Why is there so little variation in the data produced by each group?
As my students are discussing this prompt, I circle and ask probing questions, such as, "What do you know about the circles each group constructed and measured?" I want students to make the connection to the concept of similarity. All circles are similar, therefore each sector made for a particular angle is similar to all of the others for that angle (i.e, corresponding parts of the sector are in the same ratio). Many of my students think of similarity when working with polygons, but are slower to apply the concept to circles. It may take some prompting on my part with questions; it is important to be patient at this stage in the lesson.
Once students generally agree that all of the sectors for a given angle are similar I ask:
These questions lead the students to see the structure of the diagram in terms of a ratio, that yields a real number as a measurement.
Once I sense that my students understand that the arc length/radius ratio for an angle will be the same for all circles, I define this ratio as the measure of the angle in radians. I explain that mathematicians this measurement is important because there are applications for arclength that require a measurement that is not dependent on the length of the radius.
As a next step, I ask the class to explain what a 1 radian measure would mean. Another way of a starting this conversation is to ask, "In yesterday's activity, which angle was closest to 1 radian?"
As students worked on the activity yesterday, groups employed different methods for finding the length of the arc. It is helpful for students to discuss different methods, as it allows students to see different ways of thinking. It also helps groups to verify their strategies, if students ask questions about a group's method (MP3). To facilitate discussion I use the following questions:
Now as a class, we determine the number of radians in 360 degrees. I explain that we want to do this without measuring or having a specific measure for r. I use r to represent the radius and generalize. The students simplify to get 2*pi. We look at our data and see that most students are close to 2pi. This is a good time to discussion precision.
Why are the arc length in the class data not exactly 2*pi? Are there any groups who came up with exactly 2*pi? Why is that so?
I expect that many of my students will realize that the tools used caused measurement error to occur. Some may state that they were not careful when they measured. When the students only measured once if the measurement was not close to the actual amount they increased the error as they went around the circle.
Once the class has found the a 360 degree angle is the same as a 2*pi radian angle. I ask the class to begin finding other angles such as 180 degree, 90 degree, 270 degree. I encourage students to explain their reasoning as they determine the radian measure.
What other angles could you find using the information you already have? Some students will find 45 degrees since it is half of 90 degrees and others will find 30 degrees.
How many radians are in 50 degrees? The students are given 2-3 minutes to think and discuss a method with their groups. The different methods are shared with the class and analyzed. Students analyze the methods and ask questions about the methods.
The class does a few more conversions from degree's to radians before working the other way, finding the degrees when given the radian measure. Students work for 2-3 minutes then different methods are put on the board and student again analyze the processes used. Many students think all angles in radians have must have pi in the angle. To help students realize that angles can be a decimal I ask students to convert an angle that is a decimal. I usually get a question on whether this is really an angle. We discuss how using pi gives the exact answer but it is sometime better to have the angle in decimal form. The students develop a process to solve.
As class comes to an end I inform my students that I want them to know the equivalent radian measure of special angles 30, 45 and 60 degrees (along with their multiples). I say, "You should know these from memory, but, I would also like you to b able to convert from degrees to radians, and, from radians to degrees." Then each person is given the angle comparison worksheet. I assign this worksheet to be completed by the next day so we can use the information for the next lesson.
I also write two questions on the board for students to consider:
I expect my students will see the relationship between the angles in that the larger angles are multiplies of the acute angles. This will help students as we talk about reference angles and co-terminal angles later in the unit.