MYP数学 Criterion B 规律探究 一般项 证明
MYP 数学的 A 到 D 四条评估标准里，Criterion B 是唯一一条要求学生自己找规律的标准。它不考你记住了哪条公式，考的是你能不能从几个具体例子出发写出一条通项，再把它验证、论证清楚。这一项满分 8 分，和 A、C、D 三条标准同权，四条分数相加（合计 32 分）后按 IB 的分数线换算成 1 到 7 级。很多学生在 A 项拿 7 分，B 项却只有 4 分，原因通常不是数学不够好，而是把找规律当成了填几个数。下面按 IBO 官网发布的《Mathematics guide》（2020 年发布，供 2020 年 9 月 / 2021 年 1 月起使用）里 Year 1、Year 3、Year 5 三套官方描述，把这一项的评分逻辑拆开讲。
In MYP mathematics the four assessment criteria A to D carry equal weight, and Criterion B is the only one that asks students to generate mathematical structure rather than reproduce it. Work is matched against band descriptors out of 8, and the four criterion totals out of 32 are converted to a final grade of 1 to 7 using the boundaries published in the MYP general grade descriptors. Teachers see the same pattern every year; a student who scores 7 in Criterion A can sit at 4 in Criterion B, because discovery is assessed differently from recall. What follows works from the official strands and descriptors published for years 1, 3 and 5 in the IB Mathematics guide for use from September 2020.
Criterion B 在四条标准里的位置
MYP 数学用四条标准评估：A 知识与理解、B 规律探究（Investigating patterns）、C 表达（Communicating）、D 数学在真实情境中的应用，每条满分 8 分，四条合计 32 分。B 项的对象是探究，guide 把它描述成让学生体验数学发现的过程：自己选方法、生成数据、找关系、写出一般规则，再论证它。B 和 A 通常不是同一类题，教师会用不同类型的任务分别评；C 项常常和 B 或 D 捆在同一份探究报告里一起评。IBO 要求每条标准的所有 strand 在每学年至少被评估两次，所以 B 项不是期末考一道题，而是贯穿全年。Year 1、Year 3、Year 5 用的是三套不同描述，同一份作业在不同年级的评分口径并不一样。
Four criteria carry equal weight in MYP mathematics, and Criterion B is the investigative one: students select a technique, generate data, look for structure, write a general rule and defend it; candidates should expect different task types for different criteria; the guide notes that criterion C is often assessed alongside B or D in the same investigation report, so one piece of work can supply evidence for two criteria at once. The IB requires all strands of all four criteria to be assessed at least twice in every year of the programme, which means Criterion B is spread across several tasks rather than settled by a single end-of-term test. Descriptors are published separately for years 1, 3 and 5, so identical work can be judged differently depending on the year group.
三条 strand 从识别规律走到证明一般规则
三条 strand 是 B 项的骨架，各年级的描述都在它们上面加限定词。Year 1 要求用数学问题解决技巧识别规律（recognise patterns）、把规律描述成关系或一般规则、并验证它在其他例子上是否成立。到 Year 3，第一条改成选择并运用技巧发现复杂的规律（complex patterns），第三条从验证升级成验证并论证（verify and justify）。Year 5 最严，第一条仍是发现复杂规律，第二条要求写成 general rules 且与发现一致，第三条要 prove，或者 test and justify。差别全落在动词上：recognise 和 select and apply 之间隔着能不能自己挑方法，verify 和 prove 之间隔着论证覆盖了所有情形还是只覆盖了几个例子。看题目时先把这三个动词圈出来，再决定答卷要写多深。
The three strands are not three separate marks; they form a progression, and each year group's descriptor bolts qualifiers onto them. A year 1 student recognises patterns with a technique they were given, whereas a year 5 student selects and applies a technique to a complex pattern independently. The last strand carries most of the difficulty: verifying a rule means checking it against further cases, while proving it means showing that no case can escape it, and the year 5 descriptor accepts either a proof or a justified test. Examiners read the whole investigation against these verbs, so an answer that generates excellent data but never states a general rule tends to stop in the 3 to 4 band.
0 到 8 分档各档要什么
等级 0 表示未达到下面任何描述的水平，IBO 官方把 1 到 8 分压成四档，分别对应 limited、adequate、substantial、excellent。1 到 2 分档的关键词是在教师支持下和简单规律，学生能做出与规律一致的预测就算达标。3 到 4 分档要能提出（suggest）与发现一致的一般规则，注意是 suggest 而不是 describe，措辞本身还带着不确定。5 到 6 分档要求选择并运用技巧发现复杂规律、把规律描述成与发现一致的一般规则、检验这些规则的有效性，三条要同时出现。7 到 8 分档在 findings 前面加了 correct，并把第三条从 test 升级成 prove 或者 test and justify。批改中分数常卡在 4 和 6 两个台阶：能提出规则但没有检验，停在 4；检验了却没有把结论说成一般规则，或者只测了例子没有论证，停在 6。
Achievement is reported on a best-fit basis against four bands: limited (1 to 2), adequate (3 to 4), substantial (5 to 6) and excellent (7 to 8), with 0 reserved for work reaching none of them; the verb suggest in the 3 to 4 band is doing real work, since a plausible rule earns credit there, whereas the 5 to 6 band demands that the rule be described and its validity tested. The top band adds two conditions students routinely miss: the rule must be consistent with correct findings, and the testing must be raised to a proof or a justified argument; teachers describe a ceiling at four and another at six, because the jump between bands is caused by an action the student never took rather than by harder arithmetic.
太引导的任务为什么封顶 4 分
IB 在 guide 里写得很直白：一份不允许学生自己选择问题解决技巧的任务太引导了，这种任务下学生最高只能拿 6 分（Year 1 和 Year 2），Year 3 及以上最高只有 4 分。这条规则解释了一个普遍困惑，题目给了详细步骤，学生老老实实做完，分数却上不去，因为被评的根本不是计算过程。教师设计任务要在两端之间找平衡，既要给出足够方向让学生能开始，又必须留下你来选方法的空间，比如给情境但不规定用数列、代数还是图示。遇到 Step 1 到 Step 6 列得很细的探究作业，先判断哪一步是方法自选，把精力压在那一步上，通常是把具体例子抽象成一般规则的环节。如果整份任务没有这种环节，满分在规则层面就不存在，可以向老师确认这道题对应哪条标准的哪些 strand。
The guide states plainly that a task which does not allow students to select a problem-solving technique is too guided, capping achievement at 6 in years 1 and 2 and at 4 from year 3 onwards; that sentence explains a common frustration: students complete a six-step worksheet correctly and still finish on a low band, because the criterion is not measuring their arithmetic. Teachers are expected to give enough direction for everyone to start, yet leave the method open, so a strong investigation supplies a context and refuses to name the mathematics. When a task prescribes every step, the ceiling is structural rather than personal, and asking which parts carry the criterion is more useful than rewriting the answers.
造表与差分把看起来像变成可核对的数据
探究的第一步是把图或情境翻译成能比较的数字。课本里最常见的一类任务是火柴棒搭一排正方形，1 个正方形用 4 根，2 个用 7 根，3 个用 10 根。先把这三组写成有序对 (1,4)、(2,7)、(3,10)，再算相邻两项的差，得到 3、3、3。一阶差恒为 3，说明这是线性关系，通项形如 3n + c，代入 n=1 得 c=1，于是 M=3n+1。差分法在纸面上最容易执行，但它只回答这是哪一类关系，不回答为什么成立。常见的错误是把 n 直接从 0 开始，这道题里 n=0 没有意义，因为 0 个正方形不存在，定义域要写成 n 是不小于 1 的整数。图形式的探究还可以把复杂规律做成第二张表，同时记录火柴根数与正方形个数，让两条序列互相印证。
Turning a picture into a table of ordered pairs is what makes everything else arguable, because numbers can be compared and drawings cannot. A row of matchstick squares needs 4, 7 and 10 matches for one, two and three squares; the first differences are constant at 3, so the relationship is linear and the general term looks like 3n + c, with c fixed at 1 by the case n = 1. Difference tables identify the family a rule belongs to and say nothing about why the rule holds, which is the gap the third strand asks students to close. One small trap costs marks every year, because n must be defined and here it is an integer of at least 1; a row of zero squares simply does not exist.
从具体例子写出一般规则
写一般规则时，变量约定要自己说清楚：n 代表什么，取值范围是多少，公式里每一项对应图或情境里的什么。以多边形对角线为例，n 边形一共有 n(n-3)/2 条对角线。这个结论可以从数据推出来，n=4 有 2 条、n=5 有 5 条、n=6 有 9 条，一阶差是 3 和 4，二阶差恒为 1，说明是二次关系。还有一条更漂亮的路径，每个顶点能和除自己与两个相邻点之外的 n-3 个点连线，n 个顶点共 n(n-3) 条连线，而每条对角线被数了两次，所以要除以 2，这段推理本身就是可以写进答卷的论证。规则写完不要停在公式上，补一句边界说明，n 必须是不小于 3 的整数，因为 n=3 时 3(3-3)/2 等于 0，三角形确实没有对角线。
A general rule is usable only when the reader knows what the variable means and which values it takes, so the definition is part of the answer rather than housekeeping; polygon diagonals give a clean illustration; the count is n(n-3)/2, which the difference table predicts from 2, 5 and 9 diagonals for quadrilaterals, pentagons and hexagons once the second difference settles at a constant 1. A better route is open to students, since each vertex joins the n-3 vertices that are neither itself nor its two neighbours, and the n(n-3) connections count every diagonal twice; that reasoning is itself the evidence the top band wants, and it is worth writing out even when the formula feels obvious. Boundary conditions belong there too; a triangle has zero diagonals, so n must be an integer of at least 3.
验证与证明回答的不是同一个问题
verify 是拿更多例子检验规则，prove 或 justify 是说明它为什么对所有情形都成立。两者的差距可以用一个经典例子说明：n 的平方加 n 再加 41 在 n 取 0 到 39 时给出的都是质数，随手验证前十个、前二十个都会让人以为结论稳了，但 n=40 时得到 1600 加 40 加 41 等于 1681，恰好是 41 的平方，不再是质数。只做验证的答卷碰到这种前几十个都对、后面突然失效的规律，会被判为论证不足。MYP 阶段的论证手段主要是三种，代数变形（把图形拆成不变量再加起来）、分类讨论（按奇偶或按余数分情况）、以及反证（假设规则不成立推出矛盾）。阅卷时看的是这些手段有没有覆盖所有 n，而不是看了多少个例子。
Verification and proof answer different questions, and the gap between them is easy to demonstrate; the expression n squared plus n plus 41 returns a prime for every n from 0 to 39, so a student testing the first twenty cases concludes the pattern is safe; at n = 40 it equals 1681, which is 41 squared and therefore composite. Twenty further examples would not have helped, because no finite list can cover infinitely many cases, and this is exactly why the higher bands separate testing from justifying; MYP-level arguments usually rely on algebraic manipulation, case splitting by parity or remainder, or a contradiction, and the marker is reading for coverage of all n rather than volume of evidence. Spreadsheets are excellent for finding a pattern and useless as proof, which students often conflate.
结论错了还能拿 6 分吗
IB 给了一条缓冲规则，Year 3 及以上，如果学生写出的 general rule 与自己的发现一致、但发现本身算错了，最高仍可拿到 6 分，前提是这条规则的复杂度与任务相当。很多学生以为 B 项拿不到 7 分是因为数学不够难，实际上 7 和 6 的差别在有没有论证，而不在计算量。这条规则背后的逻辑是 B 项评探究过程而非答案，所以发现阶段的数据记录、差分过程、规则推导都要留在答卷上，只写一个最终公式会同时丢掉过程分和这条缓冲。反过来也有代价，如果规则和发现不一致（表里写着 4、7、10，推导却写出 3n+2），那就连 consistent with findings 都没达到，通常落到 3 到 4 分档。答卷里每一节末尾留一句过渡，说明下一步做什么，评分的人才能顺着你的推理链走；阅卷时最怕看到漂亮的表格之后突然跳出一个没有来源的公式。
The IB builds in one cushion that surprises students: from year 3 onwards a rule consistent with the findings, but built on incorrect findings, can still reach 6 provided the rule is of comparable complexity to the task; Criterion B is assessing the investigation rather than the arithmetic, which is why the data tables, the difference working and the reasoning all have to stay visible in the submitted work. Consistency runs both ways, since a table showing 4, 7 and 10 followed by a stated rule of 3n + 2 fails the descriptor outright and usually lands in the 3 to 4 band. Examiners read a chain of reasoning, so a polished table followed by an unsourced formula is the most common way to lose marks that were already earned.
Criterion C 与 B 的捆绑评分
guide 明确说 C 项常常和 B 或 D 捆在同一份作业里评，这意味着同一份探究报告会被读两遍，一次看探究质量，一次看表达质量。表达上的扣分点很具体，变量没有定义就出现在公式里，公式和文字之间没有连接，表格没有编号或标题，坐标轴没有标注，答案没有单位。这些都不是数学错误，却足以让 C 项停在 4 分档，并把四条标准的总分一起拉下来。把 B 和 C 一起处理的做法是给答卷加一个读得懂的骨架，每张表配一句说明，每一次操作写清目的，最后一节回到最初的问题并用完整句子回答它。英文写作部分注意术语统一，同一概念不要在文中换三种叫法，比如 standard form 不要中途改称 scientific notation。
Because criterion C is commonly assessed together with B or D, the same report is usually read twice, once for the quality of the investigation and once for the quality of the communication; marks disappear in concrete places; an undefined variable appears in a formula, a table has no title, axes carry no labels, a solution has no units, and paragraphs never explain why the next step is being taken. None of these are mathematical errors, yet they hold the communication criterion in the adequate band and pull the total criterion score down with it. A little discipline covers most of it, numbering each table, captioning what it shows, stating the purpose of each operation, and closing with a sentence that answers the original question in words.
一份 Criterion B 答卷的骨架
把前面的要求落成一份可执行的答卷，顺序大致如此：抄写题目给出的情境并写清自己要找什么，列出前几个例子对应的数据表，算差或找结构，写出一般规则并定义变量与范围，用更多例子检验并给出论证，末尾回到题目本身。时间分配上，找规律和写规则占一半，论证和表达占另一半，因为 7 到 8 分档的两条硬要求都落在后半段。IBO 在 MYP 第 5 年提供可选的 eAssessment，其中 on-screen examinations 每场时长在 1 小时 45 分钟到 2 小时之间，由 IB 培训过的阅卷员评分；校内 Criterion B 的分数则由任课教师按标准描述给出，两者口径一致但用途不同。平时练手可以从课本的图案序列和数列题开始，每次都用完整骨架写一遍，比只算答案有效得多。
A workable investigation follows a predictable shape; restate the context, state what is being found, tabulate the first few cases, look for structure in the differences, write the general rule with its variable and domain defined, then test and justify it before answering the original question in words; time is best split evenly between discovery and justification, because the two conditions for the top band live in the second half. Criterion B marks are awarded by classroom teachers against the descriptors; in the optional year 5 eAssessment the IB adds externally marked on-screen examinations of one hour 45 minutes to two hours, and the two routes share the same objectives even though they serve different purposes; practising with a full skeleton on textbook sequence questions builds the habit faster than racing to an answer.
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