Download SCALE CITY The Road to Proportional Reasoning: Belle of

SCALE CITY
The Road to Propor tional Reasonin
g:
Belle of Louisville Lesson
TABLE OF CONTENTS
Click 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 ................................................................................-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
BELLE OF LOUISVILE: Musical Scale
BELLE OF LOUISVILLE: MUSICAL SCALE
Grades 6-8
Essential
Question
Length:
1-3 days
What is the proportional relationship between
the length of a musical
string or pipe and the
sound it produces?
Technology
computer
Internet connection
computer projector
computer lab for individual or paired
exploration
Concept/Objectives:
Students will learn ways
that proportional
reasoning is applied in
music. Students will explore the terms frequency,
musical scale, and ratio
using skills in mathematics 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 Louisville
Online Interactive:
Musical Scales
Assessments (included in
this lesson)
Classroom Handouts
(PDFs)
All resources are
available at
www.scalecity.org.
Vocabulary
direct proportion
frequency
Hertz
inverse proportion
octave
pitch
ratio and different
methods of
expressing ratio
(1:4, 1/4 and 1 to 4)
scale
scale factor
x/y = k where x and y
are two related
variables and k is a
constant
xy = k where x and y
are two related
variables and k is a
constant
waves
Instructional Strategies and Activities
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.
BELLE OF LOUISVILE: Musical Scale 2
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.
Kentucky
Academic
Expectations
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.
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 Length
C4
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.
2.7
2.8
2.12
Kentucky
Program of
Studies
Grade 6
MA-6-NPO-U-1
MA-6-NPO-U-4
MA-6-NPO-S-NO3
MA-6-NPO-S-RP3
Grade 7
MA-7-NPO-U-4
MA-7-NPO-S-RP2
MA-7-NPO-S-RP3
Grade 8
MA-8-NPO-U-4
MA-8-NPO-S-RP1
Kentucky Core
Content for
Assessment 4.1
Grade 6
MA-06-1.3.1
MA-06-1.4.1
Grade 7
MA-07-1.3.1
MA-07-1.3.2
MA-07-1.4.1
Grade 8
MA-08-1.3.1
MA-08-1.3.2
MA-08-1.4.1
MA-08-3.1.3
MA-08-5.1.5
© KET, 2009
BELLE OF LOUISVILE: Musical Scale 3
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 targets
for 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 Length
C4
261.63
9
2354.67
D4
293.66
8.02
2355.1532
E4
329.63
7.1434
2354.678942
F4
G4
349.23
392.00
6.7425
6.0068
2354.683275
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 viewpoint, 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.
BELLE OF LOUISVILE: Musical Scale 4
Alternative Preparation for Day Two
One 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 demonstrate 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.
BELLE OF LOUISVILE: Musical Scale 5
• 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
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
* 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
BELLE OF LOUISVILE: Musical Scale 6
Support/Connections/Resources
Where Math Meets Music
www.musicmasterworks.com/WhereMathMeetsMusic.html
This web site looks at the mathematics of the musical scale relative to frequency
and ratio.
Make a PVC Flute
www.nativeaccess.com/ancestral/flute-adv.html
This web site provides step-by-step instructions on how to make a flute out of PVC pipe.
Panpipes
www.philtulga.com/Panpipes.html
This site includes virtual panpipes, instructions on how to make five- and eight-note panpipes using the Western and
pentatonic scale.
Pianos and Continued Fractions
www.research.att.com/~njas/sequences/DUNNE/TEMPERAMENT.HTML
It 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 Scale
www.midicode.com/tunings/greek.shtml
Learn more about Pythagoras and musical scales of ancient Greece.
A Pythagorean Tuning of the Diatonic Scale
www.music.sc.edu/fs/bain/atmi02/pst/index.html
This web site compares Pythagorean tuning and modern piano tuning.
Musical Intervals, Frequency, and Ratio
cnx.org/content/m11808/latest/
This site looks at the relationship of musical intervals and frequency ratios, providing examples and exercises.
Temperament and Musical Scales
hyperphysics.phy-astr.gsu.edu/Hbase/music/et.html
This 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 Experiments
www.britannica.com/EBchecked/topic-art/485171/75247/Pythagoras-coloured
Adaptations for Diverse Learners
Explore 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.
BELLE OF LOUISVILE: Musical Scale 7
Applications Across the Curriculum
Science
Frequency 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 Studies
College 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 Enrichment
Discuss 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%
BELLE OF LOUISVILE: Musical Scale 8
SMENT
S
E
S
S
A
E
C
N
A
M
R
PERFO
SCALE CITY
Prompt
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.
Directions
Use your knowledge of musical scale and proportional reasoning to complete the following chart.
Pythagoras Ratio
to First Note
1:1
Length of Pipe
Length of Pipe in
Scale Drawing
Scale:
1” = _________________
4 feet
9:8
81:64
4:3
3:2
27:16
243:128
2:1
PERFORMANCE SCORING GUIDE
4
•The student applies proportional reasoning to this task with few errors.
•The student successfully completes at least 12 answers of the chart, including converting most measurements correctly to the smaller scale.
3
•The student applies proportional reasoning to this task with few errors.
•The student successfully completes at least 12 answers of the chart, including converting most measurements correctly to the smaller scale.
2
•The student applies proportional reasoning to this task with good general understanding.
•The student successfully completes at least 8 answers of the chart, but responses indicate possible gaps in under
standing of proportions and scale or errors in computation.
1
•The student applies proportional reasoning to this task with limited understanding.
•The student makes an effort to complete the chart, but fails to complete
the answers correctly.
BELLE OF LOUISVILE: Musical Scale 9
0
•Blank or no response.
NT
E
M
S
S
E
S
S
A
E
S
N
O
OPEN RESP
SCALE CITY
Prompt
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?
OPEN RESPONSE SCORING GUIDE
4
•The writing shows outstanding compre- hension and communi-
cation of proportional reasoning as it relates to musical octaves.
•The values are correctly represented and labeled with no errors.
•The response indicates clear understanding of frequency and pitch.
•The student correctly identifies the ratio of a musical note’s frequency to the frequency of the note an octave lower and an octave higher.
3
•The writing shows good understanding of propor-
tional reasoning as it relates to musical octaves.
•The values are correctly represented and labeled with few errors.
•The response indicates adequate understanding of frequency and pitch.
•The student correctly identifies the ratio of a musical note’s frequency to the frequency of the note an octave lower and an octave higher.
2
•The writing shows general
understanding of propor-
tional reasoning as it relates to musical octaves.
•The values may have errors or be incorrectly labeled.
•The response indicates general understanding of frequency and pitch.
•The student correctly identifies the ratio of a musical note’s frequency to the frequency of the note an octave lower and an octave higher.
BELLE OF LOUISVILE: Musical Scale 10
1
•The writing shows minimal understanding of proportional reasoning as it relates to musical octaves.
•The answer indicates lack of effort or understanding.
•The response shows limited understanding of the concepts discussed in class.
0
•Blank or no response
ENT
M
S
S
E
S
S
A
E
IC
O
H
MULTIPLE C
Name:
Date:
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 pick
A. the shortest bar
B. the bar that is the middle length
C. the bar next to the shortest bar
D. the longest bar
2. The scientific measurement of frequency used for sound is
A. mph
B. Hertz
C. waves
D. 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 is
A. Archimedes
B. Thales
C. Socrates
D. Pythagoras
4. The calliope in the Belle of Louisville would play the highest pitch with the
A. shortest pipe
B. 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 Hz
B. 220 Hz
C. 440 Hz
D. 660 Hz
6. A guitar player changes the pitch by
A. holding the guitar in two hands
B. strumming with a pick
C. pressing and shortening the string
D. pushing air through the frets
BELLE OF LOUISVILE: Musical Scale 11
Multiple Choice Assessment
7. The frequency of G4 to the frequency of G3 (the G note an octave below it) is represented as the ratio
A. 1:2
B. 2:1
C. 4:3
D. 3:4
8. The string was 12 inches long, so a string that produces a sound twice its frequency would be
A. 24 inches long
B. 6 inches long
C. 4 inches long
D. 2 inches long
BELLE OF LOUISVILE: Musical Scale 12