Teacher Guidelines
Materials Needed:
This applet is fairly easy to use. The only aspect of the applet you can physically drag are the two point labelled A, B, O, and X. The points A and B, drag the radius along the circle which in turn changes the angle, while the points O and X can be used for changing the side of the radius. This allows students to play around with angle side and also with the size of the circle, which can lead into investigations about angle measures across all circles rather than a single one. The pink points on the circle represent an angle measure of one radian. That way students have a reference point to utilize when asked about radian measures. Lastly are the check boxes labeled, Ratio of Arc Length to Radius, <AOB, and Semicircle. On screen the only check box available is ration of arc lengths to radius one. By clicking this it basically shows what it says, but it also reveals the <AOB check box. The same idea goes for this check box. It shows what it says and lastly reveals the semicircle check which simply shows the degree measure and the radian measure of a semicircle.
Possible Misconceptions:
One misconception a student may have is that they might not understand that an angle measure does not have to be equal to the arc length. For instance, by looking at the diagram, it seems that the angle made by the two radii at points A and B should be equal to that of the arc made by A and B. So that could be a misconception some students may occur. Another misconception a student may come across is with the semicircle check box. Since the applet only provides the symbol for pi and not the value it may cause some confusion since some students may not realize that the symbol for pi represents a number. Also it may cause some confusion since some students may be taken back by the sudden change from number values to symbols.
Possible Guiding Questions:
A possible way to sequence the responses is to start with the definition of a radian that they created at the beginning of the activity. For this, have the class categorize themselves based off there definition. In other words, have students in groups come up and write down what the most important part of a radian. Next have a classroom discussion on which of these are the most important based off the exploration made in the activity and create a class definition. Next pull up the actual definition of a radian and compare the class definition to it. With this, focus mainly on the main ideas that are emphasized in the activity such as the relationship between the arc length and the angle and the relationship between the radius and the angle. With that in mind, continue to lead into a discussion about what each piece of the definitions represent and go further in depth with how the classroom definition compares to the actual one and how can be improved or change if it needs to be. This can lead students to build a conceptual understanding of a radian.
- Activity #1 Worksheet
- Computer/Laptop
- Geogebra Applet
This applet is fairly easy to use. The only aspect of the applet you can physically drag are the two point labelled A, B, O, and X. The points A and B, drag the radius along the circle which in turn changes the angle, while the points O and X can be used for changing the side of the radius. This allows students to play around with angle side and also with the size of the circle, which can lead into investigations about angle measures across all circles rather than a single one. The pink points on the circle represent an angle measure of one radian. That way students have a reference point to utilize when asked about radian measures. Lastly are the check boxes labeled, Ratio of Arc Length to Radius, <AOB, and Semicircle. On screen the only check box available is ration of arc lengths to radius one. By clicking this it basically shows what it says, but it also reveals the <AOB check box. The same idea goes for this check box. It shows what it says and lastly reveals the semicircle check which simply shows the degree measure and the radian measure of a semicircle.
Possible Misconceptions:
One misconception a student may have is that they might not understand that an angle measure does not have to be equal to the arc length. For instance, by looking at the diagram, it seems that the angle made by the two radii at points A and B should be equal to that of the arc made by A and B. So that could be a misconception some students may occur. Another misconception a student may come across is with the semicircle check box. Since the applet only provides the symbol for pi and not the value it may cause some confusion since some students may not realize that the symbol for pi represents a number. Also it may cause some confusion since some students may be taken back by the sudden change from number values to symbols.
Possible Guiding Questions:
- (3) What are the values that show up on the screen as you drag your point to the pink point? What do you think this measurement represents?
- (4) How does changing the angle measure affect the arc length? Is it increasing or decreasing? (Do the same for radius)
- (5) What did you notice in the previous question? How can that helps us with this question?
- (6) Drag a point around the circle. What do you notice about the values as you change the angle?
- (7) Use the same question as in question 5.
A possible way to sequence the responses is to start with the definition of a radian that they created at the beginning of the activity. For this, have the class categorize themselves based off there definition. In other words, have students in groups come up and write down what the most important part of a radian. Next have a classroom discussion on which of these are the most important based off the exploration made in the activity and create a class definition. Next pull up the actual definition of a radian and compare the class definition to it. With this, focus mainly on the main ideas that are emphasized in the activity such as the relationship between the arc length and the angle and the relationship between the radius and the angle. With that in mind, continue to lead into a discussion about what each piece of the definitions represent and go further in depth with how the classroom definition compares to the actual one and how can be improved or change if it needs to be. This can lead students to build a conceptual understanding of a radian.