Engaging students: Defining the terms perpendicular and parallel

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 Diana Calderon. Her topic, from Geometry: defining the terms perpendicular and parallel.

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

– This topic of parallel and perpendicular appears in art in the early 1900’s, late 1910’-1930’s. The movement was widely known as De Stijl, which in Dutch means “the style”. This movement had characteristics of “abstract, pared-down aesthetic centered in basic visual elements such as geometric forms and primary colors.” , the two main artists of this artistic movement were Theo can Doesburg and Piet Mondrian. The artistic movement started because of a reaction to the end of World War I, “Partly a reaction against the decorative excesses of Art Deco, the reduced quality of De Stijl art was envisioned by its creators as a universal visual language appropriate to the modern era, a time of a new, spiritualized world order”. As seen below, there are multiple lines, all of which are either perpendicular to each other or parallel, “De Stijl artists espoused a visual language consisting of precisely rendered geometric forms – usually straight lines, squares, and rectangles–and primary colors.”.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

– When we think of geometry a lot of people instantly think of triangles, SOHCAHTOA, and other 2D or 3D shapes. But when I think of geometry I think of the Greeks and Euclid, the literal father of geometry, only because I learned about him in Dr. Cherry’s class. With that being said, Euclid was one of the first mathematicians to define the term parallel, in specific, parallel lines. In 300 BCE Euclid stated some definitions for the basics of geometry, then give five postulates, “The postulates (or axioms) are the assumptions used to define what we now call Euclidean geometry.” The fifth postulate is what we want to focus on because it is called the parallel postulate, “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” He also states how to construct a perpendicular in Proposition 12, “To draw a straight line perpendicular to a given infinite straight line from a given point not on it.”, this construction states that by a given line AB and a point C not on the line then it is possible to construct a perpendicular on line AB.

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

– A good group project for the topic of parallel and perpendicular lines would be to allow the students to create a town. The requirements would be for the student’s town to be no bigger than 100 square inches, the students would have the liberty to create any quadrilateral shape as long as it meets the 100 square feet requirement. Another requirement that the project would have is that there must be at least 4 parallel streets, one perpendicular street that is only perpendicular for one of the parallel streets and finally one diagonal street that intersects 3 parallel streets. A town obviously needs to have shops so the students would be required to put shops within the town but must have an explanation as to why the shops were chosen. Finally the students must bring a physical final product, the shops must be in 3D form, the town area may be made with cardboard, cardstock or any material that would sustain the shops on top of it, the streets and corners of streets must be labeled with the corresponding angles. Finally, when they bring their final piece as a class we will walk around and allow the groups to present their product. As an exit ticket for presentation day the students must turn in the definitions of parallel and perpendicular in their owns words and how it was shown in their project product.

Citations:

o Mondrian Returns to France (Figure 1)
https://worldhistoryproject.org/1919/mondrian-returns-to-france

o De Stijl
https://www.theartstory.org/movement/de-stijl/

o The Three Geometries
https://mathstat.slu.edu/escher/index.php/The_Three_Geometries

o Euclid’s Elements I-XIII
https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html#posts

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