##
* *Reflection: Perseverance
Area Construction Challenge - Section 3: Problem #4

The first time I ever saw problem #4, I literally was confounded for two days. I couldn't stop thinking about the problem and yet I was making no progress. I had several ideas. One even woke me out of my sleep and got me out of bed in the middle of the night. But none of these initial attempts were fruitful. In hindsight, what was holding me back for so long was that I wasn't willing to imagine that the solution could be very complex and involve lots of math. I was thinking small and so I was looking for simple and quick solutions. This kept me from trying things that eventually ended up working once I let myself think bigger. I wouldn't rest until I got this problem right, and when I did get it right, it was exhilarating. This really brought home for me what it means to persevere in solving problems. There are elements of frustration bordering on suffering that almost seem to be the cost of admission to the problem. But then there is also the payoff. This experience makes me think about how I want my students to experience the same payoff that I experienced. The difficult part is getting them to persist through the "suffering" phase.

*The power of persevering*

*Perseverance: The power of persevering*

# Area Construction Challenge

Lesson 7 of 8

## Objective: SWBAT Use Classic Constructions to Solve Problems

I will typically give page 1 of Area Construction Problems, which contains only problem #1, to students before giving them page 2. That way I can make sure that they focus on one problem at a time.

I give this problem as a take-home assignment and students will have three nights to complete it. On day 2, when students return to class, some students will have the problem done already. Others will not. To provide the next level of scaffolding, I'll give students (those who want it) Construction Area Problem #1 Guidance.

On Day 3, I'll learn who the students are who have still not completed the problem even with the guidance. For those students, and for other to know if the process they used is correct, I show the following demonstration. I apologize for the video having no sound, but I wanted to allow the viewer to read on their own.

After this, students should be able to complete the construction on their own.

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#### Problems #2 and #3

*25 min*

On day 4, after students have turned in problem #1, I'll pass out page 2 of Area Construction Problems and give students 25 minutes of class time to work on problems 2 and 3.

Problem #2 relies on the fact that any two triangles with the same base and height have equal areas. Problem #3 is essentially a modified version of problem #1. So these should be within the power of students to solve.

At the end of the 25 minutes, I'll ask students to put the construction problems away and they will be due two days later.

Problem #4 is not a required problem so I will give extra credit to any student who can solve it and adequately explain the rationale for the process. More on that in the next section.

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Problem #4 is a whale of a problem. I hope you enjoy it.

I have purposely not included the solution to the problem because I wouldn't want to deprive anyone of this authentic problem-solving experience. I have, however left a clue or two in the following screencast.

*expand content*

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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