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* *Reflection: Backwards Planning
Regular Polygons and their Areas - Section 1: Modeling and Guided Process

As I'm teaching this lesson, I'm aware of what students will need to know and know how to do in the next lesson. That makes me really intentional about what things I emphasize and model thoroughly in this lesson. One example is the idea that the acute angle formed between the apothem and radius of a regular polygon is 180/n. Another is how we divide a fraction, e.g., 2/a by a constant c for example to get 2/ac quickly rather than going through all of the gyrations. Similarly the process for solving an equation in which the variable is in the denominator can be streamlined significantly if one finds the regularity in the repeated reasoning of solving such equations. So I try to share these things with students up front in this lesson so they'll have an easier go at it when they get to the next lesson, which is tough enough without having to struggle with these things.

*Getting students ready to be independent*

*Backwards Planning: Getting students ready to be independent*

# Regular Polygons and their Areas

Lesson 5 of 8

## Objective: SWBAT express the area of a regular polygon in terms of the number of sides and side length.

#### Modeling and Guided Process

*25 min*

This lesson is the first in a series of two functions-themed lessons designed for an Honors Geometry course. The lesson requires students to understand and perform challenging mathematics. They must have a good grasp on regular polygons, right triangle trigonometry, and algebra. That said, I also want to scaffold the process for students so that they have access to the lesson. That is the purpose of this section.

I start by giving each student a copy of Derive formulas for regular polygon areas in terms of n and s or r . Next I explain the meaning of the diagram at the top of the page. I explain that this is my attempt at representing all possible regular polygons. After that, I lead a discussion on how how we should label the diagram. I ask where we should put the a for apothem, why the value of the angle between the apothem and radius should be 180/n, and what we should label the side opposite the angle.

Then it's time to start on items 2 through 6. For each item, I allow students to get a head start before I write anything under the document camera. Then I'll show the answers (see Deriving Regular Polygon Area Formulas KEY.pdf) so that students know if they have completed the item correctly. When we get to item #4, I call the class to order so that I can give a more in depth explanation and modeling. I explain why we need to solve for a in terms of a and s, and then show how to do that (for example, how s/2/a = s/2a).

Item 5 is basically a substitution so I allow students to complete that on their own and then compare with their partners before I finally show the answer under the document camera. I handle item 6 in the same way as I did #5.

Finally, I give a recap of the entire process, being clear about the purpose for each step and what it achieves.

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#### Independent Challenge

*30 min*

In this section of the lesson, students will be working independently to derive a formula for the area of a regular polygon given the number of sides and the radius. The process is analogous to the process I guide and model in the previous section. So in this section, students just get a blank copy of page 2 of Derive formulas for regular polygon areas in terms of n and a or r and 20 minutes to work things out. It's not meant to be easy, so I'll let them grapple with the challenge, even if it means enduring some temporary frustration.

I walk around providing moral support and general coaching. For example, if a student is struggling to write the apothem in terms of r, I might say "When did we do something similar to this in the previous section of the lesson? How could you use a similar approach/strategy here?" Some students will ask if they can use the formula from the previous section and just tweak it. For me, the answer to that is NO. I want them to start from scratch so that they can see the process through from start to finish.

When the time has elapsed, I will collect the papers to see how students have done. As I am going through the papers, I will give positive feedback for quality attempts and I will select exemplars to share with students either online or by having the student(s) demonstrate and explain their work at the document camera.

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: A Deeper Look at Area
- LESSON 2: Proving Area Formulas for Parallelograms, Triangles and Rhombuses
- LESSON 3: Proving the formula for the area of a trapezoid
- LESSON 4: Construct Regular Polygons Inscribed in Circles
- LESSON 5: Regular Polygons and their Areas
- LESSON 6: Areas of Regular Polygons Inscribed in Circles
- LESSON 7: Area Construction Challenge
- LESSON 8: Application: Geometric Integrals