Lesson 7.1
Ratios That Refuse to Change
Focus: families of similar right triangles — invariant internal ratios — sine, cosine, tangent named — class trig table built
This lesson works best on paper with a ruler and pencil rather than at boards with thick markers. The key result — that internal side ratios are constant for a fixed angle — is obscured when measurement error is large. Thin pencil lines and careful measurement are worth emphasizing before the task begins.
Pacing: 45 minutes for the discovery task and gallery walk, 45 minutes for consolidation. The original 30-minute consolidation felt too quick in the first year. Students need time to internalize the naming of the ratios before building the class table. Do not rush the naming moment.
Three approaches emerged from groups in Year 1 and all are worth preserving. The most common: extend a leg and reconstruct similar triangles using parallel lines or angle replication, then measure all sides. This works but introduces measurement error at every step. A stronger approach, used by one group: draw one triangle, measure its sides, then multiply all sides by a constant scale factor to produce new triangles. This guarantees a constant ratio. A third approach: use the Pythagorean theorem to calculate the hypotenuse from measured legs, rather than measuring it. This also produced constant ratios because measurement error was eliminated from one side. Both of these approaches are worth seeding if groups are struggling with precision.
The consolidation message is the most important moment of the lesson. It should be spoken plainly: ‘The ratio is constant depending on the angle. We just give it a name. Sine is the name of the ratio of the opposite side to the hypotenuse. Nothing more complicated than that — just the name of the ratio you found today.’ SOH-CAH-TOA is a memory aid only, introduced after students have already discovered the ratios.
The class trig table built at the end of this lesson is not a prop. It will be used actively in Lessons 7.2 and 7.3. Post it prominently. Students should average their values across their triangle family to improve precision. The 45° row is needed for Lesson 7.4 — if no group was assigned 45°, add it during consolidation.