Lesson Plan, Extensions, and Assessments

TABLE OF CONTENTSClick on a title to go directly to the page. You also can click on web addresses to link to external web sites.

Overview of Lesson Including Kentucky Standards Addressed............................................................. 2

Instructional Strategies and Activities• Day One: Ratios and Music ................................................................................ 3-4• Day Two: Extending the Lesson ......................................................................... 5-6• Day Three: Performance, Open Response, and Multiple Choice Assessments ............................................................................. 6

Support/Connections/Resources .............................................................. 7

Adaptations for Diverse Learners/Lesson Extensions ...................... 7

Applications Across the Curriculum ........................................................ 8

Performance Assessment ............................................................................ 9

Open Response Assessment ....................................................................... 10

Multiple Choice Assessment ...................................................................... 11-12

SCALE CITY The Road to Proportional Reasoning: Belle of Louisville Lesson

BELLE OF LOUISVILE: Musical Scale

BELLE OF LOUISVILE: Musical Scale 2

BELLE OF LOUISVILLE: MUSICAL SCALE

Instructional Strategies and Activities

Length: 1-3 days

Concept/Objectives: Students will learn ways that proportional reasoning is applied in music. Students will ex-plore the terms frequency, musical scale, and ratio using skills in mathemat-ics and music. Students will learn that a 2:1 ratio is evident when notes are an octave apart.

Activity: Students will align musical notes using an online interactive. Students will calculate values for musical string and pipe lengths and frequencies using skills in proportional reasoning. Students will apply understanding of varied tuning methods and inverse proportion to answer questions and complete charts related to music.

Resources Used in This Lesson Plan: Scale City Video: Greetings from The Belle of LouisvilleOnline Interactive: Musical ScalesAssessments (included in this lesson)Classroom Handouts (PDFs)

All resources are available at www.scalecity.org.

Grades 6-8Essential QuestionWhat is the proportion-al relationship between the length of a musical string or pipe and the sound it produces?

TechnologycomputerInternet connectioncomputer projectorcomputer lab for indi- vidual or paired exploration

Vocabularydirect proportionfrequencyHertzinverse proportionoctavepitchratio and different methods of expressing ratio (1:4, 1/4 and 1 to 4)scalescale factorx/y = k where x and y are two related variables and k is a constantxy = k where x and y are two related variables and k is a constantwaves

NOTE TO TEACHER:You may want to send an email to parents to let them know about the Scale City web site and encourage them to have their children access the site at home for additional practice.

Sample email to parents

Our mathematics class is exploring proportional reasoning as it applies to real-world problems. This concept is important in algebraic thinking and mathematical reasoning. Currently, students are exploring how mathematics is applied in analyzing musical scales. Kentucky Educational Television has created online interactive learning activities and other resources for students to explore this concept.

We will be using this web site in class instruction. Your child may also access the site www.scalecitiy.org from home for additional practice.

If you are a musician and would like to play for the class, we welcome your participation.

Sincerely,

Teacher

NOTE TO TEACHER:If necessary, given time limitations, this lesson could be done in one class period.


DAY ONE: RATIOS AND MUSIC 1. Distribute “Handout 1: Scale City Calliope Video Notes.” Students will complete the blanks as they watch the video.

2. Use an Internet projector to watch the “Greetings from the Belle of Louisville” video at www.scalecity.org. Or download the video to a DVD to show to your class.

3. Review student responses to Handout 1.

NOTE TO TEACHER:The video’s brief explanation of how a saxophone works simplifies the actual mechanics of this instrument. If you would like to explore this and other wind instruments in more detail, you could start with “How Do Woodwind Instruments Work?” at www.phys.unsw.edu.au/jw/woodwind.html.

4. Using an Internet projector, go to the “Musical Scales” page at www.scalecity.org and follow the prompts to explore the concept. Instruct students to use “Handout 2: Musical Math” to take notes as they work to complete the chart in the activity.

Kentucky Academic Expectations

2.72.82.12

Kentucky Program of Studies

Grade 6MA-6-NPO-U-1MA-6-NPO-U-4MA-6-NPO-S-NO3MA-6-NPO-S-RP3

Grade 7MA-7-NPO-U-4MA-7-NPO-S-RP2MA-7-NPO-S-RP3

Grade 8MA-8-NPO-U-4 MA-8-NPO-S-RP1

Kentucky Core Content for Assessment 4.1

Grade 6MA-06-1.3.1MA-06-1.4.1

Grade 7MA-07-1.3.1MA-07-1.3.2MA-07-1.4.1

Grade 8MA-08-1.3.1MA-08-1.3.2MA-08-1.4.1MA-08-3.1.3MA-08-5.1.5

© KET, 2009

NOTE TO TEACHER REGARDING CALCULATION IN THE “MUSICAL SCALES” INTERACTIVE:

C4 is what piano players think of as “middle C.” It is the C in the middle of the piano keyboard. The chart below shows how the interactive chart for “Musical Scales” will look after it is filled in correctly online. As students measure using the ruler and compute the frequency times length, the interactive will accept and adjust answers in a close range so that the figures will appear as they do below.

When the students enter the product of frequency times length and click “Check Answers,” the value for k will round to the nearest 50, which is 2350 for all the calculations.

Note Frequency Length Frequency x LengthC4 261.63 9 2350

D4 293.66 7.99 2350

E4 329.63 7.13 2350

F4 349.23 6.74 2350

G4 392.00 6.00 2350

A4 440.00 5.35 2350

B4 493.88 4.76 2350

C5 523.26 4.5 2350

However, based on the frequency data and the first length alone, the chart might look like the second version on page 4. The length values for D4, E4, G4 and B4 are marginally different by a value from 0.01 - 0.03.

www.phys.unsw.edu.au/jw/woodwind.html


NOTE TO TEACHER (CONTINUED)If student understanding is advanced and the numbers on the chart are questioned, it is an opportunity to discuss how mathematics gives us targetsfor understanding relationships. The numbers we find in experiments may be slightly different than the theoretical number that describes the relationship. Measurement, conversions, rounding, and other issues may influence the numbers of an experiment. For example, the values for frequency are not limited to two places after the decimal, so rounding may result in different values for this variable. If students were testing frequencies and measuring pipe lengths in a laboratory setting, they would have similar results.

Note Frequency Length Frequency x LengthC4 261.63 9 2354.67

D4 293.66 8.02 2355.1532

E4 329.63 7.1434 2354.678942

F4 349.23 6.7425 2354.683275

G4 392.00 6.0068 2354.6656

A4 440.00 5.3515 2354.66

B4 493.88 4.7677 2354.671676

C5 523.26 4.5 2354.67

5. After exploring all the questions on the online interactive, instruct students to complete “Handout 2: Musical Math” individually or as guided practice.

The key ideas to develop from “Handout 2: Musical Math” are:

• Hertz is the unit of sound for frequency.• Different notes can be identified by frequency.• The higher the pitch, the greater the frequency.• A longer pipe produces a deeper sound at a lower frequency than a shorter pipe.• The notes of a musical scale can be described mathematically.• The frequency of a note an octave higher is two times greater than the frequency of the original note. This relationship can be expressed as a 2:1 ratio.

NOTE TO TEACHER:“Just intonation” is a system of musical tuning in which ratios of whole numbers describe the relationships between the frequencies of notes. It is not often used in Western music because of what composers call “wolf intervals,” intervals between notes that sound wrong to our ears because they are inconsistent with the rest the intervals in the scale. Think of wolf intervals as howling instead of singing. However, from a mathematical view-point, just intonation musical intervals can be expressed as simple ratios and thus are easy to manipulate.

“Tuning Time” gives you and your students an opportunity to briefly discuss how other types of world music have scales and sounds that are different from the Western 12-tone scale.


Alternative Preparation for Day TwoOne alternative you might consider on Day Two is to demonstrate how an instrument might be constructed or to have students try to put instruments together in small groups. Gather only one set of materials if you plan to conduct a classroom demonstration.

• 8 uniform glasses • pitcher of water• pipes cut at different lengths for percussion instruments• rubber bands or string for stringed instruments• stapler• cardboard• straws and scissors for flutes or wind instruments

DAY TWO: EXTENDING THE LESSON

1. Today, students will further explore the concept of ratio in creating music. As an informal quiz, ask students to answer the following four questions:

• What is the unit of sound for frequency? Hertz• As you go up a scale, the pitch gets higher. As the pitch of the music gets higher, what happens to the frequency? It increases.• Two pipes have the same diameter and are made from the same material, but they are different lengths. Which pipe produces the deepest sound? The longer pipe.• What is the ratio of the frequency of two notes an octave apart? 2:1

2. Review the answers with the students. Distribute “Handout 4: What Is a Pythagorean Scale?” Use this as guided practice by discussing with the students methods of calculating the ratios. The inverse proportions of the string length may be difficult at first, but it provides a concrete way for students to experience reasoning with inverse proportion. It would be helpful if one of your students who can play the violin, guitar, or some other stringed instrument could demon-strate how the length of the string affects the sound. Students could even measure the lengths of each note in a scale.

This point in the lesson would also be a good time to bring up multiplying by the inverse of the ratio or fraction to solve the inverse proportions in the handout. Students might be encouraged to discover this method, or you might demonstrate how they could solve the first problem in the table in Handout 4 by multiplying 16 inches by 8/9, the inverse of the ratio of the longer string to the shorter string.

3. Write on the board “What We’ve Learned about Mathematics and Music.” Discuss with students what they’ve learned about how mathematics is involved in what we hear and how music is played.

Possible discussion points:

• The scientific unit Hertz represents the number of electromagnetic waves or cycles per second produced in the creation of sound or electrical signals. Hertz is the unit used to describe sound frequencies or pitches. • Different notes can be identified by frequency.• The higher the pitch, the greater the frequency.• A longer pipe produces a deeper sound at a lower frequency than a shorter pipe.• The notes of a musical scale can be described mathematically.• A note an octave higher has twice the frequency of its lower counterpart. The ratio of the higher note’s frequency to the lower note’s frequency is 2:1. • The relationships between notes can be described using ratios.• The length of the bar, pipe, or string is inversely proportional to the frequency.• Pythagorean ratios indicate that with two strings an octave apart, one string is twice the length of the other and half the frequency.• As a scale goes up and frequency increases, the string gets shorter.


• As a scale goes up and frequency increases, the pipe gets shorter.• The different lengths of pipe in a pan flute result in different pitches when you blow air through them.• Musical notes have different pitches that can be measured by frequency.

4. Ask students to look over the “What We’ve Learned about Music” information and write down any facts they would use in creating a homemade instrument. If time allows, students could make instruments using these ideas and simple materials provided in

* Possible scale of drawing: 1:24 (1” = 0.5 feet or six inches). This would make sense on a piece of 8.5-inch by 11-inch paper. You might point out that it would be easier to do these computations in the metric system, and that instrument makers must use very precise measurements to make sure that the instruments work as they should.

Key to Open Response (see page 10)A5 should have a frequency of 880 Hz. A3 should have a frequency of 220 Hz. A3 would have the lowest, deepest pitch. A4 would be in the middle. A5 would be the highest pitch of the three. The ratio of a musical note’s frequency to the note an octave below it is 2:1, while the ratio of its frequency to the note an octave above is 1:2. As notes get higher, their frequency increases.

Key to Multiple Choice (see pages 11-12)1. D, 2. B, 3. D, 4. A, 5. C, 6. C, 7. B, 8. B

class. This also could be done briefly as a classroom demonstration. Challenge students to construct their own instruments individually at home and bring them in to share a scale or song.

5. You may want to extend the lesson by using “Handout 5: Fractions in Musical Notation” as class work or homework.

DAY THREE: PERFORMANCE, OPEN RESPONSE, AND MULTIPLE CHOICE ASSESSMENTS

Use the Open Response, Performance Assessment, and/or Multiple Choice to assess student understanding.

Key to Performance Assessment (see page 9)

Pythagoras Ratio to First Note Length of Pipe Length of Pipe in Scale Drawing

1:1 4 feet 8 inches*

9:8 3.5555 feet 7.111 inches

81:64 3.1605 feet 6.321 inches

4:3 3 feet 6 inches

3:2 2.6667 feet 5.3334 inches

27:16 2.3703 feet 4.7406 inches

243:128 2.107 feet 4.214 inches

2:1 2 feet 4 inches


Support/Connections/ResourcesWhere Math Meets Musicwww.musicmasterworks.com/WhereMathMeetsMusic.htmlThis web site looks at the mathematics of the musical scale relative to frequency and ratio.

Make a PVC Flutewww.nativeaccess.com/ancestral/flute-adv.html This web site provides step-by-step instructions on how to make a flute out of PVC pipe.

Panpipeswww.philtulga.com/Panpipes.htmlThis site includes virtual panpipes, instructions on how to make five- and eight-note panpipes using the Western and pentatonic scale.

Pianos and Continued Fractionswww.research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTMLIt is an old (and well-understood) problem in music that you can’t tune a piano perfectly. To understand why takes a tiny bit of mathematics and a smattering of physics (acoustics, namely). This web site explains why.

Ancient Greek Origins of the Western Musical Scalewww.midicode.com/tunings/greek.shtmlLearn more about Pythagoras and musical scales of ancient Greece.

A Pythagorean Tuning of the Diatonic Scalewww.music.sc.edu/fs/bain/atmi02/pst/index.htmlThis web site compares Pythagorean tuning and modern piano tuning.

Musical Intervals, Frequency, and Ratiocnx.org/content/m11808/latest/This site looks at the relationship of musical intervals and frequency ratios, providing examples and exercises.

Temperament and Musical Scaleshyperphysics.phy-astr.gsu.edu/Hbase/music/et.htmlThis site explains the equal tempered scale, the common musical scale used for tuning pianos and other instruments of relatively fixed scale.

Pythagoras Woodcut: A 15th-Century Depiction of Pythagoras Doing Musical Experimentswww.britannica.com/EBchecked/topic-art/485171/75247/Pythagoras-coloured

Adaptations for Diverse LearnersExplore the way that homemade instruments reinforce the basic science and math of music. Making a variety of these instruments may reinforce the uniform rules. Have students who are musicians or who have friends and family who play music provide classroom demonstrations.

Lesson Extension: Use “Handout 5: Fractions in Musical Notation” as a way of reinforcing the mathematics in the rhythm of music.

www.musicmasterworks.com/WhereMathMeetsMusic.html

www.nativeaccess.com/ancestral/flute-adv.html

www.philtulga.com/Panpipes.html

www.midicode.com/tunings/greek.shtml

www.music.sc.edu/fs/bain/atmi02/pst/index.html

http://cnx.org/content/m11808/latest/

http://hyperphysics.phy-astr.gsu.edu/Hbase/music/et.html

www.britannica.com/EBchecked/topic-art/485171/75247/Pythagoras-coloured

http://research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTML


Applications Across the CurriculumScienceFrequency alone provides a rich topic for science. Accelerated students will be able to examine what cycles per second means. A companion science exploration of sound will be very useful to students using the ratios of Pythagoras, just intonation, and measurements of frequency to create and play instruments. Terms like Hertz, sine wave, and frequency would be essential to understanding.

Practical Living Examine the frequencies commonly heard by middle school students. If possible, arrange for hearing tests for students. Discuss ways of preventing hearing loss.

Vocational StudiesCollege students often study both music and mathematics. Examine the similarities of these career fields, the career of musicologist, or the study of music theory. Students may be interested in the way music and mathematics are integrated in many careers. Examine the “Mozart Effect” to see how music influences brain activity related to spatial reasoning.

Music This lesson lends itself nicely to further exploration of instrument families and alternate music scales around the world. KET’s Music Toolkit includes a CD called World Music, which includes many different musical methods and instruments from around the world. Students can also explore world music and dance in another multimedia CD-ROM, The World of Dance and Music, available with the 2nd edition of the Dance Toolkit. For more information, go to www.ket.org/artstoolkit. At present, the Toolkits are only available to Kentucky educators, but you’ll find many other wonderful resources at the Toolkit web site that anyone can access. Teachers also might explore how people “play by ear” without reading musical notation. Examine the skills and knowledge necessary to “play by ear.” Many people who don’t read music notation have very clear understanding of chords and rhythm. Test if tuning by “ear” is as effective as tuning by a digital tuner. Discuss how “having a good ear” for music is developed through specific skills that are also related to mathematical reasoning.

Music EnrichmentDiscuss with a music teacher or musician, the mathematics of tuning. A demonstration with an instrument, digital tuner, and chart would be helpful. Some musicians say that equal temperament tuning (12-TET) is a compromise: In an effort to make all notes tuned, no note sounds very good. An octave includes 12 notes. The following chart shows the frequencies if tuned by equal temperament tuning (12-TET). While it may at first appear that the spacing between notes is not an even interval, examining the numbers shows that the frequencies increase proportionally. You might ask students to compare these data to the data in the chart in question 3 of Handout 3.

Note Equal Temperament Tuning Frequency

Difference in Frequency from Previous Note

Percent Change from Previous to Current Note

C2 65.406 Not Applicable Not Applicable

C#2 69.296 3.89 5.95%

D2 73.416 4.12 5.95%

D#2 77.782 4.366 5.95%

E2 82.407 4.625 5.95%

F2 87.307 4.9 5.95%

F#2 92.499 5.192 5.95%

G2 97.999 5.5 5.95%

G#2 103.83 5.831 5.95%

A2 110.0 6.17 5.95%

A#2 116.54 6.54 5.95%

B2 123.47 6.93 5.95%

C3 130.81 7.34 5.95%

www.ket.org/artstoolkit


PERFORMANCE SCORING GUIDE

4 3 2 1 0

PERFORMANCE ASSESSMENT

SCALE CITYPrompt Determine the length of each pipe in a PVC pipe instrument that will play an eight-note scale. Then determine a scale for a drawing of the instrument, and determine how long each pipe would be in the drawing.

DirectionsUse your knowledge of musical scale and proportional reasoning to complete the following chart.

•Thestudentappliesproportionalreasoningtothistaskwithfewerrors.•Thestudentsuccessfullycompletesatleast12answersofthechart,includingconvertingmostmeasurementscorrectlytothesmallerscale.

•Thestudentappliesproportionalreasoningtothistaskwithgoodgeneralunderstanding.•Thestudentsuccessfullycompletesatleast8answersofthechart,butresponsesindicatepossiblegapsinunderstandingofproportionsandscaleorerrorsincomputation.

•Thestudentappliesproportionalreasoningtothistaskwithlimitedunderstanding.•Thestudentmakesanefforttocompletethechart,butfailstocompletetheanswerscorrectly.

•Blankornoresponse.

Pythagoras Ratio to First Note

Length of Pipe Length of Pipe in Scale Drawing

1:1 4 feet

9:8

81:64

4:3

3:2

27:16

243:128

2:1

Scale:1” = _________________

•Thestudentappliesproportionalreasoningtothistaskwithfewerrors.•Thestudentsuccessfullycompletesatleast12answersofthechart,includingconvertingmostmeasurementscorrectlytothesmallerscale.


OPEN RESPONSE SCORING GUIDE

4 3 2 1 0

SCALE CITYPrompt Use your knowledge of frequency to predict the frequency of two notes and describe their sound.

Directions A digital tuner set at 440 Hz is used to tune an instrument. The 440 Hz is called A4 indicating where it would be on the piano keyboard. Use this information to predict what the frequencies should be for an octave higher (A5) and an octave lower (A3). How would A5 and A3 compare with the pitch of A4? What is the ratio of a musical note’s frequency to the frequency of the note an octave lower? To the frequency of the note an octave higher?

•Thewritingshowsoutstandingcompre-hensionandcommuni-cationofproportionalreasoningasitrelatestomusicaloctaves.•Thevaluesarecorrectlyrepresentedandlabeledwithnoerrors.•Theresponseindicatesclearunderstandingoffrequencyandpitch.•Thestudentcorrectlyidentifiestheratioofamusicalnote’sfrequencytothefrequencyofthenoteanoctavelowerandanoctavehigher.

•Thewritingshowsgoodunderstandingofpropor-tionalreasoningasitrelatestomusicaloctaves.•Thevaluesarecorrectlyrepresentedandlabeledwithfewerrors.•Theresponseindicatesadequateunderstandingoffrequencyandpitch.•Thestudentcorrectlyidentifiestheratioofamusicalnote’sfrequencytothefrequencyofthenoteanoctavelowerandanoctavehigher.

•Thewritingshowsgeneralunderstandingofpropor-tionalreasoningasitrelatestomusicaloctaves.•Thevaluesmayhaveerrorsorbeincorrectlylabeled.•Theresponseindicatesgeneralunderstandingoffrequencyandpitch.•Thestudentcorrectlyidentifiestheratioofamusicalnote’sfrequencytothefrequencyofthenoteanoctavelowerandanoctavehigher.

•Thewritingshowsminimalunderstandingofproportionalreasoningasitrelatestomusicaloctaves.•Theanswerindicateslackofeffortorunderstanding.•Theresponseshowslimitedunderstandingoftheconceptsdiscussedinclass.

•Blankornoresponse

OPEN RESPONSE ASSESSMENT


Name: Date:

MULTIPLE CHOICE ASSESSMENT

1. The science teacher says you’ll get extra credit if you can hit the lowest note on the xylophone on the first strike. You should pickA. the shortest barB. the bar that is the middle lengthC. the bar next to the shortest barD. the longest bar

2. The scientific measurement of frequency used for sound is A. mphB. HertzC. wavesD. Quartz

3. The Greek philosopher who may have been the first to note the relationship between math and music and who is credited with developing a method of tuning based on ratio isA. ArchimedesB. ThalesC. SocratesD. Pythagoras

4. The calliope in the Belle of Louisville would play the highest pitch with theA. shortest pipeB. middle-sized pipe.C. longest pipe. D. pipe closest to 2 feet tall

5. If the frequency of a note is 220 Hz, an octave higher should have a frequency of A. 110 HzB. 220 HzC. 440 HzD. 660 Hz 6. A guitar player changes the pitch by A. holding the guitar in two handsB. strumming with a pickC. pressing and shortening the stringD. pushing air through the frets


7. The frequency of G4 to the frequency of G3 (the G note an octave below it) is represented as the ratioA. 1:2B. 2:1C. 4:3D. 3:4

8. The string was 12 inches long, so a string that produces a sound twice its frequency would beA. 24 inches longB. 6 inches longC. 4 inches longD. 2 inches long

Multiple Choice Assessment

Lesson Plan, Extensions, and Assessments

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