Foundational Circles Vocabulary and Chord Properties

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Objective

Students will be able to prove all circles are similar and identify and describe relationships among angles and arcs.

Big Idea

Students will engage in a small group investigation in which they will distinguish between central and inscribed angles.

Warm-Up and Quick Write

15 minutes

This is the first day of the unit, so rather than bog students down with definitions they may or may not recall, I have them work through a quick matching activity and quick write.  I circulate the room as students share out their answers to the matching activity and quick write in their small groups. I take note of interesting sketches students may have drawn to make their point clear.  I debrief the answers to the matching activity by calling on student volunteers and select students to project their diagrams for the quick write.

During the similar circles discussion, I try to give ample wait time to give multiple students the opportunity to share their ideas since the Common Core calls students to construct viable arguments and critique the reasoning of others (MP1MP3).  I want students to distinguish similarity from congruence—the reason why some students might disagree that “all circles are similar”—and encourage them to bring forth evidence that will make their point clear.  Often times, this includes students showing transformations of circles on a graph; but students also draw on prior knowledge to relate circles to similar triangles (AA~) and the notion that all equilateral triangles are similar, as are all squares (and all regular polygons in general).       

Group Task: Define Angles in a Circle

20 minutes

Students need to have a strong grasp of circles vocabulary in order to talk precisely about the properties of circles they will discover (MP6).  I have found that when students examine examples and non-examples of a term (in this case, central and inscribed angles) and engage in the process of writing a “good” definition for it—one for which no one in the class can find a counterexample—they deepen their understanding. The Define Angles in a Circle worksheet includes a series of figures and prompts designed to help us to accomplish this goal.

Teacher's Note: Please see my earlier lesson, Who’s a Widget?, to get a better idea of how to facilitate this whole-class discussion.

Connect Two: Circles Vocabulary

10 minutes

In Connect Two, students can use their work from the warm-up, quick write, and defining angles in a circle activity, to explain the connections they see between the terms. I include this activity at this point in the lesson to make sure my students have individual processing time. This time is necessary for them to make sense of the circles vocabulary discussed today and connect it to their understanding of different part of a circle.  

Teacher's Note: I am grateful to my Science Department colleague, Ben Lowell, for introducing me to the idea for Connect Two.

Group Investigation: "Achording to Us…"

20 minutes

In Achording to Us…, students work in groups to explore chord properties of circles through compass and straightedge constructions and by using tracing paper. Students are expected to explore several cases, to compare their findings with their group, and to write group conjectures that they can justify. 

In this investigation students will see that congruent chords in the same circle intercept congruent arcs and that they are equidistant to the center of the circle.  Students will also discover that the perpendicular bisector of the chord passes through the center of the circle; they will be able to justify this idea by drawing on prior knowledge—earlier this year they constructed and proved that any point equidistant from the endpoints of a segment (the center of the circle in this case) must be on the perpendicular bisector of the chord.