Lesson 4 of 17
Objective: SWBAT construct regular hexagons, quadrilaterals, and triangles inscribed in a circle. Students will understand how the symmetry of a regular polygon is related to the symmetry of a circle.
Using the Slide Show, I display the warm-up prompt for the lesson as the bell rings. The prompt asks: Is it ever possible for a quadrilateral to fit inside a circle so that all four vertices lie on the circle? I push students to explain their answer (MP3). Plausible reasoning is fine at this point, since we have yet to learn formal proof.
The warm-up is an advance organizer for the lesson, which concerns constructions of inscribed polygons. It is also a good check for understanding: do students remember vocabulary like "quadrilateral" and "vertex", and do they see that the circle must intersect the 4 vertices of the polygon. Since the activity follows our Team Warm-up routine, with students sharing their answers and a randomly selected scribe writing the team's answer on the front board, this gives students an opportunity to learn from one another. Students write their answers in their Learning Journals.
As I review the team answers, I look for answers that show students recognize that the question applies to any quadrilateral, not just the one pictured alongside the prompt. At least one team should recognize that a regular quadrilateral is likely to fit inside a circle in the manner described.
Displaying the Agenda and Learning Targets, I tell the class: When a circle intersects all the vertices of a polygon, the circle is said to "circumscribe the polygon". The polygon is "inscribed in" the circle. As you have guessed, regular polygons--those with congruent sides and angles--can always be inscribed in a circle. Why? It is all about symmetry.
The goal of this activity is not just to teach a construction, but for students to analyze the symmetry of a regular polygon. There are many congruences: not just between the sides and angles of the polygon, but also between the various segments and angles that "come up" in the construction (MP7).
This activity also gives students another opportunity to apply the concept that we can only be sure that objects are congruent by superimposing one onto the other (MP6). This can be done using tracing paper or a compass--or by folding the construction (MP5). The focus is on examining the congruence of the parts of the figure (MP7).
I plan to assign problems #1 and #3, saving problems #2 and #4 for teams that are working faster than the others.
The lesson uses the Rally Coach format, because I want students to support each other in carrying out the constructions. This activity is a great opportunity to see which students can read the instructions--written in the language and notation of geometry--for understanding. Some students will want to be shown how to perform the constructions. I push them to puzzle through the instructions with the help of their partner (MP1). Try something! I will answer specific questions and let the student know if they are going down the right track.
Summarizing Regular Polygons
We summarize regular polygons with the help of the Guided Notes for the lesson.
Although we are not ready to prove that every regular polygon can be inscribed in a circle, later students will see that the rotational symmetry of a regular polygon guarantees that this must be so. In fact, the center of rotation of a regular polygon must be the center of a circle that passes through all the vertices of the polygon. The notes are intended to help students make a connection between regular polygons and circumscribing circles.
More on how I use Guided Notes can be found in my Strategies folder.
Displaying the Lesson Close prompt, I ask students to summarize what they learned from the lesson with their team-mates, then select the best answer to write on the board. This activity follows our Team Size-Up routine.
Homework Set 1 problems #12 and 13 review the constructions of inscribed hexagons and quadrilaterals students learned in the lesson. For students who did not get as far, problem #14 (and later #20) introduce the constructions of inscribed triangles and octagons. Students will be able to refer to these problems on the unit quiz.