IGCSE 0580 数学 三轮复习 代数 几何 统计
Cambridge IGCSE Mathematics 0580 是一张容易被低估的卷子。知识点本身不算深，但 2025 年起卷面结构改过，非计算器卷的权重上来了，很多按老办法刷题的学生会在考场上发现节奏完全不对。这篇文章把考纲里写清楚的规则、复习的先后顺序、以及容易丢分的地方理一遍。
The Cambridge IGCSE Mathematics 0580 paper looks forgiving until the timing is tight. The content is not especially deep, but the assessment changed for 2025 and the non-calculator weight is now real. This piece walks through what the syllabus actually says, the order in which topics repay revision, and the places where marks quietly disappear.
先确认 Core 还是 Extended
Core 和 Extended 是两个报名口径，不是两套教材。0580 考纲写得很直接：Core 内容面向目标 C 到 G 的学生，报 Paper 1 和 Paper 3，最终成绩落在 C 到 G；Extended 内容包含 Core 再加额外内容，报 Paper 2 和 Paper 4，成绩落在 A* 到 E。选错层比少做几套题更伤，因为 Extended 的卷子不会给你 Core 的简单分。
Core and Extended are entry decisions, not two textbooks. The 0580 syllabus for 2025 to 2027 states that the Core content is intended for learners targeting grades C to G, who sit Papers 1 and 3 and can be awarded grades C to G only. The Extended content contains all of the Core material plus additional topics, and those candidates sit Papers 2 and 4 for grades A* to E. A misjudged tier costs more than a few missed past papers, because the Extended paper hands out no easy marks. If a school predicts a grade D, the Core route is the honest option, and it should be settled with the teacher before any revision timetable is drawn up.
2025 版四张卷子的时长与分值
2025 到 2027 这一版把四张卷子统一成了一个对称结构。Core 的 Paper 1 是非计算器卷，1 小时 30 分、80 分；Paper 3 允许计算器，同样 1 小时 30 分、80 分。Extended 的 Paper 2 是非计算器卷，2 小时、100 分；Paper 4 允许计算器，2 小时、100 分。每张卷各占 50%。和 2023 版比，Paper 2 从 1 小时 30 分、70 分变成 2 小时、100 分，还禁用了计算器，这一条对刷旧题的学生影响最大。
The 2025 to 2027 syllabus equalises the four papers. Core candidates sit Paper 1, non-calculator, 1 hour 30 minutes and 80 marks, and Paper 3, calculator, the same length and value. Extended candidates sit Paper 2, non-calculator, 2 hours and 100 marks, and Paper 4, calculator, 2 hours and 100 marks. Each component carries 50 per cent of the qualification. Compared with the 2023 version, Paper 2 has grown from 1 hour 30 minutes and 70 marks to 2 hours and 100 marks and lost the calculator. Students working through papers from 2023 or earlier are therefore practising a different exam, and both the timing and the arithmetic load need re-calibrating. The older papers keep their value as question banks, but they no longer work as mock exams.
公式表只覆盖一部分公式
每张卷子的第 2 页都印了一份公式表，Core 卷 11 条，Extended 卷 15 条，多出来的是求根公式、正弦定理、余弦定理和 (1/2)ab sin C。圆的面积与周长、圆柱圆锥球的表面积体积、棱柱棱锥的体积都在表里。不在表里的照样要背：勾股定理、直角三角形的三角函数、梯形与平行四边形的面积、弧长与扇形面积、复利、直线的斜率与方程。很多学生以为公式表等于不用记公式，翻开卷一才发现自己背的和表上给的并不是同一批。
Every 0580 paper prints a list of formulas on page 2. The Core list has 11 entries and the Extended list has 15; the extra four are the quadratic formula, the sine rule, the cosine rule and the area formula (1/2)ab sin C. Areas and volumes of circles, cylinders, cones, spheres, prisms and pyramids are all supplied. What the list does not supply still has to be memorised: Pythagoras, right-angled trigonometry, the area of a trapezium and a parallelogram, arc length and sector area, compound interest, and the gradient and equation of a straight line. Students who assume the formula sheet removes the need to learn formulas find, on opening the paper, that their memorised set and the printed set are not the same. A quick exercise is to mark a personal formula list against the printed one in two colours, then learn only the gaps.
计算器型号有硬性限制
Paper 3 和 Paper 4 要用科学计算器，考纲明确要求带三角函数，强烈建议用函数齐全的科学计算器，代数计算器和图形计算器一律不允许带入。这一条每年都有人踩，因为学校平时用的 TI-Nspire 图形机正好在禁用名单上。平时练习就换回考场那台机器，别等到考试当天才发现按键位置不熟。计算器的另一个用法是复核，估一个量级再按一遍，答案差十倍时能立刻发现数量级错误。
Papers 3 and 4 require a scientific calculator. The syllabus recommends one with trigonometric functions and states plainly that algebraic and graphical calculators are not permitted. This catches candidates every year, because the graphing calculator a school uses in class is exactly the type on the banned list. Practice should be done on the machine that will be in the exam room, so that the key positions are automatic by the time it matters. A second use for the calculator is a magnitude check: estimate the answer roughly, compute it, and a factor-of-ten gap becomes obvious at once. Checking an answer costs far less time than losing the mark for it.
非计算器卷的算法训练
非计算器卷考的是数感和拆解，不是硬算。卷面上常见 12.5% 的折扣、3/8 这样的分数运算、以及要保留根号的答案。训练方式很土：把所有非计算器题抽出来单独成册，限时做，草稿纸只写关键中间值。常见的错误是把 7/8 提前化成小数再算，或者把根号近似成 1.41 之后继续往下用，最后一步答案对不上评分标准。剑桥的评分说明里写明，精度分要求到最终答案才舍入，中途的近似值不能再拿来用。
The non-calculator paper rewards number sense and clean decomposition rather than long arithmetic. Typical items involve a 12.5 per cent discount, a fraction such as 3/8, or an answer that should stay in surd form. A practical drill is to pull every non-calculator question out of the past papers into a separate booklet, work it against the clock, and keep the rough work to key intermediate values only. The recurring mistake is to convert 7/8 into a decimal too early, or to approximate a surd as 1.41 and then keep using it. Cambridge's notes are explicit that accuracy marks depend on not rounding until the final answer, and that a value needed later should be carried forward unrounded. Practising with a pencil and no calculator for twenty minutes a day is enough to change the arithmetic habits.
代数 因式分解 二次方程 不等式
代数部分真正决定分数的是三件事：因式分解、二次方程、不等式。因式分解有固定顺序，先看能不能提公因式，再看平方差或完全平方，实在不行才试十字相乘。二次方程优先分解，分解不动才上求根公式（Extended），动手前先看判别式判断有几个根。不等式的坑只有一个，乘除负数时不等号翻转，很多学生以为只有解方程才需要注意符号。
Three algebra skills decide most of the marks: factorising, solving quadratics and handling inequalities. Factorising follows a fixed order, taking out a common factor first, then looking for a difference of two squares or a perfect square, and only then trying the cross method. Quadratics should be factorised where possible; the quadratic formula, which sits on the Extended formula list, is the fallback, and the discriminant tells you how many roots to expect before any work begins. Inequalities turn on a single detail: multiplying or dividing by a negative number reverses the sign, and a quick number-line check catches the slip. Completing the square still turns up inside graph-transformation questions, so keeping it fluent pays off beyond the algebra section itself.
函数与图像 变换 定义域 值域
2025 版在 Extended 的函数内容里加了定义域和值域，图像部分还加了 x 的 -1/2 次方和 1/2 次方。图像变换可以压成一句话：括号里面加减管左右，符号相反；括号外面加减管上下；系数管拉伸和翻转。错得最多的是把 y = f(x+2) 说成向右移两格。定义域题先看分母和根号，再看题目给的区间，两步就够。
The 2025 syllabus adds domain and range to the Extended functions content, along with the powers -1/2 and 1/2 for graphs of the form y = ax^n. Transformations reduce to one sentence: a change inside the bracket shifts the graph horizontally against its sign, a change outside shifts it vertically, and a coefficient stretches or reflects it. The commonest error is reading y = f(x + 2) as a shift two units to the right. Domain questions are settled by checking denominators and square roots first, then applying the interval the question supplies. Sketching the original and the transformed curve on the same axes takes ten seconds and exposes most sign errors before they reach the answer line.
几何 圆定理 相似 方位角
几何在卷面上占分稳定，难点集中在圆定理和相似。圆定理要能从空白纸开始复述：同弧所对圆心角是圆周角的两倍，直径所对的圆周角是 90 度，圆内接四边形对角互补，切线与半径垂直。相似题先找对应角，再写比例式，单位统一之后再代入。2025 版把全等判定条件从 Extended 里删掉了，全等的概念还在，但 SSS、SAS 这一串缩写不再是考点。
Geometry carries a steady share of marks, with the difficulty concentrated in circle theorems and similarity. Circle theorems should be reproducible from a blank page: the angle at the centre is twice the angle at the circumference on the same arc, the angle in a semicircle is 90 degrees, opposite angles of a cyclic quadrilateral add to 180 degrees, and a tangent meets a radius at right angles. Similarity questions start with matching corresponding angles, then a ratio statement, then substitution once the units agree. The 2025 syllabus removes the congruence criteria from the Extended content, although congruence itself remains; the abbreviations SSS, SAS and ASA are no longer examinable vocabulary. Bearings and scale drawings also appear, and they are among the cheapest marks on the paper whenever a diagram is drawn to scale.
三角 正弦定理 余弦定理 精确值
直角三角形用 sin cos tan，非直角三角形用正弦定理和余弦定理，面积用 (1/2)ab sin C。Extended 新增了 30、45、60 度的精确三角函数值，答案要写成根号形式，写 0.866 会被扣分。用余弦定理求角时先把 cos A 单独移出来，先算分子再除，不要一次按完再回头找错。立体几何题只是把平面三角形藏在图里，先把这个三角形画出来再套公式。
Right-angled triangles use sine, cosine and tangent; other triangles use the sine rule and the cosine rule, with (1/2)ab sin C for area. The Extended content now includes exact values of sine, cosine and tangent for 30, 45 and 60 degrees, so the answer is expected as a surd rather than as 0.866. When the cosine rule is used to find an angle, rearranging for cos A first keeps the arithmetic visible and the sign errors down. Three-dimensional questions are two-dimensional problems in disguise: find the right-angled triangle inside the solid and draw it separately before reaching for a formula. Marks also fall away on answers that are numerically right but presented in the wrong form, so keep the exact value unless a decimal is demanded.
概率与统计 树状图 累积频率
概率的两条主线是树状图和条件概率。树状图要先写清每一层的总数，再沿路径相乘、跨分支相加。统计部分的重点在累积频率图和从中读中位数、四分位数，2025 版把箱线图从 Extended 里删了，但读图找中位数的题仍在。常见的错误是把「至少一个」当成一个结果来算，正确做法是用 1 减去「一个都没有」的概率。
Probability runs on two ideas: tree diagrams and conditional probability. A tree diagram works only when each layer states its total; after that, branches multiply along a path and add across alternatives. In statistics the emphasis falls on cumulative frequency graphs and on reading the median and quartiles from them. The 2025 syllabus drops box-and-whisker plots from the Extended content, although pulling a median off a graph remains examinable. The recurring mistake is to treat "at least one" as a single outcome instead of taking 1 minus the probability of none. Writing the complement down first turns a two-minute question into a twenty-second one.
精度 三位有效数字 上下界
0580 的评分规则写得很具体：非精确数值答案保留 3 位有效数字，角度保留 1 位小数，题目另作要求时按题目来。中间值不要舍入，后面还要用的答案带着未舍入的值走。π 用计算器里的 π 或 3.142。上界下界题要写成 2.5 ≤ x < 3.5 这样的区间，只给一个端点会丢分。开卷先把这个规则写在草稿纸顶端，能省掉一整类本来不该错的错。
The marking rules are explicit. Non-exact numerical answers go to 3 significant figures and angles to 1 decimal place, unless the question states otherwise. Intermediate values are not rounded, and any value needed later is carried forward unrounded. Pi comes from the calculator or from 3.142. Upper and lower bound questions want an interval written as 2.5 ≤ x < 3.5, and a single endpoint loses the mark. Writing the rounding rule at the top of the rough page before starting removes a whole class of avoidable errors.
真题的两遍用法
真题要分两遍用，目的不同。第一遍按知识点拆开做，不掐时间，用来暴露漏洞；第二遍整卷限时，按 1 分钟 1 分的节奏，留 5 到 8 分钟检查。分类练用 2019 年以后的卷子最合适，注意 2025 版删掉了 Core 的向量加减和 Extended 的箱线图，这几类旧题可以直接跳过。做完当天对答案，错题分三类记：概念不清、计算失误、读题错。
Past papers serve two purposes and should be used in two passes. The first pass splits them by topic and runs without a timer, so that gaps surface fast. The second pass is a full paper under exam conditions, at roughly one minute per mark, leaving five to eight minutes for checking. For topic practice, papers from 2019 onwards are the most useful, with two caveats: the 2025 syllabus removed vector addition and subtraction from the Core content and box-and-whisker plots from the Extended content, so those items can be skipped. Marking should happen the same day, and every wrong answer belongs in one of three piles: concept, arithmetic, or misreading. Whichever pile grows fastest sets the agenda for the following week.
半年时间表怎么排
距离考试 6 个月的话，前 10 周按章节过知识点，每章配 20 到 30 道分类题；中间 6 周做整卷，每周两套，周末复盘；最后 4 周只做错题和公式默写，每周留一次限时卷保持手感。每周固定 4 到 5 小时，比考前突击 20 小时有用。计划要按考试日期倒推着写进日历，先定卷子，再填章节。撑不过一个漏掉的一周的计划，本来就不算计划。
With six months to go, a workable shape is ten weeks of chapter coverage with 20 to 30 topic questions each, then six weeks of full papers at two a week with a weekend review, then four weeks of wrong-answer work and formula recall while one timed paper a week holds the pace. Four to five steady hours a week beats twenty hours crammed into the final fortnight. The plan should be built backwards from the exam dates, papers first, chapters second. A schedule that cannot survive one missed week was never a schedule, and rebuilding it costs less than abandoning it.
考场上的时间分配
进考场先扫一遍全卷，把 30 秒内能下手的题做完，再回头啃长的。每题写完立刻看单位和小数位，几何题把已知条件全部标到图上。最后 10 分钟不碰新题，只做三件事：检查计算器有没有按错、答案有没有抄错行、有没有漏做小问。空着的小问和做错的小问扣一样的分，能写一步就写一步。
Skim the whole paper first, clear the questions that yield in under thirty seconds, then return to the long ones. Check units and decimal places at the end of every question, and copy every given value onto the diagram before starting a geometry part. The last ten minutes should not open new questions; they go on checking calculator entries, transcribing answers, and confirming that no part has been left blank. A blank part scores the same as a wrong one, so writing a first line is always worth the ink. Candidates who plan the closing five minutes tend to find two or three marks that would otherwise have walked away.
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