Engaging students: Using the undefined terms of points, line and plane

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 Alizee Garcia. Her topic, from Geometry: using the undefined terms of points, line and plane.

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

There is various way I could create an activity for this topic, but I think one that would be the most successful a project for the students in which they can better understand the terms. Since all three terms are related and relatively simple to describe the project could also be an in-class activity depending on the time given. However, in this project the students would have to take pictures of real-world examples for a point, line, and plane as best as they can and describe why they chose the examples they did. It is important that when teaching geometry as well as other lessons, that real-world examples are given to help students better understand the topics. Also, students can give their best definitions of the terms as well as drawing out them. This will allow students to think about the terms mathematically and as real-world subjects too.

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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The use of undefined terms point, line and plane can be used in video games such as Minecraft and call of duty. Both games consist of a map of some sort with different coordinates of safe zones or just where the game will take place. In call of duty, using an aiming weapon allows for the player to find a point and from there to where they are aiming from is the line that will connect it. As well as in Minecraft, you are able to build off of other buildings as well as being able to connect the points in a certain grid in order to succeed. I think video games and technology would be the most common pop culture examples that this topic will appear in. Although there are far more video games that relate to the undefined terms of point, line, and plane, it is a good way to let students understand how geometry can be seen in the real world.

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How was this topic adopted by the mathematical community?

The undefined terms point, line and plane, are based off Euclidean geometry, which was brought up from Euclid of Alexandria, a Greek mathematician. This topic of the undefined terms point, line, and plane were discovered after the non-Euclidean was discovered. The topic of part of Euclidean geometry which is the mathematical system that proposing theories based off of other small axioms in which these are those small axioms. These terms are considered undefined due to the fact that they are used to create more complex definitions and although they can be described they do not have a formal definition.  Euclidean geometry was said to be the most obvious that theories brought from it were able to be assumed true. Although this is not what makes up the entire Euclidean geometry, it is what is able to allow these terms to be undefined and furthermore used to define more complex terms.


Artmann, Benno. Euclidean Geometry. 10 Sept. 2020, http://www.britannica.com/science/Euclidean-geometry.

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