Tom
Chandler
HIGHLAND HIGH ,
AULT,
CO

Geometry :
Unit #3 - Congruence and Rigid Motions :
Lesson #4

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.

1 Lesson Open -
9 minutes

**Team Warm-up**

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.

**Goal-Setting**

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.

resources

PerfectPolygons_LessonOpen.pdf

https://betterlesson.com/lesson/resource/3132807/perfect-polygons-lesson-open
PerfectPolygons_AgendaandLearningTargets.pdf

https://betterlesson.com/lesson/resource/3132802/perfect-polygons-agenda-and-learning-targets
CongruenceandRigidMotions_PerfectPolygons.pptx

https://betterlesson.com/lesson/resource/3132801/perfect-polygons-lesson-slide-show
2 Constructing Perfect Polygons -
30 minutes

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.

resources

PerfectPolygons_ConstructingPerfectPolygons_Instructions.pdf

https://betterlesson.com/lesson/resource/3132804/perfect-polygons-constructing-perfect-polygons-instructions
PerfectPolygons_ConstructingPolygons_SelectedSolutions.pdf

https://betterlesson.com/lesson/resource/3132805/perfect-polygons-constructing-polygons-selected-solutions
PerfectPolygons_ConstructingPolygons.docx

https://betterlesson.com/lesson/resource/3132803/perfect-polygons-constructing-polygons-activity
Why Teach the Construction of Polygons Inscribed in a Circle?

Student Ownership

PerfectPolygons_WhyTeachtheConstructionsofInscribedPolygons.MP4

https://betterlesson.com/lesson/section/79694/perfect-polygons
Why Learn to Construct a Polygon Inscribed in a Circle?

Intrinsic Motivation

PerfectPolygons_WhyLearntoConstructInscribedPolygons.MP4

https://betterlesson.com/lesson/section/79694/perfect-polygons
Demonstrating Properties of Congruence With a Compass

Student Ownership

PerfectPolygons_DemonstratingPropertiesofCongruenceWithaCompass.MP4

https://betterlesson.com/lesson/section/79694/perfect-polygons
3 Summarizing Regular Polygons -
10 minutes

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.

resources

CongruenceandRigidMotions_Notes_SymmetryandRegularPolygons_Key.pdf

https://betterlesson.com/lesson/resource/3132800/guided-notes-symmetry-and-regular-polygons-key
4 Lesson Close and Homework -
5 minutes

**Team Size-Up**

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**

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.

resources

PerfectPolygons_LessonClose.pdf

https://betterlesson.com/lesson/resource/3132806/perfect-polygons-lessonclose