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* *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

Lesson 5 of 5

## Objective: SWBAT construct the inscribed and circumscribed circles of a triangle.

#### Concept Development

*10 min*

Using the Inscribed and Circumscribed Concept Development slideshow, I give students examples and non-examples of inscribed and circumscribed figures. During the presentation I also develop the meanings of the prefixes in- and circum- and the root scribe.

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#### Freestyle Construction

*20 min*

For this section, I hand students a copy of Inscribed and Circumscribed Circles_Freestyle. They will be trying to inscribe and circumscribe circles using only a compass, a pencil, and their good vision. My hope is that they will begin to realize that they are being prevented from successfully completing the construction because they don't know where to locate the centers of the circles.

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At this point, I'll remind students that they have just attempted to draw the inscribed and circumscribed circles of triangles. I will also remind them that the key to succeeding in that task would have been to precisely locate the centers of the circles.

I then tell them that there is some knowledge we gained earlier in the course that might be helpful in that regard. Check out the screencast for a summary of what I share with students.

After seeing the demonstration, students know enough to complete the proof. I won't provide any more hints. It's time for students to own the constructions.

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Students have just been trying to locate the centers of the inscribed and circumscribed circles of a triangle using only their compass and good vision. They have hopefully realized that the key things they're missing are the centers of the inscribed and circumscribed circles.

At this time, I give them language to assign to the centers they were seeking. I explain that the incenter is the center of the inscribed circle and that the circumcenter is the center of the circumscribed circle.

Next I give each student a copy of Inscribed and Circumscribed Circles_Constructed. Students will be working independently for 15 - 20 minutes on these. When the time has elapsed, I'll randomly select non-volunteers to come to the front of the class to present.

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#### Extension

*20 min*

Using Exploring the Effect of Triangle Classification on Inscribed and Circumscribed Circles, students will perform classic constructions to explore the effect of triangle classification on the location of the circumcenter and incenter and make conjectures.

Students will also use Geometer's Sketchpad to verify or disprove their conjectures.

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###### Circles Unit Assessment

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras