Engaging students: Rational and Irrational Numbers

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 again comes from my former student Allison Metzler. Her topic, from Pre-Algebra: rational and irrational numbers.

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C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The video below is a scene from Star Trek. While most students will not have seen this version of Star Trek or perhaps any version at all, most are familiar with the franchise. Because the students will recognize the popular TV show, this video will immediately grab their attention and keep it for the whole video. The video clearly displays how it’s impossible for the computer to compute pi because it is a “transcendental” number. Thus, since pi is irrational, the computer will never be able to find the last digit of pi, causing it to focus on this insolvable problem forever. This video would provide the students with not only entertainment, but also a way to easily remember what an irrational number is. I would also point out that if Spock would have told the computer to compute a rational number such as any fraction or whole integer, it would have taken a matter of seconds.

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C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Rational and irrational numbers can be found in music theory which is incorporated in classical music. First, Pythagoras was credited for discovering that “consonant sounds arise from string lengths related by simple ratios: -Octave 1:2 –Fifth 2:3 –Fourth 3:4” which are all rational numbers. Rational numbers are also found in the sound frequency and the diatonic scale. In order to get an equal tempered scale, we must get from the note C to the note C’ in twelve equal multiplicative steps we must find x such that x12=2. This causes us to take the twelfth root of 2 which produces an irrational number. The benefit of tuning a piano to tempered scale is that (1) “Sharps and flats can be combined into a single note” and (2) “Performers can play equally well in any key.” Rational and irrational numbers can also be found in other areas of music as evidenced below.

”At least one composition, Conlon Nancarrow’s Studies for Player Piano, uses a time signature that is irrational in the mathematical sense. The piece contains a canon with a part augmented in the ratio square root of 42:1.”

Also, when you play a fretted instrument (i.e. guitar, banjo, balalaika, bandurria, etc.), you are playing irrational numbers. According to http://www.woodpecker.com/writing/essays/math+music.html, the reason guitars are so hard to tune is that “our ears don’t like the irrational numbers”. However, they are needed to make “complex chordal music.”

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D1. What interesting things can you say about the people who contributed to the discovery and the development of this topic?

The video below displays who discovered irrational numbers while also getting into why the square root of 2 is irrational. I would play the video until the 4 minute mark so that I can keep the attention of the students. I would then go further into who contributed to the discovery of irrational numbers.

The Pythagoreans were set on the idea that all numbers could be expressed as ratios of integers. However, Hippasus of Metapontum, a philosopher at the Pythagorean school of thought, discovered otherwise. He supposedly used the Pythagorean Theorem (a2 + b2 = c2) on an isosceles right triangle where the congruent sides were each 1 unit. Using the theorem, he found that the hypotenuse was the square root of 2 which proved to be incommensurable. The other Pythagoreans were so horrified with this discovery, that it’s said they had Hippasus drowned. They wanted to punish him while also keeping irrational numbers a secret. However, it’s hard to prove that this information is true because of the vague accounts of who discovered irrational numbers. Therefore, I would inform my students of this interesting story, but also tell them about the uncertainty of what actually happened.

References:

Discovery of Irrational Numbers (n.d.). In Brilliant. Retrieved February 7, 2014, from https://brilliant.org/assessment/techniques-trainer/discovery-of-irrational-numbers

Hippasus (2014, January 14). In Wikipedia. Retrieved February 7, 2014, from http://en.wikipedia.org/wiki/Hippasus

Pre-Algebra 32-Irrational Numbers. YouTube, 2012. Web. 7 Feb. 2014. <https://www.youtube.com/watch?v=q_wstDWjnKQ&gt;.

Reid, H. (n.d.). On Mathematics and Music. In Woodpecker. Retrieved February 7, 2014, from http://www.woodpecker.com/writing/essays/math+music.html

Shatner, William, and Leonard Nimoy, perf. Star Trek. YouTube, 2009. Web. 7 Feb. 2014. <https://www.youtube.com/watch?v=H20cKjz-bjw&gt;.

Time Signature (2014, February 6). In Wikipedia. Retrieved February 7, 2014, from http://en.wikipedia.org/wiki/Time_signature

Wassell, S. R. (2012, March 29). Rational and Irrational Numbers in Music Theory. In docstoc. Retrieved from http://www.docstoc.com/docs/117428973/Rational-and-Irrational-Numbers-in-Music-Theory

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