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 Peter Buhler. His topic, from Geometry: deriving the distance formula.

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

Although the distance formula may be introduced as part of the Geometry curriculum, it also has applications in Algebra and even Pre-calculus. This allows for many possible applications, as it can be used in various ways. One project that students could be assigned to is by modeling something in real life on a coordinate grid, and using the distance formula to calculate various distances within that real life object or place. An example of this could be to take a baseball diamond and use the fact that the bases are 90 feet apart, and calculate the distance between the corners on opposite sides. Another example could be to overlay a map of their town onto a coordinate grid and measure the distance between places that they usually visit. These students can fact check the distances by plugging them in to Google Maps. One aspect of this project to be careful of is to make sure that students are using the distance formula, and not the Pythagorean Theorem. Allowing the students to present their findings could spark curiosity into how mathematics is used in everyday life by city planners, architects, engineers, and in other careers.

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How has this topic appeared in high culture?

The following piece of artwork was created by Mel Bochner and titled, Meditation on the Theorem of Pythagoras.

While immediately this picture appears to be related to the proof of the Pythagorean Theorem, There are also applications to the distance formula. This artwork could be a great engaging activity for students as they come into class, simply by reflecting on what can be seen. A challenging question would be to ask students to guess how many hazelnuts they think the artist used to create this artwork (without counting each piece). It should be noted that each corner of the triangle consists of two corners of the squares, so the answer is not simply 9+16+25, but you must subtract off how many are shared.
We can apply this to the distance formula by asking students how to relate the Pythagorean Theorem with the distance formula. Having students compare and contrast these two mathematical equations could provide excellent discussion. As an instructor, you can also overlay this artwork onto a coordinate grid and have students use the distance formula to calculate the various side lengths and confirm that it works.

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How was this topic adopted by the mathematical community?

The three mathematics who are primarily responsible for what we know as the distance formula are: Euclid, Pythagoras, and Descartes. Euclid stated in his third Axiom that “it is possible to construct a circle with any point as its center and with a radius of any length”. This matters because the distance formula is a corollary of the circle formula. Pythagoras then took this idea, and proceeded to invent the Pythagorean Theorem, which can be easily converted to the distance formula. Later on, Descartes applied this to the coordinate system, in an event consisting of the union of algebra and geometry.
While this material may seem fairly dry to middle school or high school students who are first learning the Pythagorean Theorem, there are certainly some applications that can make the history more appealing. One such application is to ask the students to connect the formula of a circle with the distance formula, and discuss how they are related. This would provide excellent discussion about how Euclid and Pythagoras may have begun their study of the distance formula. Another application could include assigning students to study one of these three mathematicians, and having them provide several interesting facts about the person they chose to study. Consequently, when introducing the distance formula, students will be familiar with those who had a huge impact on the development of the distance formula.


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