能。将分数化为小数，用分子除以分母；例如 ¾ = 3 ÷ 4 = 0.75。要表达为百分数，将小数乘以 100%，得到 75%。
When adding or subtracting fractions, first rewrite them with a common denominator. Multiplication of fractions involves multiplying numerators together and denominators together, while division is performed by multiplying by the reciprocal.
做分数加减法时，需要先将它们改写为同分母分数。分数乘法是用分子相乘作分子、分母相乘作分母；分数除法则乘以倒数进行计算。
Finding a percentage of an amount can be done by converting the percentage to a decimal and multiplying. A percentage increase or decrease is calculated by adding or subtracting the relevant proportion to the original amount.
求一个数量的百分之几，可先将百分数化成小数再相乘。计算百分数增减，则是在原数量上加上或减去相应的比例。
3. Standard Form and Approximations | 标准形式与近似值
Standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. Large numbers such as 45000 become 4.5 × 10⁴, and small numbers like 0.0032 become 3.2 × 10⁻³.
标准形式写作 A × 10ⁿ，其中 1 ≤ A < 10 且 n 为整数。大数如 45000 可写成 4.5 × 10⁴，小数字如 0.0032 可写成 3.2 × 10⁻³。
Rounding to a given number of decimal places or significant figures allows us to give sensible approximations. The digit after the cut-off point determines whether the last kept digit stays the same or rounds up.
按指定的小数位数或有效数字进行四舍五入，可以给出合理的近似值。截断位之后的数字决定了保留的最后一位是保持不变还是进位。
Estimating complex calculations by rounding each number to one significant figure provides a quick check for accuracy. For instance, estimating 48.7 × √3.9 by calculating 50 × 2 gives a rough answer of 100.
将每个数字四舍五入到一位有效数字来估算复杂计算，可以快速检查结果的准确性。例如，用 50 × 2 来估算 48.7 × √3.9，得到大致答案 100。
4. Algebra Basics: Expressions and Formulae | 代数基础：表达式与公式
Simplifying algebraic expressions involves collecting like terms. Terms are ‘like’ if they contain exactly the same variables and powers; for example, 3x and −2x are like terms, but 3x and 3x² are not.
化简代数表达式需要合并同类项。如果项中包含完全相同的变量和幂次，它们就是“同类”的；例如 3x 和 −2x 是同类项，但 3x 和 3x² 不是。
Expanding brackets means multiplying each term inside the bracket by the term outside. For a single bracket, a(b + c) = a×b + a×c. If a minus sign precedes the bracket, remember to multiply all terms by −1 as well.
展开括号就是将括号外的项乘以括号内的每一项。对于单项括号，a(b + c) = a × b + a × c。如果括号前是负号，务必也将每一项乘以 −1。
Factorising is the reverse of expanding. You look for the highest common factor of the terms and write the expression as a product. The expression 6x + 9 factorises to 3(2x + 3).
因式分解是展开的逆运算。找出各项的最大公因数，将表达式写成乘积形式。例如 6x + 9 分解为 3(2x + 3)。
Substituting numbers into a formula or expression replaces each letter with a given value. Always follow BIDMAS after substitution to evaluate correctly.
将数字代入公式或表达式就是用一个给定的值替换每个字母。代入后始终遵循 BIDMAS 规则进行正确计算。
5. Solving Linear Equations | 解线性方程
A linear equation contains an unknown, usually shown as a letter, and can be solved by carrying out inverse operations to isolate the unknown. For a one-step equation such as x + 7 = 15, subtract 7 from both sides to find x = 8.
线性方程包含一个未知数，通常用字母表示，可通过逆运算来求解，将未知数单独留在等号一侧。对于一步方程如 x + 7 = 15，两边同时减去 7 得到 x = 8。
With two-step equations, use inverse operations in reverse order. Solve 2x − 3 = 11 by first adding 3 to both sides, giving 2x = 14, then dividing by 2 to get x = 7.
对于两步方程，按逆序使用逆运算。解 2x − 3 = 11 时，先两边加 3，得到 2x = 14，再除以 2，得到 x = 7。
When the unknown appears on both sides, eliminate the smaller number of x terms from one side. For 5x + 2 = 3x + 10, subtract 3x from both sides to obtain 2x + 2 = 10, then solve as usual.
当未知数出现在方程两边时，将较小数量的 x 项从一边消去。对于 5x + 2 = 3x + 10，两边同时减去 3x，得到 2x + 2 = 10，然后照常求解。
Forming equations from word problems requires identifying the unknown, representing it with a letter, and building an equation from the relationships described. Always check your solution satisfies the original problem.
从文字题列方程需要识别未知数，用字母表示，并根据描述的关系建立方程。务必检验所得的解是否符合原问题。
6. Sequences and the nth Term | 数列与第 n 项
A number sequence that goes up or down by the same amount each time is called an arithmetic (linear) sequence. The difference between consecutive terms is called the common difference, d.
每次以相同数量增加或减少的数列称为算术（线性）数列。相邻两项之间的差称为公差，记作 d。
The nth term of a linear sequence can be written as an expression of the form a n + b, where a is the common difference. To find b, work out what the 0th term would be, or use the formula T(n) = d n + (first term − d).
线性数列的第 n 项可以写成 a n + b 的形式，其中 a 是公差。要找到 b，可以推算出第 0 项，或使用公式 T(n) = d n + (首项 − d)。
For example, the sequence 5, 8, 11, 14, … has a common difference of 3. The nth term is 3n + 2 because when n = 1, 3×1 + 2 = 5.
例如数列 5, 8, 11, 14, … 的公差为 3。第 n 项为 3n + 2，因为当 n = 1 时，3 × 1 + 2 = 5。
Once the nth term is found, you can generate any term of the sequence by substituting the position number into the expression.
一旦找到第 n 项，就可以把位置数代入该表达式，求出数列中的任意一项。
7. Coordinates and Straight Line Graphs | 坐标与直线图像
A point on a graph is located by its coordinates (x, y), where x is the horizontal distance from the origin and y is the vertical distance. The axes are labelled x and y, meeting at the origin (0, 0).
图上一点由其坐标 (x, y) 定位，其中 x 是该点到原点的水平距离，y 是垂直距离。坐标轴分别标记为 x 和 y，相交于原点 (0, 0)。
The equation of a straight line is usually written as y = mx + c. The gradient m tells you how steep the line is, and the intercept c shows where the line crosses the y‑axis.
直线方程通常写作 y = mx + c。斜率 m 表示直线的倾斜程度，截距 c 表明直线与 y 轴的交点位置。
To draw a graph of a linear equation, choose at least three x‑values, substitute them into the equation to find the matching y‑values, plot the points, and join them with a straight line.
要绘制线性方程图像，至少选取三个 x 值，代入方程求出对应的 y 值，描点后用直线连接起来。
Parallel lines have the same gradient, m, but different intercepts. This fact can be used to identify lines that never meet.
平行线具有相同的斜率 m，但截距不同。这一性质可用于识别永不相交的直线。
8. Ratio, Proportion and Rates of Change | 比、比例与变化率
Ratios compare quantities of the same kind. The ratio 3:2 means for every 3 parts of one quantity there are 2 parts of another. Ratios are simplified by dividing by the highest common factor, just like fractions.
比用来比较同类量。比值 3:2 表示一种量每有 3 份，另一种量就有 2 份。比的化简与分数类似，除以最大公因数即可。
Sharing a quantity in a given ratio requires finding the value of one part. For a ratio 3:5, add the parts to get 8 total parts, divide the amount by 8, then multiply by 3 and 5 to find the separate shares.
按给定比例分配一个量需要先求出一份的值。对于比值 3:5，将份数相加得到 8 份，总量除以 8，再分别乘以 3 和 5 得到各份的量。
Direct proportion means two quantities increase or decrease at the same rate. The unitary method involves finding the value of one item and scaling up to find multiples.
正比例表示两个量以相同速率增加或减少。归一法先求出一个单位的量，再按比例放大求得所需数量的值。
Compound measures such as speed, density and pressure mix two different units. For uniform motion, speed = distance ÷ time, so distance = speed × time.
复合度量如速度、密度和压强涉及两种不同的单位。对于匀速运动，速度 = 距离 ÷ 时间，因此距离 = 速度 × 时间。
9. Statistics: Charts and Averages | 统计：图表与平均数
Data can be represented in bar charts, pie charts, line graphs and frequency tables. Choosing the right diagram depends on the type of data and what message you want to highlight.
数据可以用条形图、饼图、折线图和频数表来表示。选择合适的图形取决于数据类型和你希望突出的信息。
The three common averages are the mean, median and mode. The mean is found by adding all values and dividing by the number of values; the median is the middle value when data are ordered; the mode is the most frequent value.
三种常见的平均数是均值、中位数和众数。均值是所有数值之和除以数值的个数；中位数是排序后处于中间位置的数值；众数是出现次数最多的数值。
The range is a measure of spread, calculated as the largest value minus the smallest value. A small range suggests data are closely grouped; a large range indicates more variation.
极差是一种离散程度的度量，等于最大值减最小值。极差小表示数据集中；极差大则表示变异性较大。
Grouped frequency tables estimate the mean using mid-interval values because the exact individual values are not known. The impact of outliers on the mean and median should be considered when choosing which average to use.
由于不知道具体数值，分组频数表使用组中值来估算均值。在选择使用哪种平均数时，应考虑异常值对均值和中位数的影响。
10. Probability | 概率
Probability measures the chance of an event happening, given as a number between 0 (impossible) and 1 (certain). The probability of an event A is written P(A) = number of favourable outcomes / total number of equally likely outcomes.
概率衡量事件发生的可能性，用一个介于 0（不可能）和 1（必然）之间的数表示。事件 A 的概率写作 P(A) = 有利结果数 / 等可能结果总数。
The sum of probabilities of all possible outcomes is 1. Therefore, the probability of an event not happening is 1 minus the probability that it does happen.
所有可能结果的概率之和为 1。因此，某个事件不发生的概率等于 1 减去它发生的概率。
Sample space diagrams list all outcomes of combined events, helping to calculate probabilities systematically. For two fair spinners, a table showing all possible sums makes it easy to find the probability of a specific total.
样本空间图列出了组合事件的所有结果，有助于系统地计算概率。对于两个公平的转盘，用表格显示所有可能的和，可以方便地求出某一具体总和的概率。
Tree diagrams are useful for showing successive independent events. Multiply along the branches to find the probability of a sequence of outcomes.
树状图用于展示连续的独立事件。沿着分支相乘，即可求出一系列结果发生的概率。
11. Pythagoras’ Theorem | 毕达哥拉斯定理
In a right‑angled triangle, the side opposite the right angle is the hypotenuse, which is the longest side. Pythagoras’ theorem states: (hypotenuse)² = (short side)² + (short side)², often written as c² = a² + b², where c is the hypotenuse.
在直角三角形中，直角所对的边是斜边，也是最长边。毕达哥拉斯定理指出：（斜边）² =（短边）² +（短边）²，常写作 c² = a² + b²，其中 c 为斜边。
To find the length of the hypotenuse, square the two shorter sides, add them and take the square root. For a triangle with legs 3 cm and 4 cm, c = √(3² + 4²) = √25 = 5 cm.
求斜边长度时，先将两条短边平方，相加后再开平方。对于两条直角边分别为 3 cm 和 4 cm 的三角形，c = √(3² + 4²) = √25 = 5 cm。
To find a shorter side, rearrange the theorem: a² = c² − b². Always subtract the square of the known short side from the square of the hypotenuse, then take the square root.
求一条短边时，将定理重新排列：a² = c² − b²。始终用斜边的平方减去已知短边的平方，再开平方求得。
Pythagoras’ theorem only applies to right‑angled triangles. It can be used to check whether a triangle is right‑angled by testing if the side lengths satisfy the equation a² + b² = c².
毕达哥拉斯定理只适用于直角三角形。通过检验三边长度是否满足 a² + b² = c²，可以判断一个三角形是否为直角三角形。
12. Introduction to Trigonometry | 三角函数初步
In a right‑angled triangle, the three trigonometric ratios relate an acute angle to the lengths of the sides. They are: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.
在直角三角形中，三个三角比将锐角与边长联系起来：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
To remember these ratios, the abbreviation SOH CAH TOA is helpful: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
为了方便记忆，可使用缩写 SOH CAH TOA：正弦是对边比斜边，余弦是邻边比斜边，正切是对边比邻边。
When one acute angle and one side are known, you can use the appropriate ratio to find a missing side. For example, if a 40° angle and the adjacent side of length 7 cm are given, the opposite side can be found using tan 40° = opposite / 7, so opposite = 7 × tan 40°.
当已知一个锐角和一条边时，可使用适当的三角比求出未知边。例如，已知 40° 角和邻边 7 cm，可用 tan 40° = 对边 / 7，因此对边 = 7 × tan 40°。
Inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) are used to find an angle when two sides are known. Always ensure the calculator is in degree mode before evaluating.
当已知两条边求角度时，使用反三角函数（sin⁻¹, cos⁻¹, tan⁻¹）。务必确保计算器处于角度模式再进行计算。
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