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)