€¦ · web viewcreating and analyzing a euclidean polyrhythm for the machine. feedback:

Overview

Title of the Project/Task: Math and Music: Creating Geared Rhythm Machines

Essential Question:How can I use the Euclidean Algorithm to create polyrhythms for a geared rhythm machine?

Overview and Purpose:Students will work on this interdisciplinary music project in Math, Science, English, History, and Visual Arts classes. The geared rhythm machine component will be a focus in math and science class. In English, History, and Art, students will be writing songs, learning about musical genres, considering social justice issues, and creating album art. The solo math and gears component could take 3 to 5 weeks. Our school serves primarily English Language Learners, so that time frame might be longer than what other populations may need. This project will help students build persistence and problem solving skills. Students will be working with the concept of gears and how they work, rhythms and polyrhythms, and how to create polyrhythms using the Euclidean Algorithm as they build their machines. Students will demonstrate what they’ve learned by creating a podcast to summarize their work, and then presenting their project to a teacher at school who will engage in a conversation with them about what they have learned. This is to prepare students for their Senior (Capstone) individual portfolio presentations. We are partnering with BEAM center to create the project and learn how to teach students to create the gears digitally, then print them.

Standards (including technical standards):CCSS for Mathematical Practice:

1. Make Sense of Problems and Persevere in Solving Them2. Reason Abstractly and Quantitatively3. Construct Viable Arguments and Critique the Reasoning of Others4. Model With Mathematics

*other content that could be taught in this unit includes gear ratios and least common multiple, as well as different number patterns for the number of the gear teeth.

*co-taught with English, Science, Art, and History departments. The Science teacher completed a Science of Sounds unit about Frequency and Wavelength where he had the students choose a chord and did the math to cut three lengths of the pipe to make that chord using this indestructible. Tools for Ambitious Science Teaching has a helpful Physics of Sound unit.

Twenty-first Century Skills:● Collaboration● Time Use and Planning● Personal Mindset

Key Activities That Develop 21st Century Skills:

Entry Event: Exploring the components of our project online: rhythm and polyrhythm/gears

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Activity 1: Students will create a Euclidean algorithm in preparation for creating the rhythms they will use on their machine

Activity 2: Exploring the components of our project online: rhythm and polyrhythm/ gears

Activity 3: Use Euclid’s Algorithm to represent your gear teeth and the number of beats you want to hear visually

Activity 4: Build your machine and adjust where necessary

Activity 5: Creating and analyzing a Euclidean polyrhythm for the machine

Feedback:Students give each other feedback on the construction of the gear machine as well as solve problems with other groups as they prototype and build their machines. The teacher will model collaboration, help students make contracts, and lead students in discussions about what works and does not work in their groups. The teacher will provide feedback on the math of the machines before students build the machines. Personal mindset is developed through the use of a project planner and the project checklist. The teacher will provide students with feedback on the final product presentation.

About Our School:The mission of The Brooklyn International High School is to help English Language Learners (ELLs) to develop their language, intellectual, cultural, and collaborative abilities so that our students become active participants in today’s interdependent world. We strive to help every student reach his or her potential by fostering academic growth, personal growth, and tolerance. At BIHS, our teaching philosophy is focused on real-world content and project-based learning. Throughout the year, students and teachers work collaboratively on authentic, experiential, interdisciplinary projects which are presented to fellow teachers, students, and outside visitors.

Credits: Lindsay Hubert Mark Kleeb (BEAM Center partner)

Supported by Educurious

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Lesson/Steps/Activities

Entry Activity: Finding Math in Music: Exploring the components of our project online: rhythm and polyrhythm/ gears

Student Directions Teacher Notes

Question students are answering:1. What do you know about music already?2. What rhythms do you hear?3. Do different music cultures have different rhythms?4. How can you hear math in music?

Skills and Knowledge:● Problem Solving● Different types of rhythms in different types of music

21st Century Skills:● Collaboration

Task Introduction:Today we will be exploring how to create rhythms and looking at how gears work in class.

Goals/Outcomes (21st Century Skills):Exploring and thinking about how rhythms and gears might relate to each other.

Feedback and Evidence:“Examples: “Your deliverable for the day is…”or “Ask two people to review your work other than your teacher…”

Your deliverable for the day is to complete the rhythm and gears activity guide.

Resources You Will Need:● Recording Rhythm Activity Guide

What to Do:1. As a class and individually/ in groups, looking at

a. Marc DeDouvan b. N oise Machines c. software to download and run

2. Students complete a guide where they record their observations about the rhythms they create and hear. (I am making this activity in hindsight and don’t have a guide to share.) Questions would focus around what changes when students change certain aspects in each polyrhythm generator.

These could provide directions for teachers, educative features (e.g. background on the topic, additional resources), alternatives for differentiation, etc.

A video of our prototype

Another great inspirational video, different and more complex, http://vimeo.com/3958856.

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3. Students explore gearsket.ch and complete a similar guide (with screenshots) where they notice what they observe.

4. Groups report back to the class. Another way to do this would be to have each group become experts in one of the above websites and teach it to the class.

5. Students predict what the next project will be about.6. Students see inspiration video and get ready to begin the project.

Alternately, you could show this first to inspire interest.

Activity #1&3: Creating Euclidean Rhythms: Students will create a Euclidean algorithm in preparation for creating the rhythms they will use on their machine.


Question students are answering:1. What is a Euclidean Algorithm?2. How is it calculated for a rhythm?3. How can you create a Euclidean polyrhythm for your geared music

machine?

Skills and Knowledge:● Problem Solving● Euclidean mathematics applied to rhythm and polyrhythm

21st Century Skills:● Collaboration: Groups must create rhythm and plan together.

Task Introduction:“ Today you will …” work on thinking about how to create three rhythms that will work together for your music machine.

Goals/Outcomes (21st Century Skills):Working individually to understand then collaboratively with your group to create a rhythm for your machine that you agree on.


Your deliverable for this activity is to create and calculate rhythms individually and then arrive at a group consensus for the rhythm you will use on your machine.

Resources You Will Need:● An app like the iPad Rhythm Necklace or the Java Wouter

Hisschemolloer generator (old and fussy at times, but works decently with patience) is a good way for students to sound out their


This is the academic paper that talks about the links between the math and the rhythms.

This page gives some examples of Euclidean rhythms in modern music.

Further examples:http://www.salsabeatmachine.org/

Drum examples of tresillo and cinquillo

tresillo (3/8)simple tresillotresillo in Indian musicTresillo in Ewe Sohu dancecinquillo (5/8)Hound Dog clapping rhythmstreet musicians in Dominica

Useful short videos about Euclid and a basic definition of an algorithm:Euclid 1Euclid 2Algorithm (BrainPop, subscription may be required)

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rhythms. I would have students calculate and count out their own rhythms first so they don’t rely on the computer to calculate them for you.

● I used Unifix cubes in blue and red for computing the algorithms with students, with an accompanying guide for students to transfer their rhythm to.

What to Do:1. Discuss who Euclid was and what an algorithm is. Talk about how it

turns out that this algorithm happens to describe rhythms of music all over the world. Show examples.

2. Demonstrate first or show video. Do several as a class.3. Be prepared for students to have to attempt this several times.

Have students choose numbers for a rhythm and calculated it on the sheet linked above.

4. Students practice with the teacher, then as a table to create algorithms with numbers I choose, then individually.

5. Then if they want to, and maybe this should be introduced much later if you don’t want it to be a crutch, students can use the software to verify their work and figure out three rhythms that work well together in preparation for designing their gears as a kind of digital prototype.

Videos I made to explain the Euclidean Algorithm visually and with manipulatives.Video 1Video 2Photo

Activity #2: Designing My Machine: Exploring the components of our project online: rhythm and polyrhythm/ gears


Question students are answering:1. How do I design my gear machine?2. How do I use software for the laser cutter?

Skills and Knowledge:● How to create files for the laser cutter● How to change and revise your prototyped design

21st Century Skills:● Model with Mathematics● Collaboration● Problem Solving

Task Introduction (for Student):“ Today you will …” For this activity you will download Inkscape and use it to create your gears. Later we will cut them out in the FabLab.

Goals/Outcomes (21st century skills):N/A


For using an 18” by 24” sheet of wood for the gears, it would be best if students chose three gears to make rhythms. For rhythms that sound clearest, it is better both sound-wise and operation-wise to have gears with more teeth and fewer sounds playing. For example, gears with 35-50 odd (larger) teeth and 5-10 sounds played. You might not want to tell students yet as they will likely end up calculating a lot of rhythms in search of a good combination, but it will be helpful in the future.

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Feedback and Evidence (for Student):“Examples: “Your deliverable for the day is…”or “Ask two people to review your work other than your teacher…”Your deliverable for this is to have three working gears that will fit on your gear machine.

Resources You Will Need:● Inkscape (free)● CorelDraw or Adobe Illustrator to create the file that the laser cutter

can use to cut out the gears

What to Do: (This provides directions for the activity with links to the resources students will need)

1. Together with the teacher, students go through the basics of how to use the software chosen.

2. In Inkscape, we have to go over how to resize the window, how to create gears with the correct number of teeth and gear angle/ size, and how to put text with group names on the gears. We found that some fonts didn’t translate well when converting the files, so simple fonts are best.

Students must be careful NOT to resize their gears using the drag-resize tool, as it changes the tooth size AND results in ovoid, unusable gears.

Activity #3: Use Euclid’s Algorithm to represent your gear teeth and the number of beats you want to hear visually(SEE NUMBER ONE)

Activity #4: Creating our Gear Machine: Building and assembling the geared rhythm machine


Question students are answering: How do we work together to build a geared rhythm machine?

Skills and Knowledge:● Using hammer, drill, pipe cutter, and glue gun.● How to work together to assemble a machine based on a visual

example.

21st Century Skills:● Collaboration● Problem Solving

Task Introduction:

These could provide directions for teachers, educative features(e.g. background on the topic, additional resources), alternatives for differentiation, etc.

The science teacher on my time did a sound unit alongside this unit where students studied sound waves, chose three note chords, did some calculations, and cut pipe to make a three-note chord.

It is also entirely possible to do this like

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“ Today you will …”For this activity, you will work with your group to assemble and troubleshoot your gear machine.

Goals/Outcomes (21st Century Skills):N/A

Feedback and Evidence:“ Today you will …”For this activity, you will work with your group to assemble and troubleshoot your gear machine.

Resources You Will Need:● 14-hole lego pieces● Rubber bands● Glue gun● Drills with 3/16 inch bit● Lego pegs: friction● Frictionless● Long, 14 by 28” piece of plywood● A prototype for students to examine●

If adding pipes for sound:● U-nails● Pipe● Pipe cutter● String● Legs to give place for pipes to hang from

Before beginning, it is helpful for students to make a plan on paper of where the gears will go on their machine. Large construction paper is the same size as the boards we used. I had students lay their gears on the paper and decide on the locations ahead of time.

What to Do:1. Lay gears on board and mark where central holes will go. Drill with

3/16” bit if using lego pegs. Put gears on board.2. Sketch out where to put striker arms (pointing to center of gear) and

holes for pipes, if using. Assemble and hot glue striker arms to wood.

3. If using, make holes for pipes. Add legs at some point.4. Hammer in u-nails to either side of hole. Suspend pipes with holes

drilled at node for desired node.5. Drill holes with 3/16” bit ½” in from edge centered on center of gear

teeth. You can choose to do every hole to allow for maximum rhythm pattern flexibility, or leave this step until when you are making your pattern.

the inspiration video as a percussion machine.

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Activity #5: Making a Rhythm with Gears: Creating a Euclidean polyrhythm for the machine


Question students are answering:1. How do I choose a combination of rhythms that will sound pleasing?2. How do I show my mathematical knowledge in this choice?

Skills and Knowledge:● Calculating Euclidean rhythms to make a polyrhythm● Showing the connection between the calculations and the physical

gear machine.

21st Century Skills:● Collaboration● Problem Solving

Task Introduction:“ Today you will …”For this activity, you will choose a combination of rhythms and make your machine play that rhythm.

Goals/Outcomes (21st Century Skills):N/A


By the end of the activity, you will have decided on a combination of Euclidean rhythms for your gear machine and linked them to the calculations you have made.

Resources You Will Need:N/A

What to Do:1. Look at your three gears. Choose a number of beats you would like

to play for each gear. (Students can use software for this step, but be aware that some of them might then want to skip the calculation step, as the answer for the pattern is already given.)

2. Do the calculations for each gear, using red and blue cubes to calculate the Euclidean rhythm.

3. Put pegs in each gear that reflect the rhythms you have created.4. Optional: have each gear start at a marked starting point. If you like,

you can choose to offset the starting point to vary the sound.

These could provide directions for teachers, educative features(e.g. background on the topic, additional resources), alternatives for differentiation, etc.

Of course, Euclidean rhythms are not necessary for this project. I was just trying to introduce some math that was a little higher-level/ easier to teach than fractions and musical notation. It would be completely possible to change this to any kind of rhythm that you would like.

For any kind of rhythm, figuring out how many turns it would take before the rhythm starts again is a good question, especially if you don’t tell them that it’s the Least Common Multiple beforehand.

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5. Otherwise, if you don’t like the rhythm, you can change the number of pegs. Warning: if you simply move the pegs around, it will no longer be a Euclidean rhythm (as evenly distributed as possible.) Students who do this will not be able to show their gears as following a Euclidean rhythm later.

6. When you have decided on a rhythm, color your gear teeth to match the colors on your Euclidean calculation work.

7. Here is a document where you can show the different ways you have represented your Euclidean rhythms.

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€¦ · web viewcreating and analyzing a euclidean polyrhythm for the machine. feedback:

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