K20 LEARN | Designing Your Area

Summary

In this lesson, students will use their existing knowledge to find the area and perimeter of triangles, rectangles, and trapezoids to discover how to find the area of a regular polygon and apply their knowledge to real-world scenarios.

Essential Question(s)

How do you find the area of any regular polygon?

Snapshot

Engage

Students recall what they know about finding the area and perimeter of composite figures.

Explore

Students explore the relationship between the area of a triangle and the area of a regular polygon.

Explain

Students complete guided notes with the class and formalize their understanding of the formula for the area of a regular polygon and learn the definition of apothem.

Extend

Students apply what they have learned to find the area of each surface of a pool to give a cost estimate for ordering tile.

Evaluate

Students determine the quantity of materials needed to build and paint a frame for an octagonal mirror.

Materials

Lesson Slides (attached)
Composite Figures handout (attached; one per pair; printed front only)
Exploring Patterns handout (attached; one per pair; printed front only)
Guided Notes handout (attached; one per student; printed front only)
Polygon Pool Party handout (attached; one per pair; printed front/back)
DIY Mirror handout (attached; one per student; printed front only)
Pencil
Scientific calculators
Student devices with Internet access

Engage

20 Minute(s)

Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s essential question. Slide 4 identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.

Instruct students to find a partner or assign students partners. Pass out scientific calculators to each student. Go to slide 5. Give each pair a copy of the Composite Figures handout. Here students are asked to find the area and perimeter of each composite figure. Once students have completed the handout, use a modified Think-Pair-Share strategy and have students find another pair of students to discuss their results and thought process.

Teacher's Note: Guiding the Activity

Be sure not to immediately tell a student how to find the area of a figure. Ask students what shapes they see that they know how to find the area of, as there are multiple ways of finding the answer. Students will each see different arrangements of shapes, so it is important that each student divides the figure into known shapes. For example, a student may not see a trapezoid but instead see a rectangle and triangle – both are valid ways of finding the area.

All of the triangles are Pythagorean triples, so minimal calculations, if any, are needed to determine each hypotenuse length. Calculators are used as a time-saver. There are many calculations throughout the lesson, but the calculations are not the focus of the lesson.

As you show the slides for students to check their work, remind students that these are just one example of how to find each area and perimeter. Encourage students to share their differing strategies, or ask volunteers to share the thought process that they heard about in their discussions, and found interesting.

Show slides 6–9, giving students time to check their work on each composite figure. Ask volunteers to share different methods of finding the area and/or perimeter of each figure, as time allows.

Explore

20 Minute(s)

Display slide 10 and pass out the Exploring Patterns handout to each pair of students. Provide students with the link to the GeoGebra activity: https://www.geogebra.org/m/x4tsbeuz. Have students work with their partner to complete the handout. Here, students explore the relationships between the area of one triangle, the number of sides of a polygon, and the area of a regular polygon. During this exploration, students should record the area of one triangle and number of sides of each polygon in the GeoGebra activity. On the handout, students are asked to use those two given details to determine the area of each regular polygon. The GeoGebra activity demonstrates how each n-gon is made up of n congruent triangles, starting with an octagon down to a triangle.

Explain

10 Minute(s)

Display slide 11. Give each student a copy of the Guided Notes handout. Complete the handout as a class. Have students add this to their math notebook if that is a classroom norm.

Teacher's Note: Guiding the Activity

Using example 1, have students share how they think that they would find the area based on their observations from the GeoGebra activity:

divide the hexagon into six congruent triangles;
find the area of one triangle using the base of 8 and height of 4 square roots of 3;
multiply the area of one triangle, 16 square roots of 3, by 6, since there are 6 congruent triangles;
find that the total area of the regular hexagon is 96 square roots of 3.

Guide students through the formal definitions of apothem, side length, and number of sides as well as the formulas for area and perimeter of a regular polygon. It is likely that apothem is the only new vocabulary word for students.

Consider asking students if they think these formulas would work for irregular polygons or for all polygons? Why or why not?

Use example 2 as a time to practice using this newly learned formula:

identify and write down what is known: a = 2, s = 2.9, and n = 5;
use the formula, A = (½)·a·n·s, to find that the area is 14.5.

Extend

30 Minute(s)

Now it is time for students to apply what they have learned; display slide 12. Give each pair of students the Polygon Pool Party handout. Instruct students to work with their partner to find the area of each surface of the pool and record their results on the back side of the handout. Here students are asked to find the square footage of each surface of the pool to determine the cost of tiling the pool. Students are also asked to find the area of the concrete patio surrounding the pool.

Evaluate

10 Minute(s)

Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Go to slide 18 and give each student the DIY Mirror handout. Students are asked to find the area and perimeter of the frame of a regular octagonal mirror.

After students have submitted their work, unhide and show slides 19 and 20. Give students time to reflect on their thinking. Use student responses to see what misconceptions still exist.

Resources

Image source: Mithul Varshan. (October 1, 2019). Empty Gray and White Concrete Spiral Stairs [Photograph]. Pexels. https://www.pexels.com/photo/empty-gray-and-white-concrete-spiral-stairs-3023211/
K20 Center. (n.d.). Bell Ringers and Exit tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Think-Pair-Share. Strategies. https://learn.k20center.ou.edu/strategy/139
K20 Center. (n.d.). GeoGebra. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2352

Authentic Lessons for 21st Century Learning

Designing Your Area

Area and Perimeter of Regular Polygons