##
* *Reflection: Advanced Students
Radian Measure Day 1 of 2 - Section 2: Investigation

As the groups began working, one of the students ask me if he could calculate the arc length instead of measuring. I asked him what he meant. He said, "I can take the circumference, then determine the portion of the circumference the angle covers." He elaborated on this definition by using the example of a 30 degree arc being is 30/360 of the entire circumference of the circle. I was pleased that he was able to apply his prior understanding to the task. I plan to have the student explain his idea to the class.

Another student realized that she could find the measure of an 180 degree arc by multiply the 90 degree arc by 2. She continued this reasoning by saying I can add the 60 degree arc and the 90 degree arc to find the 150 degree arc. She clearly understood the concept of measurement and she applied reasoning skills developed earlier in the curriculum.

*Using a Formula*

*Advanced Students: Using a Formula*

# Radian Measure Day 1 of 2

Lesson 2 of 13

## Objective: SWBAT find the radian measure of an angle.

## Big Idea: By finding the ratio of arc length to radius, students will understand that radian measure is a real number.

#### Bell Work

*5 min*

For Bell Work, my students are asked to read over the activity Measuring Angles a New Method before starting. (Many times my students begin doing an activity before looking it over.) After the students have had time to read over the activity, I ask my students to ask questions about the activity. I want them to start thinking about the activity before they begin. To make sure the students understand the task well, I will ask the following questions before they begin if a student has not asked:

- What materials do you need?
- How are you going to begin the activity?
- How can you use a piece of string to make a circle?
- What does the word subtended mean?

Once I am sure the students understand the activity, I let my students organize themselves into groups of four to prepare to work on the activity.

#### Resources

*expand content*

#### Investigation

*45 min*

This activity takes students most of the hour to complete. It can be completed more quickly when students work as a cooperative group. Once the groups are formed I ask each group to divide their work as follows:

- one person responsible for recording the groups results
- one person responsible for measuring
- one person performs the necessary calculation
- one person takes care of materials

All students will work together to check for accuracy (**MP6**).

Some groups have students that get upset if the circle is not perfect. The students may need to draw the circle over and for some groups I have to assist the students to get beyond this part of the activity.

During the activity students will see that **the ratio of arc length/radius of a circle gives the same value for an angle no matter how big the circle** (**MP8**). Later, we will discuss how the ratio gives us a real number that can be used to describe the measure of an angle. Since students sometimes conclude that this is a redundant measure (i.e., "Why not just use degrees?"), I am ready to discuss how radians are often used when performing calculations in science and engineering, as well as when working with functions in Calculus.

As I observe my students I watch for some common errors. One error that occurs often is when measuring the arc length. Some groups will only measure the arc from the old angle to the new angle instead of measuring from point P to the end of the new angle. Since all the answers should be the same for all groups, errors are obvious.

As students work, I expect some will start to notice patterns in the length of the arcs. For example, some will realize that they can find the length of a 10 degree angle by dividing the 30 degree arc by 3. The students then use multiplication to find the length of the other arcs. This is important reasoning, which I encourage. Once it is being widely adopted, I ask a student to demonstrate how this process is done. Then, I'll ask a person not doing the calculations in the same group to explain what is being used to do the calculations. Finally, I will ask if the group has verified if this will work.

As groups complete the activity, students put their answers on a class data sheet that will be used for tomorrow's class discussion of the results.

*expand content*

#### Closure

*5 min*

As the class ends I have students who have completed the task begin to look at the class data sheet. I ask students:

**What do you notice about the results?****Is this something you expected?**

At this point in the lesson, informal conversations will help me to plan for tomorrow. They also help students to process the results of today's work.

I leave the class with a final questions to lead into tomorrows lesson:

**Why are the ratios for each angle about the same for all groups? **

Some students may be thinking about the regions being similar and consider this while others will not know why they are the same. This question will be where we start Part 2 of the lesson tomorrow.

#### Resources

*expand content*

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Is John Guilty
- LESSON 2: Radian Measure Day 1 of 2
- LESSON 3: Radian measure Day 2 of 2
- LESSON 4: Coterminal and Reference Angles
- LESSON 5: Sine and Cosine Day 1 of 2
- LESSON 6: Sine and Cosine Day 2 of 2
- LESSON 7: Developing the Unit Circle
- LESSON 8: Evaluating Trigonometric Functions
- LESSON 9: Finding the angle when given the function value
- LESSON 10: How do you find the Inverse of a Trigonometric Functions
- LESSON 11: Using Inverses to Evaluate
- LESSON 12: Review of Trigonometric Functions as Real Valued Functions
- LESSON 13: Assessment of Trigonometric Functions as Real Valued Functions