Year 9 WJEC Further Maths: Core Knowledge Review | Year 9 WJEC 进阶数学：核心知识点梳理

📚 Year 9 WJEC Further Maths: Core Knowledge Review | Year 9 WJEC 进阶数学：核心知识点梳理

In Year 9 WJEC Further Maths, students build on the foundations from Key Stage 3 and begin to explore more abstract concepts. This article summarises the core topics that every student should master, covering algebra, geometry, trigonometry, statistics and probability. A solid understanding of these areas will prepare you for GCSE Higher tier and beyond.

在Year 9 WJEC进阶数学中，学生们在Key Stage 3的基础上进一步探索更抽象的概念。本文梳理了每个学生必须掌握的核心主题，涵盖代数、几何、三角函数、统计与概率。扎实掌握这些领域将为你准备GCSE Higher tier及更深层次的学习奠定基础。

1. Algebraic Manipulation and Equations | 代数运算与方程

by collecting like terms is a fundamental skill. For example, 3a + 2b - a + 4b = 2a + 6b.

通过合并同类项来化简代数表达式是一项基本技能。例如，3a + 2b - a + 4b = 2a + 6b。

straightforward: 2(x + 5) = 2x + 10. Always multiply the term outside by each term inside.

展开单项括号很简单：2(x + 5) = 2x + 10。一定要将外面的项乘以括号内的每一项。

istribution: (x + 2)(x - 3) = x² - x - 6. Check your sign when multiplying negative terms.

展开双括号需要仔细分配：(x + 2)(x - 3) = x² - x - 6。在乘以负数项时要注意符号。

2x + 5 = 13, isolate x: 2x = 8, so x = 4. Always perform the same operation on both sides.

解线性方程如 2x + 5 = 13，分离出 x：2x = 8，所以 x = 4。始终在等式两边执行相同的操作。

ar on both sides, e.g. 3x - 7 = x + 5, collect x-terms on one side: 2x = 12, giving x = 6.

当未知数出现在等式两边时，例如 3x - 7 = x + 5，将含 x 的项移到同一边：2x = 12，得出 x = 6。

erm by the least common denominator. For x/3 + 2 = 5, multiply by 3: x + 6 = 15, so x = 9.

涉及分数的方程需要将每一项乘以最小公分母。对于 x/3 + 2 = 5，两边乘以 3：x + 6 = 15，所以 x = 9。

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2. Quadratic Expressions and Factorising | 二次表达式与因式分解

A quadratic expression takes the form ax² + bx + c. Factorising reverses the expansion of two brackets.

二次表达式的一般形式为 ax² + bx + c。因式分解是展开两个括号的逆过程。

multiply to c and add to b. For x² + 5x + 6, those numbers are 2 and 3, so (x + 2)(x + 3).

当 a = 1 时，找出两个数，乘积为 c 且和为 b。对于 x² + 5x + 6，这两个数是 2 和 3，因此 (x + 2)(x + 3)。

erence of two squares pattern: x² - a² = (x + a)(x - a). Example: x² - 9 = (x + 3)(x - 3).

平方差模式：x² - a² = (x + a)(x - a)。例如：x² - 9 = (x + 3)(x - 3)。

When a > 1, use the 'ac method' to 5 are 4 and 1, so rewrite as 2x² + 4x + x + 2 = 2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2).

当 a > 1 时，使用“ac方法”拆解中间项。对于 2x² + 5x + 2，ac = 4。乘积为 4 且和为 5 的数是 4 和 1，因此改写为 2x² + 4x + x + 2 = 2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2)。

Always expand your factorised form to verify it matches the original quadratic. This habit avoids sign and coefficient errors.

始终将因式分解的结果展开，验证是否与原始二次表达式一致。这个习惯可以避免符号和系数错误。

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3. Inequalities and Number Lines | 不等式与数轴

Solving linear inequalities is similar to solving equations, except the direction flips when multiplying or dividing by a negative number. From 2x - 3 < 7, we get 2x < 10, then x < 5.

解一元一次不等式与解方程类似，但当乘以或除以一个负数时，不等号方向要改变。由 2x - 3 < 7 可得 2x < 10，然后 x < 5。

If we have -2x > 6, dividing by -2 flips the sign: x < -3. This rule is crucial for WJEC Further Maths assessments.

如果遇到 -2x > 6，除以 -2 后方向反转：x < -3。这是WJEC进阶数学考试中的关键规则。

sed (filled) circle for ≤ or ≥. For x < 5, draw an open circle at 5 and shade to the left.

在数轴上表示解集：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。对于 x < 5，在5处画空心圆并向左阴影。

ean x is between -2 and 4, including -2 but not 4. On the number line, use a closed circle at -2 and an open circle at 4, then connect them.

复合不等式如 -2 ≤ x < 4 表示 x 在 -2 和 4 之间，包括 -2 但不包括 4。在数轴上，-2 处用实心圆，4 处用空心圆，然后连接起来。

Always write the variable on the left in your final answer: x < 5, not 5 > x. It improves clarity and matches mark schemes.

最终答案中始终把变量写在左边：x < 5，而不是 5 > x。这样做更清晰，也符合评分标准。

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4. Indices and Surds | 指数与根式

= aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. These are essential for handling algebraic forms.

指数法则可以简化幂的表达式：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。这些是处理代数形式的必备工具。

Fractional indices connect powers and roots: a^½ = √a (the square root), and a^⅓ = ³√a (the cube root). In general, a^m/n = ⁿ√aᵐ.

分数指数将幂和根式联系起来：a^½ = √a（平方根），a^⅓ = ³√a（立方根）。一般地，a^m/n = ⁿ√aᵐ。

To simplify surds like √12, look for the largest perfect square factor: √12 = √4 × 3 = √4 × √3 = 2√3. Always express answers in simplest surd form.

化简根式如 √12，找出最大的完全平方因数：√12 = √4 × 3 = √4 × √3 = 2√3。始终将答案表示为最简根式。

Rationalising a denominator means removing a surd from the bottom. For 1/√2, multiply numerator and denominator by √2: 1/√2 = √2/2. This is a required skill in WJEC Further Maths.

分母有理化是指去除分母中的根号。对于 1/√2，将分子分母同乘 √2：1/√2 = √2/2。这是WJEC进阶数学中必须掌握的技能。

When multiplying surds, combine under one root: √a × √b = √ab. For example, √5 × √10 = √50 = 5√2.

两个根式相乘时，可以合并在一个根号下：√a × √b = √ab。例如，√5 × √10 = √50 = 5√2。

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5. Coordinates and Linear Graphs | 坐标与直线图像

= (y₂ - y₁) / (x₂ - x₁). A positive gradient slopes upward, a negative gradient downward.

过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m 由公式 m = (y₂ - y₁) / (x₂ - x₁) 求得。正斜率向上倾斜，负斜率向下倾斜。

y = mx + c, where m is the gradient and c is the y-intercept (the value of y when x = 0).

直线方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距（即 x = 0 时的 y 值）。

gradient. If one line has m = 2, any line parallel to it also has m = 2 but a different c.

平行线具有相同的斜率。如果一条线的斜率为 2，那么任何与之平行的线斜率也为 2，但截距 c 不同。

y m₁ × m₂ = -1. So if a line has gradient 3, a line perpendicular to it has gradient -1/3.

垂直线的斜率满足 m₁ × m₂ = -1。因此，如果一条线的斜率为 3，与之垂直的线斜率为 -1/3。

int into y = mx + c to find c. For (1,2) and (3,8), m = (8-2)/(3-1) = 3, then 2 = 3(1) + c ⇒ c = -1, so y = 3x - 1.

通过两点求直线方程，先计算 m，再将一个点的坐标代入 y = mx + c 求出 c。对于 (1,2) 和 (3,8)，m = (8-2)/(3-1) = 3，然后 2 = 3(1) + c ⇒ c = -1，因此方程为 y = 3x - 1。

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6. Quadratic Graphs and Parabolas | 二次函数图像与抛物线

rabola, which is symmetric. If a > 0, the parabola opens upward (U-shaped); if a < 0, it opens downward (∩-shaped).

二次函数的图像形式为 y = x² + bx + c（或 y = ax² + bx + c）。其形状是抛物线，具有对称性。如果 a > 0，抛物线开口向上（U 形）；如果 a < 0，开口向下（∩形）。

0) are the x-intercepts. They can be found by factorising or using the quadratic formula.

根（方程 ax² + bx + c = 0 的解）是 x 轴截距。可通过因式分解或求根公式找到。

The turning point (vertex) is the minimum or maximum point. For a U-shaped parabola, the turning point is at the bottom; its x-coordinate is -b/(2a).

顶点（转折点）是最小或最大值点。对于 U 形抛物线，顶点在最低点；其 x 坐标为 -b/(2a)。

3. The y-intercept is c=3. The vertex is at x = -(-4)/(2×1) = 2, y = (2)² - 4(2) + 3 = -1.

画草图需要确定截距和顶点。对于 y = x² - 4x + 3，因式分解为 (x-1)(x-3)，因此根为 x=1 和 x=3。y 轴截距为 c=3。顶点在 x = -(-4)/(2×1) = 2，y = (2)² - 4(2) + 3 = -1。

The line of symmetry passes through the vertex. For the example above, the line is x = 2.

对称轴经过顶点。上例中，对称轴为 x = 2。

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7. Pythagoras' Theorem and Trigonometry | 勾股定理与三角函数

ls the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse.

在直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²，其中 c 为斜边。

own lengths and solve. If the legs are 3 cm and 4 cm, then c² = 3² + 4² = 25, so c = 5 cm.

求未知边长时，代入已知长度并求解。若两直角边分别为 3 cm 和 4 cm，则 c² = 3² + 4² = 25，故 c = 5 cm。

potenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Memorise as SOH CAH TOA.

直角三角形的三角函数比值：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。用 SOH CAH TOA 帮助记忆。

Example: if opposite = 4 and hypotenuse = 5, then sin θ = 4/5, so θ = sin⁻¹(4/5) ≈ 53.1°.

利用反三角函数（sin⁻¹, cos⁻¹, tan⁻¹）可以在已知两边时求角度。例如，若对边为 4，斜边为 5，则 sin θ = 4/5，所以 θ = sin⁻¹(4/5) ≈ 53.1°。

Exact values for key angles are often tested: sin 30° = ½, cos 45° = √2/2, tan 60° = √3. Learn these common exact trig values.

特殊角的精确值经常考到：sin 30° = ½，cos 45° = √2/2，tan 60° = √3。请熟记这些常见的精确三角函数值。

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8. Similarity and Congruence | 相似与全等

portion. The scale factor k links corresponding lengths: new length = k × original length.

相似图形对应角相等、对应边成比例。比例因子 k 关联对应边长度：新长度 = k × 原长度。

by k² and its volume by k³. A rectangle with sides doubled (k=2) has area multiplied by 4.

一个图形以比例因子 k 放大后，面积乘以 k²，体积乘以 k³。当矩形的边长变为原来的 2 倍（k=2）时，面积乘以 4。

Congruent shapes are identical in shape and size. They can be proved using four conditions: SSS (side-side-side), SAS, ASA, and RHS (right-angle, hypotenuse, side).

全等图形形状和大小完全相同。可以用四种条件证明：SSS（边边边）、SAS（边角边）、ASA（角边角）和 RHS（直角、斜边、边）。

ons carefully: AB/DE = BC/EF = AC/DF. This is essential for solving missing side problems.

处理相似三角形时，要仔细写出比例关系：AB/DE = BC/EF = AC/DF。这是求解未知边长问题的关键。

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9. Probability and Venn Diagrams | 概率与维恩图

ber of favourable outcomes / total number of possible outcomes. All probabilities lie between 0 and 1.

事件 A 的概率 P(A) = 有利结果的数量 / 所有可能结果的总数。所有概率值都在 0 到 1 之间。

same time: P(A or B) = P(A) + P(B). For a fair six-sided die, P(2 or 5) = 1/6 + 1/6 = 1/3.

互斥事件不可能同时发生：P(A 或 B) = P(A) + P(B)。对于公平的六面骰子，P(2 或 5) = 1/6 + 1/6 = 1/3。

ect the other: P(A and B) = P(A) × P(B). Tossing a coin and rolling a die are independent.

对于独立事件，一个事件的结果不影响另一个：P(A 且 B) = P(A) × P(B)。抛硬币和掷骰子是相互独立的。

A Venn diagram displays sets and their relationships. The intersection A ∩ B represents outcomes in both A and B. The union A ∪ B includes all outcomes in either A or B or both.

维恩图展示集合及其关系。交集 A ∩ B 代表同时属于 A 和 B 的结果。并集 A ∪ B 包含属于 A 或 B 或两者兼有的所有结果。

To find probabilities from a Venn diagram, count the elements in each region and divide by the total. The diagram often includes numbers outside the sets for 'neither'.

通过维恩图求概率时，数出每个区域中的元素数量，再除以总数。维恩图常常在集合外部包括“两者都不”的数字。

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10. Statistics: Averages and Range | 统计：平均数与极差

ividing by the number of values. For data set 4, 7, 9, 10, the mean is (4+7+9+10)/4 = 7.5.

平均数通过将所有数值相加并除以数值的个数来计算。对于数据集 4, 7, 9, 10，平均数为 (4+7+9+10)/4 = 7.5。

of observations, average the two middle numbers. The median of 2, 3, 5, 8 is (3+5)/2 = 4.

中位数是数据排序后位于中间的值。如果有偶数个观测值，则取中间两个数的平均值。2, 3, 5, 8 的中位数为 (3+5)/2 = 4。

t occurs most often. There can be more than one mode. Range = maximum - minimum, measuring

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