Circles and Parallelograms
Lesson 9 of 12
Objective: SWBAT discover the relationship between the area of a circle and the area of a parallelogram
As students enter the room, they will have a seat, take out their Problem of the Day (POD) sheet and begin to work on the question on the SMARTboard. The POD allows students to use MP 3 continually based on the discussions we have about the problem each day.
The POD today will be a brief, review question to focus the student’s attention on finding the area of a circle. As we discuss as a class, I will guide the discussion toward the concept of area, what it means, and how it is calculated.
What is the difference between the circumference and the area of a circle?
- Learning Target
Students will see that finding the area of a circle is related to finding the area of a parallelogram.
Today students are going to use the measurements of a circle to investigate the relationship between the area of a circle and the area of a rectangle. Students will receive two copies of the circle (each a different color) and a ruler. Students measure the radius of their circle, calculate the circumference, and record this information on their sheet. Students will then carefully cut their circle in half along one of the diameters and trade with another student so that they have two semicircles that are different colors. Students will then cut out each section of the semicircles so they have ten pieces. Take the ten pieces and arrange them to form a parallelogram with one color on top and another color on the bottom.
Students will then discuss and explore how they could develop the formula for finding the area of a circle by making a parallelogram shape using the circle pieces. Students will be asked to examine the parallelogram shape. They will discuss with a partner how the base of the parallelogram relates to the original circle. How does the height of the parallelogram relate to the circle? We will discuss their conclusions after they have had time to work through the activity. I want them to determine that the base of the parallelogram is equal to one-half of the circumference of the circle (I will point out to students that the rounded edges are actually the circumference of the circle-provided nobody makes that distinction-and only half of them are along the bottom base while the other half are along the top base, so each base is equal to 1/2 of the circumference of the original circle). The height of the parallelogram is equal to the radius of the circle. The height goes from the circumference (rounded part of the piece) or the edge of the circle to the point (center of the circle) of any section of the circle. We know that the base times the height is the area of a parallelogram. Therefore, we can find the area of a circle the same way. (1/2 C) is equal to the base and r is equal to the height, so (1/2 C) r is equal to the area of the circle. To find (1/2 C) students need to use the formula for the circumference of a circle which is either C = d or C = 2r. If students use C = d as the formula for finding circumference, they will have to realize that 1/2 of the diameter (d) is the radius (r), so 1/2 (d) is equal to (r). I will encourage students to extend their understanding of where the formula for finding the area of a circle comes from by using other circles. We will discuss how all of the area formulas they have learned so far are all based on finding the area of a rectangle.
To end class we will summarize their understanding of the relationship between finding the area of a circle and finding the area of a parallelogram. Do they recognize that the base of the rectangle is half the circumference of the circle? Do they understand that the radius of the circle is the height of the rectangle?
If the measurements of a circle are C =12 and r = 4, how can you use these measurements to find the area of a rectangle?