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

isn’t needed. And if your overview talks about z-transforms and

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

about generalized functions.

Also I like your marbles argument. Haven’t really had the chance to fully

digest it yet but I will try to incorporate it if that is ok with you?

Thanks you so much for you input and help with this. I will look over

your corrections/suggestions

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Regions of a Circle.pdf |