Lesson 8.4
Central and Inscribed Angles
From a broken shard to a theorem and back again
This is one of my favorite lessons in the unit, and also the one most worth being clear-eyed about structurally: it runs like a hybrid of a Dan Meyer three-act task and a BTC lesson. It opens with a genuine hook, an archaeologist has found a fragment of a circular pottery vessel and wants to know how large the original was, that students cannot fully crack in the moment. That stall is the point. It creates the need to solve a simpler version of the problem first, one of the most honest moves in mathematics and science. (Use the printable large-arc shard image for the launch.)
It is worth saying plainly that a three-act hook is not automatically safe in a thinking classroom. A problem that no group can enter just drops everyone out of flow. What makes this one work is how quickly you pivot. I let groups push on the real shard only long enough to surface the obstacle, then redirect to the workable question underneath it: given an arc, can we guarantee that we can find a true diameter? Watch the timing here, productive frustration curdles if you let it sit too long.
The early attempts were the best part. A few groups fit a compass to the arc to estimate the center; I told them it was clever and that it would be close enough for an actual archaeologist but this course is about what we can know with certainty, and that gap is the wedge. Others slid a ruler around hunting for a diameter. The moment I called the whole class over to see was a group who wrote, “assuming we found the true diameter by going through the center, then…” they had named their own assumption without being asked to. We voted as a class on whether we believed the reasoning, and then I asked what it would take to remove that assumption and tie it back to the axioms we agreed to accept in the fall. That set our path for the rest of the lesson to make sure we can find a true diameter.
From there students drew a central angle and an inscribed angle on the same arc, measured, and most conjectured the inscribed angle theorem quickly; a few noticed instead that the three convex angles of the quadrilateral they had formed summed to the central angle. The proof was harder than the conjecture. I offered the auxiliary line from the point through the center to create a diameter. Groups handled the isosceles triangles and their equal base angles well, but many assumed the two isosceles triangles were congruent, they were really reading their own particular drawing, and a reminder that the point can sit anywhere on the major arc undid that. No group remembered the exterior-angle result; handed it on the side, they could justify it, but they struggled to see it living inside the construction. We called the class, I proved it for them to take notes on, and asked them to reconstruct it cold the next day.
On day 2 we brought back the challenge, could the inscribed angle help us find the diameter we had been after. A single student offered extend a right angle from a point on the arc and see where its sides meet the rim. I demonstrated with the corner of a sheet of paper, and plenty of students did not follow me. Having that student come to the board and explain it to the class worked far better than my version did. We spent the rest of the period on practice problems including the inscribed-quadrilateral corollary.