IB MYP数学 Criterion B 规律探究 高分技巧
在 IB MYP 数学的四个评分标准里，Criterion B（Investigating Patterns，规律探究）是很多学生最没有把握的一项。它不像 Criterion A 那样可以靠刷题积累，也不像 Criterion C 那样只要表达清楚就能拿分。Criterion B 考的是一条完整的科学思维链：你能不能从一堆具体的数字或图形里，看出隐藏的结构，用数学语言把它写成一个通式，再用它去预测和检验。本文把这条思维链拆成可操作的步骤，配上完整的解题范例、二次规律的进阶技巧和常见失分点，帮你把 Criterion B 从 4 分档稳定推到 7 到 8 分档。
Criterion B in IB MYP Mathematics is the criterion that worries students most, because it cannot be improved by drilling alone. It tests a complete chain of mathematical thinking: can you look at a set of numbers or shapes, spot the underlying structure, express it as a general rule in correct notation, then use that rule to predict and to verify. This guide breaks the chain into concrete steps, adds fully worked examples for both linear and quadratic patterns, and lists the mistakes that cost the most marks.
1. Criterion B 到底在考什么 | What Criterion B Really Tests
MYP 数学的 Criterion B 通常占整份考评的 20% 左右，评估的是"探究规律"的能力。考官看四件事：第一，你是否系统地收集和记录数据（organised data）；第二，你是否提出了合理的猜想（conjecture）；第三，你是否把规律写成了一般的数学表达式（general rule），并且说明了变量的含义；第四，你是否检验了这条规律的适用边界（validity and limits）。很多学生只做到第二步就停笔了，把"我发现每次加 2"当成最终答案，结果只能落在中低档。真正拉开差距的，是通式的书写规范和适用范围那段讨论。
Criterion B typically accounts for around 20 percent of the overall MYP mathematics assessment. Examiners look for four things: systematically collected and well-organised data, a reasonable conjecture, a general rule written in correct notation with the variables clearly defined, and a check of where that rule does and does not hold. Many students stop after step two, treat "it goes up by 2 each time" as the final answer, and are capped in the middle band. The written general rule and the discussion of its limits are where the top-band marks actually live.
2. 四步探究框架 | The Four-Step Investigation Framework
把每道 Criterion B 题目都按同一个框架推进，可以保证不漏掉任何一个评分点。第一步，把题目给的图形或情境编号，做成数据表，例如 n = 1, 2, 3, 4, 5，把对应的数量依次写清楚，必要时画出草图。第二步，做一阶差分，看相邻两项的差是否恒定，如果恒定就是线性关系；如果差本身也在按规律变化，就继续做二阶差分。第三步，用差分结果写出通式，并代入原始数据逐项检验。第四步，明确写出 n 的取值范围，说明这条规律在哪一步开始失效，以及为什么失效。这四步在纸上写出来大约只需要六到八行，但它覆盖了几乎全部的评分点。
Pushing every Criterion B task through one fixed framework guarantees you hit each marking point: number the given shapes or situations (n equals 1, 2, 3, 4, 5), record the corresponding count in a table with a sketch if helpful, take first differences to see whether they are constant, take second differences if they are not, write the general rule suggested by those differences, substitute the original data back in to test it, and finally state the range of n for which the rule is valid and explain why it fails outside that range. Written out, the whole framework rarely takes more than eight lines, yet it covers almost every strand on the mark scheme.
3. 如何写出一条"完整"的规律 | How to Write a Complete General Rule
所谓完整的规律，必须包含三样东西：变量声明、表达式本身、以及验证过程。变量声明要写清楚"设 n 为图案的编号"，不能出现 n 却不解释它代表什么。表达式要使用课本规范符号，乘号应当省略或写成标准的点乘，而不是字母 x。验证过程不能只验第一项，至少代入三项数据，最好包括表格里的最后一项。举例来说，如果第 n 个图案需要 3n + 1 根小棒，那就把 n = 1, 3, 7 分别代入，得到 4, 10, 22，再与题目中的实际数量一一对比。这样写出来，"通式"和"有效性"两个评分点才算都有了证据。
A complete rule needs three parts: an explicit definition of the variable (for example, let n be the pattern number), the expression itself in correct notation, and the verification. Never introduce n without saying what it stands for, and never use the letter x for multiplication. A single substitution is not verification, because examiners want to see that the rule reproduces the data across the sequence. Substituting three values, including the last one in the table, and comparing each result with the collected data is the standard that earns full credit for validity.
4. 从特例到通式：差分法的实际使用 | From Cases to Formula Using Differences
差分法是最可靠的机械方法，它把"找规律"这件看起来靠灵感的事情变成了可以按步骤执行的流程。设数据为 2, 6, 12, 20, 30，一阶差分是 4, 6, 8, 10，二阶差分恒为 2。二阶差分恒定说明这是一个二次关系，二次项系数等于二阶差分除以 2，也就是 1，因此通式形如 n 平方加上一个一次项。代回 n = 1 得到 1 + b + c = 2，n = 2 得到 4 + 2b + c = 6，两式相减得 b = 1，再代入得 c = 0，于是通式为 n(n + 1)。用 n = 5 检验得到 30，与表格一致。把整个推理链条写出来，就是考官要看的"系统探究"。
The method of differences is the most dependable mechanical route, because it turns pattern hunting into a procedure. For the data 2, 6, 12, 20, 30, the first differences are 4, 6, 8, 10 and the second difference is constant at 2, which signals a quadratic relationship whose leading coefficient is the second difference divided by 2, giving 1. So the rule has the form n squared plus a linear term. Substituting n equals 1 and n equals 2 produces two equations, solving them gives b equals 1 and c equals 0, and the rule is n(n + 1). Checking with n equals 5 returns 30, which matches the table. Writing that full chain down is exactly what systematic investigation means to an examiner.
5. 二次规律的两个特征 | Two Signatures of a Quadratic Pattern
二次规律是 Criterion B 里最容易丢分的一类。两个特征可以帮助你快速确认：第一，一阶差分本身呈等差变化，例如 5, 7, 9, 11；第二，二阶差分恒定，且这个数的一半就是 n 平方的系数。以数据 3, 8, 15, 24 为例，一阶差分为 5, 7, 9，二阶差分为 2, 2，所以通式形如 n 平方加 bn 加 c。代入 n = 1 得到 1 + b + c = 3，代入 n = 2 得到 4 + 2b + c = 8，相减得 b = 2，再代回得 c = 0，通式为 n 平方加 2n，也就是 n(n + 2)。用 n = 4 检验得到 24，完全吻合。整个过程没有任何猜测成分，只要差分表写对，答案就是确定的。
Quadratic patterns are where most marks are lost, and two signatures identify them quickly: the first differences themselves change by a constant amount, such as 5, 7, 9, 11, and the second difference is constant, with half of that value as the coefficient of n squared. For the data 3, 8, 15, 24, the first differences are 5, 7, 9 while the second difference is 2, so the rule has the form n squared plus bn plus c. Substituting n equals 1 and n equals 2 gives two equations, subtracting them yields b equals 2 and c equals 0, so the rule is n squared plus 2n, that is n(n + 2). Checking with n equals 4 returns 24, exactly as observed. There is no guesswork left once the difference table is correct.
6. 高分例析：一个完整的 Criterion B 答案 | A Worked Example
题目：一个图案由若干个小正方形组成一条折线，第 1 个图案有 4 个小正方形，第 2 个有 7 个，第 3 个有 10 个，请找出规律并预测第 10 个图案的数量。第一步，列表：n = 1, 2, 3 分别对应 4, 7, 10。第二步，一阶差分恒为 3，说明这是线性关系。第三步，线性通式写作 3n + c，代入 n = 1 得 3 + c = 4，所以 c = 1，通式为 3n + 1。第四步，验证：n = 2 给出 7，n = 3 给出 10，与数据完全一致。第五步，预测：n = 10 给出 31。第六步，写出适用条件：n 必须为正整数，当图形被截断或不再按同样方式增加时公式失效。这样一份答案的每一句都对应一个评分点，稳稳落在最高档。
Task: a pattern is built from small squares arranged in a zigzag, with 4 squares in pattern 1, 7 in pattern 2 and 10 in pattern 3. Find the rule and predict pattern 10. Step one, tabulate the data. Step two, the first difference is constant at 3, so the relationship is linear. Step three, write the general linear form 3n plus c and substitute n equals 1 to get c equals 1, giving the rule 3n + 1. Step four, verify with n equals 2 and n equals 3 against the collected data. Step five, predict pattern 10 as 31 squares. Step six, state the limits: n must be a positive integer, and the rule breaks down if the figure is truncated. Every marking point has a matching sentence, which is what pushes an answer into the top band.
7. 三类常见情境与应对 | Three Common Contexts
MYP 的探究题通常出在三种情境里。第一类是图形拼接，例如用小棒摆三角形或正方形，数据往往是线性或二次的，做法是把每增加一个图形所增加的边数记录清楚，注意共享边的扣除。第二类是数表与数字序列，例如三角数、平方数、以及各种变体，关键在于差分表。第三类是现实情境建模，例如座位安排、包装盒尺寸、储蓄增长，这类题的变量往往带有单位，作答时必须保留单位并说明其实际含义。三种情境的操作流程完全一致，区别只在于数据从哪里来。
MYP investigations usually appear in one of three contexts. Shape construction, such as matchstick triangles or squares, tends to produce linear or quadratic data, and the key is recording the number of edges added each time while remembering to account for shared edges. Number tables and sequences, including triangular and square numbers and their variants, are solved through the difference table alone. Real-life modelling, such as seating arrangements, box dimensions or savings growth, carries units that must be preserved and explained in the answer. The procedure is identical in all three cases; only the source of the data changes.
8. 五个最常见的失分点 | Five Common Ways to Lose Marks
第一，只写结论不写过程，考官看不到探究是否系统。第二，变量没有定义，直接写 3n + 1 却不说 n 是什么。第三，验证时只代入第一项，无法证明规律对整体成立。第四，忽略适用范围的讨论，失去最后一大块分数。第五，用词与符号不规范，例如把乘法写成 x，或者用约等于符号代替等号。把这五点逐条对照自己的练习本，通常一次修改就能提升一到两个分数档。尤其要提醒的是第一条，很多学生以为自己"心里明白"，但考官只能给纸面上的文字打分。
First, writing only the answer gives the examiner no evidence that the process was systematic. Second, using n without defining it weakens the communication of the rule. Third, verifying with only the first term fails to show that the rule holds in general. Fourth, skipping the discussion of limits gives away a whole marking strand. Fifth, sloppy notation, such as writing x for multiplication or a rounded symbol instead of an equals sign, loses communication marks. Checking your work against these five points usually moves a response up one or two bands in a single revision.
9. 交卷前的自检清单 | A Pre-Submission Checklist
写完答案之后，用一分钟过一遍这份清单：数据表是否有编号和标题；差分表是否至少写到二阶；通式是否写成了含 n 的表达式而不是具体数字；变量是否明确定义；是否代入了至少三组数据做验证；是否写明了 n 的取值范围与规律的失效条件；单位是否保留；符号是否规范。这份清单本身也可以直接抄在笔记本的扉页上，因为 Criterion B 的评分点是跨题目重复出现的，一次整理可以反复使用。
Before handing in, spend one minute on this checklist: does the data table have numbering and labels; does the difference table reach at least the second difference; is the rule written as an expression in n rather than a specific number; is the variable defined; has at least three data points been substituted as verification; are the range of n and the conditions where the rule fails stated explicitly; are units preserved; is the notation standard. The same checklist can be copied into the front of your notebook, because the marking strands of Criterion B repeat across almost every task you will meet.
10. 备考练习与时间分配 | Practice Strategy and Timing
练习时不要只做一题就对答案，建议同一类情境连做三题：一道线性、一道二次、一道涉及几何量（周长、面积、棱数）。每道题都完整写出六步流程，坚持两周，框架就会变成习惯。考试中建议给 Criterion B 留出总时间的四分之一，其中一半时间用于列表和差分，另一半用于写通式、验证和讨论范围。不要因为想省时间而跳过列表，列表本身就是评分点之一。如果时间实在紧张，宁可将验证写得简短，也不要省略适用范围的讨论，因为后者通常是区分中档与高档的关键。
Practise in linked sets of three rather than single questions: one linear, one quadratic, and one involving a geometric quantity such as perimeter, area or number of edges. Write all six steps in full for each, and after two weeks the framework becomes automatic. In the examination, reserve about a quarter of the total time for Criterion B, splitting it evenly between tabulating and differencing, and between writing, verifying and discussing the limits of the rule. Never skip the table to save time, because the table is itself a marking point. If time runs short, shorten the verification but never drop the discussion of limits, since that is usually what separates a middle-band answer from a top-band one.
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