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 Banner Tuerck. His topic, from Precalculus: finding the equation of a parabola from the focus and directrix.

An interesting way to present the mathematics behind parabolas and their focus points is through the applications it has in science present in our everyday lives. http://spacemath.gsfc.nasa.gov/IRAD/IRAD-4.pdf

The above link includes a great engagement activity for students to do as a group activity. The first exercise presented involves the students in the design of a parabolic dish after observing the properties of a satellite dish with a radio receiver (located at the focus). Once the students have completed the design of the parabolic dish the instructor could then use the second half of the pdf from the link as an elaboration activity. The instructor could either keep the students in the groups or have them work the problems individually. Nevertheless, the second activity would be for the students to work problems one and two, which deal with aiding a bird watcher and a hobbiest in determining the focus points in order to design their parabolically shaped tools. The last problems are excellent real world examples of why one would need to know and apply the mathematics for parabolas. This will encourage students to view everyday objects with a more mathematical respect.

The understanding of the relationship between the focus point and the directrix of a standard parabola is fundamental when students extend their mathematical and science education in post-secondary courses. For example, when students reach multivariable calculus they will graph and study the properties of conic sections on a three dimensional scale. With respect to this topic the students can apply their preexisting knowledge of two-dimensional parabolas to the paraboloids presented in this course. Furthermore, if students from a pre-calculus high school course were to not keep with the theoretical study of mathematics they could benefit greatly from this topic in careers such as architecture, art, or graphic design.

As an instructor of a pre-calculus course one has many technological resources to use in order to construct an elaborate lesson on the directrix and focus of a parabola. For example, modern graphing calculators allow instructors to link their calculator to a projector and show the entire class various parabolas in order to further visualize the changing distances to these specific points. Furthermore, I believe a unique homework assignment would be for students to graph given quadratic equations with an online resource such as http://www.wolframalpha.com/. This assignment would also be a great review of how to apply the distance formula. I recommend having the students check that the points on the parabola are equidistant apart from the focus and the directrix they have already found after graphing and computing. Another idea is requesting (for full credit of the assignment) the students use the following link: https://www.khanacademy.org/math/algebra2/conics_precalc/parabolas_precalc/v/parabola-focus-and-directrix-1

to facilitate their understanding of the definition of a parabola as well as the importance of the focus point and directrix line. This is a way to involve technology while simultaneously ensuring that students review key aspects of the lesson after it was given by the instructor during class time.

References

http://spacemath.gsfc.nasa.gov/IRAD/IRAD-4.pdf