Regions of a Circle and Difference Equations

Lee Windsperger took the lead in developing this circle session on Regions of a Circle and Difference Equations for the 2011 Circle on the Road workshop.

Apprentices: Daniel Ullman, Katharine Ott

Audience: High school students and teachers

Abstract: In the sciences, observations in particular cases often lead to the
formulation of general laws. Our exploration starts with investigating a
circle in which we join each pair of n points round its boundary with a
straight line. Investigating what happens when n = 2,3,4,5 “empirical
induction” leads us to believe that there might be 2^(n-1) regions
bounded by these lines. However, as it happens quite often in
mathematics, a statement involving a natural number n might turn out false
even if it happens to be true for the first five, ten, or
million natural numbers. In order to find the correct formula for the
number of regions (which will turn out to be an integer representing
polynomial of degree four), triangular differences of sequences and the
convolution of sequences (Cauchy product) turn out to be appropriate
tools. Only a basic algebraic knowledge of how to multiply polynomials is
required as background, but the introduced methods can be used to further
explore general linear difference equations (including the Fibonacci
sequence and its generalizations) and probability distributions of sums of
independent random variables.

–Daniel Ullman wrote:

Wow, this is developing into an impressive piece of work. You’re
up to 10 pages, and (if I understand your plan) you haven’t gotten to
part 3 yet. Allow me to make a few general remarks, and then I’ll
suggest a few minor edits. I hope I’m within my place to offer these
suggestions. I figure I’m supposed to do something useful to earn my
keep around here!
I am the director of a Math Science Partnership working with inner
city (Washington DC) middle school math teachers. Since our session
in Houston is targeted toward teachers, my experience may come in
handy. We won’t have time in Houston to establish a “trust circle”,
inside which teachers can safely be wrong or be confused without
feeling deficient. Kids can shout out ideas and ask for help and
admit defeat, but teachers won’t likely feel safe doing these things
when the material we are presenting is allegedly targeting not them
but their students. They will think that the material is supposed to
be easy for them. That’s wrong, of course, but there is a certain
delicacy that is needed here. US math teachers are not always the
best mathematicians on the block, and they are rarely capable of
working at an Olympiad level. Count Jim Tanton and Zuming Feng as
real rarities; they are almost unique.
I’ve also been the director of the DC Math Circle, working with 8th
and 9th graders in the city. Your materials are ambitious for them.
I would have spent 10 sessions just doing the first part of your
materials. But on the other end of the spectrum, I’ve also coached
the US IMO team, and your materials seem to be quite nice for that
group. Or for any high school math club aimed at really talented
students. I’m not criticizing your materials; I’m just trying to make
certain that you are aware of the level. For average teachers, the
materials will be challenging, and minor points for you might be utter
stumbling blocks for them.
One way to soften the material might be to introduce a related but
easier problem as a warm-up: How about: How many regions in the
plane are formed by n lines? (Starts 1, 2, 4, . . . . Must be the
powers of 2, right?) The solution isn’t really closely related to the
regions-in-circles problem, but it’s easier.
By the way, here is my combinatorial solution to the
regions-in-circles problem. You can tell me if this is how the
problem is solved by Tanton or Gowers: Put the points around the
circle in general position. Place a marble in every region and allow
the marbles to move within their regions to the point that is closest
to the center of the circle (as if the center of the circle were a
source of gravity). Certain marbles fall to intersection points,
while others fall to points along a line segment. In fact, exactly
one marble falls onto each intersection point (there are n-choose-4 of
these), and exactly one marble lands along each (boundary-to-boundary)
diagonal (there are n-choose-2 of these). Finally, there is one
marble in the central region, which falls to the center. Hence the
total number of marbles is 1 + n-choose-2 + n-choose-4.
One final general remark: I recommend that you keep technical
terms out of the story as much as possible, introducing them only
slowly and carefully. Even the word “field” in your title probably
Cauchy products and Motzkin paths, you will have a limited audience.

— Lee replied:
I agree with you…I have experience working with teachers at both the
high school and grade school levels. We do have to be very delicate with
the way the material is presented. I think that this can be an important
aspect of math circles….teachers should come out of their comfort zones.
They should work with the students to try to make sense of good problems.
This way everyone is learning and coming as close as possible to doing
‘real’ math (no answers in the back of book). The circles I work with at
in Baton rouge are often run this way. I have had the pleasure of working
with students who are much more clever than I am (not too difficult to
find)….in these cases I really serve as a facilitator for discussion and
work to help solidify, condense and communicate these students thoughts
and ideas.

I also agree with you that this material may be ambitious for most
audiences, especially if it is just covered in 3 sessions. My aim is to
make the material flexible. Give the students something to play with and
become interested in whether it is difficult or easy for them….realizing
that there are many different approaches to the problem and many different
avenues for further exploration.

At the same time our LSU Circles try to find opportunities to expose
higher level mathematics that they won’t otherwise see in elementary/high
school. We hope that difficulties with problems make the students open to
learning about new mathematical tools to help them with the problems they
want to solve. It is a tricky balance (between
harnessing/fostering/celebrating the students own individual problem
solving ability/mathematical gifts and having them learn higher level
mathematics) which I am working towards achieving in our LSU circles.

As far as how to approach Saturday’s sessions…I am not sure yet. We
don’t necessarily have to present or have the teachers/students explore
all aspects….we could have them just look/explore/actually work on the
first part. Really having it be as much like a circle as possible and
could incorporate warm up problems like the one you mentioned. We could
conclude the session (wherever the ending points turns out to be) with an
outline the different paths that one could take the problem…using what I
am working on as a handout that they take with them. This option may be
best if we have several students in the session.

If there are more teachers, then perhaps we make the Saturday sessions as
more of an overview of what I am writing up. We will not ask them to
produce the answers and thus they won’t feel uncomfortable. The goal of
the session would be to show them a good problem and what sorts of
discussion can come out of it. At the same time they may learn a little