Two Rules Fit: An Ambiguous Four-Term Sequence
A tutor in a summer enrichment program is reviewing a four-term serial item she wrote for students practicing sequences. Each term is a rectangle containing small solid dots: term 1 has 2, term 2 has 3, term 3 has 5, term 4 has 8, and the student must pick term 5 from five drawn options. The printed key says 12, from the rule "add 1, then 2, then 3, then 4." A student named Dev chose 13 and explained that each term is the sum of the two before it: 2 + 3 = 5, 3 + 5 = 8, so 5 + 8 = 13. Both justifications fit all four visible terms. The options are 12, 13, 11, 16, and 10. Thirty copies are already printed and the session starts in ninety minutes. The tutor wants to know whether Dev should be marked wrong, what to say to the group, and how to fix the item next time.
1. Should Dev be marked wrong? Decide, and state the principle you used.
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No. A sequence item is fair only if its visible terms support exactly one continuation. Here two rules fit all four terms and neither is contradicted.
Rule A: differences of 1, 2 and 3, so the next difference is 4 and the answer is 12. Three confirmations. Rule B: each term is the sum of the two before it, since 2 + 3 = 5 and 3 + 5 = 8, so the answer is 13. Two confirmations.
Rule A has more support, but support is not the test; uniqueness is. A child of this age commonly meets both patterns, and Dev's reasoning is arithmetically correct and self-consistent. Because both 12 and 13 are printed options, the item invites exactly this conflict.
Marking Dev wrong would teach him that careful pattern-finding is punished. Accept both for this administration, record the defect, and keep the item as a teaching example rather than a scored one.
2. You have ninety minutes and thirty printed copies. Give the fastest repair that keeps the item usable, and one that fixes it permanently.
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Fast repair: extend the strip on the board before the session. Write a fifth visible term of 13 and ask for term 6. Rule A would then need a difference of 5, giving 18, while Rule B gives 8 + 13 = 21. Only one of those is in the option list, so the item becomes determinate in one written line. If time does not allow that, announce that 12 and 13 are both accepted and run the item as a discussion piece. Do not reprint today.
Permanent fix: before printing, generate the next three terms under every simple rule you can imagine for the visible terms, and keep the item only if the intended rule is the sole survivor across all of them. A concrete check for this item would have shown that the sum-of-previous-two rule reproduces terms 3 and 4 exactly, which is a red flag for any four-term sequence.
3. What is the thirty-second message to the group, and what general habit does it install?
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Something close to: "Dev found a second rule that fits. Check it yourself: 2 plus 3 is 5, and 3 plus 5 is 8. When two rules fit the same four terms, the question is underspecified, so I am accepting both answers. That is on me, not on anyone who chose 13."
Then the habit: after you find one rule, spend five seconds trying to break it by looking for a rival rule. State the rival out loud; if it also fits every term, you are looking at a flawed item rather than a wrong answer.
One honest caveat to include: on a real test, items are screened so that a single rule survives, so if two rules seem to work, recheck the arithmetic first, then check whether the second rule truly fits all terms rather than only the last two.