What is a Math Circle?
What is a math circle?
A math circle is a group of students (usually motivated high school or middle school students) led by a mathematician who get together each week to learn mathematics. Often the leader changes regularly which has a few advantages:

The leader doesn't get burned out; it is easy and fun to prepare a couple of presentations per year for motivated student.

The students see different mathematical styles and different topics.

The leader can make the same presentation at multiple circles if there is more than one circle in the area.
Math circles are different from the typical high school "math club":

Circles emphasize problem solving.

Circles do not necessarily cover material from the standard curriculum.

Circles get students to think; they are generally not designed to drill the students for mastery of a skill or topic (although sometimes they can be designed to do this and the students do not even realize that it is happening).
To assure that the circle session you lead goes as well as possible:

Circles sessions often concentrate on problem solving techniques applicable in many areas. Sample circle topics include: symmetry, the pigeon–hole principle, divisibility,, counting, probability, invariants, graphs, induction, plane geometry, or inversion in a circle.

Hand out a set of problems a week before your session. Not too many, perhaps three or four, but selective. Include an easy one and a challenge one.

Try not to lecture. Even though introducing new theory and techniques is an integral part of math circles, your sessions should be as interactive as possible. Score yourself: 1 point per minute you talk; 5 points per minute a student talks; 10 points per minute you argue with a student; 50 points per minute the student argue among themselves.

Divide students into groups of 2–4 to solve problems. Have them present their own solutions.

Be encouraging, even about wrong answers. Find something positive in any attempt, but do not be satisfied until there is a rigorous solution. Wrap up each problem by reviewing the key steps and techniques used.

If the kids cannot answer your question immediately, do not just tell them the answer; let them think. If they are still stuck, give hints, not solutions.
Variations in age, ability, motivation
Within a circle the variables of greatest significance will be ability and motivation. Though many circles are categorized by age level, they are really tied according to ability. The idea is, if a student is in your circle, then he/she can handle it (and should not be a behavior problem despite any age differences). Expect great differences in ability, even among students of similar age. Motivation may or may not be a problem, depending on your students, but different kinds of activities are more conducive to (i) different types of learners, (ii) different age groups, and (iii) different maturity levels.
What to expect
Things will not go exactly as you predict. Prepare 2 to 3 times as much material as you think you could possibly go through. Be prepared to use all or very little of it. Actually, in practice, people tend to prepare far too much material to cover in an hour (or two hours). Be prepare to go faster or slower. Predicting the appropriate pace can be the most difficult part of a lesson. Do not worry if you do not "complete" the topic or lesson. This is not regular school. There are no curriculum objectives and you may not be explicitly training our students for math competitions. Let a lesson or topic go where it may. It is OK to go off on a good tangent.
What to do when problems arise
Kids who are lost:

Go to the strategy board. Go through the list.

Pair him/her with a stronger student. Explain that they need to work as a team.

Group students who are struggling together and go over the problem again with them (as a group), addressing any difficulties. You may want to give them a simpler (version of the) problem.

Always have good hints ready. Ask for good hints from the class (starting points or ways other students have thought about the problem).

Have simpler problems ready or ways to incorporate slower students into a complex problem (keep track of what is been done, record the data, etc..)

Have struggling students explain the problem back to you. See if you can determine where he/she got lost.
Kids who are bored:

First, try to engage them. Maybe they do not understand the problem. Is it too hard/complex or is it too easy? If it is too easy, make sure they understand the process and can explain it. Right answers are not enough.

If students choose not to participate because they think it is boring, feel free to tell them that may sit quietly if they choose not to work, but may not disrupt the rest of the class.

Give them more problems and harder ones.

Ask them what they know. What have they discovered? They might just need attention.
Challenge them with questions such as "Oh, I see you have figured out this, but what about this possibility?"
Kids who do not want to be there:

There are only two rules at Math Circle: (1) that you have fun doing math and (2) that you want to be here. You should tell students this. Let them know that if they do not want to be there, then they should not be there and that is ok. Tell them we are happy to have them in our class if they wish to participate and think they can have even a little bit of fun doing math.

If it is clear a student is there by no choice of his/her own, you may want to ask to speak with the adult that he/she came with. This is a voluntary program, students who wish not to be there, take away from the classroom environment and spirit of math circles.
Kids who are disruptive (too active, usually):

Usually they are bored. Have more problems for them.

Give them something to do: tasks (passing stuff out, helping other students), challenge them!

Give them an opportunity to speak/share ideas with the class, but call on them, so that they know you are in control.

Have them sit alone.

Speak to the student individually after class. Calling them by name to inform him/her that you wish to stay has an impact all of its own.

It is not ok for kids to be disruptive, they need to know this.

You can have students sit outside the classroom quietly if they simply cannot behave.

You can also send them to another part of the circle with older students if you think they are bored.

Or you can send them to another part of the circle with younger students if they behaving like a younger student.