# Engaging students: The distance formula

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 Samantha Offutt. Her topic, from Geometry: the distance formula.

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

https://livelovelaughteach.wordpress.com/2012/01/20/travel-the-us/

I found this awesome idea to help students see the distance formula in a real world view. Students often complain that they will never use math outside of classrooms, or that it’s pointless to learn math because they want to be something completely unrelated to math (i.e. Professional Body Piercer). However, most people will take a trip, at least once in their life. That’s why I find this project to be beneficial. Students have the freedom to create a trip catered to their wants and desires, instead of doing countless problems and worksheets to drill the formula. They get the time to learn about different places and still apply mathematical concepts.

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

Because the distance formula is a variation of Pythagorean Theorem, it was quite funny to come across a scene from The Wizard of Oz. In this scene, the wizard was giving the scarecrow his Th.D, his PHD in Thinkology. Immediately after receiving this, the scarecrow recites his version of the Pythagorean Theorem.

“The sum of the square roots of two sides of an isosceles triangle is equal to the square root of the remaining side.”

This statement works if two sides of the triangle are 1 and the last angle is the sqrt(2). He states it like he’s talking about the distance formula, using square roots, but perhaps, make it a right triangle, and specify the remaining side is opposite the right angle and you can have your brain, Scarecrow! But then again, this is Dorothy’s dream, so perhaps it’s Dorothy who doesn’t know the proper way to state such a theorem!

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

The distance formula is motivated by the Pythagorean Theorem, which it’s name alone brings Pythagoras to light. Even though not many people know whether or not Pythagoras actually proved it or whether he gave it it’s definitive form, I know not. But, Pythagoras was a great philosopher and Mathematician (Mastin).

Euclid, did write proofs for the Pythagorean Theorem (I.47). In the books, Geometry: Euclid and Beyond by Robin Hartshorne, and Euclid’s Elements expansion of this concept is given. From the proof of Pythagorean Theorem, to the distance formula used to model axioms (I1)-(I3), (B1)-(B4), and (C1)-(C3) and define congruence for line segments (Hartshorne 87).

Descartes invented the coordinate system, thus his role in contributing to this formula is quite vital. He created a bridge between the worlds of Algebra and Geometry. Watch this Khan Academy Video to gain more insight about Descartes and said bridge: https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/descartes-and-cartesian-coordinates

Resources

Clark, Jeff. “Jeff Clark’s Math in the Movies (Wizard of OZ).” YouTube. YouTube, 26 Jan. 2013. Web. 09 Oct. 2015. <https://www.youtube.com/watch?v=jbvip1Ot6jQ&gt;.

Fields. “Travel the US, with the Distance Formula!” LiveLoveLaughTeach. N.p., 20 Jan. 2012. Web. 09 Oct. 2015. <https://livelovelaughteach.wordpress.com/2012/01/20/travel-the-us/&gt;.

Hartshorne, Robin. Geometry: Euclid and beyond. New York: Springer, 2000. Print.

Khan, Sal. “Introduction to the Coordinate Plane.” Khan Academy. N.p., 12 Feb. 2012. Web. 09 Oct. 2015. <https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/descartes-and-cartesian-coordinates&gt;.

Mastin, Luke. “Pythagoras – Greek Mathematics – The Story of Mathematics.” Pythagoras – Greek Mathematics – The Story of Mathematics. N.p., 2010. Web. 09 Oct. 2015. <http://www.storyofmathematics.com/greek_pythagoras.html&gt;.

“Proposition 47.” Euclid’s Elements, Book I,. N.p., n.d. Web. 09 Oct. 2015. <http://www.clarku.edu/~djoyce/elements/bookI/propI47.html&gt;.