If our ultimate goal is for
students to learn to speak the language of mathematics, then vocabulary
should be an essential part of our instruction. Therefore, I will
try to include a list of key concepts at the beginning of each
lesson. The definitions in blue are linked to the Intermath dictionary
definition.
Key Concepts
angle: determined by rotating a ray about its endpoint
initial side: starting position of the ray
terminal side: ending position of ray
vertex: endpoint of the ray
standard position: an angle situated on a coordinate system such that the origin is the vertex and the initial side lies on the positive x-axis
positive angles: generated by a counterclockwise rotation
negative angles: generated by a clockwise rotation
coterminal angles: angles which have the same terminal sides
degree: an angle equivalent to a rotation of 1/360 of a complete revolution about the vertex
central angle: an angle whose vertex is at the center of a circle
radian: the measure
of a central angle that intercepts an arc s equal in length
to the radius r of the circle
1. Begin by introducing the basic concepts: angle, initial side, terminal side, vertex, standard position, positive angles, negative angles, coterminal angles, degree, and central angle. These are concepts that are most easily understood when illustrated by the teacher.
2. Use Geometer's Sketchpad (GSP) to introduce the concept of radian. While this is a fundamental concept for future mathematics, many students do not grasp what a radian truly represents.
a. Discuss the definition of a radian (given above).
b. Students begin by constructing their own circle in GSP. It can be any size. Use GSP to construct arcs on the circle that are equal in length to the radius. Use the measure tool to measure the length of the radius and the lengths of the arcs. In the end, they should have something that looks like this:
Q1: How many radians do you think there are in one circle?
It appears that there are a little over six radians in a circle. Students should have gotten this no matter what size circle they constructed.
Q2: How do we find out exactly how many radians are in a circle?
Recall that the circumference of a circle is given by:
So, if we want to know how many r's it takes to get all the way around that circle, we simply divide the total distance around it, C, by r!
Remember that pi is approximated in many calculations by the decimal value 3.14. So, our answer is approximately 6.28 - just over 6 radians!
3. Discuss the language of radians. Radians are expressed in terms of pi.
4. Discuss the conversion of degrees to radians and vice versa.
Since we've talked about the relationship of radians to the circumference of the circle, we should discuss the conversion in terms of the circumference as well. . . Remember that there are a total of 360 degrees in a circle and and total of 2*pi radians in a circle.
We can use a ratio to solve this problem. . .
Again, we can use a ratio to solve this problem. . .
Now, we are ready to write a rule of conversion. The following ratio will always be accurate, just plug in the value that you know.
This concludes our introduction to degrees and radians. I believe that the most important part of this lesson is the students' understanding of what a radian represents. Thus, I tried to make this aspect of the lesson as detailed as possible. GSP is a powerful tool that allows students to make some conjectures about radians. However, providing some good instruction on why the conjectures are true/untrue is essential as well.