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 Banner Tuerck. His topic, from Geometry: finding the area of a circle.

How could you as a teacher create an activity or project that involves your topic?

There are many fun and exciting activities one may present to a class in order to initiate a lesson over calculating the area of a circle. An example would be to allow students to graph various size circles on a grid with squares of one unit and then have them count the number of squares contained within each circle. Obviously the students will have to deal with adding partial squares, thus resulting in an estimated area for the individual circle. Once students have calculated a few diverse areas, the instructor could then ask students to try to find a relationship between the radius of each circle and their corresponding area. Having circles of various sizes will allow student to get closer to deriving a more universal formula. For example, some students may realize that the area, when divided by the radius, is close to the radius times a number slightly greater than three, but less than four. Furthermore, if students are able to see that dividing the area by the radius leaves a remaining radius times a number greater than three , then some individuals in the class may go as far as to say that the area is three times the radius times itself. Although, this engagement activity would work fine, it may be wiser to give the students an even greater physical demonstration of where the area formula comes from. Therefore, I would recommend the specific activity provided by this link… http://illuminations.nctm.org/Lesson.aspx?id=1852

The above link leads one to a very hands-on and visual activity for students. It centers around students cutting out a specially marked circle that when folded and cut further as instructed eventually facilitates the students comprehension of the area formula as a direct relationship as seen with shapes like the square or rectangle (i.e. Area = Base * Height) except, with respect to the circle, the base and the height are now the radius (base) and the product of the radius times pi (height) or vice versa. Either of these activities along with the appropriate guidance should aid in getting students to become enthusiastic about the topic before attempting to apply the formal formula to given problems. Nevertheless, as stated earlier, it is my opinion that the illuminations activity may provide a more direct approach to a solid understanding and acceptance of the formula for the area of a circle.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

In relation to the area formula for a circle appearing in high culture, one could look at many architectural designs. However, I would like to briefly review the architectural design of a rather popular city structure that is the Logan Circle. The Logan Circle is a historical district in Northwest Washington, D.C. that remains one of the only circularly designed downtown districts occupied solely by residents instead of businesses. Furthermore, in relation to geometry, this historical landmark has a total area of .17 square miles. Architectural structures and designs such as the Logan Circle are a great way to get students involved in applying what can sometimes be considered dry mathematical formulas to real world situations. For example, an instructor could easily make the Logan Circle’s area the basis for an elaboration activity requiring students to work backwards in finding a potential radius one could realistically measure.

What are the contributions of various cultures to this topic?

Many ancient cultures contributed to facilitating the official area formula we use today. For example, before pi was even established or discovered as a constant representing the ratio of the circumference to the diameter of a circle, Euclid had already derived that the area was a product of the radius squared times a constant. However, it was not until Archimedes’ proof, which used the preexisting geometric properties of other shapes, did we arrive to our current formula (with an exception being made for the Archimedes notation of pi). Nevertheless, without straying from the topic of calculating the area of a circle, it should be noted that many cultures contributed to furthering the area formula by furthering their approximations and formulas for the mathematical constant pi. An example of one culture, as opposed to the more commonly referenced Greek mathematicians, would be ancient Chinese mathematicians such as Lui Hui, Zhang Heng, and Wan Fran. Each of these individuals had opposing views on the true value of pi. It is my belief that these opposing views occurred globally throughout history and led to the continuing examination of the ratio that is pi. Therefore, furthering the development of the area formula.

References

http://illuminations.nctm.org/Lesson.aspx?id=1852

http://en.wikipedia.org/wiki/Logan_Circle,_Washington,_D.C.#Geography

http://www.ams.org/samplings/feature-column/fc-2012-02

http://en.wikipedia.org/wiki/Liu_Hui’s_%CF%80_algorithm

http://en.wikipedia.org/wiki/Pi

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 Andy Nabors. His topic, from Geometry: defining the terms parallel and perpendicular.

How could you as a teacher create an activity or project that involves your topic?

One of the most appealing things, to me, about geometry is the amount of real world examples you can find that relate to the material. While some topics are easier to find (shapes), sometimes it is not clear why they are chosen. For example, it is easy to say “a stop sign is an octagon”, but much harder to answer “why are stop signs octagons?” This activity would explore that and have the students use characteristics of parallel and perpendicular lines to explain why they are used in the real world.

This would start by reviewing the definitions of parallel and perpendicular lines. Then the students would come up with and write down three varied examples each of real world occurrences of parallel and perpendicular lines. Then the student would write a two-to-three sentence explanation of why they occur, citing specific characteristics that make sense. For example, a two lane highway, while not fully parallel, has segments of road where the northbound and southbound lanes are parallel to each other. If the lanes were not parallel to each other than the lanes would intersect and the cars would hit each other. The class would have a discussion of what the students came up with, allowing for volunteers to share, then they would turn in what they had written so the teacher could check for students’ recognition and understanding of parallel and perpendicular lines.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Parallel and perpendicular lines have been used for… a long time probably, only no one had invented the terms parallel and perpendicular yet. The man that did bring these terms about in concise definitions was Euclid. In his Elements, Euclid clearly defines the terms and proves how to construct them with only a straight edge and compass. He also proves certain characteristics these lines have, like the angle relations when parallel lines are intersected by a line. Then he proceeds to use those relations to prove bigger and more complicated geometrical instances. If I was to include Euclid in a lesson, I would give a little biographical information about him, and then see if the students could do some of Euclid’s parallel and perpendicular straight edge and compass constructions and prove that they work. Then I would go over them with the class.

How can technology be used to effectively engage students with this topic?

Students would use graphing calculators for this activity. This would come after the definitions of parallel and perpendicular lines had been gone over. The students would be given a worksheet with two columns of linear equations, and some blank graphs. They would be told that each equation in one column corresponded with an equation in the other by being either parallel or perpendicular. The students would use the graphing calculator to check the equations to find which lines look parallel and perpendicular. When they find a match, they would graph the lines on a blank graph, write the equations underneath, and say whether they were parallel or perpendicular. Hopefully the students would pick up on the rules of looking at slope to find whether or not two lines are perpendicular or parallel. Graphing the lines by hand would show the students whether or not they are correct, as it may be easier to discern graphing by hand. Once all the equations had a match, the student would make a conjecture about how the slopes of parallel lines and perpendicular lines relate to each other.

Resources:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html (Euclid’s Elements)