The concept of proportional reasoning is one of my all time favorites to teach. Teachers across the country approach the topic in different ways. Some use scale figures of photographs, maps, or animals while others discuss unit rates of movie tickets or grocery items. My approach to introducing the concept of proportionality is to use music. Students love listening to music. Many of them seem to have their headphones permanently attached to their heads! So when one of our math lessons features music, they are all ears (sorry, I couldn’t resist). The lesson consists of making students DJs for a day.
Professional disk jockeys are masters of proportionality. In order to layer songs to play back-to-back seamlessly, DJs have to align their beats per minute (BPM) so that they match. When done well, dancers at weddings or senior proms can barely tell that one song has ended and another has begun. It truly is a beautiful thing to experience. For two songs to play well one after the other, they need to be within 10 BPM of each other. That proximity allows DJs to speed up or slow the tempo enough to overlap the beats. Once the songs are set to the same tempo and the BPMs align, DJs will listen to the next song with one headphone while shifting the turntable to play the first beat of that song over and over, attempting to match it to the tempo of the current song. Once the beat is aligned to the tempo of the current song, the DJ lets go of the turn table and starts the new song, hopefully with minimal distraction to the audience.
The process of mixing requires DJs to have to know the BPM for every song in their music libraries. A typical DJ can have over 3,000 songs in his or her collection. Today’s DJs have sophisticated computer programs that analyze the BPM during the first few seconds of any song and display it next to the song’s title, doing much of the work for them. But before these programs existed, DJs would have to calculate the BPMs themselves. Considering that counting the beats for a minute for every song in a 3,000 song music library would take 50 hours to accomplish, proportional reasoning presented a very attractive alternative. Since the music had a constant rhythm, DJs would often only count beats for 10 seconds and then multiply by 6. Following this process saved DJs considerable time.
Once I tell students the basics of the song mixing process, I point out that as beginning DJs they do not have the sophisticated computer programs to be able to calculate the BPM for songs. As a result, they’ll have to rely on their math skills and their ears to calculate the BPMs by hand. I play a popular song they all love and assign groups of students different intervals of time to count beats for the song. Students count for 10, 15, 30, 45, or 60 seconds. I demonstrate the beat for the song as it begins playing by tapping my foot and encouraging the students to follow. Once they all feel the rhythm, I start a timer and announce “Start.” I announce when the time intervals we are using are completed and students confer with their groups to agree on a number of beats. We then form ratios of their results (beats/time) on the board. I ask students to consider the different ratios and tell me if they notice anything interesting. Students soon realize all of the ratios are equivalent (or close to it). I ask them why that is and many will intuitively understand that it is because the tempo of the song never changes. I introduce the concept of proportionality at this point. I also ask which of the ratios would be easier to find for each song in a 3,000-song library. Students quickly discover that counting for a minute per song is not optimal. Most choose to count for 10 – 15 seconds and then convert their ratio to the true BPM. At this point, students are allowed to choose 6 of their favorite songs (or choose songs from a list I provide) and are asked to make a playlist that would lend itself for song mixing.
Our school has a two channel mixer that can play two songs simultaneously and switch from one to the other with a fader. (You can also achieve this with two CD players or Mp3 players hooked to different speakers.) We use this to analyze the results of our students’ work. If done correctly, you should be able to tap your foot to one song, and not have to change the tempo of your tapping when a new song is introduced. Our students play their songs and fade between them, testing to see if they have to change the tempo. When it is done correctly, they are thrilled and really enjoy the musical transition. When the alignment is off, even by a small margin, they pick up on it quickly. That usually leads to the question, what do DJs do when they want to drastically change tempos? The answer is that they usually speak on the microphone to make an announcement.
Discovery Education’s Math Techbook often incorporates the theme of music into the curriculum. For fun examples your students will enjoy, check out the looping problem, examine musical storage, or plan a radio show!