Download lesson 40 proportions

Tucson GEAR UP Project
Math through Mariachi
Lesson 40:
Phi, Phi, Pho, Proportions!
Math Strand: Patterns, Algebra, and Functions Geometry and
Measurement
POS 3.3.11: Solve proportion.
Objectives:
Students will identify what proportions are in
various contexts and apply them to music and
their worlds.
Teacher Directions:
A few simple proportions will be introduced and then an extension of a special
proportion (phi) and its relationship to music, the world, and even the students themselves will
be introduced.
Important Definitions:
 a part is considered with relationship to the whole or two numbers described by a
quotient, colon, or the word “to” (ratio)
o Example ¾, 3 to 4, 3 : 4,
 two ratios that are equivalent (proportion)
o Example: ¾ = 9/12, 4/8=2/4 = ½
Proportions are written as fractions and come in various mathematical problems.
Ask the students if they know of any proportions in their lives before we begin with the lesson.
Sample Responses:

Toy replica and a real car.

Dollhouse furniture and life size furniture.

Miniature dogs and the normal sized dogs.

Model Airplanes.

Regular size and mini Reese’s Peanut Butter Cups!
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Example 1: Calculate the Ratio to solve the Proportion
Maria has to draw two triangles. She knows the following information below. Help her find the
missing sides so she draws the triangles appropriately.
Now that we have some information, set up some
proportions and see how Triangle B relates to
Triangle A.
Ratio between Triangle A and Triangle B:
3 inches = 3.
1 inch
Therefore the sides of Triangle A are THREE
times bigger than the sides of Triangle B.
Warning! Do not confuse this with Triangle A is three time greater in size than Triangle B. (We
would have to compare their areas to do that ratio.)
Now we know the rule to move back and forth between Triangle A and Triangle B.
Proportion 1:
Step 1: 3 inches = 1 inch
4 inches Y inches
Step 2: Cross multiply: 3Y = 4.
Step 3: Solve for Y. Y = 4/3.
Therefore, the side Y in Triangle B is 4/3 inches.
Proportion 2:
Step 1: 3 inches = 1 inch
X inches
5/3 inches
Step 2: Cross multiply: 3 ( 5/3) = X*1
5 = 1X
Step 3: Solve for X. X= 5.
Therefore, the side X in Triangle A is 5 inches.
Notice how if you have three missing pieces of the proportion A = B then you can solve for
C
D
the missing side.
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Example 2: Use a given Ratio to Solve a Proportion
What if we knew that the ratio was 3 between the sides of Triangle A and Triangle B, but we
only knew Triangle A’s measurements? Could we still find all of the sides for Triangle B?
Sure could!
Set up our proportions again, but instead add in
the proportion that accounts for our given ratio
between Triangle A and Triangle B:
Proportion 1:
Step 1:
Sides
Given ratio
4 inches =
3
Y inches
1
Step 2: Cross multiply: 4* 1 = 3Y
Step 3: Solve for Y. 4 = 3Y
4/3 = Y.
See how we got the same answer as above!
Based on the remarkable website http://GoldenNumber.net, we see that there is a special
proportion in nature called Phi (pronounced Fi with a strong i (or sometimes Fee with a strong e
sound). With its many pronunciations, it has many names: Golden ratio, golden proportion,
golden number, etc.
It appears most popularly in architecture with the golden rectangle. If the length and width of a
rectangle have a ratio of 1.618 (approximately), the rectangle is said to be “golden.” For the
Fibonacci sequence ( 1,1,2,3,5,8,13,21…), as we take the ratio of two consecutive terms in the
sequence, it approaches…. 1.618! The golden ratio! No way.
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Good thing you asked! We find this Golden ratio (or set of equivalent ratios that create
proportions as well) in music with the keys on a piano and how a violin is created!
Our website states that on a piano:
There are 13 notes in the span of any note through its octave (from one C to the
next higher C). A scale is comprised of 8 notes, of which the 5th and 3rd notes
create the basic foundation of all chords, and are based on whole tone which is 2
steps from the root tone, that is the 1st note of the scale.
Note too how the piano keyboard scale of C to C above of 13 keys has 8 white
keys and 5 black keys, split into groups of 3 and 2.
In addition, “Fibonacci and phi relationships are found in the timing of musical
compositions. As an example, the climax of a song is often found at roughly the phi point
(61.8%) of the song, as opposed to the middle or end of the song. (61.8% is derived from
1.618=1+.618 = 100% + 61.8%)
Finally, “Fibonacci and phi are used in the design of violins and even in the design of
high quality speaker wire!” Below is a snapshot of the Medici Violin that follows the golden
ratio provided by our website with a few additions.
It is up to the teacher’s discretion as to whether
you want to spend some time measuring the
instruments to determine this proportion. This
website is an EXCELLENT source of everything
phi! Phi is even in the ratio of our teeth, arms/legs,
and facial features! Spend some time on the website
and try to incorporate any measurements the
students could do themselves if time allows.
Additional websites for Phi:
http://cuip.uchicago.edu/~dlnarain/golden/activity8.htm
http://www.math.uiuc.edu/~gfrancis/math306/math306web/GoldenRatioPeteWintermute.htm
http://goldennumber.net
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