AQA GCSE数学 代数 一次方程 二次方程 图像
在 AQA 的 GCSE 数学 8300 里，代数单独占 3.2 一整节，从 A1 编到 A25，是整份考纲条目最密的一块。三份卷子都是 1 小时 30 分钟、80 分、各占总分三分之一，其中 Paper 1 不允许使用计算器。Ofqual 规定的主题权重里，代数在基础档占 20%，在高等档占 30%，比数字与比例以外的任何一块都重。下面按 AQA 官网 2026 年 9 月版 1.1 考纲的条目顺序，把最常考的十几个动作走一遍。
AQA GCSE Mathematics 8300 devotes a whole section, 3.2, to algebra, and it runs from A1 to A25. All three papers last 1 hour 30 minutes, carry 80 marks and count for one third of the total, and the first paper is non-calculator. Ofqual fixes the topic weighting, and algebra takes 20 per cent at Foundation and 30 per cent at Higher. The order below follows the specification published on the AQA website, version 1.1 dated September 2026.
代数记号与最简写法
考纲 A1 要求会读也会写代数记号。ab 表示 a 乘 b，3y 表示 y 加 y 加 y，也等于 3 乘 y，a 的平方是 a 乘 a，a 除以 b 写成 a 除以 b 的分数形式。很多学生以为 3y 与 3 加 y 是一回事，代入 y 等于 4 时前者是 12，后者是 7，一步就走偏。系数要写成分数而不是小数，因为 AQA 考纲的 Notes 里明确要求答案以最简形式呈现，不写分数形式常被扣分。写 2x 时不要写成 x2，写 x 乘 x 时不要写 x2 以外的花样，答题卡上的批改只看标准记号。
A1 asks students to use and interpret algebraic notation, which catches people out more often than the algebra itself; the convention is that ab means a multiplied by b, that 3y stands for y + y + y as well as 3 times y, and that a divided by b is written as a fraction. Many students lose the mark here because they read 3y as 3 plus y; substitute y = 4 and the two readings give 12 and 7; AQA notes that answers are expected in their simplest form even without an instruction, and coefficients are written as fractions rather than decimals.
同类项合并与指标运算
同类项指的是字母部分完全相同的项。3x 加 5x 得 8x，4a 的平方加 a 的平方得 5a 的平方，而 2x 的平方与 3x 不是同类项，合并就等于把两种单位硬加在一起。指标运算有三条固定规则：同底数相乘，指数相加；同底数相除，指数相减；幂再乘方，指数相乘。常见的错误是 (2x) 的平方写成 2x 的平方，正确结果是 4x 的平方，括号里的系数也要平方。任何非零数的零次幂等于 1，这一条在化简里经常单独占一分。
Like terms share exactly the same letters and powers, so 3x plus 5x is 8x and 4a squared plus a squared is 5a squared; a term in x squared and a term in x are not like terms, and adding them is a common slip that examiners see every session. The index laws are short: add powers when multiplying, subtract when dividing, multiply when raising a power to a power; watch the bracket, since squaring 2x gives 4x squared rather than 2x squared, and anything non-zero to the power zero is 1.
展开括号：单项式与双括号
单项式乘括号时,括号里的每一项都要乘到。3 乘 (2x 减 5) 得 6x 减 15，2x 乘 (3x 加 4) 得 6x 的平方加 8x，第二项漏掉符号是阅卷时见得最多的失分。两个括号相乘有四项，可以用网格法把每一项对齐：x 乘 x、x 乘负 4、3 乘 x、3 乘负 4，合起来是 x 的平方减 x 减 12。三个括号相乘时先算前两个，把结果当成一个整体再乘第三个，不要一次铺开九项。展开完回头数项数，是控制错误最省时间的动作。
Expanding a single bracket means multiplying every term inside, so 3(2x - 5) becomes 6x - 15 and 2x(3x + 4) becomes 6x squared plus 8x; dropping the sign on the second term is the single most frequent mistake in this topic. Two brackets give four products, and a small grid keeps them in order: x times x, x times -4, 3 times x and 3 times -4 add up to x squared minus x minus 12; when three brackets appear, multiply the first two and treat the result as one expression.
提取公因式与平方差
因式分解是展开的逆过程，第一步永远是找公因式。6x 的平方加 9x 的系数有公因数 3，字母有公因式 x，所以写成 3x 乘 (2x 加 3)。提出公因式后括号里剩下的项要能与外面相乘还原，写不回去就说明提错了。平方差的形式是 a 的平方减 b 的平方，等于 (a 加 b) 乘 (a 减 b)，4x 的平方减 25 就是 (2x 加 5) 乘 (2x 减 5)。很多学生以为平方差只能处理 x 的平方减 9 这类整数，系数是平方数时一样适用。
Factorising runs the expansion backwards, and taking out the highest common factor comes first every time; in 6x squared plus 9x the numbers share 3 and the letters share x, so the answer is 3x(2x + 3), and you can always check by expanding back. The difference of two squares, a squared minus b squared, factorises to (a + b)(a - b), which turns 4x squared minus 25 into (2x + 5)(2x - 5); many students assume this pattern only works with plain numbers, yet it applies to any pair of square terms.
二次式因式分解 x2+bx+c 与 ax2+bx+c
首项系数是 1 的二次式,找两个数,乘积等于常数项,和等于一次项系数。x 的平方减 7x 加 12 的两个数是负 3 与负 4，乘积 12、和负 7，所以写成 (x 减 3) 乘 (x 减 4)。常数项为正、一次项为负时两个数都取负号，符号判断错是这一步最常见的翻车点。高等档要处理首项系数不是 1 的情况，2x 的平方加 7x 加 3 拆成 (2x 加 1) 乘 (x 加 3)，把 2x 的两个候选拆法与常数项的因子逐一配上，再用中间项验证。写完一律展开验算中间项。
When the coefficient of x squared is 1, look for two numbers whose product is the constant and whose sum is the coefficient of x; for x squared minus 7x plus 12 those numbers are -3 and -4, giving (x - 3)(x - 4), and the signs have to be settled before anything else. At Higher tier the leading coefficient can be something else, as in 2x squared plus 7x plus 3, which factorises to (2x + 1)(x + 3); pair the possible splits of 2x with the factors of 3, then expand to confirm the middle term.
一元一次方程与两边含未知数
解方程的核心是等式两边同时做同一件事。5x 减 3 等于 2x 加 9，两边同减 2x 得 3x 减 3 等于 9，两边同加 3 得 3x 等于 12，最后除以 3 得 x 等于 4。目标是把未知数集中到一边、把数字集中到另一边，中间步骤写清楚，跳步最容易丢过程分。带括号时先展开，带分数时两边同乘分母的最小公倍数，一步消掉全部分母。很多学生以为把项从左边搬到右边只要抄过去，其实移项等于两边同加减，符号必须变。
Solving a linear equation is about doing the same thing to both sides; start from 5x - 3 = 2x + 9, subtract 2x to get 3x - 3 = 9, add 3 to reach 3x = 12, then divide by 3 for x = 4; collect the unknown on one side and the numbers on the other, and keep the working visible because method marks live in those lines. Clear brackets first and, with fractions, multiply through by the lowest common multiple; moving a term across the equals sign is not copying, it is adding or subtracting on both sides, so the sign has to change.
公式变形与改变主元
考纲 A5 要求会把公式变形、更换主元。v 等于 u 加 at，要解出 a，先两边同减 u 得 v 减 u 等于 at，再同除 t 得 a 等于 (v 减 u) 除以 t。做逆运算的顺序与原来计算时相反：原来是先乘再加，变形就先减再除。P 等于 2 乘 (l 加 w)，解 l 时先除以 2 得 P 除以 2 等于 l 加 w，再减 w。常见的错误是整式除以 2 时只除了第一项，括号里每一项都要除，这一步最容易在分数与括号混合时出错。
A5 also covers rearranging formulae to change the subject; from v = u + at, subtract u on both sides and then divide by t, which gives a = (v - u) / t. Undo the operations in reverse order: if the original multiplied and then added, the rearrangement subtracts and then divides; with P = 2(l + w), divide by 2 first and subtract w afterwards; the recurring error is dividing only the first term when the whole right side is divided, so check that every term has been treated the same way.
一元一次不等式与数轴表示
解不等式与解方程几乎一样，唯一的差别在乘除负数上。3x 减 4 小于 11，两边同加 4 得 3x 小于 15，两边同除 3 得 x 小于 5，方向不变。负 2x 大于 6，两边同除负 2，不等号必须翻成 x 小于负 3，忘记翻转是阅卷时错得最多的一类。数轴上的画法考纲 A22 有明确约定：严格不等式用空心圆，含等号的用实心圆，再向解集一侧画箭头。写答案时把端点单独标出来，别只写一条线。
Inequalities follow the same steps as equations, with one exception; solve 3x - 4 < 11 by adding 4 and dividing by 3, and the direction of the sign is unchanged; divide -2x > 6 by -2 and the sign must flip, giving x < -3, which is the mistake examiners flag most often. The AQA specification sets the drawing convention in A22: an open circle marks a strict inequality and a closed circle marks an included boundary, with the arrow pointing to the solution set.
两个未知数的联立方程
联立方程的两种解法都要练熟。2x 加 y 等于 11 与 x 减 y 等于 1，两式相加时 y 与负 y 抵消，得 3x 等于 12，x 等于 4，代回任一式得 y 等于 3。相加或相减之前先看哪个字母的系数相同或互为相反数，这一步决定了要不要先乘一个常数把系数配平。代入法适合已经解出某个未知数的情况，比如 y 等于 2x 减 1，直接代进另一式。解完把 x 与 y 同时代回两个原方程，只验一个方程会漏掉算错的那一支。高等档还会出现直线与抛物线联立，先用代入法化成二次方程。
Both standard routes into simultaneous equations deserve practice; with 2x + y = 11 and x - y = 1, adding eliminates y and gives 3x = 12, so x = 4 and y = 3. Look at the coefficients first, because equal or opposite coefficients mean you can add or subtract straight away, while anything else needs one equation scaled up. Substitution suits cases where one unknown is already isolated, such as y = 2x - 1; always check the pair in both original equations, and at Higher tier expect a line meeting a curve, where substitution turns the system into a quadratic.
二次方程的三种解法
能因式分解就先因式分解，x 的平方减 5x 加 6 等于 0 拆成 (x 减 2)(x 减 3) 等于 0，得到 x 等于 2 或 3。拆不动时用求根公式，x 等于负 b 加减根号下 b 的平方减 4ac，再除以 2a，2x 的平方加 5x 减 3 等于 0 代入得负 5 加减 7 再除以 4，两个根是 0.5 与负 3。高等档还要会配方，把 x 的平方加 6x 加 2 写成 (x 加 3) 的平方减 7，顶点与最值一并读出。很多学生以为公式法最保险，其实要先确认方程已经整理成一般式，常数项搬到左边漏改符号会让两个根一起错。
Factorise when the factors are obvious: x squared minus 5x plus 6 equals zero becomes (x - 2)(x - 3) = 0, so x is 2 or 3; when nothing factors neatly, use the formula x = (-b plus or minus the square root of b squared minus 4ac) / 2a; applied to 2x squared plus 5x minus 3 = 0 it gives (-5 plus or minus 7) / 4, so x = 0.5 or -3. Completing the square is Higher tier work and rewrites x squared plus 6x plus 2 as (x + 3) squared minus 7, which also reveals the turning point; many students reach for the formula too early, before the equation is in the standard form with zero on one side.
直线图像 y=mx+c 与平行垂直
直线的方程写成 y 等于 mx 加 c 时，m 是斜率、c 是纵截距。y 等于 3x 减 2 的斜率是 3，与 y 轴交于负 2，画图时先点截距再按斜率走一步。两点求斜率用纵坐标之差除以横坐标之差，顺序在上下要一致，反了会得到相反数。平行的两条直线斜率相同，垂直的两条直线斜率乘积为负 1，负 1 这个结论属于高等档内容。考纲 A10 还要求能读懂图像上的斜率与截距，读到的是变化率，单位常常是每单位时间多少。
In y = mx + c, m is the gradient and c is the intercept on the y-axis, so y = 3x - 2 rises 3 for every 1 across and crosses at -2; plot the intercept first, then use the gradient to step to a second point; the gradient between two points is the change in y divided by the change in x, taken in the same order both times. Parallel lines share a gradient, and perpendicular lines multiply to -1, which is Higher tier content; A10 also asks students to read gradient and intercept as a rate, so the units usually say how much per unit of time.
数列第 n 项与二次数列
线性数列的第 n 项靠公差。4、7、10、13 每步加 3，先写 3n，n 等于 1 时应为 4，所以补 1，第 n 项是 3n 加 1，再用 n 等于 4 验一次。二次数列看第二层差：3、8、15、24 的一阶差是 5、7、9，二阶差恒为 2，说明第 n 项里含 n 的平方，配上 2n 正好得 n 的平方加 2n。斐波那契型与等比数列在考纲 A24 里也有要求，等比数列的相邻项之比固定，写第 n 项时把首项与公比都写清。常见错误是把第 n 项写成第 10 项，n 是位置不是数值。
A linear sequence is pinned down by its common difference: 4, 7, 10, 13 steps up by 3 each time, so start with 3n and adjust to 3n + 1, then test n = 4 to be sure. For a quadratic sequence look at the second differences; in 3, 8, 15, 24 the first differences are 5, 7, 9 and the second difference is a constant 2, so the nth term carries n squared, here n squared plus 2n; A24 also includes Fibonacci-type and geometric sequences, where consecutive terms share a fixed ratio; keep n as a position rather than a value.
文字题建模与检验
考纲 A21 要求把情境翻译成式子、解出来再解释答案。三个连续整数之和是 72，先把最小的设为 n，另两个是 n 加 1 与 n 加 2，方程是 3n 加 3 等于 72，解得 n 等于 23，三个数是 23、24、25。设未知数时把单位与含义写在同一行，后面解释答案才有依据。算完回到原题核一遍：23 加 24 加 25 等于 72，答案落在题目要求的形式上。常见的错误是解出方程就交卷，没有把数值还原成题目问的量，长度题甚至忘了写单位。
A21 asks for a situation to be turned into an expression or equation, solved, and then interpreted; three consecutive integers adding to 72 start as n, n + 1 and n + 2, giving 3n + 3 = 72, so n = 23 and the numbers are 23, 24 and 25. Define the variable together with its meaning and its unit, because the final sentence has to answer the question that was asked. Check the arithmetic against the original wording, and remember that a length without a unit is not a finished answer.
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