Engaging students: Deriving 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 Emma White. Her topic, from Geometry: deriving the distance formula.

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

In previous courses, it’s safe to say around 8^th grade, students learn the Pythagorean Theorem ( $a^2+b^2=c^2$ ). This deals with the sides and length of a triangle. The Distance Formula is the same concept but with coordinate values and finding the length of a so-called “distance”. We could go as far as to say that the formula can use earthly coordinates, such as North, South, East, West, and all that fall in between. Since the students are familiar with the Pythagorean Theorem, introducing the Distance Formula is a small step up. Another concept that is extended is building on the idea of coordinate points and understanding word problems. As stated earlier, the Distance Formula uses point on a coordinate graph and this can be transformed into a mapping concept, with compass directions. With this topic, students must extend their knowledge on word problems talking about “45 degrees south of east” and “30 degrees north of west” and how to apply this to coordinates.

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

Wow, the people who contributed to the discovery and development of the Distance Formula might as well be some of the biggest nerds Math majors know. A man by the name of Euclid (known as the father of Geometry) is who started the foundation for this formula. Euclid, as stated in his third Axiom, said it is “possible to construct a circle with any point as its center and with a radius of any length” (also Postulate 3 in “Euclid’s Elements: Book I”). Comparing the Distance Formula to a circle may seem a little confusing but let me challenge you to think again. Look at the standard form of the equation of a circle below:

$r^2 = (x-h)^2+(y-k)^2$

Now look at the Distance Formula:

$d = \sqrt{(x_1-x_0)^2+(y_1-y_0)^2}$

There are some similarities, right? Pretty close similarities too! A traveler, scientist, and philosopher by the name of Pythagoras took this idea from Euclid and ran with it, essentially being the man who invented the Distance Formula, or what is called the “Pythagorean Theorem. What interests me the most about this man is that he was a traveler, and he created the “Distance Formula” (get it, because he traveled distances…I thought that was ironic). Lastly, we must recognize Renee DesCartes (he developed the coordinate system which is connected to geometry and the Distance Formula uses these coordinates). Euclid, Pythagoras, and DesCartes contributed to the discovery of the Distance Formula and the development was so exemplifying that many, many, many occupations use it today!

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I believe technology is close to vital resource when teaching the Distance Formula to students. I say this because the topic is having to do with “going/finding a certain distance” per say. Having access to visuals helps the students put an idea to a tangible concept they experience every day, traveling. The resource below from Desmos is a prime example of how teachers can use technology to teach a lesson and make it interactive. Khan Academy also has some videos in which students can watch and follow along. Even more so, Khan Academy took a scenario from an athlete perspective and answered his question using the Pythagorean Theorem and Distance Formula. Having real life scenarios is what draws students to be engaged. If a student walks into a lesson not knowing the “why”, why are they going to want to sit through your class with a topic they see as useless? Therefore, I think technology, especially visuals (such as Desmos) and the Khan Academy example, would be beneficial for teachers to use in their classrooms when teaching the Distance Formula.

Reference(s):

http://harvardcapstone.weebly.com/history2.html

https://mathcs.clarku.edu/~djoyce/elements/bookI/post3.html

https://www.desmos.com/calculator/s7blqjtusy

https://teacher.desmos.com/activitybuilder/custom/5600a868e795241d06683511

https://www.khanacademy.org/math/algebra-basics/alg-basics-equations-and-geometry/alg-basics-pythagorean-theorem/v/soccer-thiago

https://www.chilimath.com/lessons/intermediate-algebra/derivation-of-distance-formula/