##
* *Reflection: Developing a Conceptual Understanding
Radians Day 1 of 2 - Section 3: Radian Investigation

I got a lot of good feedback from this activity. The students really liked the physical connection and other teachers at my school who also used this lesson said that it will well with their students.

I do have a couple of pointers to keep this lesson running smoothly. Accuracy in measurement was a big issue. If the students weren’t super careful when measuring the radius (which happened more than I would have liked), their measurements were really off. Also, they wanted to put the pipe cleaner around the outside edge of the circle. This also made the measurements incorrect. It is better to measure a bit on the inside than around the outside. A fellow teacher used string and had good luck with it. I chose pipe cleaner due to its ability to hold its shape but both tools can be used successfully. As we went to each step, I would hold up student work as a model. This really seemed to help keep everyone on track. When we went to measure the 0.28 radians (Student Picture), the student's percentages were really spread. I wrote down each pairs’ percentage on the white board so everyone could see the range and then gave them the actual measurement. This lead to a great discussion on accuracy and human error.

*Developing a Conceptual Understanding: Making This Activity Successful For All Students*

# Radians Day 1 of 2

Lesson 4 of 19

## Objective: Students will be able to use radians to measure angles.

## Big Idea: The stars in the sky are not necessarily the best way to measure angles, this lesson explains why.

*55 minutes*

#### Warm up and Homework review

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Ups- Radians, which asks students to identify which quadrants have the same cosine.

I also use this time to correct and record the previous day's Homework.

*expand content*

#### History of Degrees

*5 min*

This is a historical introduction to our lesson on radians. This introduction is based off of information obtained **here**. This goal is for students to see why we use a method to measure angles other than degrees. I highly recommend reviewing the information in the article before teaching this lesson. Any time that history or other connections are included in a lesson, the teacher has a lot of power over whether this will go over well or not. I find that the students will connect better to the lesson if I use some flare and enthusiasm. This is a story and goes better when told as such. These slides are just a guide I use to keep the flow, I tell the story and not just read the slides.

#### Resources

*expand content*

#### Radian Investigation

*10 min*

The next portion of this lesson is a guided investigation that uses the physical representation of a radian and connects it to π.

My personal preference, when it comes to investigations, is to do a whole class guided investigation rather than having them do it individually or in pairs. I find that I can keep the pacing appropriate and we can stop together to talk about the key points. If you and your students do well with independent investigations, please feel free to put the next slides into a sheet for that purpose. Please watch this short video on **Guided Investigations.**

I pass out a circle to each student and a ruler, protractor, pipe cleaner and scissors to each pair of students. There should be a variation in the sizes of the circles and the centers should be marked. It wouldn't be a good use of time in an already full lesson to have the students draw their circles and cut them out. The best scaffolding for this lesson is to model the activity yourself at the front of the room.

All the instructions are prepared in the PowerPoint. The students measure the radius of their circle with a length of pipe cleaner. They then lay the pipe cleaner around the circle, marking that length as they go. Next, they measure their pipe cleaner in cm and use that to find the percent of the remaining portion (which should be around 0.28).

Some scaffolding may be necessary for some students when it comes to finding percent. I walk around the classroom and give individual help. If too many are struggling, I may address the class and ask if someone can explain how to find the percent.

I have them identify to total number of radians, including the percentage, as a decimal. I then write their numbers on the whiteboard. They will be somewhere around 6.3ish. There will obviously be some variation in the numbers. This is an excellent time to discuss **Math Practice 6**. Why didn’t we all get the same answers? What could we have done to make our solutions more accurate?

Before pulling up the 2π slide, I see if any students can figure this out by discussing it in pairs and then as a class.

The final portion of this investigation has them measure the angle that cuts of the arc that is one radian. They will get an angle around 57^{o}. To many this will seem like a crazy number to be a unit, so I remind them that degrees are the arbitrary ones and not radians.

#### Resources

*expand content*

#### The Unit Circle

*10 min*

I pass out the Radian Measure and the Unit Circle sheet. The students will fill in the radian and degree measures today and put in the coordinates and discuss the trig ratios in the next lesson.

*expand content*

The final goal is to give the students the opportunity to figure out a method for finding radians from angles. The question I asked is "What if I wanted to know how many radians are equal to 140^{o}?". If time is short, this can be done with a class discussion, otherwise, I have them work as pairs and then ask for volunteers to share out. I am careful not to just give them the formula but build it so the it makes sense to them why it works. This is an excellent opportunity for students to write out an explanation (**Math Practice 3**). For example, they could say that you find what percent the degrees are to 180 and then multiply that by π. Expect that there will be a number of students that will end up just plugging numbers in but they should have the opportunity to understand why it works.

Next the students convert 55^{o}, 110^{o}, and 440^{o} into radians to practice the skill. Notice that these are multiples of each other. Some students may catch on to this and use it to find the radian measure (**Math Practice 7**). After all three problems have been completed, I see if I can get someone to bring it up by asking a leading question like "Did anybody notice anything interesting about these three problems?". The better students can connect the fractional parts of a circle to radians, the better off they are going to be.

The final portion of this lesson asks students to figure out how to convert radians to degrees. This may prove more challenging and require more scaffolding. I may ask the students guiding questions like "How would you undo what you did in the last type of problem?" (**Math Practice 2**) Again, I have them write a quick explanation of the process rather than writing a straight formula. There are a few practice problems to finish off this lesson.

*expand content*

#### Exit Ticket

*5 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's Exit Ticket asks the students to find the radian measure given degrees.

#### Resources

*expand content*

*Responding to Kimberlie Mattern*

Thank you! Â I really like this lesson as well. Â History is such a powerful way to connect our students to the mathematics. Â Somehow, it makes it more real. Â

| one year ago | Reply

This week will mark the second year that I will be using the radian activity. It went very well. I like being able to provide the history behind the math that I present.

Â

| one year ago | Reply

This is awesome! What a great tie to the real world! I'm definitely going to use this. Thanks!

| one year ago | Reply

My students really got into this lesson. They enjoyed the connection of the unit circle and quadrants to history and rotation of "the big dipper" counterclockwise.Â Each student had a circle of different size so it was interesting to listen to their discourse in the discovery of the radian measure. I did the activity along with them using the document camera. Only a few were able to get the .28 measure due to our human error in laying the pipe cleaner round the circle, but most were close. Thanks so much for the clever way of introducing the radian measure.

| 2 years ago | Reply*expand comments*

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
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- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Angle and Degree Measure
- LESSON 2: Trigonometric Ratios
- LESSON 3: Trigonometric Ratios of General Angles
- LESSON 4: Radians Day 1 of 2
- LESSON 5: Radians Day 2 of 2
- LESSON 6: The Unit Circle Day 1 of 2
- LESSON 7: The Unit Circle Day 2 of 2
- LESSON 8: Graphs of Sine and Cosine
- LESSON 9: Period and Amplitude
- LESSON 10: Period Puzzle
- LESSON 11: Transformations of Sine and Cosine Graphs
- LESSON 12: Graph of Tangent
- LESSON 13: Model Trigonometry with a Ferris Wheel Day 1 of 2
- LESSON 14: Model Trigonometry with a Ferris Wheel Day 2 of 2
- LESSON 15: Modeling Average Temperature with Trigonometry
- LESSON 16: Pythagorean Identity
- LESSON 17: Trigonometric Functions Review Day 1
- LESSON 18: Trigonometric Functions Review Day 2
- LESSON 19: Trigonometric Functions Test