Musical Math Project
For this project, we modified the lyrics of Jack Johnson's Banana Pancakes to reflect the math concept of the unit circle. This project was based on Calculus the Musical - for more information, visit their site. See below for the song and the lyrics.
Photo cred: http://fc.dekalb.k12.ga.us/~Michael_L_Washington/S13728FDC.7/UnitCirclePancake.gif?src=.PNG
For this project, we modified the lyrics of Jack Johnson's Banana Pancakes to reflect the math concept of the unit circle. This project was based on Calculus the Musical - for more information, visit their site. See below for the song and the lyrics.
"Unit Circle
Pancakes"
Can't you see
that it's just radians?
Ain't no need to use degrees...
Baby, it’s the unit circle
All the way is two pi
One pi is one eighty
One quarter is a one point six.
Thirty is one third of this
Two times that is pi thirds
Forty five’s in the middle
Pretend like it's the pi fourths now
Radians are the radius
They go around the unit circle
Ain't no need to use degrees
But just maybe, try to see the cosine
Thirty, forty-five, sixty
Square root three over two
And two below square root of two
One half is for sixty
'cause cosine zero is one
Cosine pi is minus one
Add for correspondin’ measures now
Radians are the radius
They go around the unit circle
Ain't no need to use degrees
Ain't no need to use degrees...
Baby, it’s the unit circle
All the way is two pi
One pi is one eighty
One quarter is a one point six.
Thirty is one third of this
Two times that is pi thirds
Forty five’s in the middle
Pretend like it's the pi fourths now
Radians are the radius
They go around the unit circle
Ain't no need to use degrees
But just maybe, try to see the cosine
Thirty, forty-five, sixty
Square root three over two
And two below square root of two
One half is for sixty
'cause cosine zero is one
Cosine pi is minus one
Add for correspondin’ measures now
Radians are the radius
They go around the unit circle
Ain't no need to use degrees
For the trig, for
the trig, mmm, mmm,
Use the pi, use the pi
Use all times, I don’t mind.
The opposite of cosine
Is sine
And the tangent
Sine on cosine
We need to use the decimals
We need the fractions
And the square roots
Just so easy
When the whole circle fits inside of two pi
Don't really need to pay attention to the degrees
Tangent sine, mmmm and cosine
Baby, it’s the unit circle
All the way is two pi
One pi is one eighty
One quarter is a one point six.
Thirty is one third of this
Two times that is pi thirds
Forty five’s in the middle
Pretend like it's the pi fourths now
Use the pi, use the pi
Use all times, I don’t mind.
The opposite of cosine
Is sine
And the tangent
Sine on cosine
We need to use the decimals
We need the fractions
And the square roots
Just so easy
When the whole circle fits inside of two pi
Don't really need to pay attention to the degrees
Tangent sine, mmmm and cosine
Baby, it’s the unit circle
All the way is two pi
One pi is one eighty
One quarter is a one point six.
Thirty is one third of this
Two times that is pi thirds
Forty five’s in the middle
Pretend like it's the pi fourths now
Radians are the
radius
They go around the unit circle
Ain't no need to use degrees
They go around the unit circle
Ain't no need to use degrees
For the trig, for
the trig
Use all times and I really really, really don't mind
Use the pi, use the pi
Tangent sine cosine
Use all times and I really really, really don't mind
Use the pi, use the pi
Tangent sine cosine
Sequences and Series Fractal Project
Modified Snowflake
Modified Snowflake
2D Analysis
Recursive Definition:
For each iteration, add a new triangle with a side length a third of the previous, to the middle of each new side.
Data:
nth Iteration
|
0
|
1
|
2
|
3
|
4
|
nth Iterative/Explicit
|
nth Recursive
|
As
|
Length of One Side of One New Triangle
|
1
|
1/3
|
1/9
|
1/27
|
1/81
|
an=(1/3)n
|
an=an-1(1/3)
a1=1/3
|
0
|
New Perimeter Added
|
3
|
1
|
2/3
|
4/9
|
8/27
|
an=(2/3)n-1
|
an=an-1(1/3)(2)
a2=1
|
0
|
Perimeter Added
|
3
|
4
|
14/3
|
46/9
|
146/27
|
an=3(1-(2/3)n)+3
|
an=an-1+(2/3)n-1
a1=4
|
6
|
Area of One
|
1
|
1/3
|
2/27
|
4/243
|
8/2187
|
an=(1/3)(2/9)n-1
|
an=an-1(2/9)
a1=(1/3)
|
0
|
Area of All
|
1
|
4/3
|
38/27
|
346/243
|
3122/2187
|
an=(3/7)(1-(2/9)n)+1
|
an=an-1+((1/3)(2/9)n-1)
a1=(4/3)
|
10/7
|
Iterations 0, 1, 2
Modified Snowflake
3D Analysis
Recursive Definition:
For each iteration, add a new tetrahedron with a side length a third of the previous, to the center of each new face, keeping the same orientation. Only the edges that stick out are considered perimeter.
nth Iteration
|
0
|
1
|
2
|
3
|
4
|
Explicit
|
Recursive
|
As n-> infinity, the sequence -> ?
|
Length of Side of One
|
1
|
1/3
|
1/9
|
1/27
|
1/81
|
an=(1/3)n
|
an=an-1*(1/3)
a1=1/3
|
0
|
New Perimeter Added
|
6
|
4
|
4
|
4
|
4
|
an=4
1
|
an=an-1
a1=4
|
4
|
Total Perimeter
|
6
|
10
|
14
|
18
|
22
|
an=4n+6
|
an=an-1+4
a1=6
a2=4
|
Infinity
|
New Area
|
4
|
8/9
|
8/27
|
8/81
|
8/243
|
an=8*3n-3
1
|
an=an-1*3*(1/9)
a1=8/9
|
Infinity
|
Total Area
|
4
|
44/9
|
140/27
|
428/81
|
1292/243
|
an=-4(1-3n)
|
an=an-1+8*3n-3
a1=44/9
|
Negative Infinity
|
New Volume
|
4
|
4/9
|
4/243
|
4/729
|
4/6561
|
an=4*3n-3
1
|
an=an-1*3*(1/27)
a1=4/9
|
Infinity
|
Total Volume
|
4
|
40/9
|
1084/243
|
3256/729
|
29308/6561
|
an=-2(1-3n)
|
an=an-1+4*3n-3
a1=40/9
|
Negative Infinity
|
3D Model
Project Reflection
I found this project to be very fun, because we took a concept that we learned through the sierpinsky's triangle, then we got to apply it to our own recursive model. With recursive equations, you can create any number of strange patterns with your data, like with the sierpinsky tetrahedron, where the surface area stays the same, the volume approaches zero, and the perimeter approaches infinity. This kind of thing is mind boggling, but is so cool and strange that you want to learn more about it, and about more things like it. I initially chose to do my project on the modified snowflake because of how cool it looked, but it doesn't have as many cool patterns as a fractal that has things taken away from the inside. I think that the most challenging thing was finding the equations for the patterns in my fractal, because in order to figure them out, you had to break it down quite a bit.
Quote Reaction
J.H. Poincare (1854-1912), (cited in H.E. Huntley, The Divine Proportion, Dover, 1970) "The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."
I think that this quote is something that most students should understand about math. Most people just see the boringness and tedium of math, but once you get deeper, you begin to understand the beauty of it. This year, I have enjoyed math more than I ever have before, because of the content that we are studying. The main concept that I have enjoyed the most is matrix transformations/fractals and strange attractors. In fact, matrices are the one thing that I have never understood before, but this year, I learned about a way to create fractals with matrix transformations. With this skill, you can create sierpinsky's triangle, or a tree, or any number of other fractals on your calculator. I think that once you see what cool things you can do with math, you then begin to see the beauty and fun in it.
