Engaging students: Defining intersection

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 Ethan Gomez. His topic, from Geometry: defining intersection.

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How can this topic be used in your students’ future courses in mathematics or science?

In geometry, students gain a better conceptual understanding of what an intersection is in mathematics. Particularly, by the end of geometry, students should be able to understand that different figures in mathematics can intersect, and depending on the nature of those figures, could intersect at more than one place. In Algebra II, students begin learning about rational polynomials. Often, the graphical representation of rational polynomials contains either vertical, horizontal, or slant asymptotes (these are the common asymptotes in Algebra II). Students could make a connection between what an asymptote is and the definition of intersection. Namely, an asymptote is some sort of “invisible line” that a function cannot intersect. Thus, by understanding what intersections are in geometry, they are able to better understand the idea of a lack of intersection. This characteristic of asymptotes should then be intuitive by students, so all they would need to learn is that the functions approach the asymptotes but never cross it, i.e., intersect it. This is the new knowledge they can add to their prior knowledge.

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How does this topic extend what your students should have learned in previous courses?

Students will most likely have taken Algebra I before Geometry. Thus, students should have discussed solving systems of linear equations. Visually, they should understand that the solution to the system of equations should be a single point in the cartesian plane, particularly a point of intersection. So, students are aware that figures in mathematics can intersect. In geometry, we introduce more figures instead of just dealing with lines. Thus, these figures can intersect, and depending on the figures, they may intersect in more than one point. Up to this point, students have not seen figures in mathematics that could intersect in more than one point, thus extending their idea of what intersections may look like.

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How can technology be used to effectively engage students with this topic?

I wasn’t able to find anything online; if I had time, I’d create a Desmos activity that reflects the ideas I’m about to propose (since I know Desmos has a lot of cool features). On Desmos, your can use sliders to adjust different variables. Thus, I would write two slope-intercept linear equations with slider-variables for the slopes and the y-intercepts. Additionally, I would write an equation for a circle with sliders-variables for the radius and center coordinate. Student would then be able to manipulate the location of the two lines and the circle, and they will be able to see the different kinds of intersections — intersections that they may not have seen in Algebra I. For example, a line can either intersect a circle at two points, one point, or no points; students would be able to visually see what each of those cases looks like. Additionally, students could make the lines perpendicular and make the circle tangent to both lines just to get them thinking about different theorems of circles and lines.

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