Really Making Music


David Macmorine

Unit Title:

Really Making Music





Estimated Duration:

2 Weeks – 15 60 minute class periods

Unit Activities:

Background Knowledge: 

In order for students and teachers to be successful in this unit, both must have some familiarity with music and simple musical instruments, such as flutes and xylophones and drums.  An interest in music would be very beneficial.  


July 2014

The Big Idea (including global relevance)

Music is an art form which crosses many cultural and societal boundaries. As a math teacher at a performing arts school, it is often my job to relay the connections between mathematics and other subjects, namely music. Many students have heard of Pythagoras and the theorem attributed to him – most students don’t know he also came up with our commonly used musical scale.  

The Essential Question
How was the musical scale created and how is it related to math?
Justification for Selection of Content

The study of sound through music is an excellent way to introduce the mathematics of waves. Students, when presented with trigonometry, are often confused and sometimes turned off by the subject. If sound is studied (using a free program such as Audacity) by looking at its wave format, using somewhat familiar terms such as amplitude and frequency, students may become more appreciative of sine and cosine functions. 

The Challenge

Grades 10-12: Each group of students will make a musical instrument that can be tuned to play a song, such as ‘Mary Had a Little Lamb’.

Grades 7-9: Each group of students will make a musical instrument that can be tuned to play one note tuned to a required pitch. Groups will be paired in threes to produce a major chord C –E-G. (See

The Hook

Do you like music? Do you want to make your own music? Do you want to be creative and unique – to do something in music no one else has done? Do you think you can make a musical instrument? Can what some people call ‘garbage’ be used in making musical instruments? Think about the people you know in music – why do you like them? Because they sound like everyone else? Do you know that the root of music is mathematics?  

Teacher's Guiding Questions
  1. What is sound? 
  2. What is music?
  3. Why does some music sound better to us than other?
  4. What does it mean to be in tune?
  5. What is harmony?
  6. Sound is a wave – what’s a wave?
  7. What are the different ways to produce sound?
ACS (Real world applications; career connections; societal impact)

Students will learn generally about waves, and how mathematics is used to describe waves. The real world applications are vast. Waves, and the mathematics used to describe waves have applications in acoustics, noise control and abatement, electrical engineering, geology, and the list goes on and on.

Engineering Design Process (EDP)

The Engineering Design Process will be followed when the students make their musical instruments. 

Students will identify and define the type of musical instruments with which they are familiar. Then they may use various sources to research what types of musical instruments are out there, and which ones are feasible to make, such as a xylophone seems feasible, whereas a saxophone is not. Then they may identify which types of materials are out there, hopefully focusing on recycled or inexpensive materials such as PVC pipe for flutes, or if making a guitar, should the strings be metal wire or plastic fishing line.

For students in PreCalculus AA, students will then play the note A (440 hz) and be within 5 hz range, that is 435 hz<A<445 hz.  If ‘being in tune’ is not achieved, students will alter and redesign their instruments to achieve ‘in tune’. Students will then tune the instrument to play an 8 note scale.

For students in Precalculus or Algebra II, students will be assigned one of three notes to produce C, E, or G.

‘C’ is 261 hz, ‘E’ is 330 hz, and ‘G’ is 392 hz. (see

3 groups will then play their notes simultaneously to produce a major chord.  

Unit Academic Standard

Create equations that describe numbers or relationships.

Linear, quadratic, and exponential (integer

inputs only); for A.CED.3 linear only A.CED.1, 2, 3, 4

Model periodic phenomena with trigonometric functions.  F.TF.5


HS-PS4-1 Use math representations to support a claim regarding relationships among frequency, wavelength, and speed of waves traveling in various media.

HS-PS4-5. Communicate technical information about how some technological devices use the principles of wave behavior and wave interactions with matter to transmit and capture information and energy. 

Where the CBL and EDP appear in the Unit

Acivity 4: What Makes Music?


Students sometimes are confused by the difference between amplitude and frequency. Additionally, students may not see the connection between period and frequency.

Also, a math concept that students traditionally get confused is the connection between triangles, the unit circle, and the graph of the sine and cosine functions. There are three concepts occurring on the unit circle which are isolated two at a time – namely, x coordinate vs angle for cosine, etc.

Additional Resources

Pre-Unit Assessment Instrument
Post-Unit Assessment Instrument
Results: Evidence of Growth in Student Learning
Students experienced a 19.54% increase from pre-test to post-test, that is, 47.13 % on the pre-test to a 66.67% on the post-test.  
How to Make This a Hierarchical Unit

The challenge of making a musical instrument that plays an 8 note scale can be simplified by limiting the type of instrument made. Stringed instruments were the most difficult to produce and could be eliminated from the possibilities allowed to be made.  Students can be limited to making a woodwind ( pan pipe or flute ) or percussion (thumb piano or xylophone).

Another option is to have the students only produce one specified note, as noted above E – C – G, and playing this major chord in unison. If a group is particularly ambitious, they could then continue to complete the instrument and an 8 note scale.  


The unit went very well overall. Once the students got ‘hooked’ into the lesson, the groups mostly were self-driven. The group that made the violin was quite ambitious, and required outside assistance to fabricate some of the parts. The pan flute group required little assistance once they agreed on their plan. Most groups went through the EDP process a few times. The group that made the xylophone originally wanted to make a drum, but then quickly realized they would have to make 8 drums to play a scale. The thumb piano group first built a box, but then found out the wood used would not support the bobby pins and simplified the plan by using just a piece of plywood.

The overall project is broad enough to be used at grade levels from 7 to 10th. I believe it would require a limited scope on the type of instrument allowed to be built. The pan flute and thumb piano are relatively easy, whereas the violin was decidedly a project requiring some specialized tools and skills.

The next time I teach this unit, I will have more recycled materials available and also more small hardware like thumb screws and eyelets.