##
* *Reflection: Coherence
Converting between Degrees and Radians - Section 2: Deriving the Formulas

At some point in my teaching career, I suppose I wouldn't have thought the formula for the area of a sector had anything to do with the formula for converting degrees to radians. But now, having taught it, I understand that the big idea is not so much the formulas as it is the idea of proportional reasoning.

Looking at it in this way has changed my approach significantly. If my only objective was for students to know the formulas, it could be a 10 minute lesson. But since I am trying to develop a bigger and more important idea of proportional reasoning, I am more concerned with their knowing how to build the formula using proportional reasoning and this takes effort. Especially because students like formulas. And if they know the formula already, good luck trying to get them to listen to the whole spiel on how the formula comes to be. It's a worthwhile struggle though, I think.

I think that once I figure out how to assess derivation of formulas better, it will make students a lot more receptive to the teaching. Otherwise, they tend to view it as just a long drawn out way of telling them the formula which they'll never need to derive again.

*Making Connections: Proportional Reasoning*

*Coherence: Making Connections: Proportional Reasoning*

# Converting between Degrees and Radians

Lesson 4 of 5

## Objective: SWBAT derive and apply formulas for converting between degrees and radians.

#### Activating Prior Knowledge

*20 min*

The main goal of this section is for students to analyze the structure of formulas in order to gain conceptual understanding from these formulas.

Each student will get a copy of APK_Convert Degrees and Radians.

The first seven items ask students to differentiate between 1-D, 2-D and 3-D measurements. Starting with the first, one by one I call out the items , asking students to hold up either one, two, or three fingers to indicate the number of dimensions to the measurement. Then after each vote, I talk about the correct answer.

The next item, #8, asks students to determine the number of non-overlapping squares with area r^2 could fit inside a circle with radius r. I give students some time to think about this one because I want to see who and how many can think this way without my assistance. Then I give an explanation of how the formula A=pi*r^2 indicates, structurally, that we have a certain number of r^2's...and that certain number in this case is pi. This, along with the other items on the handout, is designed to prepare students to understand why 2pi radians is equivalent to 360 degrees.

After I explain #8, I gradually release control to the students, letting them figure out the answer and calling on random non-volunteers to give a thorough explanation of how they analyzed the structure of the formula in order to figure out the answer.

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#### Deriving the Formulas

*25 min*

Conceptually, this lesson is very similar to an earlier lesson in this unit on deriving sector area and arc length formulas. So in the first part of Converting Between Degrees and Radians students are taken back to the context of that lesson in order to connect to what they learned in that lesson.

There are two sets of blanks for students to fill in. For the first set of blanks, I ask the students to fill in the blanks on their own and then to compare/discuss with their partners. Finally, I'll show the correct answer and then ask students to share with the class why they feel the formula makes sense. After we get a two or three quality answers, I ask students to write in the space where it asks them to explain why the formula makes sense.

Then we'll repeat this process for the second set of blanks.

Next we'll read the bottom of page 1 together as a class. When we come to a blank, I'll ask students to do a quick share before I call on a non-volunteer to say what goes in the blank. Then we'll read through the paragraph again with the blanks already filled in. Finally, I'll give students a couple of minutes to respond to the prompt at the bottom of the page.

Turning to page 2, I have students work with their partners to figure out what goes in the blanks at the top of page 2. I announce that I'll be calling a student randomly to come to the document camera to show and give a rationale for what they've written. This usually improves the quality of collaboration.

After that, I'll call a student to the front to share and explain what they've written. Hopefully the quality of presentation is good enough to benefit the class. Otherwise, I may call another student or re-summarize myself to make sure that the correct information is available to the class.

Next, each student will work independently for 5 minutes determining the first formula, providing a rationale, and showing an example problem. After the 5 minutes, I'll have the students compare their work with their partners.

Finally, They'll repeat the same cycle for the second formula.

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#### Using the Formulas

*20 min*

When I go to make photocopies, I copy pages 3 and 4 of Converting Between Degrees and Radians separately. The answer to page 3 are on page 4. In any case, for this section, I'll pass out page 3 and instruct students to work independently, without a calculator, to complete the unit circle with its radian measures. When ten minutes have passed, I'll collect all of the page 3's. Then I'll hand out page 4 which asks students to convert radians to degrees. Again, I give students 10 minutes to see how much they can finish correctly. At the end of the 10 minutes, I'll return the page 3's to students so that they can check their work on both pages 3 and 4.

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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: Deriving Formulas for Sector Area and Arc Length
- LESSON 2: Define Radian Measure
- LESSON 3: Using the Unit Circle to Prove the Law of Sines for Obtuse Triangles
- LESSON 4: Converting between Degrees and Radians
- LESSON 5: Introduction to Trigonometric Functions and their Graphs