Review: Arcs, Angles, Chords, Tangents, and Proof
Lesson 6 of 8
Objective: Students will be able to solve problems and write proofs using a variety of circles properties.
Arcs and Angles Menu
In this menu activity, I like to give students choice in terms of how they work (individually, in pairs, in groups) and the problems they would like to work on. At the same time, it is essential to make the expectations for how students work, how they show their work, and the amount of work they must be do, clear to students. I make sure to write my Expectations for Student Work on the whiteboard at the start of the lesson:
- Write the problem number, include the diagram, and show high quality work (MP1)
- Justify your reasoning by naming the relationships/concepts used and why they are important
- Do a minimum of 8 problems
I also make sure to list problem solving strategies that students can reference and use throughout the activity:
- “Add a line”: drawing in the radius, diameter, and connecting points is often a good strategy for seeing relationships (isosceles triangles, special quadrilaterals, etc.)—this is a way that students can look for structure (MP6)
- Break down a given diagram into inscribed angles
- Look for relationships between inscribed angles, central angles, and arc measures
In debriefing this activity, I project the Arcs and Angles Menu Answer Key and give students time to make corrections, explain errors, ask questions, and present their work to their peers. This is an ideal time to call on student volunteers to share out the concepts they used to justify their reasoning (MP3). In my experience, students who are able to explain their reasons today, will have confidence using the concepts and skills moving forward.
On this review sheet, students will solve several problems that use inscribed and central angles, arcs, and tangents. The sheet is intended for students to work independently, and, to complete outside of class as preparation for the assessment.
On the worksheet, students will work on proofs, trying to get better at identifying what their goal (and corresponding sub-goals) are when trying to construct the proof (MP2). For example, if students are trying to prove that a triangle is isosceles, under GOAL they should write “prove that the base angles are congruent” or “prove these two sides are congruent” and under SUB GOALS, they should write “find congruent triangles and use CPCTC” or “show the arcs are congruent, which would make either the chords congruent or the inscribed angles that intercept those arcs congruent”. The work is challenging.