A Heuristic Approach to Angles in Circles

09 Feb

I am taking a qualitative research class right now, and my mind is full of lots of new-to-me words (many of which my spell checker doesn’t know, either): hermeneutics, phenomenology, ethnography, ethnomethodology, interpretivism, postpositivism, etc. One that has struck me is heuristic, the definition of which I can actually remember because I try to teach heuristically. (The word does not yet roll off of my tongue, but the definition, I get.)

On Monday, our content was G-C.A Understand and apply theorems about circles

1. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

We started with a Quick Poll. I asked students for their best guess for the angle measure. I showed the results without displaying the correct answer, noting the lowest and highest guesses.

Students moved to the technology. What happens to the angle measures as you move the points on the circle?

They moved to the next page, which revealed more information. What happens to the angle measures as you move the points on the circle?

I sent the poll again. There was one team who hadn’t answered yet, so I made a brief stop by their table. Last semester, I remember reading something about how a certain example might give students the eyes to see what you’re trying to get them to see. So we moved the points around to look something like this.

If you have 49 and 43, how can you get 46?

Changing the numbers purposefully helped them see.

I sent one more poll before we talked about why.

So we gave our best guess, and then we used technology to explore. Students practiced MP8 I can look for and express regularity in repeated reasoning as they noticed what stayed the same and what changed with an angle whose vertex is in the center of the circle. They generalized the result. But we hadn’t yet discussed why that happens.

Students practice MP7 I can look for and make use of structure. By now they know our mantra for MP7: What can you make visible that isn’t yet pictured?

I saw a line constructed parallel to the given line, which made alternate interior angles visible.

I saw a chord drawn that made a triangle visible.

I asked students to write down everything they knew about the angles in this diagram.

They made suggestions about what we know. They didn’t say the relationships exactly like I would. I wrote them down anyway. They didn’t recognize the exterior angle of the triangle and so ending up proving the Exterior Angle Theorem again off to the side. I wrote it down anyway.

And so the journey continues, always trying to enable my students to discover or learn something for themselves (and sometimes succeeding) …