📚 Year 10 WJEC Further Mathematics: Core Topics Overview | Year 10 WJEC 进阶数学核心知识点梳理
Year 10 is a critical stage in the WJEC Further Mathematics course, where you build the foundational skills needed for advanced GCSE and A‑level study. This article summarises the core topics you will encounter, from algebra and trigonometry to the first ideas of calculus and matrices. Mastering these concepts now will give you confidence in tackling more complex problems later.
十年级是 WJEC 进阶数学课程的关键阶段，在这一年里你将建立起 GCSE 高难度内容与 A‑level 学习所必需的基础技能。本文梳理了你需要掌握的核心知识点，从代数、三角学到微积分和矩阵的初步概念。现在把这些概念学扎实，将为以后解决更复杂的问题带来自信。
1. Algebraic Manipulation and Polynomials | 代数运算与多项式
A polynomial is an expression built from variables and constants using addition, subtraction, multiplication and non‑negative integer exponents. In Year 10, you will handle polynomials up to degree 3, such as 2x³ – 5x² + 3x – 7. The degree is the highest power of the variable.
多项式是由变量和常数通过加、减、乘和非负整数指数构成的表达式。十年级你会处理最高三次的多项式，比如 2x³ – 5x² + 3x – 7。次数是指变量的最高次幂。
Adding and subtracting polynomials means collecting like terms – those with exactly the same variable part. For instance, (3x² + 2x – 1) + (x² – 4x + 5) simplifies to 4x² – 2x + 4. Always be careful with signs when subtracting.
多项式加减就是合并同类项——变量部分完全相同的项。例如 (3x² + 2x – 1) + (x² – 4x + 5) 化简得 4x² – 2x + 4。做减法时要特别注意符号。
Multiplying polynomials involves distributing each term in the first bracket across every term in the second. For two binomials, the FOIL method (First, Outer, Inner, Last) is a handy tool.
多项式乘法是把第一个括号中的每一项分别乘以第二个括号中的各项。对于两个二项式，FOIL 法则（首项、外项、内项、末项）是一种便捷的方法。
(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6
Expanding brackets often generates new terms that can be simplified. Mastering this skill underpins factorisation, equation solving and later work with functions.
展开括号常常会产生可合并的项。掌握这一技能是学习因式分解、方程求解以及后期函数变换的基础。
2. Factorisation and Solving Equations | 因式分解与方程求解
x. For example, x² + 7x + 10 factorises as (x + 2)(x + 5) because 2 × 5 = 10 and 2 + 5 = 7.
因式分解是展开的逆过程。对于二次式，找出两个数使其乘积等于常数项、和等于 x 的系数。例如 x² + 7x + 10 分解为 (x + 2)(x + 5)，因为 2×5=10，2+5=7。
Always check for a common factor first. In 3x² – 12x, the common factor 3x can be taken out to give 3x(x – 4). This simple step makes more complex factorisation much easier.
务必先检查公因子。对 3x² – 12x，提取公因子 3x 得 3x(x – 4)。这个简单步骤能让更复杂的分解变得容易。
ic is factorised, you can solve the equation using the zero product property: if (x + 2)(x – 5) = 0, then x = –2 or x = 5. When factorisation is not possible, use the quadratic formula.
二次式一旦分解，就可以用零积性质求解方程：若 (x + 2)(x – 5) = 0，则 x = –2 或 x = 5。当无法因式分解时，使用求根公式。
x = (-b ± √b² – 4ac) / (2a)
The expression under the square root, b² – 4ac, is called the discriminant. It tells you whether there are two real roots, one repeated root, or no real roots.
根号下的式子 b² – 4ac 叫做判别式。它能告诉你二次方程有两个实根、一个重根还是没有实根。
3. Coordinate Geometry and Straight Lines | 坐标几何与直线
o points (x₁, y₁) and (x₂, y₂), the distance between them comes from Pythagoras: d = √[(x₂ – x₁)² + (y₂ – y₁)²]. The midpoint is simply the average of the coordinates: ((x₁+x₂)/2, (y₁+y₂)/2).
坐标几何把代数与图形联系起来。对于两点 (x₁,y₁) 和 (x₂,y₂)，距离由勾股定理得出：d = √[(x₂–x₁)² + (y₂–y₁)²]。中点就是坐标的平均值：((x₁+x₂)/2, (y₁+y₂)/2)。
e gradient (slope) of a line is the ratio of vertical change to horizontal change: m = (y₂ – y₁)/(x₂ – line rises, a negative gradient means it falls, and zero gradient gives a horizontal line.
直线的梯度（斜率）是纵坐标变化量与横坐标变化量之比：m = (y₂ – y₁)/(x₂ – x₁)。正梯度表示直线上升，负梯度下降，零梯度表示水平线。
f a straight line is usually written as y = mx + c, where m is the gradient and c is the y‑intercept. You can also use the point‑gradient form y – y₁ = m(x – x₁) when you know a point and the slope.
直线方程通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距。若已知一点和斜率，也可以用点斜式 y – y₁ = m(x – x₁)。
f they have the same gradient. They are perpendicular if the product of their gradients is –1. This gives a quick check: m₁ × m₂ = –1.
两条直线平行的条件是斜率相等；垂直的条件是斜率乘积为 –1。一个快速检验法则是 m₁ × m₂ = –1。
4. Functions and Transformations | 函数与图像变换
) = 2x + 3 means multiply by 2 and add 3. Always remember that f(a) means replacing x with a.
函数接受输入值 x 并产生唯一输出值 f(x)。函数记号是描述关系的有力工具；例如 f(x) = 2x + 3 表示乘以 2 再加 3。始终记住 f(a) 意味着将 x 替换为 a。
Transformations let you shift or stretch graphs. Adding a constant to the function, f(x) + a, moves the graph vertically. Subtracting a constant inside, f(x – a), shifts it a units to the right.
图像变换可以平移或拉伸图形。在函数后加常数 f(x) + a，图像垂直移动。在括号内减去常数 f(x – a)，图像向右平移 a 个单位。
y = f(x) + 2 (up by 2), y = f(x – 3) (right by 3)
Multiplying the function by a constant, k f(x), stretches the graph vertically by a factor of k. Multiplying the input, f(kx), compresses the graph horizontally by factor 1/k if k > 1. These ideas are vital for sketching curves without plotting every point.
将函数乘以常数 k，即 k f(x)，会使图像在竖直方向上拉伸 k 倍。若对输入作乘法 f(kx)，且 k>1，则图像在水平方向上压缩为原来的 1/k。这些概念对快速绘制函数草图至关重要。
5. Trigonometry | 三角函数基础
Trigonometry begins with right‑ t/hypotenuse, tan θ = opposite/adjacent. You will use these to find missing sides and angles.
三角学从直角三角形开始。三个基本比为正弦、余弦和正切，常用 SOH CAH TOA 来记忆：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。你将用它们求未知的边长和角度。
° and 60° must be memorised. They often appear without calculators: sin30° = 1/2, sin45° = √2/2, sin60° = √3/2; cos30° = √3/2, cos45° = √2/2, cos60° = 1/2; tan30° = 1/√3, tan45° = 1, tan60° = √3.
30°、45° 和 60° 的精确值必须熟记，考试时常不给计算器：sin30°=1/2，sin45°=√2/2，sin60°=√3/2；cos30°=√3/2，cos45°=√2/2，cos60°=1/2；tan30°=1/√3，tan45°=1，tan60°=√3。
mental. It allows you to find one ratio when the other is known, and it is the basis for many proofs and equation solutions. For any acute angle, this relationship always holds.
恒等式 sin²θ + cos²θ = 1 是基础。它可以从一个比值求出另一个，也是许多证明和解方程的依据。对于任意锐角，这个关系始终成立。
6. Introduction to Calculus – Differentiation | 微积分入门——微分
s for any real power, but in Year 10 you focus on positive integer powers and simple polynomials.
微分是求曲线斜率的方法。对于 y = xⁿ，导数为 dy/dx = n xⁿ⁻¹。这个简单规则适用于任何实指数，但十年级主要关注正整数幂和简单多项式。
y = x³ → dy/dx = 3x²
ms, differentiate each separately. A constant multiplier stays: d/dx (5x²) = 5 × 2x = 10x.
常数项的导数为零。对和差形式的函数，逐项求导即可。常数乘子保留：d/dx (5x²) = 5 × 2x = 10x。
Once you have the derivative, you can find the gradient at any point by substituting the x‑value. This lets you write the equation of the tangent line to a curve at a given point – a key skill that bridges algebra and geometry.
求出导数后，代入 x 的值就能得到该点的斜率。进而可以写出曲线在某点处的切线方程——这是连接代数与几何的关键技能。
7. Matrices | 矩阵基础
on, subtraction and scalar multiplication of 2×2 matrices, matching corresponding entries.
矩阵是一个矩形数字阵列，由其大小（行数×列数）来描述。你需要进行 2×2 矩阵的加法、减法和标量乘法，对应位置元素运算。
3 × [1 2; 3 4] = [3 6; 9 12]
orresponding elements of column j of the second matrix and add them. For two 2×2 matrices, this gives four entries.
矩阵相乘较为复杂：乘积中第 i 行第 j 列的元素，将第一个矩阵第 i 行元素与第二个矩阵第 j 列对应元素相乘再求和。对于两个 2×2 矩阵，这样会得到四个元素。
verse, a matrix A⁻¹ such that A × A⁻¹ = I. The inverse exists only when the determinant ad – bc ≠ 0.
单位矩阵 I = [1 0; 0 1] 的作用类似于数字 1：A × I = A，I × A = A。有些矩阵存在逆矩阵 A⁻¹，满足 A × A⁻¹ = I。只有当行列式 ad – bc ≠ 0 时逆矩阵才存在。
8. Vectors in 2D | 二维向量
n as a column (x y) or using unit vectors i and j, e.g. 3i + 4j. The magnitude (length) is |v| = √x² + y², which comes directly from Pythagoras’ theorem.
向量描述大小和方向。在
Published by TutorHao | Year 10 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图