Lesson 6.3
When Is Similarity Guaranteed?
Focus: counterexample investigation — AA, SAS∼, SSS∼ discovered — the limits of inductive evidence
This lesson mirrors the structure of Lesson 3.2 (triangle congruence shortcuts) but for similarity. Students try to break proposed shortcuts by constructing counterexamples. If they can construct two non-similar triangles that satisfy the given condition, the condition fails. If they cannot, despite serious effort, the condition may be sufficient. The same mathematical norm from Unit 2 applies here and should be named explicitly: the inability to find a counterexample does not prove a claim. It is evidence, not proof.
The scenarios to present, in order: one angle congruent (fails easily — one angle is clearly insufficient), two angles congruent (AA — most groups will struggle to find a counterexample, though they may not be convinced), one side in proportion (fails easily), two sides in proportion (fails — SS alone is not enough), two sides in proportion with included angle congruent (SAS∼ — groups should fail to find counterexamples), all three sides in proportion (SSS∼ — groups should fail).
An important note on Day 3 from the first year: the lesson opened with homework review before the task, and students were noticeably more passive in the task that followed. The BTC research is explicit on this point: beginning class by reviewing answers shifts students into a passive receiving mode that is hard to reverse. If there is homework to address, consider whether it can wait until after the task or be handled differently.
Also from Day 3: students who were allowed to use grids were less engaged in the genuine geometric thinking. Next time, require students to measure angles and compute side ratios using their tools. This makes the task a better thinking exercise and reinforces the proportional reasoning from Lessons 6.1 and 6.2. The goal of the task is to build and test conjectures with appropriate tools — grids allow shortcuts that bypass that reasoning.
The shortcuts are accepted as postulates after the investigation, as the congruence shortcuts were in Unit 3. The note that these can be proven but are being accepted without proof — for the same pedagogical reasons as SSS, SAS, ASA, and AAS — should be made honestly.