Logarithms and Music

1. Logarithms and Music Christina Colangelo

2. Lesson Plan • Introduction • Justification for Lesson • Description of Population • Prerequisite Knowledge • Major Math Content • NCTM Standards • Length of Time • Lesson Goals • Instructional Aids • Lesson Plan • Evaluation • Positives/Negatives/Effectiveness • Personal Response

3. Lesson Plan • Students tested vaguely remember learning anything about logs • Couldn’t recall why logs were used or what they were used for • This is lesson is a great way for students to conceptualize logs and see them used in a real life situation • 3 day lesson, each class 60 minutes

4. Lesson Plan • Major Math Content Covered • Functions • Linear, Exponential, Logarithmic, Sine/Cosine, Bounded, Damping • Graphing & Manipulations • Ratios

5. Lesson Goals • Understand a real life application of logarithmic functions • Learn the basics of notes, how they are arranged on a piano, and their frequencies • Understand how pitch and frequency are related to logarithmic functions • Learn what damping functions are and how they are applicable to music • Realize how mathematics fits into the real world and possibly inspire someone to explore the field of math and music

6. Part 1: 3-5 Songs • What did you notice about how the song ended? • What was different between each song’s ending, if anything? • How did the song end? Fade out? Abrupt stop? Other? • Think about the functions we have discussed throughout Algebra I and II (linear, exponential, polynomial, rational, logarithm, periodic). If you were to pick one of those functions to describe the way in which the music ends, what function would you choose? Why?

7. Brief Topics • Frequency/Pitch & their relationship • Damping functions • Bound functions • Sound waves = Sine waves • Use damping function to bound the end of the sound wave • Listen to sound clips that have damping functions applied to them (the will fade out the sound) • Linearly, Exponentially, Logarithmically

8. Conclusion • Students start to hear the logarithmic fade sounds the best and most natural • We also find out that music programs adjust the base of a logarithmic to determine how fast or slow a song fades out

9. Conclusion • After the lesson, students will listen to the 5 songs again • Students get to reevaluate what they said and write a short composition on how each song fades out and why they think that • New Found Glory – “On My Mind” • Computerized linear fade with extremely small slope

10. Negatives • Timing for questions or confusion (originally underestimated so I had to readjust time) • Math journal for opening questions or other exercises • 5 songs were too many for part 1, maybe cut it down to 3 • Some students thought thinking of functions off the top of their head were too hard when relating them to the songs…draw them on the board?

11. Negatives • I never allocated time to go over homework during class • I never had a closing for each day • Could students figure out frequency or pitch equations themselves? • Not enough time or knowledge • Frequency/Decibel parts a bit confusing for some student

12. Positives • Opening with music clips: “real attention grabber” • Connection I was drawing between functions and music was great, although in the beginning they didn’t see how it was possible • The use of the piano was a big hit! • Tissue box demonstration was thrilling • Different size rubber bands stretched across opening to show students the differences in pitches

13. Positives • Continuously recapping • Use of technology (music, sound clips, sound wave examples, piano, graphing calculator projections, etc) • “Autumn Fell” example was great! • Sound Clip: faded out linearly, exponentially, logarithmically • Reference Sheet & Handout • Composition was a great conclusion

14. Effectiveness • Extremely interesting to students • Students able to connect with material • Although some students didn’t completely understand all the music material they enjoyed learning about it • Students said they learned a lot