Engaging students: Finding the area of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Rebekah Bennett. Her topic, from Geometry: finding the area of a circle.

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Culture:

The area of a circle is used in our everyday life. Landscaping uses this topic quite a bit. Suppose a person wants to put a circular pool or even a fountain in their yard. The landscaper needs to know the area of the basic circle that is being used so that they can make sure there is enough land to build on. We also know contractors use this everyday too. When building a circular building, the contractor needs to know the area of the base of the building so that he/she can clear a big enough area. They also use this when building circular columns, such as the ones you would see on a big, fancy building. The contractor must know how much area the base (circle) takes up to see how much of the platform they have left to work with. Then he/she can now see how many evenly spaced columns will fit on the platform. A room designer also comes to mind. Let’s say if someone wanted a circular table placed in their living room, the designer needs to know how much space (area) the table takes up in order to figure out how much area is left in the room to fit other items comfortably. These are all instances where someone in the artistic world would need to use area of a circle.

 

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Application and Technology:

To explore this topic, I would give each student a cut out of a circle, each circle having a different size. Then I would tell them to figure out the area of the circle. I would give them hints as to how would you use the radius, diameter, and circumference within a formula. I would suggest the idea of splitting the circle into even pieces, and then ask the students if there is a way that they can transform the pieces of a circle into a more familiar shape. The students would have about 5 minutes to experiment on their own and then I would show them this video.

This video shows the students a more in depth definition of area of a circle. The video actually derives the formula from the normal area formula of a parallelogram (base x height). Here we pull the whole circle apart, piece by piece to create a parallelogram. The video relates height to radius and base to ½ of the circumference. These are both previous terms that the student already knows. The guy in the video manipulates the area formula for a parallelogram to derive the area of a circle. This video is a great way to show students that there is more than one way of solving for the right answer but also more importantly, it shows where the formula for area of a circle actually comes from. This gives the student a justification as to how and why we created this formula, relating back to the exploration.

After watching the explanation from the video, the students would now have a chance to replicate the demonstration with their original circle. By having the students recreate the video demonstration themselves, it gives them a better understanding as to why the formula works like it does and they can see how the formula works with a guided hands on approach.

 

 

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Curriculum:

From previous math courses, the student should already know the terms of a circle such as; radius, diameter and circumference. The student should know how to find the radius given the diameter, vice versa. The student knows that the circumference is the perimeter of a circle and how to find it, given the radius or diameter. They should already know the term area: space that an object takes up. The student should know how to find the area of a rectangle and parallelogram: (length x width) or (base x height). This activity shows how to relate the area of a circle to the area of a rectangle, given the radius and height, which is the same thing. The student can now create a formula for the area of a circle by using the same method as solving for area of a rectangle or parallelogram. The area of a circle extends the previous knowledge that every student should learn in algebra before entering a geometry class.

 

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