Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Monthly Update sign up
Mailing List signup
Teaching Math Home   Sitemap
Session Home Page
Reasoning and ProofSession 04 OverviewTab atab btab ctab dtab eReference
Part A

Observing Student Reasoning and Proof
  Introduction | Inscribed Triangles | Problem Reflection #1 | Inscribed Triangle Continued | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal


Ms. Saleh's tenth-grade geometry class is working on an inscribed triangle problem. Students had previously worked with angles, triangles, and circles and had been introduced to the idea of a proof.

Ms. Selah's instructions:

"Work in your groups to construct an inscribed triangle in the top of a semi-circle. Here's an example:

Inscribed Triangle

You'll need to use your compass, protractor and straightedge to do so, and you may choose any random point C on the semi-circle, you don't have to make the triangle look like the example. After you have your triangle, measure angles A, B, and C for at least five different cases and record your findings in a table. I'm particularly interested in what you find out about angle C. In a few minutes I'll come around to each group to hear your observations."

One group of students created this chart.

Student's Inscribed Triangle

Here is a dialogue between Ms. Saleh and members of one group:

Ms. Saleh: Can you describe how you worked on the problem?

Ben: We made a circle with the compass, and then drew lines.

Ms. Saleh: How did you draw the lines?

Ben: We found a place on the top of the circle and drew lines to either corner.

Sonia: See these are our points. {she points to the top of the circle] we actually tried a bunch of different points for C.

Ms. Saleh: What about that point in the middle, O?

Grant: We thought we'd need that because last time we did circles we used it. But we didn't use it to draw with.

Ms. Saleh: Okay, we may want to come back to it later, so it wasn't a mistake to put it in. Now tell me about your chart. Can you draw any conclusions about angle C?

Grant: Well, we measured it in five positions on the circle. It is always close to 90 degrees, although it's never exact.

Ben: No, it was exact once.

Grant: Oh, I see. Yes, but it was just once.

Ms. Saleh: Do you have any ideas about angle C?

Sonia: Yes, it's always the top of the triangle and C always looks like a right triangle. So I think C is a right triangle, no matter what.

Ms. Saleh: Yes, it certainly looks that way. But does the table support the idea that angle C will be a right angle in every case? We have some data, 94 degrees for instance, that isn't a right angle.

Grant: That's just cuz we measured with a protractor and maybe it was off a little.

Ms. Saleh: If we had a more accurate protractor and many more triangles how could we show that angle C is always a right triangle? How many measurements would it take?

Ben: You'd have to do every single triangle ever. And that's not possible, right?

Ms. Saleh: Sonia what do you think? Could we check every possible angle C?

Sonia: No, I couldn't at least. Maybe I could make an argument about why it needs to be 90 degrees.

Ms. Saleh: That's great. Please work on an argument you think will convince one another that angle C is always a right angle. Then I'll come back and you can try to convince me!

Next  Reflect on the Inscribed Triangle problem

    Teaching Math Home | Grades 9-12 | Reasoning and Proof | Site Map | © |  

© Annenberg Foundation 2017. All rights reserved. Legal Policy