Pile Splitting

James Tanton and Anna Burago took the lead in developing this circle session on Pile Splitting for the 2011 Circle on the Road workshop.

AUDIENCE: Middle and High-School Students

DESCRIPTION: Start with a collection of nine coins and split them into two piles in any way you like. Note the size of the two piles you created and write down on a piece of paper the product of those two sizes. (So, for example, if you split the coins into a pile of 6 and a pile of 3, write down the number 6×3 = 18.) Now split each of the two piles into smaller piles and record the product of their sizes as well. Keep doing this until you have nine piles of one coin each. Add up all the products you recorded. This is your magic number. Regroup the nine coins and repeat this activity again, making different choices for the splits along the way. Do you see why your number is magic?

Let’s spend some time together creating and discovering a whole host of magic numbers hidden in the pile splitting game!

Apprentices: Michael Nakamaye, Dan Finkel, Ruilin Wang

A detailed lesson plan for this session may be found below and at:

Pile Splitting Lesson Plan

G’Day All:

Apologies for being a tad slow to chime in.

I believe the session is 90 minutes long, will end up being a mix of middle and high school students (it is easier if it is one or the other, my approaches would be different), and these students will almost certainly be new to the math circle experience.

1. I personally think that 90 minutes is too long. In my work I usually keep things at the 50-55 minute mark, but that is how my style of things tends to work best. But there is nothing saying that we have to go the full 90 minutes!

The most important thing is to go by the read of the kids. If things are “working” then go with the flow, no matter what that flow is. If not, then CHANGE SUBJECT! It is completely okay to completely drop an idea that is not working and go a different route or even change the topic completely. (Well, I usually end up by doing something that actually is related, but doesn’t seem like it at the time.) For example, we might be counting strings among N dots and the kids are clearly disengaged and uninterested. Then I’d say something, okay, “I’m bored. Let’s do something completely different. My name is JIM. How many ways to rearrange the letters of my name? Your name is SUZY. Your name? What about HARRY here? What if someone’s name was DOODLEDOODDLY?” I’d massage the conversation into combinatorics and ask about arranging the letters CCNNNN? Oh, C stands for chosen and N for not chosen. We’re can now count strings among dots, making some joke about me not actually changing the subject at all. I try to do this in a way that is not too forced, but I am manipulating things.

2. Although I used combinatorial coefficients and functions in my write up, I would never use those words in a class. A kid might bring them up, and that is okay, but I would always be talking around these topics and never about them directly, In drawing, it is better to focus on the space around an object than the object itself. Focussing on the names of things and the jargon in mathematics gets in the way of the mathematical experience. By playing a simple “rearrange the letters” game we can get to formulas for counting rubber triangles – if that is where we end up going – without me ever have to give a spiel on C(n,k).

If I were to go the function route with this activity, highschoolers might be fine with that terminology if it feels right to use it. If not, I would probably talk about your favourite list of numbers, A_1, A_2, A_3, … and work with A_(a+b) – A_a – A_b. (But discussed in a more kid freindly way). But I would probably only choose to go this route with the problem if the crowd were highschoolers. For the middle school crowd, factor trees are good.

3. Please don’t feel locked into having to get somewhere specific. It is okay to map out in one’s mind possible sequences of events, but also be willing to completely abandon things and follow the route the kids go. For example, kids might find the magic numbers are 1+2+3+4+…+N and then someone might mention that 36 is also a square number and then want to know if other magic numbers are also square, and then … well, off you go, thinking on your feet. What fun!

Or maybe things are just going to be slower than you planned. Having kids figure out how many strings there are among N dots might be juicy and interesting enough for them and the rest of your time is taken up just doing that. That’s okay.

4. With new students I do always start the session with a “crowd pleaser” just to ease nerves and set the playful tone. My standard is the “math salute” which MSRI has video-taped me doing and has put up on its site somewhere. It works brilliantly.

5. I know the “function ending” to the activity comes out of the blue, but I did actually have a student in a session once suggest something very much along these lines and that led the group to actually come up with basically the same idea! The point is to be flexible and listen carefully to what kids are saying and be very willing to follow their paths. If you can see on the spot that some of their paths will lead to the same sort of place you have in mind, bingo! If not, don’t force it.

6. Math Circle work is so very different from “teaching.” It’s hard to stand up in front of a crowd and not know where you are going to end up nor exactly what you will be going to do. There is a lot of thinking and reacting on the spot as you work on the kids’ lead , and not your own, or, if not quite that, assessing the mood and flow of things and making decisions about what alternative routes are most likely to work best right at that moment, or when it is best to just put a halt on things and try something completely different, etc. Hard! (And way fun!)

Just FYI I’ve attached a write-up I’ve done on how I approach combinatorics with highschool kids and highschool teachers. These notes do address what is done in the standard curriculum, but it is my attempt to bring a math circle approach to the highschool classroom, a half-way point. I attach it here so that you can see how my approach to rearranging letters leads to all the usual perms and combns stuff.

Looking forward to meeting you all. It is going to be jolly fun and the point is to just have fun. Let the math be organic and playful and free.



From: Daniel Finkel (finkelitis@gmail.com)
Sent: Monday, March 07, 2011 2:56 AM
To: Anna Burago
Cc: Michael Nakamaye; Tanton, James; Mei-Ling Wang; Ruilin Wang; David Auckly
Subject: Re: Session planning

Hi All,

I’m looking forward to working with you all at the conference! And Jim, beautiful problem. Really lovely ideas at play there.

Certainly 90 minutes would be too long if the students were just to sit and listen the whole time, but my sense is that the students would spend the bulk of that time working on their own and in groups. I’m imagining maybe 5-10 minutes to introduce the problem and make the rules of the game clear, and then the “Your Turn” piece on the first page taking up the next 15 minutes, as the kids work through the different possible ways to break up 9, and come to realize that they always get the same number. If they get hooked into the bigger question of why it works, and start exploring smaller and bigger numbers, I imagine that’s another 15-25 minutes minimum. Depending on how long we let the flailing go, I could see this circle easily going 90 minutes or more. Of course, there’s got to be that student buy in, but if they’re there because they’re interested in seeing something cool in math, and we can encourage them in their flailing so they don’t get discouraged if answers aren’t immediately forthcoming. The mystery of why this thing works is certainly intriguing.

I do think it’s likely that many of them won’t be comfortable with combinations, or be ready to tackle the rubber triangle problem (though it still might be worth mentioning). The first ending is pretty wild, though it is a little out of the air. My sense is that I’d prefer the second ending when the time comes. I also think the Evenstown example is great. It leaves them, potentially, with a new mystery that they won’t be able to solve during the circle.

In any case, I’m looking forward to the pile splitting circle. Just to check: will this be run once with the apprentices as spectators and once with us doing the running?


On Sat, Mar 5, 2011 at 2:38 PM, Anna Burago wrote:

Here are several moments that I want to put up to the discussion.

1. 90 minutes is a LONG time. It is difficult for the students (especially, unprepared ones and middle-schoolers) to listen though a class that is so long. Even that our presentation is math-circle style (questions-driven, students involved) it is still a long session.
It would be nice to provide some change in activities. In addition, it would be helpful to have some back up in case the students run out of steam or the teacher runs out of material that is suitable for the audience.
– some light warm up for the first 5 minutes – just while we wait for the late students to trickle in.
– a few additional problems for independent work ( easy – to more difficult, math-Olympiad style, rely not on specific knowledge, but creativity))
Any other ideas?

2. The Possible Ending One will, most probably, not work out. Our students will be math circle newbies, some even not at high school. There is a chance that not only they would not now what C n_k is, but that they even would not know what a function is.

3. Suppose that we follow the Possible Ending Two. How would you predict the approximate time breakup for the main part and for the ending?
My feeling is that the main part should take not more that 40 minutes (again, I skip the optional “rubber triangles” section). Then we have the Ending Two that discusses the properties of factors tree in a very nice format – as a set of problems to post to the students. However, it might happen that Ending Two will take not more that 30 minutes. In this case we are left with 20 minutes that we have to fill with some kind of activities.
James, what is your opinion? You have actually taught this lesson.

4. There is the notion of invariant that the students are not familiar with. We can plan on spending some time on discussing the invariants, and how useful the invariants technique is for various types of problems.


UNIT 16_Counting Principles_July 2010
UNIT 17_A Grid of Numbers