WJEC数学 代数 因式分解 二次方程 公式法
很多学生把 WJEC 的数学当成一套卷子准备，等到翻开 Unit 1 才发现不能用计算器，整块代数题的计算节奏全乱了。威尔士的 GCSE 数学其实有两个资格证书，考试拆成两张卷子。代数在两个资格里的分量不一样，在三个层级里的深度也不一样。这份考纲从 2015 年 9 月开始施教，2017 年做过一次修订，现行版本沿用至今。下面按 WJEC 官网考纲与 Guidance for Teaching 的顺序，把代数部分会被考到的内容逐节拆开。每一节都标清楚 Intermediate 与 Higher 的分界，也标出这道题会落在哪张卷子上。
WJEC splits its GCSE mathematics offer into two separate qualifications, and the algebra expectations are not identical in either. Numeracy stays tied to everyday life and the workplace, while Mathematics is the route into scientific, technical and further mathematical study. This walkthrough follows the current specification and its teaching guidance, marking the boundary between Intermediate and Higher content at every stage. It also flags which paper each technique is likely to appear on, because Unit 1 forbids calculators outright.
两个资格证书：代数只多不少的那一份
WJEC 在威尔士发两份数学 GCSE，一份叫 GCSE Mathematics，资格代码 3300QS，另一份叫 GCSE Mathematics 与 Numeracy，代码 3310QS。考纲写得很直白，GCSE Mathematics 包含 Numeracy 的全部内容，再加额外的代数、几何与概率。两份资格的分工也因此清楚，Numeracy 面向日常生活与职场里的数学，Mathematics 面向要继续读数学、科学或技术类课程的学生。这条差别决定了一件事，代数里最难的那部分只会出现在 Mathematics 的卷子上。很多学生以为两份资格只是难度不同，其实内容范围本身就不一样。选错资格，复习的代数深度会差一整层。
Only one of the two Welsh mathematics GCSEs carries the full algebra load. GCSE Mathematics absorbs every topic from the Numeracy qualification and then adds further algebra, geometry and probability on top. Numeracy remains anchored in daily life and workplace contexts, whereas Mathematics prepares candidates for scientific, technical and further mathematical study. That split is structural rather than a question of difficulty, so the qualification a centre enters a candidate for changes how much algebra they will ever be examined on.
两张卷子的分工与三层分级
GCSE Mathematics 只有两张卷子，Unit 1 是非计算器卷，Unit 2 允许使用计算器，各占 50%。Higher 与 Intermediate 两个层级都是 1 小时 45 分、80 分，Foundation 是 1 小时 30 分、65 分。层级用报名代码区分，Foundation 是 3300PF，Intermediate 是 3300PN，Higher 是 3300PH。考纲附录里把计算器的边界写清楚了，带符号运算、能存公式或存文字的机型一律不许用。成绩等级是 A* 到 G，不是英格兰那套 9 到 1。JCQ 官网 2026 年的威尔士新闻稿里，GCSE Mathematics 拿到 A* 与 A 的比例是 13.9%，C 及以上是 48.3%。WJEC 官网写明 2026 年夏季是 GCSE Mathematics 最后一次完整考试，2026 年 11 月与 2027 年 6 月的重考视需求开放。
Both papers carry equal weight, with Unit 1 banning calculators and Unit 2 permitting one that satisfies the specification. Higher and Intermediate candidates sit 1 hour 45 minutes for 80 marks, while Foundation candidates get 1 hour 30 minutes for 65 marks. Grades run A* to G in Wales, and the JCQ figures for 2026 put 13.9 per cent of mathematics entries at A* or A, with 48.3 per cent at C or above. WJEC states that summer 2026 is the final full assessment opportunity, with resits in November 2026 and June 2027 subject to demand.
附加公式表上有什么，没有什么
Higher 层级的考生在考场上会额外拿到一张 Additional Formulae 表，卷面代码是 C300UA0。这张表印着梯形面积、棱柱体积、球与圆锥的体积和表面积、正弦定理、余弦定理、三角形面积公式，还有二次方程的求根公式。考纲附录 A 里另有三份公式表，分别对应 Foundation、Intermediate 与 Higher 三个层级。表上没有的东西恰恰是代数最吃功夫的部分，完全平方、平方差、配方的步骤、因式分解的拆分思路、nth term 的构造方法，全都得自己掌握。常见的错误是拿到求根公式就直接套数字，忘了先把方程整理成 ax² + bx + c = 0 的标准形。这一步最容易丢分，整理时移项与符号变化都要重新做一遍。
The Higher tier paper comes with a separate Additional Formulae sheet, printed under code C300UA0. It supplies the area of a trapezium, prism volume, the sphere and cone results, the sine and cosine rules, and the quadratic formula itself. Appendix A of the specification holds three further formula lists, one for each tier. Everything absent from those sheets is what genuinely has to be learned, since completing the square, the difference of two squares and the reasoning behind factorising or building an nth term are never printed for candidates.
展开两个线性式与完全平方
考纲在代数里点名要求展开 a(bx + c) 这样的单括号，也要求展开两个线性式相乘，(ax + by)(cx + dy) 与 (ax + by)²。完全平方错得最多，很多学生把 (2x + 3)² 直接写成 4x² + 9，中间的两倍乘积整块消失。稳妥的写法是先拆成 (2x + 3)(2x + 3)，再逐项乘出四项，最后合并同类项，得到 4x² + 12x + 9。四项的顺序可以固定成第一项平方、两项交叉乘积、第二项平方，这样不容易漏。符号也要跟着括号走，减号后面的每一项都要变号。阅卷时看到的扣分多数集中在这两处，漏项与符号。展开完再数一次项数，两个二项式相乘应该出四项，合并以后才变少。
Expansion starts with a single bracket such as a(bx + c) and moves on to the product of two linear expressions. The classic slip is writing (2x + 3)² as 4x² + 9 and dropping the cross term entirely. Writing the bracket twice, multiplying term by term and then collecting like terms keeps the middle term where it belongs. Treating the four products in a fixed order, squaring each first term, taking both cross products, then squaring each second term, gives a structure that is hard to break under time pressure.
因式分解：从 x²+ax+b 到 ax²+bx+c
Intermediate 层级的因式分解停在 x² + ax + b 这种首项系数为 1 的二次式，思路是找两个数，相乘得 b，相加得 a。Higher 层级往上加两样，一是 ax² + bx + c 这种首项系数不为 1 的式子，二是平方差。首项系数不为 1 时可以先用 ac 法，把 ac 算出来，再找两个数相乘得 ac、相加得 b，最后把中间项拆开分组。以 6x² + 11x - 35 为例，ac 是 -210，要凑 11 的两个数是 21 与 -10，拆成 6x² + 21x - 10x - 35，前两项提 3x，后两项提 -5，得到 (2x + 7)(3x - 5)。平方差要看结构，两个完全平方相减才成立，4x² - 9 分解成 (2x - 3)(2x + 3)。动手前先提公因式，3x² + 6x 应写成 3x(x + 2)，漏这一步后面的分解会白做。
Intermediate factorising stops at quadratics whose leading coefficient is one, where two numbers must multiply to the constant and add to the coefficient of x. Higher tier adds the general ax² + bx + c case and the difference of two squares. Multiplying the first and last coefficients, then splitting the middle term and grouping, handles 6x² + 11x - 35 cleanly. Common factors should always come out before anything else, because 3x² + 6x has to become 3x(x + 2) before the remaining bracket can be read.
二次方程的三种解法怎么挑
Higher 考纲的原话是把因式分解、图像法与公式法三种解法都列出来，还要求考生 selecting the most appropriate method，也就是自己判断该用哪一种。能分解的题优先分解，速度快，也不会有舍入误差。x² - 5x - 14 = 0 分解成 (x - 7)(x + 2) = 0，两个根是 7 与 -2，一步到位。分解不出来的题才轮到求根公式，2x² + 3x - 4 = 0 套进去得到 x = (-3 ± √41) / 4，题目通常会指定保留几位小数。图像法的作用是读根的个数与近似值，题目写 use your graph 就必须从图上读，不能改用代数法。常见的错误是方程还没整理成标准形就套公式，常数项搬错一边，整道题的两个根都会错。
Higher tier candidates are expected to choose between factorising, reading a graph and using the formula. Factorising is fastest and leaves no rounding error, so it comes first whenever the numbers allow it. The formula takes over when a quadratic will not factorise, and a graph is used when the question asks for the number of roots or an approximate value read from the curve. Rearranging into the standard form ax² + bx + c = 0 before substituting is not optional, because a constant moved to the wrong side produces two wrong roots.
带分母的方程与代数分式
Higher 考纲单列了一条，要求解带线性分母的方程，整理之后可能变成二次方程，也可能变回线性方程。做法是两边同乘最小公分母，每个项都要乘到，漏一项就前功尽弃。2/(x + 1) = x/3 两边乘 3(x + 1)，得到 6 = x(x + 1)，整理成 x² + x - 6 = 0，分解得 (x + 3)(x - 2) = 0。求出根以后必须回代检查分母，让分母变成零的解要排除，这一条在考纲里写得很清楚。同一层级还要求化简代数分式，分子分母先各自分解再约，(x² - 9)/(x + 3) 化简成 x - 3，条件是 x 不等于 -3。很多学生把约分当成消项，直接把分子分母里的 x 划掉，这是最常见的错误。
Equations with linear denominators sit in the Higher tier content and can resolve into either a linear or a quadratic equation. Multiplying every term by the lowest common denominator is the reliable first move, since one missed term ruins the rest of the work. Any root that makes a denominator zero has to be rejected afterwards, and the same discipline applies when simplifying algebraic fractions by factorising first. Cancelling a bare x from a numerator and a denominator, rather than cancelling a whole factor, is the mistake that costs the most marks here.
联立方程要同时给出两个未知数
考纲要求用图像法与代数法解两个线性方程的联立，两种方法都要会，题目也可能先给一张画好的图。给图的时候读交点坐标即可，坐标要写成一对数值，漏掉其中一个就是半分。不给图就用代入或消元，系数整齐时消元更快，一边已经写成 y 等于某式时代入更快。现实情境题的难点在列式，把两句话各自翻成一个方程，再回到联立这一步。解完还要把答案放回原题，问的是票价就写票价，问的是数量就写数量，单位也要带上。常见的错误是算完 x 就停笔，忘了代回去求另一个未知数。
Simultaneous linear equations have to be solved both graphically and algebraically, and some questions hand candidates a ready-drawn pair of lines. Reading the intersection gives the solution directly, provided both coordinates are stated as a pair. Without a graph, substitution or elimination works, and the real difficulty in context questions is turning the two sentences into equations before solving anything. Returning to the original context, with the right units, is what converts a correct pair of numbers into full marks.
不等式与直线图上的区域
考纲要求形成并处理简单的线性不等式，也要求处理带分数系数的线性不等式与方程。Higher 层级再加一条，要用直线图定出线性不等式给出的区域，这一步在卷面上通常是两三分的小题，但很容易全错。画边界线时注意不等号带不带等号，带等号画实线，不带等号画虚线。判断取哪一侧，挑一个不在线上的点代进去，代入原点最省事。多个不等式要取公共区域，把每条线的可行侧都涂一次，重叠的地方才是答案。很多学生会在这里写错不等号方向，两边乘除负数时不等号要翻转，这条规则年年有人忘。
Linear inequalities appear from the Intermediate content onwards, and Higher tier adds regions defined by several of them on a straight-line graph. Solid boundaries mark an included line and dashed ones exclude it, while a test point decides which side satisfies the inequality. Where two or more inequalities apply, only the overlap of the shaded sides counts. Multiplying or dividing by a negative number flips the inequality sign, which is the step most often lost under exam pressure.
变形公式：subject 出现两次怎么办
Intermediate 的要求是变形公式时 subject 只出现一次，Higher 的 Guidance for Teaching 把它推到更远，subject 出现两次、含简单幂、含根号、含倒数的式子都要会处理。思路是逆运算一层层剥，两边始终做同样的操作，等式才不会破。有分母先把分母乘掉，有根号先把两边平方，有平方最后开方。以 A = πr² 求 r 为例，两边除 π 得 A/π = r²，再开方得 r = √(A/π)，指引里给的正是这个例子。subject 出现两次时先把它挪到同一边，再提公因式，最后整体除。常见的错误是只对一边动手，或者把 1/x 当成 x 直接移项。
Rearranging a formula begins with the case where the new subject appears only once, and the teaching guidance extends it to formulae where the subject occurs twice or sits inside a power, a root or a reciprocal. Undoing operations in reverse order, while treating both sides identically, keeps the equation balanced. Taking A = πr² and making r the subject gives r = √(A/π), the example the guidance itself uses. When the subject appears twice, gathering those terms on one side and factorising them out turns the problem back into a simple division.
数列的 nth term 与两种规则
考纲要求找出线性或二次数列的 nth term，也要求反过来按给定的 nth term 生成数列。Guidance for Teaching 把规则分成两类，term-to-term 是逐项递推，position-to-term 是直接写第 n 项，考卷上问 nth term 要的是后者。线性数列看公差，5, 9, 13, 17 每次加 4，n 的系数就是 4，再把第 1 项对回去得到 4n + 1。二次数列看二阶差，3, 8, 15, 24 的一阶差是 5, 7, 9，二阶差恒为 2，n² 的系数取二阶差的一半即 1，再减出余下的 2n。指引还点名要认得三角形数、平方数、立方数、Fibonacci 型与等比数列，遇到图形数列先数出前三项再套方法。常见的错误是把 n 从 0 开始算，常数项整体偏一位，后面每一问都跟着错。
Sequence questions ask for the nth term of a linear or quadratic pattern, and also expect candidates to generate terms from a given rule. The guidance separates term-to-term rules, which step from one term to the next, from position-to-term rules, which give the nth term directly. For a quadratic sequence, halving the constant second difference fixes the coefficient of n², and substituting the first term supplies the rest. Triangular numbers, square numbers, cube numbers, Fibonacci-type sequences and geometric progressions are all named in the guidance and are worth recognising on sight.
三次方程的试算与图像识别
考纲要求用 trial and improvement 解一批三次方程，还要求 justifying the accuracy of the solution，光写答案不给夹逼过程拿不到方法分。做法是两次试算把根夹在中间，再比较两次结果离目标哪个更近。比如解 x³ - 2x = 15，试 2.7 得 14.283，试 2.8 得 16.352，根落在两者之间，离 15 更近的是 2.7，取到一位小数就是 2.7。Higher 层级还要画与识别 y = ax³ + bx² + cx + d 这类三次图像，以及 y = k/x 这种反比例图像。辨认靠形状与转折的个数，三次曲线最多两个转折点，反比例曲线分成两支，而且永远不穿过坐标轴。很多学生把反比例图像画成一条穿过原点的直线，这是典型的形状混淆。
Cubic equations are solved by trial and improvement, and the specification asks for the accuracy of the answer to be justified rather than simply asserted. Bracketing the root between two trials and comparing which result sits closer to the target does that job. Higher tier also requires sketching and recognising curves such as y = ax³ + bx² + cx + d and the two-branch shape of y = k/x. A cubic turns at most twice, while a reciprocal curve never touches either axis, which is enough to tell the two families apart.
2026 年之后的考纲变化
WJEC 官网写得很清楚，从 2025 年 9 月起，学生不能在 10 年级注册旧的单科 GCSE Mathematics，要注册 Made for Wales 的 GCSE Mathematics and Numeracy 双奖。新资格从 2025 年 9 月开始施教，2026 年 11 月首次颁奖，三个单元的分工是 Unit 1 财务数学与数值应用占 30%，Unit 2 非计算器占 30%，Unit 3 可用计算器占 40%。Unit 2 的说明里明确写着以代数、概率与几何为重点，代数在非计算器卷上的分量被写进了资格说明。分数结构也变了，Unit 1 的 Higher 是 1 小时 45 分 80 分，Unit 3 的 Higher 拉长到 2 小时 90 分。等级仍是 A* 到 G，Foundation 只能给到 C 到 G。这份新资格不提供给英格兰的考试中心，打算在 2026 与 2027 年补考的考生用的还是旧考纲那一套代数要求。
From September 2025 no new Year 10 cohort can be entered for the single-award GCSE Mathematics, and centres move to the Made-for-Wales Mathematics and Numeracy Double Award. Its three units weight 30, 30 and 40 per cent, with the non-calculator unit described as emphasising algebra, probability and geometry. Higher tier candidates sit 1 hour 45 minutes for the first unit and 2 hours for the final calculator-allowed unit, and grades stay on the A* to G scale with Foundation capped at C to G. The qualification is not available to centres in England, and candidates resitting the older specification in 2026 or 2027 still meet the algebra content set out above.
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