Page 185 - IM_FL_Geometry_Print Sample

2. Make a conjecture about pairs of opposite angles (for example, angles and ). Consider testing whether they may be congruent, supplementary, or complementary, whether the measure of one is a multiple of the other, or some other relationship. Does this conjecture seem to be true in your partner’s quadrilateral as well? Be prepared to share your thinking.
Student Response
1. Response should be any quadrilateral with vertices on the circle.
2. Opposite angles appear to be supplementary. This seems to be true with every quadrilateral formed from points on a circle.
Activity Synthesis
Ask groups to share their ideas and push them to explain why they think those ideas are true for di erent cyclic quadrilaterals. This might involve using tracing paper or some other means to experimentally con rm that opposite angles are supplementary.
11.3 Prove It
20 minutes
In this activity, students prove that the opposite angles of a cyclic quadrilateral are supplementary. Look for students who use these strategies:
• Decompose the cyclic quadrilateral into isosceles triangles and use arguments from the Triangle Angle Sum Theorem to show that opposite angle measure sum to radians.
• Argue from the Inscribed Angle Theorem that opposite angles trace out the entire circumference of the circle, and thus must measure half of radians. For more detail about these strategies,
Instructional Routines
• Anticipate, monitor, select, sequence, connect
What: Fans of 5 Practices for Orchestrating Productive Mathematical Discussions (Smith and Stein, 2011) will recognize these as the 5 Practices. In this curriculum, much of the work of anticipating, sequencing, and connecting is handled by the materials in the activity narrative, launch, and synthesis sections. But teachers will need to take this ball and run with it by developing the capacity to prepare for and conduct whole-class discussions. The book itself would make excellent fodder for a teacher PLC or study group.
Why: In a problem-based curriculum, many activities can be described as “do math and talk about it,” but the 5 Practices lend more structure to these activities so that they more reliably result in students making connections and learning new mathematics.
112
Teacher Guide