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* *Reflection: Vertical Alignment
Areas of Regular Polygons Inscribed in Circles - Section 1: Making Preliminary Predictions

One of the crazy things about the Common Core transition is that, inherently, students are going to come to us with gaps. The ninth graders I got this past year, for example, missed out on the entire elementary school progression of the CCSS. Not only that, as teachers are struggling to teach everything in the new standards and teach it correctly, students come in with gaps from the classes they have taken.

To give an example, my students this past year came to Geometry (having taken Algebra 1), but not having a really basic understanding of what a function is. I would argue that the idea of function is one of the most important in high school mathematics. So when I realized this, I had to think of how I would address this gap while still teaching Geometry. That is what spawned this lesson. The content is Geometry, but the big idea is really all about functions. Even if they forget the geometry in this lesson, I will be happy if when they get to Calculus functions are a very familiar thing to them.

*Shoring up the gaps during transition to CCSS*

*Vertical Alignment: Shoring up the gaps during transition to CCSS*

# Areas of Regular Polygons Inscribed in Circles

Lesson 6 of 8

## Objective: SWBAT express the ratio of an inscribed regular polygon's area to the area of its circumscribed circle as a function of the number of sides.

In this section, students will be getting familiar with the context of the lesson and making some preliminary predictions. Each student receives Inscribed Regular Polygon Area Ratios_Predictions.

Item 1 asks students to estimate the ratio of a regular heptagon to that of its circumscribed circle and express that ratio using only a single number. I've found that my students are sometimes not sure how to express a ratio as a single number so I explain how 3:2 can be expressed as 1.5, for example. Once students have made their estimates, I'll do a quick whip around to hear students estimates and how they arrived at them.

Item 2 requires students to do some visualization in order to understand the question. I have the students do a quick pair share to discuss what it means to vary the radius. Then I'll have them discuss their answers to the question. After that, I'll call on some non-volunteers to share what they've concluded.

Next we'll proceed to work through item 3 in the same way we did for item 2.

Item 4a asks students to identify the practical domain of a function. After students have had a chance to discuss their answers with their partners, I'll call on students randomly to share their answers. As they do I'll provide the necessary coaching to make the answers more precise. For example, if a student says, "All numbers from 3 to infinity...," I"ll ask, "Does that include 3? How about 5.7?"

Finally, it's time for students to sketch their conception of the function graph. I require that this be a totally independent task for each student. This way, each student will communicate his or her understanding, which will be important for them to reflect upon later in the lesson.

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#### Group Activity

*30 min*

In this lesson, students will be working in groups of four. The goal is for students to come up with a general formula of the ratio of a regular polygon's area to the area of its circumscribed circle as a function of the number of sides the polygon has. In order to reach this goal, each student in a group of four will be closely working with one regular polygon (square, pentagon, hexagon or octagon) and then the group members will be coming together to abstract to the general case based on their findings in the specific cases.

Before the lesson, I'll photocopy the first four pages of Area Ratio of inscribed regular polygon to circumscribed circle, each with the fifth page (The General Case) photocopied on the back. This way each student in a group will have his or her specific case on one side and the general case on the back.

Before getting students into group (i.e., while they are still facing forward in rows) I'll want to do some basic orientation to the task. First I call attention to the Long Term Goal and have students discuss what they think it means. Then I'll have a couple of share outs, clarifying or elaborating as necessary to make sure students understand the goal.

Then I'll explain that we will be writing areas in terms of r since the area of a circle is already in terms of r. Students have had a good deal of practice writing areas of regular polygons in terms of different variables in the previous lesson so I feel comfortable turning them loose on this task.

I have the students get into groups of four and I hand each student in a group a different specific case. Then I direct the students to start working. I explain that they are all working on similar but slightly different problems so they will be able to get general guidance, but not specific answers, from their partners.

When students have had enough time to complete the work for their specific cases, I'll direct the groups to turn their focus to the back side which asks them to record the four specific cases and determine the rule for the general case. At this time, I'll walk around the classroom to see students having aha! moments or having questions. If groups finish early, I'll have them get to work on the Extension.

Finally, I'll demonstrate the process and rationale for re-writing the function using the double angle formula (See Ratio of Inscribed to Circumscribed_KEY)

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#### Plenary Discussion

*20 min*

Now that the groups have finished their work, I have the students return the desks to row seating formation. Then I show and discuss the work for one of the specific cases and re-cap how all of the specific cases turned out and how we used those to abstract to the general case.

Then all that's left to do is to determine what happens with the function (i.e., the ratio between the area of a regular polygon and the area of its circumscribed circle) as n approaches infinity. I do that via a demonstration on DESMOS.

Check out the screencast to get an feel for the content and style of that demonstration.

After seeing this demonstration, we'll have some discussion about limits, asymptotic behavior, discrete vs. continuous functions, etc. Not that these are the learning goals of the lesson, but it doesn't hurt to expose students to important concepts and language at this early stage.

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#### Reflection and Closure

*10 min*

Now that students have been through the lesson, it's time for them to reflect on what they learned and make any necessary revisions to their original thinking based on what they've learned. Page two of Inscribed Regular Polygon Area Ratios_Predictions, provides a space for students to do just that. So at this point, I give students 7-10 minutes to complete the Post-Activity Reflection and then prepare to turn in their work.

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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