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 Sarah Asmar. Her topic, from Algebra II: deriving the distance formula.

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

Many high school students complain about why they have to take a math class or that math is not fun. Deriving and even learning the distance formula is not interesting for very many students. One way that I would engage my students would be to take the entire class outside to teach this lesson. We will walk down to the football and I will have a three students go to one corner of the football field while the rest of the class stands at the opposite corner diagonally. I will then hand a stopwatch to three other students. Each of them will have one stopwatch. The three students on the opposite corner will be running to the corner where the rest of the class is standing. The students holding a stopwatch, will each be timing one of the students running. I will ask one student to run horizontally and then vertically on the outrebounds of the football field, one student will run vertically and then horizontally, and the last student will run diagonally through the football field. Once all three students have made it to the corner where the rest of the class is, I will then ask everyone “Who do you think made it to the class the fastest?” I will allow them to say what they think and why, and then I will ask the students with the stopwatches to share the times of each of the students that ran. At the end, this will get the students to conclude that the student that ran diagonally got to the entire class the fastest. This is a short activity, but it changes the atmosphere for the students by taking class outside for a little, and it is fun.

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

There were three main mathematicians/philosophers that contributed to the discovery of this topic. Pythagoras, Euclid and Descartes all played a roll in deriving the distance formula. Pythagoras is a very famous mathematician. At first, he saw geometry as a bunch of rules that were derived by empirical measurements, but later he came up with a way to connect geometric elements with numbers. Pythagoras is known for one of the most famous theorems in the mathematical world, the Pythagorean Theorem. The theorem touches on texts from Babylon, Egypt, and China, but Pythagoras was the one who gave it its form. The distance formula comes from the Pythagorean Theorem. Euclid is known as “The Father of Geometry.” He has five general axioms and five geometrical postulates. However, in his third postulate, he states that you can create a circle with any given distance and radius. This is represented by the formula x²+y²=r². The distance formula comes from this equation as well. Last but not least, Descartes was the one who created the coordinate system. When finding the distance between two points on a coordinate plane, we would need to use the distance formula. All three of these men helped form the distance formula.

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

Students find everything more interesting when they are able to use technology to learn. There is a website that allows students to explore math topics using what is called a Gizmo. A Gizmo can be used to solve for the distance between two points. The students are allowed to pick what their two points are and then use the distance formula to find the distance between the points they chose. When students have control over something, they tend to do what they are supposed to do without any complaints. The Gizmo allows students to explore on their own without the teacher having to tell them what to do step by step. I can even ask the students to plot three points that form a right triangle and have them find the distance of the points that form the hypotenuse. This can allow the students to make the connection between the distance formula and Pythagorean Theorem. There are many applications out there, but I remember using Gizmos when I was in high school and I loved it. It is a great tool to explore a mathematical topic.

References:

http://www.storyofmathematics.com/greek_pythagoras.html

http://www.storyofmathematics.com/hellenistic_euclid.html

http://www.storyofmathematics.com/17th_descartes.html

Engaging students: The distance formula

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>.

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/>.

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>.

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>.

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

Engaging students: Deriving the distance formula

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 Michelle McKay. Her topic, from Algebra II: deriving the distance formula.

C. How has this topic appeared in pop culture?

Numb3rs is a relatively popular TV show that revolves around the character Dr. Charlie Eppes, a mathematician. The show’s plot is primarily centralized around Dr. Eppes’ ability to help the FBI solve various crimes by applying mathematics.

In the pilot episode, Dr. Eppes uses Rossmo’s Formula to help narrow down the current residence of a criminal to a neighborhood. Rossmo’s Formula is a very interesting in that it predicts the probability that a criminal might live in various areas. In the Numb3rs episode, Charlie manipulates the formula and projects the results onto a map to show the hot spot, or rather, the location where the criminal is most likely to be living in.

Rossmo’s Formula, however, would not be complete without including what we know as a Manhattan distance formula, which is just a derivation of the Euclidian distance formula.

From the distance formula we can derive…

The distance formula is a byproduct of Pythagorean’s Theorem. By examining any two points on a two dimensional plane, x and y components could be observed and used to calculate the distance between the points by forming a right triangle and solving for the hypotenuse. Later in time, the distance formula has been adapted to fit many different situations. To name a few, there is distance in Euclidean space and its variations (Euclidean distance, Manhattan or taxicab distance, Chebyshev distance, etc.), distance between objects in more than two dimensions, and distances between a point and a set.

E. Technology

The best way for students to really understand the distance formula is to allow them to make it their discovery. We can handle this in many ways. One of the more obvious explorations is to give them a piece of graph paper and have them plot points. However, this is an instance where technology can serve a great purpose in the classroom. There are vast amounts of apps online that will allow students to manipulate two points on a grid. After looking at several different apps, I find the one I have listed in the sources to be great for a few reasons. First, students can move two points around a virtual grid. This is a “green” activity and saves paper. Second, while students move the points, a right triangle is automatically drawn for them. Depending on the level of the class, students can make connections between the Pythagorean Theorem and how it leads to the distance formula. Third, above the grid is an interactive equation. It automatically plugs in the values of the points on the grid and finds the distance between them. What is even more impressive is that it solves the equation in steps.