Lesson 2.1
Inductive Reasoning
Focus: patterns, conjectures, the limits of inductive reasoning, the circle task gut-punch
This lesson does two things: it gives students the vocabulary for something they have been doing since Unit Zero, then shows them why it is not enough. Let them experience the three tasks without telling them that they will rank their results by trustworthiness. The ordering becomes the lesson during consolidation.
The handshake problem is the most important task in the lesson, and not for the reason students expect. Most groups will find the formula n(n−1)/2 and be satisfied. The important move comes after: ask whether they can explain why the formula must continue to work, not just that it has worked so far. Groups that connect it to the structure of the problem — each of n people shaking n−1 hands, each handshake counted twice — have crossed from pattern recognition into something more powerful. They have a reason. Hold that distinction. It is the pivot of the consolidation.
The circle task is one of the most important moments in the unit. A pattern holds for one case, two, three, four, five. Students are confident. They have checked five cases. Then they check six, and the pattern fails. The result is not 32. It is 31. That is not a counting error — every careful recount will still produce 31. The pattern was wrong.
Do not soften this moment. The failure is the lesson. Students who feel the confidence collapse — who genuinely believed the pattern and then watched it break — have a visceral understanding of inductive reasoning’s fundamental weakness that no amount of explanation could produce. The rest of the unit follows from that feeling.
Consolidation: Gallery Walk and the Ordering
After the gallery walk, ask the whole class: all three tasks used the same method — we looked at examples and found a pattern. But not all three results are equally trustworthy. Which do you trust most? Which least?
Let students argue. Then guide toward the ordering: most trustworthy is the handshake problem, because some groups can explain why the formula must continue — not just that it has. Middle is the visual pattern, where the rule seems clear but nothing rules out a change at the 7th term. Least trustworthy is the circle task, where five clean confirmations were not enough. Name this explicitly: this is inductive reasoning. One counterexample disproves a conjecture, no matter how many cases confirmed it.
Discovering Geometry: An Investigative Approach — Michael Serra (Key Curriculum Press). The visual pattern image and handshake diagrams used in Tasks 1 and 2 are sourced from this textbook. Serra, Michael. Discovering Geometry: An Investigative Approach. Key Curriculum Press. Cite this source on any printed task materials.