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 comes from my former student Daniel Herfeldt. His topic, from Geometry: finding the area of a circle.

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Learning area of a circle as well as the circumference in geometry is beneficial for anyone who is looking to pursue any career involving math. This includes anything from a math teacher to an architect. This helps with most future courses in mathematics. With this said, it will be very beneficial when going into pre-calculus. This is because in pre-calculus you will deal a lot with trigonometry. This includes such things as the unit circle, which is a great deal to pre-calculus itself. Being familiar with the equations of a circle helps to understand why things work in a unit circle. It can help with simple things such as why x2+y2=r2. Knowing the area of a circle will make the class easier to understand in all. This topic is also very important to future architects. The reason for this is because if an architect doesn’t know the area of a circle or any other shape, it would be very difficult to construct a building. If one cannot figure the dimensions of a pillar to help support the ceiling of a building, the building will have a possibility to collapse. This causes the structure of a building to rely highly on the dimensions, area, and volume of all shapes including the circle. This proves that the importance of the area of a circle to be very high. Most students will not know that everywhere they go, circles are needed. Informing them about these small details could have the students more eager to learn. Giving them great real world examples might also help the students understand and grasp the knowledge that you are trying to teach them because it relates to them.

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Circles will be anywhere you go. They are in your everyday TV show, video games, and movies. Although at first glance you might not actually see them, they really are there. When creating a character for an animated movie or a popular video game, artist first start to draw with simple circles and lines. They need to figure out a certain area of the character’s face to be able to fit the facial features. For example, they need to be able to fit eyes, a mouth, a nose and a few more features. From this, they will go on to the animation of the character. This also includes circles because in an animation, when you are wanting to move one object, you have to move it all. The same process applies when working on the landscape properties. It will mostly start with simple lines, circles, and boxes. From there, it will progress into more advance steps, putting more and more detail into it. When moving a character, it is also necessary to move the landscape and surroundings as well. This would be great to tell a class because students will be able to relate to the subject. Most kids in the high school level will play video games. Whether the game is on their phone or a gaming console, they still require the beginning steps. If the student doesn’t play video games, they can relate to it due to watching an animated movie. This will be a great way to engage the class in the first few minutes of class. Below is a picture of the progression of drawing a Pokémon.


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Many ancient civilizations have been fascinated with circles. Circles can be seen in many ancient structures and buildings from the Roman Coliseum to Stonehenge.

One ancient civilization fascinated with circles were the Greeks. The believed the circle to be a sacred divine shape mostly based on its multiple points of symmetry.  The Greeks also invented a puzzle called squaring the circle, in which the person had to construct a square with the exact area of a circle a compass and a straight edge. This puzzle has been proved mathematically impossible.

Other instances in which circles played an important role in history were the circles that appeared in the crops in different areas of the world. These crop circles have been argued to be a hoax while others indicate it is not possible for the crop circles to be the work of humans.  Regardless, of their origin, these crop circles continue to fascinate us.

Circles continue to have significance today. They are used in logos and other things usually to signify unity and harmony. Even the Olympic symbol is made up of five interlocking colorful rings. The circle is still found today enclosing the all seeing eye over the pyramid in the dollar bill on the US currency.

The significance of presenting this information to students, especially high school students, is to give them background information. I think many high schoolers will be interested to learn how circles have been significant in other cultures throughout history.  Students can be given a short introduction in the subject and asked to look for more instances in which circles played a role in an ancient civilization and then bring that trajectory to modern times.


Website: https://nrich.maths.org/2561






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