Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Observing Student Reasoning and Proof
|Introduction | Inscribed Triangles | Problem Reflection #1 | Inscribed Triangle Continued | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal|
After all the groups had made their triangles and tables, Ms. Saleh drew another circle on the board, this time with point O labeled, and gave the following challenge to the class.
"Okay, there has been a lot of good investigative work about angle C going on. If I were a detective presented with the evidenced you've just shown me I would say that angle C has a likelihood of being 90 degrees in every case. So here is my conjecture, the angle inscribed in a semicircle, that is our angle C, must be a right angle. Our job is to prove it.
As a help, I noticed that some groups labeled the origin of the circle O, so I have done that up here. It may make working on this easier. Another thing that will be useful is recalling characteristics of triangles. Now work in your groups on developing a convincing argument about angel C."
Ms. Saleh circulated among the groups and here are some examples of the dialogue she heard and participated in.
Eliza: I think we should measure a bunch more and see if there are patterns or variations. We could also draw different size circles and see if that shows anything.
Katoisha: Everybody's measured a lot, though, and the answers all are close to 90 degrees. That's the pattern already. I don't know about the different size circles though.
Alex: I just tried a tiny size circle and the angle measures aren't different it's just harder to measure with the protractor. Besides, we know it looks like 90 degrees no matter how many times we measure. I don't think we should measure any more.
Ms. Saleh: Hi, I see you've looked at a few more cases, are there some things about triangles you know that might help with this.
Eliza: I'm not sure; they all have 180 degrees.
Ms. Saleh: What do you mean?
Eliza: If you add up the angles in a triangle you get 180.
Alix: That's right, so if C is always 90 degrees then angle A + angle B has to be 90 degrees too.
Katoisha: That's gotta help prove it.
Ms. Saleh moved on to another group, as Katoisha began to draw.
Booker: Okay, if we draw out from O to the edge of a circle we get a radius, and it's equal no matter what.
Irene: So, segments OA, OC, and OB are all the same length.
Booker: These two triangles I just made look the same.
Irene: That's just because our C is at the top. What if you drew it way over here close to A. (Irene makes a drawing.)
Booker: Look, though OA, OC, and OB are still the same even in this one. They'll always be the same. It's an isosceles triangle.
Ms. Saleh: What have you found?
Irene: I think we are getting close. If you divide the big triangle into two triangles you get isosceles ones.
Ms. Saleh: How do you know it's an isosceles triangle? Do you have evidence.
Booker: Yes, since the two segments are both on a radius, but since they are on the same circle, they have to be the same length. That's why it's isosceles.
Ms. Saleh: That's good reasoning. Is there a characteristic of isosceles triangles that can help you prove that angle C is a right angle?
Booker: No. It's just that they have two sides that are the same. That's not going to help.
Ms. Saleh: Is that the only thing about isosceles triangles?
Booker: Well, the angles at the bottom are the same.
Irene: Hey that could help a lot. Let's measure them and see.
Ms. Saleh: If you know they are isosceles triangles, do you need to measure them to know they are the same? Could you just write that somehow?
Booker: Let's use x for one of them and y for the other. Then we've got our whole triangle covered.
Irene: Just adding up all the angles? That would be 2 x's and 2 y's.
Ms. Saleh: That sounds like a good plan. Work that out a little more and then all the groups will report.
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