Music and Mathematics
Music and Pre-Calculus
Unit topic: Pre-Calculus Analytic Trigonometry Grade level: 11
Duration of unit: 5 weeks Designer: Laura Granger
The Music Project and Problem:
You are a mathematician with a strong appreciation for the beauty of mathematics. You have found that you cannot share that appreciation with anyone other than another mathematician. Because you are so anxious to share your feelings about the beauty and wonder of mathematics, you are looking for a scheme to represent it in a musical way. Music seems to be a medium that most people can enjoy and understand at some level. You believe that if a person understands the music it will help them understand the mathematics. In order to show this beauty in math and its connection to music you will build a homemade musical instrument and tune the instrument using trigonometric graphs. Others might be skeptical so you will also create a presentation that will explain the mathematical connection between your graphs and the tune your instrument plays, and defend the fact that your trigonometric graph shows that your instrument is tuned.
Rigor, relevance and relationships, 21st century skills and best practices are all developed and used in this project.
- Understands characteristics and uses of basic trigonometric functions (e.g., the sine and cosine functions as models of periodic real-world phenomena
- Understands the effects of parameter changes on functions and their graphs
- Understands the graph of a function of the form f(t)=Asin(Bt+C)+D or g(t)=Acos(Bt+C)+D, in terms of amplitude, frequency, period, and vertical shift, and phase shift
- Understands properties of graphs and the relationship between a graph and its corresponding expression (e.g., maximum and minimum points)
- Understands periodicity in the trigonometric functions and their graphs
- Understands the basic concept of inverse function and the corresponding graph
- Uses the GDC to graph functions What is this? I tried to look it up but no success?
- Uses GDC to analyze functions
- Uses Geometer's Sketchpad to build and label graphs of function This I am familiar with ☺
Student Friendly Standards:
- Understands characteristics the sine and cosine functions as models of periodic real-world phenomena
- Understands the effects of number changes on functions and their graphs
- Understands the graph of sine and cosine functions in terms of amplitude, frequency, period, and vertical shift, and phase shift
- Understands properties of graphs (e.g., maximum and minimum points)
- Understands periodicity in the trigonometric functions and their graphs
- Uses the GDC to graph functions
- Uses GDC to analyze functions
- Uses Geometer’s Sketchpad to create professional looking graphs
Can mathematicians and musicians live and work together in harmony?
Student “I Can” Statements:
- I can use the sine and cosine functions to model real-world applications
- I can understand how the changes in the numbers in my functions effects their graphs
- I can understand the graph of the function of the form f(t)=Asin(Bt+C)+D or g(t)=Acos(Bt+C)+D
- I can understand the terms amplitude, period, frequency, phase shift and vertical shift
- I can understand the properties of graphs such as maximum and minimum points
- I can understand that trigonometric graphs are periodic
- I can calculate and draw the inverse trig functions
- I can graph trig functions by hand
- I can use the GDC to graph and analyze trig functions
- I can use Geometer’s Sketchpad
Students will know:
- Know how each parameter in a trigonometric equation affects the graph
- Recognize the parent trigonometric graphs
ƒ(x) = sin x
ƒ(x) = cos x
ƒ(x) = tan x
ƒ(x) = cot x
ƒ(x) = sec x
ƒ(x) = csc x
- Know the relationship between the trigonometric functions and their inverses
- Know the definition of a periodic function
- Know the period of basic trigonometric functions
Students will understand
- Students will understand how the graphs of trig functions can be used in the real world, specifically in tuning a musical instrument This is very clear for your connection to the essential question.
Students will do
- Find a trigonometric equation that fits a set of data by adjusting the parameters
- Use the graph of the function to make predictions
- Graph trigonometric functions using changes in parameters
- Graph the inverses of trigonometric functions
- Collect, organize and analyze data to test a hypothesis
- Find the period of a function given the graph
- Find the period of a function given the equation
- After you explain CDC to me, you will have mastered writing this section ☺
Project or Problem to Be Solved: Students will be presenting a Music Project to explain how trigonometric graphs are related to music.
They will do this by building a homemade musical instrument and tuning their instrument by analyzing the trigonometric graphs formed by the sound of their instrument. In their presentation students will defend why they know their instrument is in tune.
This is project shows rigor and relevance and is engaging as well.
This project in itself uses best practices and therefore brain based due to the complexity and challenge of the project. It is also brain based in that it is authentic to the learner because they are learning to graph trig functions for the purpose of tuning an instrument. 21st Century skills include collaboration and presentation. Rigor, relevance, and relationships are also evident in this project.
Students will complete a 10 question ALEKS quiz on their knowledge of graphing non-trig functions
This is as intended to be. Formative Assessments:
PTT (prime time task) quizzes at the start of each day on material that should have ben mastered by that point
Periodic ALEKS quizzes. If you use these formatively, you do not want to include them in the final evaluation.
Continual checks for understanding during mini lessons and work time – observations, use of individual white boards, thumbs up, individual voluntary and cold call responses, choral responses, etc. These are very effective informal methods. Summative Assessment:
Students will be assessed on their Music Project using an analytic rubric with the following criteria: mathematical reasoning and problem solving, graphs, modeling, mathematical concepts, technology, neatness and organization
See rubric for evaluation of this section.
Sequence of Instruction
Invite a piano teacher to speak with the students about the process of tuning a piano.
Show the video of the Tacoma Narrows Bridge
Activities and Assignments:
1. Connect the unit circle to the graphs of sine and cosine. This will be differentiated in three ways:
• A hands on spaghetti activity (best practices due to non-linguistic representation) where students graph the sine and curve with spaghetti https://www.tes.co.uk/teaching-resource/spaghetti-trig-graphs-6197457
• Applet investigation on how the unit circle maps into the sine and cosine curve (non-linguistic representation) http://www.intmath.com/trigonometric-graphs/trigo-graph-intro.php
• Teacher directed mini lessons, which will also be uploaded to Google Drive. Best practices and brain-based learning will be used in all mini lessons such as: note taking, homework, feedback, authentic learning, etc.)
2. Investigate parameter changes; period, phase shift, and vertical shift on trig functions. This will be differentiated two ways:
• Math practice tutorials https://www.khanacademy.org/math/trigonometry/trig-function-graphs
• 5 teacher directed mini lessons on period, phase shift, vertical shifts and families of graphs which will be uploaded to Google drive.
3. CBR (motion detector) Investigation to generate data that has a sinusoidal shape to connect the real world with trigonometric graphs.
• Resources found at http://education.ti.com/en/us/activity/detail?id=D5F0704EF1AA476C9CAF471410629E1B
4. Assign and discuss the Music Project to connect trigonometric waves, particularly sine waves to musical harmony. Skills developed and used will be creativity (design and construction of the instrument), critical thinking (connect sound to trigonometric graphs), communication (presentation), collaboration (group work), technology skills (use of CBR and graphing calculator), flexibility and adaptability (novelty in the type of project), productivity and accountability (time management and final product completed), and social skills. Differentiate by providing some resources for students who need help finding definitions:
• Resource #1: http://cnx.org/content/m12373/latest/
• Resource #2: http://www.sciencefriday.com/pages/2006/Apr/music/
• Resource #3: http://books.google.com/ and enter “The Physics of Music”.
• Resources for making the instrument: http://www.philtulga.com/MSSActivities.html#09
• Directions for learning about sound and making wind chimes
• Other wind chime resources:
• http://home.fuse.net/engineering/Chimes.htm (Includes downloadable freeware called Wind Chime Designer to help determine the notes and tube lengths)
5. Investigate inverse trigonometric functions. This will be differentiated in two ways:
• Tutorial; http://www.themathpage.com/atrig/inverseTrig.htm
• http://www.sascurriculumpathways.com>Mathematics>Trigonometry > Inverse Functions > Web Inquiry 114
• Teacher directed mini lesson
Mathematical Reasoning and Problem Solving Graphs Modeling Mathematical Concepts
Technology Neatness and Organization
20% 20% 20% 20% 10% 10%
Music Project Rubric
4 Our musical instrument contains a minimum of 5 notes that can be used to play a tune accurately. Our presentation includes accurate trigonometric graphs that represent the actual sound of each note our instrument plays. Our presentation includes accurate equations that model the graphs of each note our instrument plays. Our presentation includes a detailed explanation, using correct mathematical terminology, of the relationship between each graph and its corresponding note. Our final product shows evidence that advanced features of multiple technology tools and electronic resources were used. Our work is presented in a neat, clear, organized fashion that is easy to understand.
3 Our musical instrument contains less than 5 notes but they are all tuned accurately. Our presentation includes accurate trigonometric graphs that represent the actual sound of 3 or 4 notes our instrument plays. Our presentation includes somewhat accurate equations that model the graphs of each note our instrument plays. Our presentation shows substantial understanding of the relationship between the graph and the musical note. Our final product shows effective use of several technology tools and electronic resources. Our work is presented in a neat and organized fashion that is usually easy to understand.
2 Our musical instrument contains a minimum of 5 notes but they are not tuned accurately. Our presentation includes partially accurate trigonometric graphs that represent the actual sound of some of the notes our instrument plays. Our presentation includes somewhat accurate equations that model the graphs of 3 or 4 notes our instrument plays. Our presentation shows some understanding of the relationship between the graph and the musical note. Our final product shows some evidence of effective use of technology tools and electronic resources. Our work is presented in an organized fashion but may be hard to understand at times.
1 Our musical instrument contains less than 5 notes and they are not tuned accurately. Our presentation includes inaccurate graphs or few graphs for the notes our instrument plays. The graphs in our presentation are inaccurate or incomplete. Our presentation shows very limited understanding of the underlying concepts needed to find the note from the graph. Our product shows little evidence of effective use of technology tools and electronic resources. Our work appears sloppy and unorganized. It is hard to know what information goes together.