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Teaching
Guide |
This teaching guide is
designed to complement the 20-minute video, Go Figure. Click here to request
the video. Please note that video supplies are limited and may no longer be
available.
It's All Up in the Air:
The Mathematics of Juggling
Have you ever juggled your schedule? Maybe you’ve heard of a
shady accountant who juggled the books. If you’re really stressed, you might say
you’re trying to keep all the balls in the air.
These are just a few modern metaphorical uses for an old art. Juggling is an activity
that most people associate with circuses and magicians, but it can have applications in
the classroom as well.
One Ball, Two Balls,
Three Balls…
There’s evidence that juggling (the manual kind) was going on in ancient Egypt,
Greece and Rome. In fact, the term comes from the Latin joculare meaning, "to
jest." By medieval times in France, the word had become jongleurs. These were
entertainers who combined music, storytelling, acrobatics and recitations with juggling as
they traveled the countryside, performing at festivals, fairs and in marketplaces.
Eventually the craft of juggling spread to our shores. Traveling tent shows, circuses
and eventually vaudeville acts weren’t complete without at least one juggler on the
bill. In the 1950s, you could count on Ed Sullivan to bring plate spinners and knife
tossers into your living room. And today, the ancient art has gotten glitzy, featured in
splashy Las Vegas shows and on Broadway.
Juggling Math,
Science and Research
The Go Figure video features a man juggling balls and demonstrating how the ball
tosses correspond to time sequences. If he seems adept at keeping all those objects
rhythmically floating, he should. He’s Ron Graham, a man who juggles much more than
balls.
Graham is chief scientist at AT&T Labs Research. He also teaches part-time at
Rutgers University; lectures and presents seminars around the world; and is very active in
the American Mathematical Society. In his spare time, he sits on the editorial board of 40
(you read that right) mathematics and computer journals. And he’s past president of
the International Jugglers Association.
Graham is interested, to say the least, in number theory and other areas of
mathematics. His work in these fields earned him the prestigious Polya Prize in 1972 and
membership in the National Academy of Sciences in 1985. He also served for two years on a
high-profile National Research Council committee on cryptography, which recommended that
the United States ease restrictions on encryption regulations.
Problem Solving
Graham is generous with one of the secrets of his success, and it’s applicable to
just about any area of life. "The best way to solve a complex problem is to break it
down into component parts, learn each of the parts, and learn how the parts go
together," he notes.
Math is connected to many of his activities. Juggling is a prime example. Many of the
3,000 members of the International Jugglers Association are interested or employed in
mathematics or computers.
"It’s difficult to explain this connection in any straightforward way,
but…juggling combines abstract patterns and mind-body coordination in a pleasing
way," Graham says.
A Matter of Math
As Graham demonstrates in Go Figure, he connects math and juggling through the
pattern of the tosses or "site swaps." He shows a three-ball cascade (a juggling
pattern) along a number line, drawing loops from number to number to show from where each
ball leaves and to where each ball will fall. In the case of three balls, each ball will
fall every third number.
Or as he puts it: "The three-ball cascade is perhaps the most basic juggling
trick. Balls are thrown alternately from each hand and travel in a figure-eight pattern.
"This pattern has a natural generalization for any odd number of balls but
can’t be done in a natural way with an even number of balls. Even if simultaneous
throws were allowed, in a symmetrical cascade with an even number of balls, there would be
a collision at the center of the figure eight," says Graham.
Juggling to Keep Kids
in School
Besides the practical applications with math, juggling has proved to be a winner with some
students in unexpected ways. As one school discovered, tossing beanbags around can keep
some at-risk kids involved.
Teachers at H. D. Perry Middle School in south Florida found that kinesthetic learners
(you know, the hands-on, physical types) usually were found in the school’s dropout
prevention program. Putting balls in these students’ hands and challenging them with
juggling soon followed. It turns out that most of the kids were naturals at the art, often
excelling while the teachers struggled. It was a role reversal of expert and novice that
changed everyone’s perspective.
For three years, the faculty also used juggling as a service learning project. Students
who formerly often were truant surprised the staff by staying late after school four to
five days a week to practice. Some of the least confident learners became mentors to grade
school children in math and juggling, making a difference in the lives of younger
students.
Sources
Buhler, Joe, David Eisenbud, Ron Graham, and Colin Wright.
"Juggling Drops and Descents." Online. America Online. March 1997.
"Edupreneurship." H.D. Perry Middle School. Online. America Online.
March 1998.
Horgan, John. "Ronald L. Graham: Juggling Act." Scientific American.
March 1997.
O’Neill, Darragh. "Juggling. 3 Ball Cascade." 28 August 1997.
Internet.
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Last Updated: 02/16/03
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