Circles: CCSS-M G-C.A 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.
4. (+) Construct a tangent line from a point outside a given circle to the circle.
How intuitive is the relationship between an angle with a vertex inside the circle and the intercepted arcs of the angle and its vertical angle?
We started our lesson with a Quick Poll.
About one-third of the students intuited the relationship.
Students interacted with the Geometry Nspired activity Secants, Tangents, and Arcs.
What happens as you move point A?
What’s the least amount of information needed to calculate the angles and arcs? If I give you the measures of two the intercepted arcs, how can you determine the measure of an angle?
I sent the poll again. Do you want to keep your answer or change it based on what you observed with the dynamic geometry software?
So why is the angle measure half the sum of its intercepted arcs?
We practiced look for and make use of structure. What do you see that isn’t pictured? What do we know so far about circles?
Students thought about what auxiliary lines might be helpful for proving this relationship.
And then we looked together at some of their ideas for proving the result. Which of these would be helpful for our proof?
What about when the vertex of the angle is outside the circle?
This time only one person got the correct answer. I initially thought that he had intuited the relationship, but after talking with him about how he got 70, I realized that wasn’t true.
We went back to the TNS document. What happens when you move point P? How can you determine the angle measure when given the two intercepted arcs?
A Class Capture gave me evidence that most students were making observations and testing their conjectures.
The bell rang as one student shared his conjecture: subtract the arcs, and divide by 2.
Jill Gough challenges us to provide Ask-Don’t-Tell learning opportunities for our students. What Ask-Don’t-Tell learning opportunities are you already providing for your students? What new Ask-Don’t-Tell learning opportunities can you provide your students this week?
And so the #AskDontTell journey continues … one lesson at a time.