**LESSON 7 (Approximately 2 hours)**

**STRING INSTRUMENT LAB, GRAPHING EXPONENTIAL & LOGARITHMIC FUNCTIONS AND DEVELOPING ROBUST MATHEMATICAL MODELS**

*Overview*- Exploring intersections between math and music
- Observing and measuring the fretting patterns of professionally built guitars, basses, and ukuleles
- Graphing data using a digital graphing program: ‘Demos’
- Representing the relationship between fret number & fret length as well as note & frequency through two mathematical models
- Using tuning forks and sound to access physics concepts
- Developing a robust mathematical model for a future string instrument design & build project

*Materials*Class set of meter sticks or yard sticks Projector screen

Class set of rulers Student and teacher laptops with internet access

One electric/accoustic guitar, bass, or ukulele per 4 students ‘Desmos’

Tuning forks Blank white paper

Digital projector Graphing Paper

*Key Words*Intersections of math & music | Raw data in a mathematics classroom | Digital graphing | Exponential functions | Logarithmic functions | Rate | Growth | Decay | Best-fit curves | Mathematical models | Fret | Bridge | Nut | Waves | Pitch | Frequency | Hertz | Octave | Chords | Harmonics | Consonance | Dissonance

__TEACHING PLAN__

*Introduction*Last class we learned about the mathematical underpinnings of exponential and logarithmic functions. Today, we will be looking into some of the many intersections of mathematics and music! First, we will be using professionally built guitars, basses, and ukuleles borrowed from the music class and thanks to the generosity of those of you who brought instruments from home. We are going to learn about how math informs where frets are placed so music sounds great and also how musical scales relate to mathematics. I know many of you are musicians and I hope everyone enjoys what we are about to explore! Our work today will be hugely important to our work beginning next class. We are going to be designing and building our own string instruments (which you will be encouraged to keep). We will use math concepts and the engineering design process to drive outstanding, creative, beautiful instruments. I hope everyone is excited!

__ASK Questions and discuss as a class__(7 minutes)

- Based on some quick observations, how would you describe the fretting pattern on the guitars, ukuleles, or basses on loan from the music class and your homes?
- What sort of mathematical relationship (if any) do we predict as we go up a scale of notes?
- Do you think people who build instruments utilize mathematical models? If so, how do you think they use them?

*Task*- Place your name on a corner of blank white paper. We are going to
random string instrument teams. Your team will complete the string instrument lab together and will also go on to make an original string instrument! (3 minutes)*form* - Taking your portfolio and a pencil with you,
your team.*find* - (
*Distribute ‘Specific Roles For Helpful Team Members: Project-Based Learning’*) (1 minute) What I have just passed out is a second copy of something you should already have from our solar oven projects: Specific Roles For Helpful Team Members. Take a moment with your new team towhich role you will take on. While it doesn’t need to be finalized right now, you will need to have this figured out by the end of class. In the meantime, please elect who will take on what role at least for today. (3 minutes)*consider* - Please
your Materials Team Member up to collect a meter stick and a string instrument. Remember to take plenty of photos and notes today. You will also be printing out some graphs later. (*send**Distribute String Instrument Lab*) (3 minutes) - At this point, please
the String Instrument Lab. In your teams, complete Part 1 and Part II. Each of you will be expected to complete your own copy but you are welcome to collaborate on ideas so long as they are put into your own words. I will be walking around the room to help. Let’s have some fun! (80 minutes)*begin*

*Wrap Up*__Discuss__

- What value, if any, was there to devising mathematical models in this lab? (3 minutes)
- Who (besides instrument builders) might use mathematical models? How might they be utilized? What value do these sorts of mathematical models hold? (5 minutes)
- How can we use our mathematical models as we prepare to create beautiful, nice-sounding string instruments? (3 minutes)
- What lessons did you learn from creating parabolic solar oven that you might apply to imagining, planning, creating, experimenting with, and improving a tunable string instrument? (3 minutes)
- Please take a moment to submit your String Instrument Labs. (1 minute)
- Review homework deliverables due next class: (5 minutes)
- Decide your official team role (e.g., Materials, Schematics, Coordinator, Recorder/Photographer)
- ‘I Used to Think…, Now I Think…’ (Harvard Project Zero, 2015) regarding the intersections of math and music, mathematical models, and exponential & logarithmic functions
- Develop a schematic drawing of the neck of a string instrument that showcases an exponential decay pattern from bridge to fret number
*or*do so in ‘Rhinoceros’. Be sure to include measurements and print before class next time. - Create three schematic drawings for a complete, tunable string instrument you would like to build
*or*digitally render a design for a string instrument using ‘Rhinoceros’. Be sure to include measurements and print before class next time.

__Distribute__

- "Specific Roles For Helpful Team Members: Project-Based Learning"
- "Algebra II String Instrument Lab"
- Link to Video: ‘Desmos Table Instruction’

__Web Resources__

1. ‘Desmos’ graphing tool

https://www.desmos.com/

2. Video: ‘Desmos Table Instruction’ (begin at 0:27)

https://www.youtube.com/watch?v=L5spXQ7YoEY

3. Video (optional for viewing outside class): ‘A Visual Representation of Consonance and Dissonance’

https://www.youtube.com/watch?v=tafPfbm4jug

4. Video (optional for viewing outside class): ‘Frequencies and Sound Explained—Harmonics and Harmonic Distortion’

https://www.youtube.com/watch?v=FzeZbJceKZE

5. Artful Thinking Routines: ‘I Used to Think…, Now I Think…’ (Harvard Project Zero, 2015)

http://www.pz.harvard.edu/resources/thinking-routines/artful-thinking/i-used-to-think-now-i-think

__Formative Assessment__

- Student Responses to ASK Questions
- Teacher observations during the lab activity
- Algebra II String Instrument Lab
- ‘Wrap Up’ Discussion questions two and three

__Summative Assessment__

- ‘Wrap Up’ Discussion questions one and four
- ‘I Used to Think…, Now I Think…’ (Harvard Project Zero, 2015)
- Schematic drawings (4) or digital renderings completed on ‘Rhinoceros’ (2)