In this whole-class discussion, I want to formalize the numerous discoveries students made about chords in the last lesson. We take notes in our note takers, sketching, constructing, and using tracing paper to make sense of the concepts. In this discussion, an explicit goal of mine is to make sure that students see that adding a line or line segment can often allow them to see certain relationships better—this is a specific way students can look for and make use of structure (MP7).
I have students talk in small groups about how they are trying to understand why congruent chords in the same circle will intercept congruent arcs—students try to convince themselves and others by drawing different sized chords and their corresponding intercepted arcs, using proportional reasoning, and considering real-life situations like cutting out slices of pizza with the same central angle and considering the amount of crust they will receive.
Throughout the whole-class discussion, we focus on the use of congruent triangles, CPCTC, and properties of isosceles triangles as ways to justify how we know.
In (Don't) Go Off on a Tangent, students work in groups to investigate and conjecture about two properties of tangents. By considering several cases of their own constructions, students discover that:
(1) A tangent to a circle is perpendicular to the radius drawn to the point of tangency
(2) Tangent segments to a point outside a circle are congruent
While students work in groups, I circulate the room, encouraging them to justify how they know tangent segments to a point outside a circle are congruent. I ask groups to prove their conjecture, which requires them to use ideas about the perpendicular bisector of a chord, properties of isosceles triangles, congruent triangles and CPCTC, and even kite properties. I encourage a variety of pathways for writing the proof so students can see how others made sense of the problem, make connections between the ideas being presented, and critique the reasoning of others (MP1, MP3).
Like most of my other lessons, I plan to formally debrief the big ideas of this lesson using our note takers. I select one or two groups to present their proof for their “tangent segments” conjecture so others in the class can get a better sense of how to justify their reasoning.
The tangent conjectures we discuss are:
I want to give an exit ticket to hold individual students accountable for the investigation they did in their group. For this reason, I write up a problem on the whiteboard for students to solve individually and turn in before they leave class that day.