**Circles: G-C**

**Understand and apply theorems about circles**

2. 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 used part of the Geometry Nspired activity Secants, Tangents, and Arcs for this lesson.

I am not convinced as to whether we should still discuss and make sense of angles formed by two chords, two secants, a secant/tangent, and two tangents. Part of me thinks maybe this is one of those topics in our infamous mile-wide-inch-deep curriculum. We had a lesson on it anyway. For two reasons. 1. CCSS-M is the floor of our curriculum, not the ceiling. 2. I asked my junior and senior students whether I include it, and they said yes.

In the spirit of deliberately having my students estimate before we explore, I sent a poll.

23% of the students got it correct.

So maybe this is another reason to include this topic: the calculations are apparently not intuitive.

We moved to our dynamic geometry software to explore. Was anyone correct in their thinking?

After students explored for a few minutes on their own, I sent another poll.

One minute in, 68% of my students have the correct answer.

But neither show the students nor stop the poll. I wait. Technology helps me to “ease the hurry syndrome”. (Or at least makes me).

One minute and 40 seconds in, 90% of my students have the correct answer.

We formalize our conjecture. And then I ask the question a different way.

What happens when an angle is formed by two secants? Before our exploration, I sent a poll.

Again, not very intuitive as to how to calculate the angle measure. 3 students have it correct, 10%.

This time I asked the students who got it correct to tell us what they did.

One said that he realized the angle would be smaller than that formed by two chords with the same intercepted arcs. Instead of adding the two arcs, he subtracted and then divided by 2. Did anyone else work it a different way? Another student excitedly said that he had the same reasoning as the first student.

We moved to the technology to verify the conjecture and formalize the results.

And a final poll, to check for understanding. 81% correct.

Of course, by this point, class was ending. I looked back through my poll results after class to see whether the same students were missing polls all day … to determine who is going to need extra support for this lesson. We had spent the class period exploring only two types of problems. 15 years ago, I would have “gotten through” all of my examples. But I know for sure, because of my assessment results, that not as many students owned the material as I had thought while they were politely nodding their heads “in understanding”. And I know for sure, because of my juniors and seniors who assured me that I should continue to teach this lesson, that learning by exploring helps them remember.

And so the journey continues …