IGCSE数学0580 非计算器卷 根式 精确三角值
0580 的卷面在 2025 年改过一次，Extended 的两份卷子各占 50%，Paper 2 变成 2 小时 100 分的非计算器卷，Paper 4 仍然可以用计算器，Core 层则是 Papers 1 与 3 的对应版本。这意味着手算能力和计算器操作各占一半分数，只练一种的复习方式已经不划算。下面十二个点按卷面顺序排开，每一条都落到具体知识点和考场上能执行的动作，口径以 Cambridge International 官网的 0580 考纲（2025-2027，首考 2025）为准。
The 0580 papers changed for 2025: the two Extended papers now weigh 50% each, Paper 2 is a two-hour non-calculator paper worth 100 marks, and Paper 4 still allows a calculator, with the Core tier mirrored by Papers 1 and 3. Hand arithmetic and calculator work therefore carry equal weight, and revising only one of them no longer pays. The twelve points below follow the order of a typical paper, and each one names a specific topic with a check that can be carried out in the exam; the Cambridge specification for 2025 to 2027 is the reference throughout.
分数、百分数与小数的手算等值
2025 年起 0580 的 Extended 两卷各占 50%，Paper 2 不再允许使用计算器，2 小时 100 分。这条变化最先考的是手算等值，分数、百分数、小数三套写法要能随时互换。先记住 1/8 是 0.125，也就是 12.5%，3/8 是 37.5%，1/6 是 0.1666 循环。很多学生以为这些等值只在低年级用，实际上化简比例、算概率、检查答案都靠它。常见的错误是把 0.3% 读成 0.03，百分号前面是 0.3，换成小数要点两位，得到 0.003。做分数运算时先约分再乘，比最后再约省一半时间。题目要求 exact 时写分数，不要写 0.33。
Since 2025 the Extended papers are split evenly, and Paper 2 is a non-calculator paper worth 100 marks in two hours; the arithmetic has to be done by hand. Memorising the standard conversions helps far more than it looks, because 1/8, 3/8 and 1/6 reappear in ratio, probability and estimation questions. A slip that costs marks every session is reading 0.3% as 0.03, when the decimal point has to move two places to give 0.003. When a question asks for an exact value, leave the answer as a fraction rather than a rounded decimal.
根式的化简与分母有理化
2025 考纲把根式加进 Extended 内容，Paper 2 会直接要求化简和分母有理化。化简靠找平方因子，√72 拆成 √36 乘 √2，得 6√2，√50 是 5√2，√18 是 3√2。分母带根号时上下同乘，1/√3 写成 √3/3，而 (2+√3) 与 (2-√3) 相乘得 1，这对共轭式出现频率很高。学生最常把 √(a+b) 拆成 √a 加 √b，写成 √(9+16) 等于 3 加 4 就错了，实际结果是 5。加减只能合并同类根式，√2 加 √8 先化成 √2 加 2√2 才有 3√2。最后检查分母是否还留着根号，Extended 卷的答案一般要求有理化。
Surds were added to the Extended content for 2025, so Paper 2 now tests simplification and rationalising denominators directly; the method is to hunt for square factors, which turns √72 into 6√2 and √50 into 5√2. Expressions such as 1/√3 are cleaned up by multiplying top and bottom, and the conjugate pair (2+√3)(2-√3) = 1 appears often enough to be recognised on sight. Splitting a root over a sum is a common trap, since √(9+16) is 5 rather than 3+4. Only like surds combine, so √2 + √8 has to become √2 + 2√2 before the terms can be added.
精确三角值 0 到 90 度
精确三角值与根式同批进入 2025 考纲，Extended 卷要求不查表、不用计算器写出五个特殊角的正弦余弦正切。sin30 是 1/2，cos30 是 √3/2，tan30 是 √3/3，sin45 与 cos45 都是 √2/2，tan45 是 1，sin60 是 √3/2，cos60 是 1/2，tan60 是 √3。这九个值可以靠两个三角形推出来，30-60-90 的边比是 1 : √3 : 2，45-45-90 的边比是 1 : 1 : √2。学生最常把 30 度与 60 度的正弦余弦对调，写成 sin60 等于 1/2。检查办法是记住角度变大时正弦变大、余弦变小，0 到 90 之间不会反过来。tan90 不存在，答题时写 undefined，不要写 0。
Exact trigonometric values sit alongside surds in the 2025 additions, and candidates are expected to quote them without a calculator; the nine values for 0, 30, 45, 60 and 90 degrees are all recoverable from two triangles with sides in the ratio 1 : √3 : 2 and 1 : 1 : √2. Mixing up sine and cosine at 30 and 60 degrees is the error examiners see most often, usually written as sin 60 = 1/2. A quick check is that sine rises and cosine falls as the angle grows between 0 and 90 degrees. Tangent at 90 degrees is undefined rather than zero.
标准形式与数量级估算
标准形式写成 a 乘 10 的 n 次方，a 落在 1 到 10 之间，考试里常要求把 0.00042 写成 4.2 乘 10 的 -4 次方。数指数时看小数点移动的位数，向左移得到正指数，向右移得到负指数。乘法时系数相乘、指数相加，除法时系数相除、指数相减，(3 × 10^4) 乘 (2 × 10^-6) 得 6 × 10^-2。计算器卷上先用标准形式估一个数量级再按键，答案差十倍的情况基本能当场发现。常见的错误是 0.00042 少点一位，写成 4.2 × 10^-5。同一个数在物理和化学题里也会出现，写法保持统一，不要中途换一种表达。
Standard form means a × 10^n with a between 1 and 10, so 0.00042 becomes 4.2 × 10^-4; the index counts the places the decimal point moves, positive to the left and negative to the right. Multiplying adds the indices while dividing subtracts them, which is why the product of 3 × 10^4 and 2 × 10^-6 collapses to 6 × 10^-2 in one step. On the calculator paper an estimate in standard form catches the answer that is out by a factor of ten before it is written down. Dropping a place value when converting small numbers is the usual cause of that slip.
展开、因式分解与配方
代数变形在两张卷子上都占分，展开、因式分解、配方是三套固定动作。展开 (x+3)(x-5) 得 x² - 2x - 15，中间项是 3x 与 -5x 相加，符号错在这里最常见。因式分解先提公因式，2x² + 7x + 3 拆成 (2x+1)(x+3)，交叉相乘的中间项要正好等于 7x。配方只处理二次项系数为 1 的情况，x² + 6x + 1 写成 (x+3)² - 8，顶点坐标是 (-3, -8)。很多学生以为配方只是换一种写法，其实顶点、最值和判别式都从这里读出来。检查时把结果乘回去，比重新推一遍快。
Expanding, factorising and completing the square are the three moves that carry algebra marks on both papers; expanding (x+3)(x-5) gives x² - 2x - 15, and the sign on the middle term is where most of the lost marks sit. Factorising starts with the common factor, so 2x² + 7x + 3 becomes (2x+1)(x+3) once the cross terms add to 7x. Completing the square on x² + 6x + 1 produces (x+3)² - 8 and hands over the turning point at (-3, -8) without any drawing. Multiplying the answer back out is faster than re-deriving it.
百分数乘数、复利与反向百分数
百分数题几乎每场都出现，乘数写法比列方程省时间，加价 15% 就是乘 1.15，降价 20% 就是乘 0.8。复利用幂，2000 按每年 4% 增长三年是 2000 乘 1.04 的三次方，取精度只在最后一步做。反向百分数是丢分最集中的地方，打八折后是 64，原价不是 64 乘 1.2，而是 64 除以 0.8 得 80。很多人以为减价后加回去就还原了，实际要除以剩下的比例。比例题先统一单位再交叉相乘，速度、密度、压强都是同一套除法结构。答案带回原题检查一次，单位不对就是公式用反了。
Percentage questions appear in almost every session, and a multiplier is quicker than setting up an equation: a 15% rise is a multiplication by 1.15 and a 20% fall by 0.8. Compound growth simply repeats that multiplier, so 2000 growing at 4% for three years is 2000 × 1.04³, rounded only at the end. Reverse percentage is the classic trap, because a price of 64 after a 20% discount comes from 64 ÷ 0.8 rather than 64 × 1.2. Ratio questions work best once the units match, and speed, density and pressure all follow the same dividing structure.
斜率、中点、长度与垂直关系
坐标几何的分数集中在斜率、中点、长度和垂直关系四个公式上。斜率是纵坐标之差除以横坐标之差，(2, 3) 与 (6, 11) 之间的斜率是 8 除以 4 得 2。中点把两个坐标各取平均，长度用勾股定理算，两条直角边就是坐标差。垂直直线的斜率乘起来等于 -1，已知斜率 2，垂线斜率是 -1/2，写成 -2 是最常见的错。求直线方程先算斜率，再代一个已知点进 y = mx + c 解出截距。坐标几何的答案可以代回原式验算，两个点都满足才说明方程对。
Coordinate geometry marks cluster around four formulas: gradient, midpoint, length and perpendicularity; between (2, 3) and (6, 11) the gradient is 8 divided by 4, which is 2. The midpoint averages the two coordinates, and length comes straight from Pythagoras with the coordinate differences as the legs. Perpendicular gradients multiply to -1, so a line of gradient 2 is met by one of -1/2, and writing -2 instead is the error that turns up most. Finding the equation means working out the gradient first, then substituting a known point into y = mx + c.
相似形与圆定理的理由分
相似形的长度比是 k，面积比是 k 的平方，体积比是 k 的立方，题目给出面积比 9 : 25 时长度比是 3 : 5。圆定理题要写理由，角度题的步骤分挂在定理名称上，半圆所对的圆周角是 90 度，圆心角是同弧圆周角的两倍，圆内接四边形对角互补 180 度，弦切角等于同弧上的圆周角。阅卷时常见的失分是只写角度不写定理名，那一步就不给分。切线题先连半径，半径与切线垂直是几乎所有切线题的起点。角度推理一步一步写，跳步的答案很难拿满。
Similar shapes scale lengths by k, areas by k squared and volumes by k cubed, so an area ratio of 9 : 25 means a length ratio of 3 : 5. Circle theorem questions award method marks for the reason, which means quoting the theorem and not only writing the angle: the angle in a semicircle is 90 degrees, the angle at the centre is twice the angle at the circumference, opposite angles of a cyclic quadrilateral add to 180 degrees, and the alternate segment theorem links a tangent to a chord. Drawing the radius to the point of contact is the usual first step on tangent questions. Working line by line is what keeps those marks reachable.
正弦定理、余弦定理与三角形面积
三角形题先看有没有直角，有直角就用正弦余弦正切，没有直角才考虑正弦定理和余弦定理。正弦定理写成 a 除以 sinA 等于 b 除以 sinB，已知两个角一条边，或者两条边与其中一条边的对角时使用。余弦定理 a² = b² + c² - 2bc cosA 处理两边夹角的情况，面积公式是 ½ ab sinC。常见的错误是两边夹角已知却用正弦定理，这时解不出唯一答案。角度求出后检查三角形内角和是否 180 度，能立刻发现公式用错。答案保留三位有效数字是默认要求，中途不要提前四舍五入。
The first decision on any triangle is whether it contains a right angle; if it does, the basic ratios are enough, and if not the sine and cosine rules take over. The sine rule suits two angles and a side or two sides and a non-included angle, whereas the cosine rule and the area formula ½ ab sinC handle the included angle case. Reaching for the sine rule when the known angle sits between the two known sides is the mistake that produces no unique answer. Checking that the angles add to 180 degrees catches a wrong rule almost immediately.
累计频率图、树状图与事件关系
统计部分的读图题要看清坐标轴，累计频率图的纵轴是累计频数，中位数在 50% 对应的横坐标上，四分位数在 25% 与 75%。2025 考纲删掉了箱线图，累计频率曲线本身仍然要会读。概率树状图沿分支相乘、在不同分支之间相加，不放回抽取时第二次的分母要减一。很多学生以为两次抽取的分母一样，题目写了 without replacement 就不一样了。独立事件的概率相乘，互斥事件的概率相加，两个词混用是失分主因。写概率答案时检查有没有落在 0 与 1 之间，超过 1 一定是加法用错了地方。
Reading the axes correctly decides most statistics marks: on a cumulative frequency curve the vertical axis is cumulative frequency, so the median sits at 50% and the quartiles at 25% and 75%. Box plots were removed from the 2025 syllabus, although the curve itself is still examined. Tree diagrams multiply along a branch and add across branches, and a without-replacement draw changes the denominator on the second pick. Independent events multiply while mutually exclusive ones add, and confusing the two is where probability answers go wrong.
函数记号、定义域与值域
函数记号 f(x) 表示输入与输出的对应，f(3) 是把 3 代进去算，不是 f 乘 3。定义域是允许的输入，值域是可能的结果，这两项在 2025 年进入 Extended 考纲。反函数写成 f^-1(x)，做法是把 y = f(x) 中的 x 解出来再交换字母，f^-1 不等于 1 除以 f。复合函数 gf(x) 先做 f 再做 g，顺序换过来结果通常不同。很多学生以为 gf 与 fg 一样，代一组具体数字进去就能看出差别。图像题先找截距和顶点，再判断开口方向，比逐点描图快。
Function notation f(x) describes a mapping, so f(3) means substituting 3 rather than multiplying f by 3. Domain and range entered the Extended content in 2025: the domain lists the inputs allowed and the range the outputs that can follow. An inverse is found by rearranging y = f(x) for x and then swapping the letters, and f^-1 is never the same thing as 1/f. In a composite such as gf(x) the inner function runs first, and swapping the order usually changes the result.
有效数字、单位与方法分
有效数字和单位是最后一道关，题目没有特别说明时答案保留三位有效数字，钱数保留两位小数，角度通常一位小数。中间步骤不要提前四舍五入，把完整数值留在计算器里，最后一步再取精度。常见的错误是面积写成厘米而不是平方厘米，单位错了整题不给分。过程分比答案分更稳，写出公式、代入数值、给出结果，三步齐全的答案即使算错也能拿到方法分。计算器卷上把每一步的中间值记在草稿上，检查时能顺着找错。时间不够时先跳过卡住的题，把能拿的分先收进来。
Precision is the last place where marks disappear: unless a question says otherwise, answers go to three significant figures, money to two decimal places and angles usually to one. Intermediate values should stay unrounded until the final step, because rounding early drifts the answer outside the accepted range. Writing the formula, substituting the numbers and stating the result keeps the method marks safe even when the arithmetic slips. Units are part of the answer, so an area given in centimetres rather than square centimetres loses the mark.
TI-Nspire CX II CAS 图形计算器 AP/IB/A-Level¥1320📱 去京东购买京东精选
需要一对一辅导？Need One-on-One Tutoring?
A-Level / GCSE / IB / KS2-KS3 全科辅导，按考试局大纲逐项过关
16621398022 同微信
Visit aleveler.com
或截图后用「扫一扫 → 相册」选择本图