Engaging students: Parallel and perpendicular lines

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 Emma Sivado. Her topic, from Algebra: parallel and perpendicular lines.

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D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

I would take my students back to the time of Euclid of Alexandria, around 300 B.C., and his great book The Elements. Little is known about Euclid except the book he left behind which is the foundation of geometry, algebra, and number theory, still to this day. Euclid wrote this book in an axiomatic way, this means that he assumes common notions, definitions, and postulates to be true and then bases all his propositions and axioms on these assumptions. Does this sound like the way that we do mathematics today? To understand how influential and enduring the Elements is I would present this incredible fact; other than the Bible, Euclid’s Elements is the most published, translated, and studied of all books in the world.

Now we would put on our Euclid caps and turn to Proposition 12 and Proposition 31. These propositions tell us how to draw parallel and perpendicular lines based only on the definitions, common notions, and axioms of Euclid. We would do the constructions step by step, straight out of Euclid’s Elements.

euclid1 euclid2






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A.2: How could you as a teacher create an activity or project that involves your topic?

To engage the students in the lesson on parallel and perpendicular lines, instead of sitting in class and listing real world examples of parallel and perpendicular lines, I would take the students out of the classroom and take a tour through the school like a bird watching group except our goal is to list all the parallel and perpendicular lines inside and around the school. We could go to the cafeteria, the gym, and walk around the outside of the building. When we got back to class we could create a long list of all the parallel and perpendicular lines that we see to hang on the wall during this unit. After we list the examples, I could ask some thought provoking questions:

“Why are these parallel and perpendicular lines important?”
“How would the world be different without parallel and perpendicular lines?”



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E.1: How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

A great activity I found on parallel and perpendicular lines involves using a graphing calculator to discover the similarities in slope between parallel and perpendicular lines. First, you give the students a list of equations to graph on their calculator. Next, you ask them to compare the graphs and identify which lines are parallel and which are perpendicular. Last, you ask them to compare the slopes of the parallel and perpendicular lines. Hopefully, they will discover that parallel lines have the same slope and perpendicular lines have the opposite reciprocal slope. This activity can be done easily because the students should already be familiar with graphing calculators, slope, and y-intercept. The activity would not take much time and can easily be differentiated based on the skill level of the students in your class. You can give some students difficult numbers or more lines to analyze if they finish the initial activity quickly. Also, you could take this one step further and give the students large sheets of graph paper and let them draw the lines and present their findings in front of the class.




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