Maxfield Palmer
Lesson Planning Template for INSYS 297A/101
Learning Goal: To show students the relation between
trigonometry and music.
Learning Objectives: To show the students that the concepts
of the trigonometric unit circle and sinusoidal waves are the basic components
of sound and music. Using math we can also discover intonation and scales used
in most types of music.
Audience: Students in a trigonometry mathematics course
-Full References listed in APA format at the bottom
|
Event of Instruction
|
Time
|
Detailed Content
|
Materials needed
|
Media Utilized
|
||
|
Gaining Attention
.
|
2 minutes
|
Hit a tuning fork to make a ringing sound. After striking
the fork, ask the students how this sound relates to mathematics,
specifically trigonometry.
|
Tuning Fork
|
None
|
||
|
Informing the learner of the objective
|
2 minutes
|
Afterwards, draw a picture of a sinusoidal wave on the
board (this will be well known to the students). Say that this is how the
tuning fork relates to mathematics. Then, describe how you are going to talk
about sound waves in music and how they relate to the unit circle. Describe
how they equally have a peak and trough and every sound (even our voices) are
made up of many waves that are working together to create the sound we hear
in our ears. And say that in fact the very music we listen to in our iPods
can be broken down using basic math and trigonometry. (Ahlstrom, 2009).
|
White board and marker
|
White board and marker
|
||
|
Stimulating Recall
|
2 minutes
|
Ask the students if they have heard of decibels, hertz,
frequency or amplitude. Most of them will have heard of these terms,
considering that they are commonly used in regards to music and they also
have been used in class. Tell them that so far we have talked about amplitude
and frequency in terms of hypothetical waves but that music is a literal
transfer of these concepts in the real world. This will spark their interest
and stimulate recall of these past subjects that we discussed previously in
class. (Kessler, 2007).
|
None
|
None
|
||
|
Presenting the material
|
8 minutes
|
Say to them; So now that we have remembered a lot of these
basic wave concepts, let’s hear what these mathematical terms actually sound like. Then play a basic sine
wave from the wave table provided in the Massive program. Play the note
Center A and explain how this note is 440 hertz and then explain how that is
just like the amount of cycles around the trigonometric unit circle that this
note makes in one second. Play this for them at the standard preloaded
amplitude. Then tell the students that you are going to increase the amplitude.
You will all hear an increase in volume. Then turn it back down to normal.
Make a quick side-note that volume is the music/sound equivalent to the word
amplitude. Then tell the students that you will play the frequency two times
440Hz or 880Hz. You then will play A4 (the A directly above center A [A3]). (Hewitt, 2008). Explain how this is called an
octave and to us if one doubles the frequency of a sound, it sounds like the
original sound just a little higher in PITCH. You can use this to transition
into explaining that pitch is
actually our musical terminology for the mathematical term, frequency. You
can tell them that this multiplication of frequencies can be broken down into
ratios. 2:1 is an octave, 3:2 is what we call a fifth in music, and a musical
third is 5:4. Tell them that these ratios, when multiplied to the original
frequency, create the foundation for the musical chord and also the musical
scale and that they are called Just Intonation (Lindley,
2013).
|
A computer for the teacher.
An oscillator a program synthesizer called Massive by
Native Instruments
|
A synthesizer in order to demonstrate the mathematical
concepts
|
||
|
Providing learning guidance
|
5 Minutes
|
Ask if there are any questions about the ideas that were
just discussed before starting this next activity. Afterwards, I have rented
the computer cart for the day and at this point give the students each a
computer (if there are more students than computers, have some pair-up). Then
send them to this site: http://onlinetonegenerator.com/ This will provide
them with a way to create different frequencies and test out the ratios that
we discussed. Ask if there are any more questions.
|
Computers
|
Online tone generator
And computers
|
||
|
Eliciting performance
|
4 Minutes
|
Have several students play, through their speakers, two
notes of an octave. The next student can play a fifth (following the given
ratio). Have each student that has volunteered (or if you have to select
volunteers yourself) to play each of the intervals by typing in different
values on the website.
|
Computer Speakers
|
Online tone generator
|
||
|
Providing Feedback
|
2 Minutes
|
Give them congratulations for not only seeing how their
math is applicable in the real world but also that many of them must have
also learned a thing or two about music as well.
|
None
|
None
|
||
|
Assessing performance
|
3 Minutes
|
Then tell the students about several more concepts that
they can learn if they decide to pursue math and about utilizing it in the
real world of music. Discuss how pianos are tuned by playing frequencies a
little off of the desired frequency because of a concept called “beatings”.
Also tell them that guitars and almost all concerts are tuned to the already
discussed frequency of 440Hz. Tell them that many orchestras use the very
close frequency 444Hz when they want some extra flare or spice (Ballora, 2003). Then proceed to ask if there are
any questions about how math is tied to music and specifically trigonometry.
|
None
|
None
|
||
|
Enhancing retention and transfer
|
2 Minutes
|
Now have them discuss among themselves, or talk to you,
about what they learned today and they can continue to master the materials
by looking around on the online tone generator website. However, do not let
them leave that site! Try to make sure that their conversations are remaining
on topic by walking around (if you are not already engaged with a student)
and prompting the students about their new area of expertise.
|
Computers
|
Online tone generator
|
||
References:
Ahlstrom, Dick.
October 13, 2009. How trigonometry makes waves and lets you hear music on your
iPod. Retrieved from
http://www.lexisnexis.com.ezaccess.libraries.psu.edu/hottopics/lnacademic/?verb=sr&csi=142626&sr=HLEAD%28How%20trigonometry%20makes%20waves%20and%20lets%20you%20hear%20music%20on%20your%20iPod%29%20and%20date%20is%20October%2013,%202009
Ballora, M. (2003). Essentials of music technology. Upper
Saddle River, NJ: Pearson Education, Inc.
Hewitt, M.
(2008). Music theory for computer
musicians. Boston, MA: Course technology: Cengage learning.
Kessler, B. (2007). A “sound” approach to
Fourier Transforms: Using music to teach trigonometry. Western Kentucky University. Retrieved from http://works.bepress.com/bruce_kessler/28/
Lindley, M. (31 May, 2013). Just intonation. Grove Music Online. Retrieved from http://www.oxfordmusiconline.com.ezaccess.libraries.psu.edu/subscriber/article_citations/grove/music/14564
