**Circles: CCSS-M G-C.A Understand and apply theorems about circles**

- 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.
- (+) Construct a tangent line from a point outside a given circle to the circle.

We started our unit on circles looking at a diagram with a right triangle both inscribed in a circle and circumscribed about a circle. What do you notice? What do you wonder?

By the end of the unit, we will be able to generalize the relationship between the sides of the right triangle and the radii of the inscribed and circumscribed circles.

My students don’t come to me knowing all of the vocabulary associated with circles, but the longer we teach with our new standards, the more I am convinced students **can** learn vocabulary through the modeling of using it properly and by practicing using it properly. Geometry vocabulary doesn’t have to be reduced to copying definitions from the glossary of the textbook onto a notecard (an apology those former students who had me before I figured this out).

For example, we started with a brief look at the Geometry Nspired activity Circles – Angles and Arcs.

Before generalize the relationship between a central angle and its intercepted arc, I sent a Quick Poll. The wording of the Quick Poll added “major arc” to students’ vocabulary.

For an inscribed angle, I started with a poll just to see how intuitive the relationship is between the angle measure and intercepted arc before any kind of learning episode to explore the relationship.

About one-third of the students intuited the relationship.

I didn’t show the results. Instead, we looked at another page in the TNS document. What do you notice as you move point A or C?

I sent the poll again, and we used their results to generalize the relationship between the measure of an inscribed angle and its intercepted arc.

And then we thought about why.

We checked again to be sure that everyone was getting what they needed to about central angles, inscribed angles, and intercepted arcs.

And then we looked at cyclic quadrilaterals. Without me telling them anything, 12 answered correctly before the bell rang.

And so the next lesson began with the results from this question. Which answer is correct? And why?

And so the #AskDontTell journey continues … one lesson at a time.

howardat58

February 23, 2015 at 7:32 pm

Have you tried the cardboard angle and two thumbtacks investigation? The result is always surprising, even though it shouldn’t be.

jwilson828

February 24, 2015 at 10:25 am

Is this like the string-art-inscribed-angle? That all angles inscribed in the same arc will be congruent?

howardat58

February 24, 2015 at 10:57 am

That is the conclusion, but arrived at from the other direction. Push the cardboard angle up between the pins and slide it round, touching the pins all the time. The fun happens when you follow the point of the angle with a pencil, and get – “But it looks like part of a circle “. And if the angle is a right angle the two pins are a diameter.

Also, if you cut the supplementary angle and do it from the other side you get the rest of the circle.

The other one, well known to younger kids, is the loop of string around the two pins for drawing an oval (looks like an ellipse- is it ?). A bit of algebra sorts that one out. Your kids can do that, but don’t tell them how to !

I did this one on my geometry program.

Hyperbolas are not so cooperative.

howardat58

February 24, 2015 at 11:08 am

Second paragraph is about conics – you can save this for a rainy day !