Larry Lesser
“Math Song Sing-Along:
Formula for a Hit!”
join me on the choruses
& become a choral-Larry!
UTEP’s Bhutanese
architecture was
‘Math Lens’ in Sept. 2008
Mathematics Teacher!
for lots of math & music resources, lyrics, MP3s, etc.,
just Google me!
calculus co-inventor
Gottfried Wilhelm von Leibniz
“Music is the pleasure
the human soul experiences
from counting
without being aware
that it is counting.”
Connections with structure of music:
demos with acoustics, music theory, etc.
(CCSSM Mathematical Practice standard #7: “look for and make use of structure”)
e.g., see Sept. 2009
Mathematics Teacher for
how group theory
and geometric transformations
connect to music
MATH
MUSIC
Ordinal numbers
naming intervals (e.g., fifths)
Least common multiple
rhythm patterns;
harmony (from pitches w/ low LCM)
Fractions
time signature;
interval is a ratio of frequencies
Geometric shapes
instrument shapes (e.g., triangle)
Arithmetic sequence
overtones (f, 2f, 3f, 4f, …)
Geometric sequence
chromatic scale (in equal temperament)
Sine function
graph of (pure tone) sound wave
Graph (pitch over time)
musical notes on a staff
Geometric transformations
melodic transformations
Permutations,
Probability
“change ringing” of bells;
Mozart’s “Musical dice game” (1793)
Statistics
DATASOUND: sonification
SOUNDDATA: traits of hit songs
Group theory, modular arithmetic
analyze the set of pitches in a scale
Fourier series, partial differential equations
musical sound, acoustics
math in lyrics of “regular songs”…
• “One is the loneliest number” – Three Dog Night
• “if I could count infinity” – Christine Kane
• “100% chance of rain” – Gary Morris
• “reduce me to the mean” – Richard Shindell
• “random sample, hold the one you need” – Rush
• “at the edges of the bell-shaped curve” – David Wilcox
• “it’s like a Möbius strip” – Nelly Furtado
…and of Math Songs!
Sometimes, the title is already a math word, like Taylor Swift’s “Mean”!
Other times, I have to change it:
Don McLean’s “American Pie” became “American Pi”
3 Dog Night’s “One is the Loneliest Number” became
“One is the Likeliest Number” (HW: look up “Benford’s Law”)
Okay, let’s start singing
with a song that helps students
recall the 2 main circle formulas
using the tune that helped us all
learn the alphabet!
“Circle Song” lyric © 2004 L. Lesser
Take your finger ‘round a jar -Circumf’rence equals 2 pi ___;
For area, you multiply
R squared by that number ___.
Twinkle, twinkle, you’re a star
Knowing math will take you___!
to recall (6 sig. figures of)
Chorus of “American Pi” (lyric © L. Lesser):
Find, find the value of pi, starts 3 point 1 4 1 5 9
A good ol’ fraction you may hope to define
But the decimal never dies,
The decimal never dies….
(1st verse of) “American Pi”
lyric  L. Lesser
In the Bible we do see
the circle ratio appears as 3,
or a little more....
That genius Archimedes found
with polygons, an upper bound
Of 22 / 7 for sure!
The Chinese got it really keen: 355 over 113!
More joined the action with series and continued fractions.
In the 1700’s, my oh my, the English coined the symbol π,
Then Lambert showed it was a lie to look for rational pi.
He started singing ..........
Chorus of “American
”
lyric © L. Lesser
SING ALONG!
Find, find the value of pi, starts 3 point 1 4 1 5 9
A good ol’ fraction you may hope to define
But the decimal never dies,
The decimal never dies….
find this song on GoogleVideo, YouTube,
and my webpage for “Pi Day”(3/14)!
Just like history teachers use
“we didn’t start the fire…”
In preparation for PI DAY,
I share the lyrics of “American Pi”
with my Algebra 1 students
who spend part of a week researching topics
& making a PowerPoint, book or short story.
-- email from Melissa S.,
HS teacher in Nokomis, IL
Math club (from Shaker Heights, OH)
wrote a song from first 50 digits of pi:
B
middle C D
E
F
G
A
B
C
D
0
1
3
4
5
6
7
8
9
2
(see April 1987 Mathematics Teacher)
Capo followup
When we shorten the string length that’s free to
vibrate, the note’s pitch (frequency) goes:
a) up
b) down
String length L is _______ proportional to
frequency F.
a) directly
b) inversely
L=kF
L=k/F
Wait! My guitar’s 6 strings have same length
but different notes!
Mersenne’s Laws include
length L, weight W, force F:
Frequency =
a more concrete way to perceive
frequency(pitch) (thanks, Leon Harkleroad!)
As turning blades of “siren whistle”
speed up, then slow down,
do you hear
frequency (cycles/second)
rise, then fall?
pick a frequency to graph and play
possible benefits of M&M:
Multidisciplinary connections,
Memory aid,
Multiple intelligences,
Motivation,
Making community,
Math anxiety reduction,
Mashing stereotypes (content, class, teacher),
Modelling stretching, etc.
a classroom study (VanVoorhis, 2002)
2 equal-GPA sections of stats
Section R read 3 definitions aloud;
Section S sang 3 stat jingles (for same concepts)
S did better than R on relevant test items
p < .05
S had high correlation between test score and
self-rated jingle knowledge
p < .05
“What p-Value Means”
lyric © 2005 L. Lesser
tune: “Row, Row, Row Your Boat”
It is key to know
What p-value means –
It’s the chance
(with the null)
you obtain
data that’s
At least that extreme!
QUIZ: a “work problem”
I sang that song by myself in 10 seconds,
so how long would it take
for a group of 5 singers?
Georgia’s gift when I became a dad
after 90s “Mozart effect” studies
lotta lottery outreach
• adult ed. courses
• media coverage (newspapers,
radio/TV from local to CNN)
• features in teacher-friendly journals
(e.g., Sept. 2012 Mathematics
Teacher)
• Lottery Literacy webpage;
YouTube video
• a song: #1 country song
whose title begged to be made into a
lottery education song!
1
/
/
/
On a | warm summer’s evenin’
on a|
4
/
1
/
| train bound for nowhere,
I|
/
/
/
/
| met up with a gambler-- we were|
5
/
/
/
| both too tired to sleep,
so|
example of “Nashville numbering
system” for chord charts
“The Gambler” © 2001, 2009 L. Lesser
On a warm summer’s evenin’,
on a train bound for nowhere,
I met up with a gambler -we were both too tired to sleep.
So he told me how he planned
winnin’ lottery prizes
‘Til, as a math teacher, I just had to speak:
“Son, you track those draws,
you say ya got a system–
You call some numbers “hot”,
you deem others “due”;
But I insist,
they each have the same chance–
If you’re gonna play the game, boy,
Ya gotta know what’s true!”
“The Gambler” © 2001, 2009 L. Lesser
SING ALONG: You gotta
Know when you pick ‘em,
What’s superstition,
Know what is strategy
And know when there’s none!
You never try to learn this
At the 7-11:
Take the time right now for learnin’
When the singin’s done!
[now, watch my capo do a math-&-music “translation”…]
“The Gambler” © 2001, 2009 L. Lesser
Now all sets of numbers
are equally unlikely,
More rare than death by lightning,
still there’s somethin’ you should know;
If you should happen
to win that big jackpot,
You’ll win more money
if you picked it all alone!
So avoid those numbers
that more folks are playin’:
Like 7’s and birthdays
and sequences, too.
‘Til this song gets famous,
you’ll have the advantage–
Maybe you’ll thank me
with a share of your loot!”
“The Gambler” © 2001, 2009 L. Lesser
SING ALONG : You gotta
Know when you pick ‘em,
What’s superstition,
Know what is strategy
And know when there’s none!
You never try to learn this
At the 7-11:
Take the time right now for learnin’
When the singin’s done!
[now, watch my brief ‘steel guitar solo’….]
‘harmonics’ in my ‘steel guitar solo’
I didn’t press any strings all the way down,
just touched some at their midpoint,
making those strings vibrate in halves
Halving wavelength doubles frequency since
wavelength = (speed of sound)/frequency
from the 2001 play PROOF
(that won Tony & Pulitzer Awards),
here’s a song about
-1
QUIZ
what happens if you
give this hotel sign
a quarter-turn?
the “Hilbert Hotel”
is no Hilton Hotel:
infinite # of rooms always full,
but always room for more guests!
David Hilbert
lyric © 2000-2012 L. Lesser
On a dark desert highway –
not much scenery
Except this long hotel
stretchin’ far as I could see.
Neon sign in front read
But it was late and I was tired,
so I went inside to plea.
The clerk said, “No problem.
1 2 3 4 5 6 ….
Here’s what can be done–
We’ll move those in a room
1 2 3 4 5 6 ….
to the next higher one.
That will free up the first room
and that’s where you can stay.”
I tried understanding that
as I heard him say:
“Hotel (Called) Infinity”
lyric © 2000-2012 L. Lesser
CHORUS (sing along!)
“Welcome to the Hotel called Infinity
Where every room is full
(every room is full)
Yet there’s room for more.
Plenty of room at the HOTEL called Infinity
Move ‘em down the floor
(move em’ down the floor)
To make room for more.”
“Hotel (Called) Infinity”
© 2000-2012 L. Lesser
My mind got more twisted
when I saw a bus without end
1 2 3 4 5 6 ….
With an infinite number of riders
coming up to check in!
1 2 3 4 5 6 ….
“Relax,” said the nightman.
“Each one will move
to the double of their room number
to free the odd-numbered rooms.”
Last thing I remember at the end of my stay–
It was time to pay the bill, but I had no means to pay.
The man in room #3 smiled, “Your bill is on me.
#4 pays mine, and so on, so you get yours for free!
“Hotel (Called) Infinity”
lyric © 2000-2012 L. Lesser
CHORUS (sing along!)
“Welcome to the Hotel called Infinity
Where every room is full
(every room is full)
Yet there’s room for more.
Plenty of room at the HOTEL called Infinity
Move ‘em down the floor
(move em’ down the floor)
To make room for more.”
QUIZ
Speaking of infinity,
what’s the world’s longest math song?
Hint: it uses “aleph null” (‫ א‬0),
which is the cardinality of {1,2,3,…}
“World’s Longest Math Song” (author unknown)
‫ א‬0 bottles of beer on the wall
‫ א‬0 bottles of beer
Take one down,
Pass it around….
‫ א‬0 bottles of beer on the wall!
(repeat over & over & over & ….)
(see Sept. 2010 Math Horizons for more variations!)
World’s Longest Math Song: “all-ages version”
‫ א‬0 bottles of  beer on the wall
‫ א‬0 bottles of  beer
Take one down,
Pass it around….
‫ א‬0 bottles of  beer on the wall
QUIZ
TRUE or FALSE?
Fractions are challenging
not only for students,
but also for 5 out of 4 adults!
math & music connection: Cuisenaire rods
named for conservatory composer
Georges Cuisenaire
who became an ES principal
seeking fresh ways to teach math!
¾ = ¼+1/8+1/8+¼ = ½ + ¼
songs can help make connections to:
language and culture
On Top of a Fraction
lyric © 2004, 2006 L. Lesser
(tune: “On Top of Old Smokey”)
On top of a fraction
is the numerator-It is the number
of pieces you store.
But we call the bottom
denominator-It is the kind of
each piece, señor.
3
4
“On Top of a Fraction” lyric © 2004, 2006 L. Lesser
Like denomination
of money or faith,
Like 3rds or 7ths or halves or 8ths.
So denominators can be explained
Because in Spanish, nombre means
name!
“On Top of a Fraction” lyric © 2004, 2006 L. Lesser
While on top of a fraction,
we enumerate!
That means to count up
those of the same trait.
& when we add fractions, make
bottoms the same.
Add apples to apples;
add 8ths to 8ths.
“On Top of a Fraction” © 2004, 2006 L. Lesser
Think about music: the time signature
Is really a fraction – you can be sure
That fraction for this tune
is three-quarter time –
So in each measure, 3 quarter-notes
we find!
from PBS-TV kids show “Blast Beyond”
(episode 29 of season 1 archived at kcostv.org)
Speaking of fractions, do ratios
of “nice” numbers sound “nice”?
Compare sound of 2:1 to √2:1 using a guitar
….or a monochord (buy one or make one
(see 3/82 MT) like the first Mathemusician,
Pythagoras!
•
QUIZ: Which radio wave is FM?
(hint: FM stands for “_______ modulation”)
(another hint: same answer for “vibrato”)
Classroom demo with a tuning fork,
blowing across soda bottle, etc.
(mikeCBLGC)
if not yet ready to try melody….
Give rap a chants chance -Rap still has rhythm ‘n’ rhyme
(shortest example: SOH-CAH-TO-A)
Try this (algo)rhythm:
Stomp-stomp-clap-(rest)
1
2
3
4
For graphing y = m x+b:
L. Lesser adaptation of 1998 version by former IL teacher-of-the-year & NCTM Board member John A. Carter
Buddy you’re tryin’ to draw the line
For the m and b we’re assigned:
Ain’t hard you see, put your pencil on the b,
One more point will guarantee that….
“We Will Graph You!” ©1998 Carter/Lesser
sing-along:
We will, we will
Graph you!
We will, we will
Graph you!
“We Will Graph You!” ©1998 Carter/Lesser
So now you know the y-intercept;
Here’s how we find the rest of it:
Up with rise, across for run,
Plot another point & when they’re
strung, sing:
“We Will Graph You!” ©1998 Carter/Lesser
sing-along:
We will, we will
Graph you!
We will, we will
Graph you!
3/31/02 [Melbourne] Herald Sun
QUIZ: Which tune(s) has/have been
used for the quadratic formula?
a) Pop Goes the Weasel
b) Frère Jacques
c) Battle Hymn of the Republic
d) Amazing Grace
e) ‘Gilligan’s Island’ theme
f) Macarena
g) A school ‘fight song’
h) All of the above
song traits are basis of music
recommendations (Pandora, Mufin, etc.);
is there a “formula” for a hit?
(e.g., scoreahit.com; uplaya.com)
“but Larry, I have no talent!”
• “Hey, it didn’t stop me!”
• “If you can walk, you can dance. If you can
talk, you can sing.” – Zimbabwean proverb
• Do we let students with ‘fear of no talent’ give
up on math class?
• Good to recall what it’s like to learn new things.
• There’s a continuum of options! Spotlight
student talent, collaborate with school’s music
teacher, press “play” for a soundfile from
singaboutscience.org, etc.
“but Larry, I have no class time to spare!”
• Some raps/songs are as short as 10 seconds
• Song can help retention more than saying
things, and that saves time!
• A little does a lot: if a minute
of fun halves anxiety and
doubles attention the rest
of the hour, isn’t it worth it?
in the big picture…
• Song is just 1 of 20 ways to make class “fun”
(Lesser & Pearl, Nov. 2008 Journal of Statistics Education)
• “Fun” is just one of many ways that may
increase student engagement, learning,
attitude, or motivation (I also incorporate
“serious” contexts such as equity)
• Fun is FUNdable: Dennis Pearl (Ohio State),
John Weber (Georgia Perimeter College), and I
got a 2012-14 NSF grant on use of fun items
some thank-you’s before my last* song
• GMC conference team: Carole, Denise, etc.
• my AASU friends (Jane Barnard, etc.)
• Art Benjamin, Dane Camp, Dennis Pearl, etc.
• Mom & Dad for the first guitar & lessons!
• Pete at
(Conyers)
* barring an “encore” 
what if things were run by
math teachers, not politicians?
The Day Math Teachers
Took Over the World!
(© 2008, 2012 L. M. Lesser)
No one got lost in Boston again:
The teachers mapped the streets
into a nice Cartesian grid,
And the road-coloring theorem
was known by every kid
The day math teachers
took over the world!
The Day Math Teachers
Took Over the World!
(© 2008, 2012 L. M. Lesser)
All the politicians had to clean up their stats
And forgo call-in polls and misleading graphs,
And approval voting kept elections on track
The day math teachers took over the world!
The Day Math Teachers
Took Over the World!
(© 2008, 2012 L. M. Lesser)
Yeah, math teachers fin’ly decided to uprise—
‘Cause their functions were underutilized!
They divided-and-conquered, got radical, too:
They brought integration to every school!
∑ ∫
∆=
The Day Math Teachers
Took Over the World!
(© 2008, 2012 L. M. Lesser)
NUMB3RS was the TV show on top of the list,
And the math that’s in music gave everyone bliss
And consumer scams all took hits
The day math teachers took over the world!
The Day Math Teachers
Took Over the World!
(© 2008, 2012 L. M. Lesser)
That day brought peace – no axes to grind:
Everyone spoke rationally, with a logical mind,
And fair division solved disputes for all time
The day math teachers took over the world!
have a great conference
& enjoy Rock Eagle!
Professor Lesser
Mathemusician of UT-El Paso
my version of “Mean”,
2012 Grammy-winning country song of the year
“Mean” (lyric © 2011 L. Lesser)
But it can get pulled off
by just one single point,
And you don’t know what you don’t know…
“Mean” (lyric © 2011 L. Lesser)
Sometimes, data have a real asymmetry
Or values that are really quite extreme.
In those cases, better use the median
If what’s typical is what you need –
That’s not gonna be the mean!
That won’t be the mean with outliers, just
forget it, do you see just what I mean
& mean & mean & mean….
“Mean” (lyric © 2011 L. Lesser)
Sometimes, data have a real asymmetry
Or values that are really quite extreme.
Yeah, yeah!
In those cases, better use the median
If the middle is what you need –
Not gonna be the mean!
my followup song to “We Will Graph You!”
(lyric © 2005 L. Lesser)
The math that I knew – I’d memorize
To get me through
what teachers devised!
And mistakes, we’ve all made a few
But I learned the reasons
behind the math that we do
And I went on & on & on & on…..
“We Are the MATHLETES!” © 2005 L. Lesser
SING ALONG!
We are the MATHLETES,
my friend,
and we’ll keep deriving ‘til the end
We are the MATHLETES,
We are the MATHLETES,
No time to sell fries,
‘Cause we are
the MATHLETES
of the world!
thanks, ya’ll !
Professor Lesser
Mathemusician of UT-El Paso
That’s Odd! (© L. Lesser 2000-2012)
1, 3, 5, 7, or 9
Is the end of a number of what kind?
THAT’S ODD!
THAT’S ODD!
When there’s no Olympics, winter or summer,
That year is what kind of number?
THAT’S ODD!
THAT’S ODD!
That’s Odd! (© L. Lesser 2000-2012)
If you pick a book and open wide,
What page number’s on the right-hand side?
THAT’S ODD!
THAT’S ODD!
An interstate goin’ north and south
Has what kind of number for the route?
THAT’S ODD!
THAT’S ODD!
That’s Odd! (© L. Lesser 2000-2012)
Do a cascade so nothin’ falls
Takes what number of juggling balls?
THAT’S ODD!
THAT’S ODD!
Stayin’ with 5
lyric © 2006, 2007 Larry Lesser
Well you can tell by the way I use my chalk
That I teach math & I love to talk
With my class usin’ rhyme
About a number that is prime!
Now it’s all right, it’s okay-Use 5 senses every day.
We can try to understand
The Roman “V” on my hand…
CHORUS: Whether it’s a wonder, this Fibonacci number,
I’m stayin’ with 5, stayin’ with 5.
Symmetry flows in an apple or a rose:
Stayin’ with 5, stayin’ with 5.
Ah,ha,ha,ha, stayin’ with 5, stayin’ with 5.
Ah,ha,ha,ha, stayin’ with 5…….
Stayin’ with 5
lyric © 2006, 2007 Larry Lesser
Platonic solids – there are 5,
Like the vowels – not countin’ y.
Starfish arms, Olympic rings,
Playin’ hoops on a team:
You know it’s all right, it’s okay
Muslims pray 5 times a day.
We can try to understand
Patterns in a pentagram…
CHORUS: Whether it’s a wonder, this Fibonacci number,
I’m stayin’ with 5, stayin’ with 5.
Symmetry flows in an apple or a rose:
Stayin’ with 5, stayin’ with 5.
Ah,ha,ha,ha, stayin’ with 5, stayin’ with 5.
Ah,ha,ha,ha, stayin’ with 5…….
Gonna celebrate Cinco de Mayo: we’re gonna “take 5”!
5 is everywhere, somebody give me,
somebody give me a “high 5”!
QUIZ
Before Caesar added July(“Julius”) and
August(“Augustus”) to the calendar,
October was the _____th month.
“8 is so neat!”
lyric © 2007 by L. Lesser
Oh I really love 8, that Fibonacci cube!
8 scale notes make octaves I need for a tune.
Spider, scorpion, pints in a gallon:
I just can’t help but love 8 -- 8 is so neat!
“8 is so neat!”
lyric © 2007 by L. Lesser
Love to play some ‘8 ball’: always I'm behind,
Thinking of umbrellas and big red stop signs.
Octopus, tug-of-war, cylinders in a big car:
I just can’t help but love 8 -- 8 is so neat!
“8 is so neat!”
lyric © 2007 by L. Lesser
8, you're so neat -- I love you!
Pieces of 8 are not enough to show I care
Ooh, I feel like countin’ corners of a cube—
It’s a fast boat’s number & name of its crew.
Chessboard, jellyfish, a byte is 8 bits:
Ancient Greeks and Romans had 8 days a week,
8 days a week,
‘cause 8 is so neat!
QUIZ:
What’s the title of Paul Simon’s only #1 hit
as a solo artist?
hint: there’s a number in it
the “Lesser-known” origin of ‘50 Ways…’
1st verse begins:
“The problem is all inside your head,”
she said to me.
“The answer is easy if you take it logically…”
Chorus begins:
“Just work from slip out the back, Jack.
Make a new plan, Stan…”
50 Ways to Work a Problem
(lyric  1998 L. Lesser)
Just say what’s known, Joan,
And what’s to find, Caroline.
What rings a bell, Nell,
To what we’ve done?
Make a plan, Stan,
Follow it through, Sue,
Try to extend, Ken,
And we’ll be done!
Simon says…
"the more you are familiar with problem solving,
you recognize variations
on problems that you've solved before....
The typical problems, everyone learns to solve in
the early years of your songwriting.
But with the atypical problems,
experience and technique become invaluable."
-- Paul Simon, interviewed in
Sept./Oct. 2006 Performing Songwriter
parallels: doing math & songwriting
• “Flashes of inspiration” may come anytime, but
are usually preceded by much reflection, false
starts, and prior effort of developing a toolkit and
repertoire
• Essence comes fast, details take longer
• Versions can have same “content”
but differ greatly aesthetically
• Some are steps toward something “bigger”
• Useful to try varying structures or starting points
• Peer feedback/collaboration helpful!
• Cultivate “whole-brain”, not just left or right-brain
QUIZ
When working a problem,
what do your students
most often forget to do
when they think they’re done?
“Check Your Work” © 2001 L. Lesser
I’ve been workin’ on a problem
seems like all the day!
Now I fin’ly got a number,
so I’ll put my book away—
But then I hear my
teacher calling:
“Don’t rise up so soon!
You need to check your work for errors –
No one is immune!”
“Check Your Work” © 2001 L. Lesser
[ (Dinah won’t you check)3x your work? ] 2x
Somethin’ went wrong for NASA
Near Mars in ’99:
They lost a pricey spacecraft
From units not aligned!
“Check Your Work” © 2001 L. Lesser
So check your units now
and the answer’s magnitude
Does good reasoning
back what you conclude?
QUIZ
Are we 100% sure
there’s a birthday match (of month & day)
among all the people in this room?
QUIZ
So how many people are needed
to be 50% sure
of at least one
?
“Birthday Song” lyric © 2001 L. Lesser
Happy birthday to you—
Bring another 22:
Then we’ll have even chances
Of a match in this room….
Or many more!
to learn why, see my article in
May 1999 Mathematics Teacher