Answer Sheet for Activity #1
Question #1:
The purpose of this question is to first have the students play with the applet. That way they can familiarize with it and be comfortable with how the applet works. Secondly, this question is supposed to be thought provoking. By providing a vague question, it allows students to really think about what the answer really is and in turn write down some really great ideas on what a radian could be. Thus the solution of this question is that there will be multiple. The point is that students should see from the applet that a radian relates to the circle and might only list the characteristics that they on screen or if they have heard of radians before then they may answer this by saying that is is a unit of measure for an angle.
Question #2:
The purpose of this question was for students to focus on how a radian changes relative to the circle and to also see that as the radian increases so does the arc length, but the radius stays constant. Thus the solution of this question is that there will be multiple. In essence, a solution would have to showcase that as the angle and the arc length have a proportional relationship, while no matter how the radian measure changes, the radius stays constant.
Question #3:
The purpose of this question is for students to realize that the pink points represents 1 radian. In other words, the distance from one pink point to another is exactly 1 radius length. Thus from this the solution of this question is that there will be multiple. Some students may not see the relation between the radian measure in the green section and the length of the actual arc length at each pink point, while others may see this. In essence, a solution would have to show or state that the pink points represents a distance of 1 radian.
Question #4:
The purpose of this question is to reinforce the previous question or to provide more hints toward the solution for the previous question. In other words, this question provides more insight and foundation for students to build towards the idea that a radian is represented by the ratio of the arc length to the radius. The only difference is that instead on focusing on strictly the radian value, the students are now focusing on where that value came from. Thus for this question the solution would have to be a variation or very close to that the arc length changes with the angle measure but is much larger than the angle itself, while the radius still stays constant.
Question #5:
The purpose of this question is for students to start seeing that a radian is the measure of the ratio between the arc length and the radius. By asking this question students have to look back and see where and how the arc length, radius, and radian relate to each other. This eventually will help them build the formal definition of a radian. Thus for this question the solutions must show or say that the radius, arc length, and radian each represent the measure or a combination of radii measures.
Question #6:
The purpose of this question is to take what students have investigated in the previous questions and solidify it. In other words, this question gives the students a way to see what they have investigated more clearly by the formula and angle measure. Thus by answering this question, students will be able to put together the formal definition of a radian through the explanation of why the ratio between arc length and radius is equal to the value of <AOB. Thus the solution of this question should be that these values are the same since they represent the same measure or a radian.
Question #7:
The purpose of this question is for students to come up with the definition of a radian based of the investigations they made in this activity. In essence, this is the question that ties the whole lesson together. Thus the solution should be based around the formal definition of a radian. In other words, the solution should have the around the statement of a radian is measured a the ratio of the arc length to the circles radius.
The purpose of this question is to first have the students play with the applet. That way they can familiarize with it and be comfortable with how the applet works. Secondly, this question is supposed to be thought provoking. By providing a vague question, it allows students to really think about what the answer really is and in turn write down some really great ideas on what a radian could be. Thus the solution of this question is that there will be multiple. The point is that students should see from the applet that a radian relates to the circle and might only list the characteristics that they on screen or if they have heard of radians before then they may answer this by saying that is is a unit of measure for an angle.
Question #2:
The purpose of this question was for students to focus on how a radian changes relative to the circle and to also see that as the radian increases so does the arc length, but the radius stays constant. Thus the solution of this question is that there will be multiple. In essence, a solution would have to showcase that as the angle and the arc length have a proportional relationship, while no matter how the radian measure changes, the radius stays constant.
Question #3:
The purpose of this question is for students to realize that the pink points represents 1 radian. In other words, the distance from one pink point to another is exactly 1 radius length. Thus from this the solution of this question is that there will be multiple. Some students may not see the relation between the radian measure in the green section and the length of the actual arc length at each pink point, while others may see this. In essence, a solution would have to show or state that the pink points represents a distance of 1 radian.
Question #4:
The purpose of this question is to reinforce the previous question or to provide more hints toward the solution for the previous question. In other words, this question provides more insight and foundation for students to build towards the idea that a radian is represented by the ratio of the arc length to the radius. The only difference is that instead on focusing on strictly the radian value, the students are now focusing on where that value came from. Thus for this question the solution would have to be a variation or very close to that the arc length changes with the angle measure but is much larger than the angle itself, while the radius still stays constant.
Question #5:
The purpose of this question is for students to start seeing that a radian is the measure of the ratio between the arc length and the radius. By asking this question students have to look back and see where and how the arc length, radius, and radian relate to each other. This eventually will help them build the formal definition of a radian. Thus for this question the solutions must show or say that the radius, arc length, and radian each represent the measure or a combination of radii measures.
Question #6:
The purpose of this question is to take what students have investigated in the previous questions and solidify it. In other words, this question gives the students a way to see what they have investigated more clearly by the formula and angle measure. Thus by answering this question, students will be able to put together the formal definition of a radian through the explanation of why the ratio between arc length and radius is equal to the value of <AOB. Thus the solution of this question should be that these values are the same since they represent the same measure or a radian.
Question #7:
The purpose of this question is for students to come up with the definition of a radian based of the investigations they made in this activity. In essence, this is the question that ties the whole lesson together. Thus the solution should be based around the formal definition of a radian. In other words, the solution should have the around the statement of a radian is measured a the ratio of the arc length to the circles radius.