Arc Length and Area of a Sector
This lesson is about teaching students to use proportions and GSP to see that the ratio of the arc length to the circumference is equal to the ratio of the area of the sector to area of the circle. The lesson will begin with a review of how to find the area and circumference of a circle and a review of the concept of central angles. The overall objective is for students to realize that the ratio of the arc length to the circumference is equal to the ration of the area of the sector to the area of the circle, regardless of the measure of the central angle
The primary learning outcome is for students to know that the ratio of the arc length to the circumference is equal to the ratio of the area of the sector to the area of the circle
Additional Learning Outcomes:
For students to achieve the primary outcome, they must first be able to perform a number of simpler skills. These would include setting up a proportion, finding the circumference of a circle, finding the area of a circle, and identifying a central angle. Students will also need to be familiar with GSP so they can draw the required sketches and use the measure and compute features of GSP.
Review the following concepts, demonstrating examples as necessary:
Circumference of a circle
Area of a circle
Central angle of a circle
Ideally, this step would only take five to ten minutes.
Instruct students to use GSP to create a sketch of a circle. There are no conditions on the radius. Students may draw circles of various sizes. Hide the point on the perimeter of the circle that is left from doing this.
Students should select two points on the perimeter of the circle. Put a third point in between these two points Use GSP to create a sector from these two points.
Draw two segments connecting the points on the perimeter to the center of the circle. Do this by selecting the point at the center of the circle and one of the two new points. From the construct menu, select line segment. Repeat this process for the other segment on the perimeter of the segment (make sure that the only item selected is the center point and the second point on the perimeter). See the image below for an example of such a sketch created in GSP.
Find the length of the sector and the circumference of the circle using GSP.
Find the area of the sector and the area of the circle using GSP.
Use the calculate feature of GSP to find the ratio of the arc length to the circumference.
Use the calculate feature of GSP to find the ratio of the area of the sector to the area of the circle.
The image below shows how these images would appear in GSP along with the image of the circle.
Click on either of the two points ( the points at the ends of the segment, not the point in the middle) placed on the perimeter of the circle. Drag the point around the circle. As the point is dragged around the circle, observe how the ratios changed. Make notes using paper and pencil about observations made. Make a general conjecture about how the three ratios are related.
Click here to explore this topic in GSP prior to attempting this lesson.
Students will be assessed based on the observations they submit. Teachers should look for evidence of recognition that the two ratios will always be equal regardless of the measure of the central angle. Hopefully, students will observe that each ratio is equal to the ratio of the measure of the central angle to 360.
As an extension, students could be asked to create congruent sectors. They could be asked how many congruent sectors can be created. They could also be asked to determine the measure of the central angle that would give the largest and/or smallest ratio of the area of the sector to the area of the circle.
Students that struggle with this concept could break the concepts into fewer pieces. For example, the students could first find only the arc length and circumference. They could set up a proportion comparing the ratio of the arc length to the circumference to the ratio of the measure of the central angle to 360. Once the students are able to recognize that these two are equal, then they could move forward to work with the area of the sector.
Students with exceptional needs may be given a peer helper to answer questions they may encounter. The instructor could also set up the sketch, measurements, and calculations in GSP ahead of time for some students. This would still provide the students the opportunity to work with GSP. Another possible would be to provide the vision impaired students with a magnifying program so they can still use GSP.
Possible modifications could include offering a peer helper for assistance. Assessment could also be modified so that these students were only expected to notice that the ratios were equal for a particular example rather than make a conjecture about all examples.
Title: Intermath Investigations for Geometry
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