Part+IV+-+Tasks

Task 1 - Developing a Definition of a Circle

To show the definition of a circle being an infinite number of points at a fixed distance away from a center, we can have students approach the board where the teacher will have set up the following: The teacher should create a center on the board, just placing one point. They should then express to the class that this is a center point; the teacher should then get a ruler, or some measure of the "radius" which is a fixed distance. Now, one at a time, the students should approach the board and place a point wherever they feel like at the fixed distance from the center.

After many of the students place their point around the center, they should begin to notice something forming. Allow all of the students to place their point on the board and this should create a circle. At this point the instructor should ask the students to define a circle based on the previous activity. Discussion should follow immediately after and should lead to ideas about circumference and the properties of a circle such as it being composed of 360 degrees.

An alternative activity could make use of the entire classroom setting. With all of the desks cleared out of the way, and the teacher standing in the middle with a string of fixed length. As students walk into the classroom, they should use the other end of the string for reference and pick a location around the teacher to stand.A great follow up would be to use a measure of string that each student holds (making up the circumference); later on in the class after students have undertaken some learning about circles and pi, we can bring back this circumference string. Now the teacher should lead the class in measuring this string, and also measuring the "radius" string. At this point, the students should have at least seen the formulas for circumference and area, so you should be able to take these amounts, and using arithmetic, students calculate and end up with something fairly close to pi, 3.14. This should be a defining moment in their fluency of circle.

At first, students may be uncertain of what activity is taking place, but as more of their classmates join and fill in the perimeter, it should begin to sink in. This is a great hook activity since students will get to see the formation of a circle from the very core of its definition. They will also being engaged in the activity since they will get to see a circle form from just a center point, to a complete circle with all its points an equal distance from the original center point. Discussion should follow this activity where the teacher focuses on the definition of a circle and reviews the parts of the circle. The students should realized the string that held the marker from the center point represents the radius while the points they made form the circumference.

Task 2 - Finding the relationship between circumference, diameter, and pi

In this task, students will learn the circumference of a circle divided by the circle's diameter will always be pi. Teachers will need to explain that pi is a very specific number that will always be the solution to a circle's circumference divided by its diameter. One way to show students how specific pi actually is, they can have the number pi printed on a scroll of paper that two students can hold and roll out to see the first part of pi. Since pi has an infinite number of digits, after the students roll out enough of the number, the teacher will need to tell the students that was only part of the number and that pi's digits go on forever without repeating. The students can use this information along with their prior knowledge of circles to develop the formula for circumference. In this activity, students will have a table they need to fill out. To fill out the table the students should work in pairs and go around the room and measure circular objects. For each circle they measure, they need to include what the circumference and diameter were and then divide the circumference by the diameter. Students should get that every circle's circumference divided by diameter is 3.14159 (pi). For students who don't get exactly pi, teachers should be prepared to explain that student's measurements could affect the final number. The student might also round their measurement which would affect their final answer. Now that the students know pi's relationship with circles, we can use the formula we just made to develop the formula for a circles circumference. (Table came From Circumference of a Circle lesson)
 * Circle || Circumference || Diameter || Circumference/Diameter. ||
 * 1 ||  ||   ||   ||
 * 2 ||  ||   ||   ||
 * 3 ||  ||   ||   ||
 * 4 ||  ||   ||   ||
 * 5 ||  ||   ||   ||

After the class finds the formula C/D = pi, the teacher can lead a class discussion on using this formula to develop the formula for circumference. To do this, the teacher could ask students to solve for "C" or to make "C" alone on one side of the equation. At this point, students can work together and come up with the formula Circumference= Diameter x pi. The teacher could ask how they got this formula which they should respond with multiplying diameter to both sides. If the students have trouble find the formula by themselves, the teacher could put the original formula on the board and as a class work out solving for circumference. The teacher can also ask what part of the circle is also related to the diameter to make sure students see the connection between C = D x pi and C= 2r x pi. Once the students have found the formula for circumference, they can apply the formula to find the circumference given a diameter or radius of a new circle.

Task 3 - Finding a Circles Area

For this lesson, begin with students estimating a circles area with a given radius. They can do this multiple ways. Some students might draw squares around the circle with side lengths equal to the diameter of the circle. Other students might draw square units inside the circle and then count how many full square units fit inside the circle. If students are having trouble thinking of ways to estimate the circle's area, the teacher can tell the students to think of partitioning and tiling or to try using area formulas they already know to help them find the area of parts of the circle. Once the students have had time to make estimates have the students call out their guesses. The teacher could also have some students who had good methods come up to the board and show the class how they came up with their estimate. The teacher can then tell the students there is a formula that will give them the precis area of any circle. In today's activity, students will develop the formula for area of a circle which they use to find the actual area of the circle they were estimating and see which students were the closest.

For this activity, have students use their compass and scissors to draw and cut out a circle on a sheet of white paper. Once the students have their circles cut out, they should take a color marker and trace the circle's circumference in one color and the radius in a different color. After that, tell the students to cut their circles like they would cut a cake. At this point, teachers can have students cut different amounts of pieces. For example, one student could cut their circle into 8 slices while another cuts theirs into 16. After the students have their pieces, they should arrange them on their desks so the edges of the slices match up so their is no spaces in between the slices. (See fig. 12. 80) At this point, the teacher should point out that the total area of the circle is represented in the slices. The teacher should ask the students if the new shape they made resembles any other shape that they know the are of. Students who cut more pieces might be able to see it more easily. Students should say that it resembles a rectangle. Students should also notice that all of the color they marked the circumference with is on the bases of their rectangle and the sides are the radius of the circle. If students don't make these connections, teachers can have them cut their pieces in half so the rectangle shape is more apparent and they can ask the students where the marks they previously made are on their rectangle. After the students notice the circumference is the base and the radius is the side, they can use the circumference formula and the formula for the area of a rectangle to derive the area of a circle. Teachers need to point out that the circumference of the entire circle is shared between the bases of the rectangle so students know to only use pi x r instead of 2 x pi x r in the area formula. Students should come up with the area formula of a circle is pi x r squared. The teacher can work through finding the formula for students who might have trouble finding the formula or have a student who found the area come to the board and show their work.

(Beckmann pg. 533//)//