Engaging students: Introducing the parallel postulate

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 Eduardo Torres Manzanarez. His topic, from Geometry: introducing the parallel postulate.

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A2) How could you as a teacher create an activity or project that involves your topic?

The Parallel Postulate is an interesting statement that intertwines line segments and angles. This postulate states that if a straight line intersects two straight lines and the interior angles on the same side add to less than 180 degrees, then those two straight lines will intersect on that side if the lines are extended. Simply, if a straight line intersects two other straight lines and the interior angles on the same side add up to 180 degrees then the two lines are parallel. One activity that can get students to understand this axiom how test the validity would be to provide sets of straight-line segments and ask students to form interior angles and find their measurements. This would be particularly best to be done with technology such as a software like GeoGebra. Students would be given a set of line segments. First, provide nonparallel line segments such as the ones below.Next, ask students to draw any line segment such that it intersects the two previously given. Letting students make their own particular line segment can suggest that the validity of the statement is universally true.

Now students can use the angle tool to measure the interior angles on both sides. The pictures below are an example.

So, in this example, the right-side interior angles add up to less than 180 degrees and so the given two lines will intersect on the right side. Students can check that the lines segments intersect by placing lines over these segments and check for an intersection. The following image provides evidence as to this being the case for the example.

Hence, this example shows some truth to the postulate. This activity can be further enhanced and propelled by giving students lines that are already parallel and checking any set of interior angles made by a third line segment. Students will find that any segment created will result in the interior angles on both sides to add up to 180 degrees exactly. Such an activity like this would be useful as an introduction to the Parallel Postulate.

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D1) What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid, a Greek mathematician, came up with the Parallel Postulate in his discourse titled Elements which was published in 300 BC. Elements is made up of 13 books that contain definitions, theorems, postulates, and proofs that make up Euclidean Geometry. The reason Euclid wanted to accomplish this was to ascertain all of geometry under the same set of axioms or rules so that everything was related to one another. Euclid’s accomplishment in doing this has resulted in him being referenced as the “Father of Geometry”. There is not that much information on Euclid’s life from historical contexts, but he did leave an extensive amount of work that propagated many fields in math such as conics, spherical geometry, and number theory. Elements is estimated to have the greatest number of editions, second to the Bible. The Parallel Postulate by Euclid led to many mathematicians in the 19th century to develop equivalent statements within the contexts of other geometries. Hence Euclid was able to propagate geometry even further, way after he passed away.

 

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D2) How was this topic adopted by the mathematical community?

Ever since Elements was made known through the mathematical community, many individuals tried to prove the Parallel Postulate by using the other four postulates Euclid wrote. There is evidence to suggest that Euclid only wrote this particular postulate when he could not continue with the rest of his writings. So, the mathematical community sought out to find a proof for it since the postulate was not clear to be trivially true, unlike the other postulates. Some mathematicians such as Playfair wanted to replace the Parallel Postulate with his own axiom. It was finally shown in 1868 that this postulate is independent of the others and therefore cannot be proven by the other postulates by Eugenio Beltrami. There has been development in a specific type of geometry known as absolute geometry which actually derives geometry without the Parallel Postulate or any other axiom that is equivalent to it. This shows how much the community has been up to challenging the postulate but also how to proceed without it to see if Euclid could have done the same.

 

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