Reflection: Intervention and Extension Inscribed and Circumscribed Circles - Section 4: Formal Constructions with Explanations

I had an 'Aha!' moment in this lesson when I realized that I could address the needs of advancing students and struggling students by helping them to see connections.

I had some students in this lesson who were stuck and didn't know which constructions they should perform in order to locate the circumcenter, for example. With these students, I engaged them in a round of dialogue that went something like this:

Ok, so you're trying to construct this circumscribed circle. What points do you definitely want it to pass through?

And they'd say something like "It has to go through all of the vertices A, B and C."

Then I'd ask them to draw a circle and then inscribe triangle ABC in it. I'd then ask them what has to be true about the relationship between the center of the circle and the points A, B, and C. (Follow-up: What's the definition of a circle?)

After they realize that the center has to be the same distance from all three points, I ask the question how could we use what we did in the Activating Prior Knowledge part of the lesson to find a point that is equidistant from all three vertices of the triangle?

Then I would walk away and let the student have some more independent thinking time to move forward on their own.

Moving around the classroom, I would also encounter students who finished both constructions and were sitting there looking like "What's next?" For these students my focus was to make sure they understood why these were the appropriate constructions. I'd start by having them explain why constructing the perpendicular bisectors helped us to locate the circumcenter. I did not accept "Because you can see it right here" as a valid explanation. When pressed, most of these students could explain why the perpendicular bisector construction was the appropriate construction for the circumcenter. Still, though, I had to coach these students in order to get them to explain more precisely. For example, I wanted them to reference the perpendicular bisector theorem. And I wanted them to talk about all points on the perpendicular bisector of segment AB being equidistant from A and B and the points on the perpendicular bisector of segment BC being equidistant from B and C. And finally I wanted them to explain that the point at the intersection of the perpendicular bisectors was equidistant from all three vertices. But all of this was mostly coaching on precision.

Where students found more of a challenge was when I asked them to explain why the angle bisector construction was the appropriate one for the incenter. For this, the answer really tended to be "Because it worked...look." The real explanation would require students to go back and consider what we'd learned in the lesson on constructing tangents to a circle from a point outside the circle. So, by asking these students to go a little deeper in explaining why constructions work, I've found it's possible to engage these students at a level that will give them opportunity to grow.

Helping students make connections
Intervention and Extension: Helping students make connections

Inscribed and Circumscribed Circles

Unit 8: Circles
Lesson 5 of 5

Big Idea: Stay calm and find your center. In this lesson, students learn to construct the incenter and circumcenter.

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Standards:
Subject(s):
Math, Geometry, Incenter, Circumcenter, Inscribed, Circumscribed, circles
95 minutes

Anthony Carruthers

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