MYP数学Criterion B探究规律高分攻略 | MYP Maths Criterion B: How to Score High on Investigating Patterns

MYP数学Criterion B探究规律高分攻略

在IB MYP数学课程中，Criterion B（Investigating Patterns）是最能体现学生数学思维能力的评估标准之一。它不仅考察你是否能"算出答案"，更关注你如何发现规律、用代数语言描述规律，并用逻辑推理验证你的发现。对于许多MYP 4-5年级（相当于国内初三至高一）的学生来说，Criterion B既是提分利器，也是失分重灾区：因为它的评分逻辑与传统数学考试完全不同。本文将系统拆解Criterion B的评分标准、解题步骤、常见规律类型和高分技巧，帮你从"会算题"进阶到"会探究"。

In the IB MYP Mathematics curriculum, Criterion B (Investigating Patterns) is one of the assessment criteria that best reflects a student's mathematical thinking ability. It evaluates not just whether you can "get the right answer," but how you discover patterns, describe them using algebraic language, and verify your findings through logical reasoning. For many MYP Year 4-5 students, Criterion B is both a high-scoring opportunity and a common pitfall -- because its scoring logic is fundamentally different from traditional math exams. This article systematically breaks down the assessment rubric, step-by-step process, common pattern types, and high-scoring strategies to help you move from "solving problems" to "investigating mathematics."

一、什么是MYP数学Criterion B？| What Is MYP Maths Criterion B?

Criterion B的全称是"Investigating Patterns"（探究规律），是MYP数学四项评估标准之一（另外三项是A: Knowing and Understanding, C: Communicating, D: Applying Mathematics in Real-Life Contexts）。Criterion B的核心任务是：给定一组数学现象（图形、数字序列或表格数据），要求学生观察并归纳其中的数学规律，将规律用代数关系式表达出来，然后验证该关系式的正确性，最后探讨规律的推广或局限性。在MYP eAssessment电子考试和校内评估中，Criterion B通常以一道独立的探究大题出现，满分8分。

The full name is "Investigating Patterns," one of the four MYP Mathematics assessment criteria (the others being A: Knowing and Understanding, C: Communicating, and D: Applying Mathematics in Real-Life Contexts). The core task of Criterion B is: given a set of mathematical phenomena (diagrams, number sequences, or table data), students observe and identify a mathematical pattern, express it as an algebraic relationship, verify its correctness, and finally discuss its generalization or limitations. In MYP eAssessment electronic exams and internal assessments, Criterion B typically appears as a standalone investigation question worth 8 marks.

二、Criterion B四档评分标准详解 | The Four Achievement Levels Explained

IB将Criterion B分为四个成就等级（0分到8分），每个等级对应不同的数学能力表现。0分代表未达到任何标准；1-2分（Level 1-2）仅需要学生识别出简单的规律并用语言或数学符号描述；3-4分（Level 3-4）要求用代数表达式描述规律；5-6分（Level 5-6）需要验证规律的正确性，即代入具体数值检验你的代数表达式是否产生正确的结果；7-8分（Level 7-8）是最高等级，要求学生不仅验证规律，还要探讨其推广性（generalization）：例如，当参数变化时规律是否仍然成立，或者规律是否存在边界条件。理解这四档要求的关键洞见是：Criterion B的分数是"累积性"的，高分必须建立在低分能力之上，你不能跳过验证直接谈推广。

IB divides Criterion B into four achievement levels (0 to 8 marks), each corresponding to different mathematical competencies. 0 indicates no standard met; 1-2 marks (Level 1-2) only require identifying simple patterns and describing them in words or mathematical symbols; 3-4 marks (Level 3-4) require expressing the pattern with an algebraic expression; 5-6 marks (Level 5-6) require verifying the pattern's correctness by substituting specific values to check whether your algebraic expression produces the correct results; 7-8 marks (Level 7-8) is the highest level, requiring students not only to verify the pattern but also to discuss its generalization -- for example, whether the pattern still holds when parameters change, or whether boundary conditions exist. The key insight is that Criterion B marks are cumulative: high scores must build upon lower-level competencies -- you cannot skip verification and go straight to generalization.

三、第一步：如何从图表中识别数学规律？| Step 1: Identifying Patterns from Diagrams and Tables

MYP的Criterion B题目通常以"图形序列"（diagram sequence）或"数值表格"（table of values）的形式呈现。比如题目可能给出第1个图形有3个点，第2个图形有7个点，第3个图形有13个点……要求学生找出第n个图形的点数公式。识别规律的关键技巧是"差分法"：先计算相邻两项的差值（一阶差分），如果一阶差分为常数，则规律是线性的（一次函数）；如果一阶差分不是常数但二阶差分为常数，则规律是二次的（quadratic）；如果差分的差分仍然是变化的，规律可能是更高次的或指数型的。对于图形序列，将图形分解为"不变部分"和"变化部分"，分别分析它们如何随n变化，是最高效的策略。

MYP Criterion B questions are typically presented as diagram sequences or tables of values. For example, a question may show that the 1st diagram has 3 dots, the 2nd has 7 dots, and the 3rd has 13 dots -- asking students to find a formula for the number of dots in the nth diagram. The key identification technique is the "method of differences": calculate the differences between consecutive terms (first differences). If first differences are constant, the pattern is linear (a first-degree function); if first differences are not constant but second differences are constant, the pattern is quadratic; if even the differences of differences keep changing, the pattern may be of higher degree or exponential. For diagram sequences, the most efficient strategy is to decompose the diagram into "unchanging parts" and "changing parts," analyzing how each varies with n independently.

四、第二步：写出准确的代数表达式 | Step 2: Writing an Accurate Algebraic Expression

一旦通过差分法确定了规律的类型（线性、二次或其它），下一步就是用数学语言将规律表达出来。对于线性规律（一阶差分为常数），公式形式为 an = dn + c，其中d是公差（一阶差分），c可以通过代入n=1求出。对于二次规律（二阶差分为常数），公式形式为 an = an² + bn + c（这里的a是二阶差分的一半）。写代数表达式时最常见的错误是：学生在代入n验证后发现结果不对，于是反复调整常数项却忽略了n²系数的正确计算。一个实用的检查方法是：将你的表达式分解为"变化项"加"初始值"，分别验证它们对n=1, n=2, n=3是否都成立：三个点足够确认一个二次函数。

Once you have determined the pattern type via the method of differences (linear, quadratic, or other), the next step is to express it in mathematical language. For linear patterns (constant first differences), the formula takes the form an = dn + c, where d is the common difference and c is found by substituting n=1. For quadratic patterns (constant second differences), the formula is an = an² + bn + c (where a equals half the second difference). The most common mistake when writing algebraic expressions is that students repeatedly adjust the constant term after checking with n=1, n=2, and n=3 while overlooking the correct calculation of the n² coefficient. A practical checking method: decompose your expression into "changing terms" plus "initial value," and verify them separately for n=1, n=2, and n=3 -- three points are sufficient to confirm a quadratic function.

五、第三步：验证你的规律：不只是"代入检查"| Step 3: Verifying Your Pattern Goes Beyond Substitution

许多学生认为验证就是"把n=1,2,3代入公式看看对不对"，这种理解在Level 3-4层面尚可接受，但到了Level 5-6就会被扣分。真正的验证（verification）需要两层工作：第一，代入至少三个不同的n值，展示你的代数表达式对已知数据点产生正确结果；第二：这也是最容易被忽略的：用你的公式预测一个未知的数据点（如n=10或n=50），然后通过继续绘制图形或扩展表格来"独立验证"该预测值是否正确。这第二层验证展示了你的规律具有"预测能力"（predictive power），而不仅仅是"拟合"了已有数据。在答题时，明确写出"代入n=1, n=2, n=3均符合已知值，使用n=10预测得X，通过扩展图形验证预测正确"这样的完整推理链，才能确保Level 5-6的分数。

Many students think verification simply means "plugging n=1, 2, 3 into the formula to check if it's right." This understanding is acceptable at Level 3-4, but you will lose marks at Level 5-6. True verification requires two layers of work: first, substitute at least three different n values to show that your algebraic expression produces correct results for known data points; second -- and this is the most commonly overlooked part -- use your formula to predict an unknown data point (such as n=10 or n=50), then independently verify that prediction by extending the diagram or table. This second layer demonstrates that your pattern has predictive power rather than merely fitting existing data. When answering, explicitly write the complete reasoning chain: "Substituting n=1, 2, 3 matches known values; predicting n=10 yields X; extending the diagram confirms the prediction." Only this will secure Level 5-6 marks.

六、第四步：阐述推理过程与推广规律 | Step 4: Justifying Reasoning and Generalizing the Pattern

Level 7-8的高分核心在于"justification"（论证）和"generalization"（推广）。论证意味着你需要解释为什么你的代数表达式是正确的：这不仅仅是列出一串代入验证，而是从图形的几何结构或数列的递推关系中推导出代数规律。例如，如果图形由"中心的正方形"和"四个角的三角形"组成，你应该分别写出正方形点数和三角形点数的公式，再合并，从而"证明"你的表达式而非仅仅"猜测"它。推广则要求你探讨：当图形的初始条件改变时（如改为中心五边形而非正方形），规律如何变化？或者给定的规律是否有适用范围（如n必须是正整数）？写出这些讨论，即使简短，也能展示你的数学洞察力达到最高等级。

The core of Level 7-8 high scores lies in justification and generalization. Justification means you need to explain why your algebraic expression is correct -- this is not merely listing a series of substitution checks, but deriving the algebraic pattern from the geometric structure of the diagram or the recursive relationship in the sequence. For example, if a diagram consists of a "central square" and "triangles at the four corners," you should separately write formulas for the square's dots and the triangles' dots, then combine them, thereby proving your expression rather than merely guessing it. Generalization requires you to explore: when the initial conditions change (such as making the center a pentagon instead of a square), how does the pattern change? Or, does the pattern have a domain restriction (such as n must be a positive integer)? Writing these discussions, even briefly, demonstrates mathematical insight at the highest level.

七、常见规律类型：线性与二次数列 | Common Pattern Types: Linear and Quadratic Sequences

线性数列是MYP Criterion B中最基础的规律类型。识别方法：一阶差分为常数。典型例子包括等差数列（如3, 7, 11, 15...，公差为4，第n项公式为4n-1）以及图形中"每次增加固定数量"的模式（如图形的边数随n线性增长）。二次数列是MYP 4-5年级最常见的探究题型。识别方法：二阶差分为常数。典型例子包括三角形数（1, 3, 6, 10...，第n项公式为n(n+1)/2）、正方形数（1, 4, 9, 16...）以及组合图形（如阶梯状图形中正方形数量等于n²）。掌握差分法+公式模板（an=dn+c用于线性，an=an²+bn+c用于二次），就能处理至少80%的Criterion B题目。

Linear sequences are the most basic pattern type in MYP Criterion B. Identification: constant first differences. Classic examples include arithmetic sequences (such as 3, 7, 11, 15... with common difference 4, nth term formula 4n-1) and patterns where "a fixed amount is added each time" in diagrams (such as side lengths growing linearly with n). Quadratic sequences are the most common investigation type for MYP Year 4-5. Identification: constant second differences. Classic examples include triangular numbers (1, 3, 6, 10..., nth term formula n(n+1)/2), square numbers (1, 4, 9, 16...), and composite diagrams (such as staircase diagrams where the number of squares equals n²). Mastering the method of differences plus formula templates (an=dn+c for linear, an=an²+bn+c for quadratic) enables you to handle at least 80% of Criterion B questions.

八、常见规律类型：几何图形与复合模式 | Common Pattern Types: Geometric Diagrams and Composite Patterns

除了纯数值序列，MYP特别喜欢考"图形序列"类题目：给你一串由点、线段、正方形或火柴棍组成的图形，让你找出第n个图形中的元素个数。这类题目的解题策略是"结构化分解"（structural decomposition）：不要试图一眼看出整个公式，而是将图形拆解为若干可识别的子结构。例如，一个由"底层矩形"+"上层三角形"组成的图形，可以分别求出矩形点数公式f(n)和三角形点数公式g(n)，总点数公式为f(n)+g(n)。这种"拆分-求解-合并"的策略不仅正确率高，而且自然地为你的论证（justification）提供了结构：每个子结构的公式都来自其几何意义，这使得证明过程清晰有力。

Beyond pure numerical sequences, MYP particularly favors diagram sequence questions -- presenting a series of diagrams made of dots, line segments, squares, or matchsticks, requiring you to find the number of elements in the nth diagram. The solution strategy for these is structural decomposition: do not attempt to see the entire formula at a glance; instead, break the diagram into identifiable substructures. For example, a diagram composed of a "base rectangle" plus an "upper triangle" can be solved by finding the rectangle's dot-count formula f(n) and the triangle's dot-count formula g(n) separately, with the total formula being f(n)+g(n). This decompose-solve-combine strategy not only has a high accuracy rate but also naturally provides structure for your justification -- each substructure's formula comes from its geometric meaning, making the proof clear and compelling.

九、高分秘诀：如何稳定拿到7-8分 | High-Scoring Secrets: How to Consistently Achieve Level 7-8

稳定拿到7-8分的关键不是"做更难的题"，而是"把每一个评分要素都写到"。IB考官评分时按照明确的 checklist 逐项核对：你是否识别了规律？（1-2分）你是否写出了代数表达式？（3-4分）你是否用具体数值验证了表达式？（5-6分）你是否解释了为什么表达式成立，并讨论了推广性？（7-8分）很多能力强的学生卡在5-6分，不是因为他们不懂推广，而是因为他们没有"显式地写出来"：他们觉得"显而易见"的推广，考官不会替你补充。实际操作中，一条经过验证的高分公式是：用完整段落写出"规律识别→表达式推导→代入验证→几何或代数论证→条件讨论"，每个环节至少3-4句话，不留空白环节。

The key to consistently scoring 7-8 is not "doing harder problems" but "writing out every scoring element." IB examiners mark against a clear checklist: Did you identify the pattern? (1-2 marks) Did you write an algebraic expression? (3-4 marks) Did you verify the expression with specific values? (5-6 marks) Did you explain why the expression holds true, and discuss generalization? (7-8 marks) Many capable students get stuck at 5-6, not because they cannot generalize, but because they did not write it out explicitly -- examiners will not fill in generalizations that students considered "obvious." In practice, a verified high-scoring formula is: write a complete paragraph chain covering "pattern identification → expression derivation → substitution verification → geometric or algebraic justification → condition discussion," with at least 3-4 sentences per section, leaving no blank links.

十、常见失分陷阱：五个MYP学生最常犯的错误 | Common Pitfalls: Five Mistakes MYP Students Make Most

第一个陷阱：跳过验证环节。很多学生写出公式后直接下结论，但Level 5-6明确要求"验证规律"。第二个陷阱：混淆"验证"与"举例"。验证必须用具体数值代入并展示计算过程，而不是说"该公式对所有n都成立"。第三个陷阱：结论过于笼统。写"这个规律适用于所有n"没有得分价值；应该写"因为n代表图形的序号，所以n必须是正整数，当n=0时公式不适用"。第四个陷阱：误用差分法。当一阶差分不是常数时，不要强行用线性公式拟合：这会导致后续所有环节全部错误。第五个陷阱：用中文答题时缺少数学符号。Criterion B的论证需要用代数表达式、等式和变量符号，纯文字描述无法达到Level 7-8。

Pitfall 1: skipping the verification step. Many students jump directly to conclusions after writing a formula, but Level 5-6 explicitly requires pattern verification. Pitfall 2: confusing verification with exemplification. Verification must involve substituting specific values and showing the calculation process, not stating that "the formula holds for all n." Pitfall 3: writing overly vague conclusions. Writing "this pattern applies to all n" earns no marks; instead write "since n represents the diagram's index, n must be a positive integer, and the formula does not apply when n=0." Pitfall 4: misusing the method of differences. When first differences are not constant, do not force a linear fit -- this will make all subsequent steps incorrect. Pitfall 5: lacking mathematical symbols when answering. Criterion B justification requires algebraic expressions, equations, and variable notation -- pure verbal description cannot reach Level 7-8.

十一、实战例题解析：从题目到满分的完整路径 | Worked Example: A Complete Path from Question to Full Marks

我们以一道典型MYP Criterion B题目为例：观察下列图形序列：第1个图形由1个正方形和4个三角形组成（共5个图形单元），第2个图形由4个正方形和8个三角形组成（共12个单元），第3个图形由9个正方形和12个三角形组成（共21个单元）。第一，识别规律：正方形数量为1, 4, 9：这是平方数序列，公式为n²；三角形数量为4, 8, 12：这是公差为4的等差数列，公式为4n。第二，写出表达式：总单元数T(n) = n² + 4n。第三，验证：n=1得1+4=5（符合），n=2得4+8=12（符合），n=3得9+12=21（符合），预测n=10得100+40=140。第四，论证：正方形数量=n²是因为图形排列成n×n的正方形网格，三角形数量=4n是因为每个正方形的四条边各外接一个三角形。第五，推广：如果改为每个正方形有k种附属图形而非三角形，则公式推广为T(n) = n² + kn。

Consider a typical MYP Criterion B question: observe the following diagram sequence -- diagram 1 consists of 1 square and 4 triangles (5 units total), diagram 2 consists of 4 squares and 8 triangles (12 units total), diagram 3 consists of 9 squares and 12 triangles (21 units total). Step 1, identify the pattern: square counts are 1, 4, 9 -- this is the square number sequence, formula n²; triangle counts are 4, 8, 12 -- an arithmetic sequence with common difference 4, formula 4n. Step 2, write the expression: total unit count T(n) = n² + 4n. Step 3, verify: n=1 gives 1+4=5 (matches), n=2 gives 4+8=12 (matches), n=3 gives 9+12=21 (matches), predict n=10 gives 100+40=140. Step 4, justify: square count = n² because the diagrams are arranged as n×n square grids; triangle count = 4n because each square's four sides each have one attached triangle. Step 5, generalize: if each square had k types of attached figures instead of triangles, the generalized formula would be T(n) = n² + kn.

十二、备考建议：Criterion B的日常训练方法 | Exam Preparation: Daily Practice Strategies for Criterion B

提升Criterion B能力最有效的方法是"微探究"训练：每天用10-15分钟独立完成一道小型规律探究题，而不是在考试前突击刷题。建议的资源包括：MYP官方教材中的"Investigation"章节练习题、IB官方发布的specimen paper（样卷）中的Criterion B题目、以及Kognity和InThinking等MYP在线平台中的探究任务。训练时特别注意"写"的质量：把自己的答案用完整的段落写出来，然后对照评分标准自评，找出漏写的环节。一个实用的自我检查口诀是："看到了吗？（识别）写出来了吗？（表达式）验证了吗？（代入）解释了吗？（论证）推广了吗？（条件讨论）"：五问全部通过，你的Criterion B就离满分不远了。

The most effective way to improve Criterion B ability is "micro-investigation" training -- independently complete one small pattern investigation question per day for 10-15 minutes, rather than cramming before the exam. Recommended resources include: the investigation chapter exercises in official MYP textbooks, Criterion B questions from IB specimen papers, and investigation tasks on online MYP platforms such as Kognity and InThinking. During practice, pay special attention to the quality of your writing -- write out your answers in complete paragraphs, then self-assess against the rubric to identify missing elements. A practical self-check mantra: "Did I see it? (identify) Did I write it? (expression) Did I verify it? (substitution) Did I explain it? (justification) Did I generalize it? (condition discussion)" -- if all five checks pass, your Criterion B is very close to full marks.

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