##
* *Reflection: Developing a Conceptual Understanding
Construct Regular Polygons Inscribed in Circles - Section 2: Student Constructions

I've often wrestled with how I should teach constructions. Should I just give the students the steps and have them memorize? Should I show them the steps and require them to explain why they work? Or should I try to give them the knowledge they need to actually come up with the steps on their own. I've tended to go back and forth between the last two choices. In this lesson, for example, I could give the students the steps and I'm sure they'd have them memorized in less than 15 minutes. But there is so much embedded math practice in performing constructions that would be lost using that approach.

For example in this lesson, students will need to understand the idea that a compass is useful for making markings at equal distances from some point. They also will need to think about central angles of inscribed regular polygons and properties of equilateral triangles and rhombuses.

So, in the end, I just couldn't justify passing up the opportunity to have students to get practice really engaging with mathematics, not just the steps, of construction.

*Constructions: To give the steps or not to?*

*Developing a Conceptual Understanding: Constructions: To give the steps or not to?*

# Construct Regular Polygons Inscribed in Circles

Lesson 4 of 8

## Objective: SWBAT construct equilateral triangles, squares, and regular hexagons in circles

In this lesson, students will be performing classic constructions using only compass and straightedge. My goal in this section is to use Sketchpad to show them the constructions they'll be doing and to introduce some important * concepts *that will come in handy when they have to problem-solve and figure out how to perform the constructions.

Check out this screencast to learn more about the demonstration I do for my students.

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#### Student Constructions

*20 min*

In the previous section, I've given students all of the hints they're going to get for now. In this section they'll be working on Student Constructions. I pass out the resource and the construction challenge is on. I do my best to create an atmosphere of challenge, puzzlement, problem-solving, etc.

As students are thinking and working, I'll walk around and see how things are going. I evade the "Is this right?" question, but I will stand and listen to students' reasoning then most often offer a cryptic "Hmmm..."

This is also a time for me to identify some student exemplars for later when I have students come to the document camera to demonstrate and explain their work.

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#### Student Presentations

*20 min*

In this section, I will call students who I have identified as exemplars to present each of the constructions, explain the steps they took and the geometry that makes them achieve the desired results. I will also ask if there are any students who performed the constructions in a different way and still got good results. It's possible that I have failed to identify a good exemplar and I don't want to miss out on discussing an innovative, non-typical, approach.

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#### Transfer Task

*10 min*

To see if students can apply what they've learned in this lesson to a slightly novel situation, I'll have them construct a regular octagon inscribed in a circle. Students will get 15 minutes alone with the Transfer Task to see what they can produce and explain.

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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