MYP数学 Criterion B 规律 一般式 验证 证明
MYP 数学的平时分由四项标准组成。很多学生把力气放在算对答案上，忽略了 Criterion B 要的是从几个特例里找出一般规则。这篇把 B 的三条 strand 拆开，用一张差分表走完整条路。先列数据，再求含 n 的一般式，然后拿表外一项验证，最后说明这条规则为什么对所有 n 成立。每一步都有可以对照的写法，也有学生最常踩的坑。
Criterion B in MYP Mathematics is not a test of arithmetic speed. Students are asked to choose a problem-solving technique, find the regularity underneath a set of results, express it as a general rule and then defend that rule. The marks follow the reasoning, not the speed at which the numbers appear. What follows works through one investigation in full and then looks at the descriptors that decide the band.
四项标准的分值：Criterion B 占 8 分
IBO 的 MYP Mathematics guide 把数学评估分成四项标准。A 是知识与理解，B 是规律探究，C 是数学表达，D 是真实情境应用。每项满分 8 分，四项等权。学年总评不是把四项相加得 32 分就结束，学校会按公布的换算表折成 1 到 7 的等级。B 的评分分 0、1-2、3-4、5-6、7-8 五档。每一档的描述都逐条对应三条 strand。很多学生以为 B 只是附加题，其实它和 A 一样重。丢掉一档就是两分，期末等级常常差在这里。
Criterion B carries the same weight as knowledge, communication and real-life application. Each criterion is marked out of 8, and schools convert the four totals into a 1 to 7 grade using their own published boundaries. The descriptors are written in bands, so a piece of work sits at 3-4 or 5-6 depending on which strands it actually shows. B is therefore not an optional extra, and a single missing strand moves the grade in the same way as a lost mark in Criterion A.
Year 1 到 Year 5 的三条 strand 写法
IBO 对 B 的表述按年级分三套。Year 1 要求 apply 技巧去 recognize patterns。第二步是描述与正确发现相符的关系或一般规则。第三步只要求 verify 规则在别的例子上是否成立。Year 3 把 apply 换成 select and apply，把简单规律换成 complex patterns，第三步升级成 verify and justify。Year 5 的第三条写成 prove, or verify and justify, general rules。三个年级的动词不一样，评分表也跟着变，拿旧年级的标准去看新年级的作业很容易高估或低估。
The three strands shift with the year level instead of staying fixed. In year 1 students apply techniques to recognise patterns, describe them and check whether the rule works elsewhere. By year 3 the wording becomes select and apply, the patterns are expected to be complex, and the last strand asks for verification and justification. Year 5 adds proof, or verification plus justification, which is why a correct formula on its own rarely reaches the top band. Reading the descriptor for the right year matters more than reading the task sheet twice.
什么叫 pattern：从特例推到一般
guide 在注释里把 pattern 解释成数学系统里元素之间的内在顺序与可预测性。重复出现的特征可以被写成关系或一般规则。做 B 的任务时，学生先算前几项，观察每步的变化，再猜一个含 n 的式子。这个 n 通常表示第几个图形或第几项，也可以是边长、层数。写清楚 n 代表什么，是整份报告里最容易被跳过的一步。把 n 与结果写成一对一对的数对，规则才站得住。常见的错误是只写每次加 3，这只是描述变化，不是一般规则。老师无法用它直接算出第 20 项，strand ii 就卡住了。
A pattern is the order or predictability that sits underneath a set of results, and the guide treats repeated features as something that can be written as a relationship or a general rule. Students usually start by calculating the first few cases, watching how the numbers move, then proposing an expression in n. The letter n normally stands for the position of the figure or term, though it can also be a side length or a number of layers. Saying only that the total rises by three each time describes a change but gives no rule, because it cannot produce the twentieth term directly.
建表与一阶差分：先看每一列怎么变
把序号写在左列，结果写在右列。两列对齐之后逐项相减，得到一阶差分。差分全部相等，说明结果随序号线性增长。差分自己在变，就再对差分做一次减法，得到二阶差分。这张表是整个探究的证据，写报告时要把原始数据与两行差分一起放进去。只写最后那条公式，阅卷老师看不到推理过程，很难给出高档。表格不必画得漂亮，但 n 与结果的对应关系要一眼能读出来。差值写成一行，正负号一并保留，负数说明结果在下降。
Lay the position numbers in one column and the results in the next, then subtract neighbouring entries to build a first-difference row. If those differences are constant the growth is linear; if they change, subtract again and look at the second differences. This table is the evidence for the whole investigation, and examiners expect the raw data beside both difference rows rather than a lone formula at the end. Neatness matters less than alignment, since the reader has to match each n with its result without guessing.
一阶差分常数：写出 an + b 并解出系数
看一组数，n 取 1、2、3、4 时结果分别是 5、8、11、14。一阶差分都是 3，所以一般式应该是 3n + b 的形状。把 n = 1 与结果 5 代入，3 + b = 5，得到 b = 2。规则写成 3n + 2。由差分求系数比直接猜公式稳，因为差分把趋势与常数项分开了。回头用 n = 4 检查，12 + 2 = 14，与表里一致。很多学生以为差分等于 3 就可以直接写 3n，忘了常数项。这样第 1 项立刻对不上，后面整列都偏。
Take the results five, eight, eleven and fourteen for positions one to four. The first differences are all three, so the rule has the form 3n + b, and substituting the first pair gives b = 2. Testing position four returns fourteen, which matches the table. Dropping the constant is the usual slip: 3n alone gives three at the first position and the rest of the column no longer fits.
二阶差分常数：二次式 an^2 + bn + c 的系数
再看 3、8、15、24 这一列。一阶差分是 5、7、9，二阶差分是 2、2。二阶差分恒定，说明结果由二次式生成。二次式的首项系数 a 等于二阶差分的一半，这里 a = 1。把 n = 1 与 n = 2 分别代入 an^2 + bn + c。两式分别是 1 + b + c = 3 与 4 + 2b + c = 8。相减得 3 + b = 5，于是 b = 2。回代得 c = 0，规则是 n^2 + 2n。用 n = 3 检查，9 + 6 = 15，与数出来的结果吻合。如果二阶差分也不恒定，就再作一次差，考虑三次式；多数 MYP 任务停在二次。
For the sequence three, eight, fifteen and twenty-four the first differences are five, seven and nine, and the second differences are all two. A constant second difference means a quadratic rule, and the leading coefficient is half that value, so a takes the value one. Substituting the first two pairs produces two equations, and subtracting them gives b = 2 with c = 0. The rule n^2 + 2n returns fifteen at position three, which agrees with the original count.
verify、justify、prove 三种收尾写法
verify 是拿表外的一项去试。例如算 n = 6 的值，与公式给出的值比较是否相同。justify 要说明这条规则为什么成立，二次式可以用差分恒等式解释，也可以用图形拼接的面积来解释。prove 要把推理写成对任意 n 都成立的代数过程。常见做法是把相邻两项作差，证明差值正好等于观察到的那条规律。Year 5 的 7-8 档要求 prove，或者 verify 与 justify 两件都做到。只做一个会停在 5-6 档，这一档的差别就在最后那一步推理上。
Verification means testing a case that was not used to build the rule, such as position six. Justification goes further and explains why the rule must hold, either through an algebraic identity or through a geometric argument about how the shapes fit together. A proof has to cover every n, and subtracting consecutive expressions is the standard route because the difference should reproduce the observed step. The top band in year 5 expects proof, or verification and justification together, and one checked case will not carry the work past the middle bands.
散点图判断形状，差分表求系数
把数据点画在坐标平面上，形状会立刻显现。线性关系是一条直线，二次关系是一条弯曲的抛物线，指数关系上升得越来越快。图像能帮学生判断该试哪一种式子。系数仍然要靠差分或联立方程求出来，图像本身给不出精确的 a 与 b。有些任务允许用图形计算器做散点图与回归，这时把回归结果写进报告即可。任务说明限制使用技术工具时，就回到手算差分。两条路的结论应当一致，不一致说明某一步算错了。读图时注意坐标轴的起点与刻度，纵轴不从零开始时，直线的斜率看起来会比真实斜率大。
Plotting the points shows the shape quickly: a straight line points to a linear rule, a curve that bends once suggests a quadratic, and values that climb faster and faster hint at a geometric sequence. The graph guides the choice of expression, but the coefficients still come from differences or from solving simultaneous equations. Where a graphing calculator is permitted, a regression line can be quoted as supporting evidence, and the numbers should agree with the hand calculation. When the task restricts technology, the difference table remains the reliable route.
扣分最多的三处：规则没有 n、起点错位、没验证
阅卷时最常见的问题是规则里没有 n。只写每增加一个图形就多 3 根小棒，这样的描述停在 3-4 档。第二个问题是起点错位。图 1 有 5 根，学生却按图 0 的 2 根列式，整条公式平移了一位。第三个问题是求完公式就交卷，没有拿另一个例子验证，strand iii 空着。补一句用 n = 7 检验，得到 23，与数出来的结果相同，这一档就补回来了。这些都不是算错，而是没有把推理写出来。
The most common loss is a rule with no n in it, which leaves the work in the middle bands however careful the counting was. A second problem is an off-by-one start, where the first figure already contains the constant and the formula is shifted by one place. The third is stopping at the formula without testing a further case, which leaves the last strand empty. One extra line, substituting position seven and matching the count, is usually enough to recover that band.
任务设计的红线：太引导会压到 4 分
guide 在 Year 5 的注释里把这条写得直接。任务如果不让学生自己选择解题方法，就属于引导过度，最高只能给到 4 分。老师仍然要给出足够的方向，保证每个学生都能开始动手。红线在起点提示与操作配方之间，把固定算法拆成步骤发给学生就越线了。另一条注释同样值得注意。学生写出的规则与错误的发现相符时，只要规则的复杂度相当，最高可以到 6 分。这两条说明评分看重方法的选择与规则的复杂度，不只是答案对不对。老师把提示写成一两句开放问题，比直接给出公式更能让学生展示 strand i。
The notes in the year 5 section are blunt about guided tasks. If students are not allowed to select a problem-solving technique the task is too guided and the maximum achievement level is four, even when every answer is correct. Teachers are still expected to give enough direction for everyone to begin, so the line sits between a starting prompt and a recipe. A second note allows up to six when a rule matches incorrect findings, provided the rule is of comparable complexity, which shows how much the descriptors reward the reasoning.
报告四块：数据表、一般式、验证、证明
一页报告按四块写就够了。第一块是数据表，放原始数据与两行差分。第二块是一般式，写出含 n 的表达式，并注明 n 与每个字母的含义。第三块是验证，给出表外一项的代入过程与结果比较。第四块是证明或论证，说明规则为什么对所有 n 成立。Criterion C 单独评表达，但 B 的分也受表达影响。阅卷老师要能在文字里找到那三条 strand，写成一大段散文、公式藏在句子中间，读者往往找不到验证那一步。四块之间留出小标题，核对时按顺序走一遍就行。
Four blocks cover the report: a data table with differences, the general rule with each letter explained, a verification using a case outside the table, and a proof or justification for all n. Communication is marked separately under Criterion C, yet clarity still affects B, because the reader has to locate the three strands in the writing. A formula buried in a paragraph of prose often hides the verification step. The band then reflects what could be found rather than what was actually done, which is a harsh but consistent outcome across schools.
从 MYP 3 到 MYP 5 的练习顺序
MYP 3 的复杂规律多是二次式与简单的递推。MYP 5 会出现含负指数的序列、分式形式的规则，还有需要作差两次以上的题型。练习时先固定一种方法，把差分表做到不用想就能列，再换题型。常见的错误是跳过线性与二次直接做指数规律。差分表立刻乱掉，学生误以为这组数没有规律。把相邻项的比值算出来，比值恒定就说明是指数关系。这条判断与差分是同一套思路的两个版本，一个看加法，一个看乘法。练到后期，可以把两种方法写进同一份报告互相印证。
Complex patterns at MYP 3 are usually quadratic or based on a simple recurrence, while MYP 5 adds negative powers, fractions and sequences that need more than one round of subtraction. Practice should lock one method first, building difference tables until the layout is automatic, then switch the type of pattern. Skipping straight to exponential rules is a frequent mistake, because the difference table stops behaving and students conclude there is no pattern at all. Taking ratios between neighbouring terms settles that question, and it is really the same idea applied to multiplication rather than addition.
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