Summary
Students will explore the relationships between exterior angles in regular and irregular polygons to determine the sum of exterior angles, and they will apply that knowledge to solve problems. To complete this lesson, students need to know the interior angle sum theorem. See the "Department of the Interior Angles" lesson for the prerequisite content.
Essential Question(s)
How do we solve problems using exterior angles?
Snapshot
Engage
Students recall properties of interior and exterior angles of a polygon to solve a problem as a group, using a modified Chain Notes strategy.
Explore
Students discover the sum of the exterior angles of a polygon.
Explain
Students formalize their understanding of finding the sum of the exterior angles of a polygon. Students also learn how those angles relate to interior angles and to the number of sides a polygon has.
Extend
Students apply what they have learned to a Polygon Puzzle that requires the use of interior and exterior angle sums, results from two parallel lines cut by a transversal, and other prior knowledge of angles.
Evaluate
Students reflect on their thinking processes for solving multi-step problems and use the Exit Ticket strategy to compare their approaches with those of their peers.
Instructional Formats
The term "Multimodality" refers to the ability of a lesson to be offered in more than one modality (i.e. face-to-face, online, blended). This lesson has been designed to be offered in multiple formats, while still meeting the same standards and learning objectives. Though fundamentally the same lesson, you will notice that the different modalities may require the lesson to be approached differently. Select the modality that you are interested in to be taken to the section of the course designed for that form of instruction.
Materials
Lesson Slides (Face to Face) (attached)
Pass It On handout (attached; one per group; printed front only)
Playing With Polygons handout (attached; one per student; printed front only)
Polygon Patterns (Face to Face) handout (attached; one per student; printed front/back)
Polygon Puzzle handout (attached; one per student pair; printed front only)
Polygon Puzzle (Sample Responses) (attached; for teacher use)
Perspectives handout (attached; one half-sheet per student; printed front only)
Paper
Pencil
Colored pencils
Straight edge
Scissors
Engage
10 Minute(s)
Introduce the lesson using the attached Lesson Slides (Face to Face). Display slide 3 to share the lesson’s essential question with students, and move to slide 4 to go over the lesson’s learning objectives.
Display slide 5 and place students in groups of 3–4. Give each group a copy of the attached Pass It On handout. Using a modified Chain Notes strategy, have students view the image on the handout and take turns identifying and writing or labeling what they know about the given polygon.
Ask students to select one member of each group to go first—this person should record one observation on the paper and pass it to the person on their right, repeating until all group members have recorded their observations. Remind students that they need to justify what they write.
Have students continue passing the paper and adding information until they have recorded everything they can think of related to the given polygon.
Once all groups have finished their lists, give them time to discuss what they have written. Bring the class back together and have students share out what their groups observed about the polygon.
Explore
30 Minute(s)
Display slide 6 and give each student a copy of the attached Playing with Polygons handout to each student.
Remind students to carefully read all the directions on the handout before using scissors. Then, have students follow the step-by-step directions for cutting out a polygon’s exterior angles and putting their vertices together.
Once students have completed the activity using the triangle on the handout, have them share out what shape the exterior angles created.
After discussing the shape formed by the triangle’s exterior angles, ask students, "How could this change if the number of sides changes?" Have students discuss their predictions with their groups from the previous activity. After students have taken time to discuss, ask for volunteers to share their predictions.
Move to slide 7 and assign each group a type of polygon: a quadrilateral, a pentagon, a hexagon, etc. Pass out paper for each student to create their own regular or irregular polygon. (It is unlikely that two students in one group create the exact same polygon, but you may encourage students to vary their drawings if you wish.)
Have students follow the handout’s step-by-step directions for creating a polygon, cutting out its exterior angles, and putting their vertices together. These steps are also on slide 7 for easy reference.
Once students have completed the activity using their assigned polygon types, ask students to display their final products at the center of their work stations or in another location you have previously designated. Have groups title their collection of work with the name of their assigned polygon type.
Move to slide 8 and pass out a copy of the attached Polygon Patterns (Face to Face) handout to each student. Have students participate in a Gallery Walk to look at four other groups’ sets of exterior angles. Ask students to record their observations in the tables on their handouts.
When students have finished the Gallery Walk, have them rejoin their own groups to discuss their observations and notes.
Explain
10 Minute(s)
As groups wrap up their discussions from the Gallery Walk, ask students to answer questions 1–4 on the back of the Polygon Patterns handout. Have students write their own definitions of exterior angles based on:
Their observations,
The relationship between the number of sides a polygon has and the number of exterior angles it has,
The relationship between corresponding exterior and interior angles, and
The relationship between the number of sides a polygon has and the sum of its exterior angles.
When students are finished writing, ask for volunteers to share their conclusions from the handout.
Display slide 9 and play the following video, titled "Exterior angles, by magic pi - math animations," so that students can see why a polygon’s exterior angles always have a sum of 360°.
Extend
25 Minute(s)
Have students work in pairs for this activity.
Display slide 10 and pass out one copy of the attached Polygon Puzzle handout to each pair of students.
Evaluate
5 Minute(s)
Display slide 11 and have each student find a new partner (not the person they worked with on the Polygon Puzzle).
Inform students they are going to use the Exit Ticket strategy to assess their understanding of the lesson. Give each student a half-sheet from the attached Perspectives handout.
With their new partners, ask students to discuss how they each solved the Polygon Puzzle and compare their strategies for solving it. Have students record the similarities and differences between their approaches on the handout.
Resources
Brzezinski, T. and Eike, M. (n.d.). Department of the exterior angles. GeoGebra. https://www.geogebra.org/m/we6ww7cz
K20 Center. (n.d.). Bell ringers and exit tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Chain notes. Strategies. https://learn.k20center.ou.edu/strategy/52
K20 Center. (n.d.). Gallery walk / carousel. Strategies. https://learn.k20center.ou.edu/strategy/118
K20 Center. (n.d.). Desmos classroom. Tech Tools. https://learn.k20center.ou.edu/tech-tool/1081
K20 Center. (n.d.). GeoGebra. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2352
Odintsov, R. (2020, September 16). Black and brown floral glass ceiling [Photograph]. Pexels. https://www.pexels.com/photo/art-pattern-architecture-window-5668546/
VanHattum, S. (2019, April 14). Exterior angles, by magic pi - math animations [Video]. YouTube. https://www.youtube.com/watch?v=VYbiE9sDXXk
