Saturday, October 22, 2016

10/18 More divisibility

After a slow start last week, this time almost all of the kids in the Math Club finished the problem of the week.  (See the end of this post)  I had one of the boys demonstrate his solution on the board. He's one of the slower but more careful writers and almost always shows something interesting. So I tend to ask questions to the room to keep everyone involved while he gets his thoughts written down.

In point of fact, I don't really have any kids that are good at talking simultaneously while they write on the board. That appears to be a learned skill. So in the beginning of almost all student work on the whiteboard I usually choose between narrating what's being written or asking background questions to prep the room.  One of these days I'd love to see what other people do.

As an amusing side note I found a very similar variant on the puzzle in the Moscow Math Club diary book I was rereading this weekend.  (See below)

Besides the odd coincidence, two ideas stuck out for me in the book.
  • The importance of emphasizing games for younger students in a math circle. In the context of this book, younger meant around 5th grade as opposed to High School students. I continually find evidence that supports this idea in the various sessions I run.
  • The focus on working problem sets every week with relatively low student to  facilitator ratios. This was my vision of how a Math club would work going into the process. I don't really follow this model very often though.  So I decided to go with a more pure problem set based day to see where we're at.

To start up with, I decided to begin with the game of 100.  This is a fairly simple two player game with the rules being:
  • The score starts at 0.
  • The goal is to be the first person to reach 100.
  • Each player takes turn picking a number between one and ten and adding it to the score.

I was a little unclear in my first explanations so I ended up doing a few whole classs demos where I played against one of the kids and walking around the room to make sure everyone had understood the rules. After that startup activity, I let everyone play with the goal of finding a strategy to always win.

Gratifyingly, after about 5-10 minutes most groups had discovered part if not all of the structure of the game.  [If you reaches 89 you can always win, which means you need to reach 78, which means you need to reach 67 .... all the way back to 1]  I then had the group talk about their discoveries.

For the middle of the day, I returned to divisibility rules. I started with having everyone name all the rules for 2,4,5,6, etc. through 9. We then spent some time talking about adding multiples again and I brought up the question "How does this relate to adding odd and even numbers?" I then went over the logic behind the nine rules again on the board.  I had the kids pick random digits for our sample number which worked well.  Then I had them pick (mostly) random digits again so we could work on the rule for 11's.  My key question here was if "9 = 10 -1" led to the rule for nines what relationship would help with 11's i.e. 11 = 10  + 1?  We then did the follow exercises:
  1. Find a pattern for nearest multiple of 11 to a power of 10 and figure out why.
  2. Using that rule, and the distributive law breaking out the sample number to see how the divisibility rule worked.

This is all a bit trickier since for 11's the numbers alternate between positive and negative i.e.

10 =  11 - 1
100 = (11 - 1)^2   =  11^2 - 2 * 11 + 1
1000 = (11 - 1) ^ 3 = 11^3 ....  -1

Which was a pattern I wanted to emphasize without the benefit of knowing the binomial theorem.

We had about 18 minutes left at the end of this work and I gave the kids a choice between a worksheet with some extension problems about divisibility or a sample AMC 8 set of problems. Interestingly,  everyone seemed to prefer the random set. I spent this period walking around, checking on  progress and answering questions.

The most interesting moment was helping a student who had forgotten how to multiple decimals. I went through the idea of treating the decimal like this:

    1 2 3 4 . 5 6
x           7 . 8 9

=  $123456\times\frac{1}{100}\times789\times\frac{1}{100}=123456\times789\times\frac{1}{10000}$

Unfortunately, I didn't have enough time to see how well that explanation resonated. Overall everyone worked fairly diligently through this experience but I was still left not feeling sure about the structure. I'd like to figure out how to get more insight into what everyone's doing and to keep the interest level high.

Idea: I saw another teacher post about doing problem sets and stipulating all work must be done on the whiteboard to facilitate discussion. This seems interesting and I think I have enough space to make it practical.


I went with a scaffolded version of the divisibility problem I found online:
10.18 Problem of the Week

I also at the very end asked if the kids would like a demo of the rule for divisibility by sevens. About half were interested so I may do so next week even though I don't expect them to remember it. This is more aimed at impressing that complex divisibility rules exists and there are patterns that extend up through the integers.


  1. Procedure ideas:
    - When kids are explaining their reasoning, we serve as a scribe at the board. That helps address the difficulty of writing and talking at the same time. Also, we can model close listening.
    - for games, we play a couple rounds of leader vs class before breaking into pairs/groups. It is almost always easier for people to see/experience playing a game rather than having the rules explained abstractly.

    I am excited that you might be using more games and am looking forward to the resources/references you use. FWIW, is my top pick.

    The table tennis problems are nice and especially timely. My two sons and I have recently begun playing ping pong and squash together, sometimes following this format, so they have been building experience about how this works.

    If we use these with a class, I wonder what sequencing is better:
    (a) give the IVA puzzle as a starter, then the ABC puzzle
    (b) give the ABC puzzle, then IVA for kids who need some scaffolding/hints

    Another idea: I wonder if these would work well for practicing the technique of looking at simpler cases? In particular, it isn't obvious how to simplify until you already know key ideas in the solution, especially the ABC puzzle. Taking IVA as the example: does a simpler case involve Victor playing 11 more games than Ivan (no) or 2 * I + 1 (yes)?

    This could be a case where the benefit of the problem solving technique is the idea to look for critical components of the structure to figure out how to simplify.

  2. Hi Joshua,
    Thanks for the suggestions. The idea about scribing is particularly interesting. I'll have to experiment.

    My only issue with games is finding enough new ones that don't require a lot of materials and I like mathpickle as well although I haven't used it much recently.

    For table tennis, I agree the IVA variant is a bit easier. My instinct nowadays is to always sequence if I have the option and let the kids build up to the harder variant. It tends to lead to better flow rather than trying to have an entire room look to make a larger intuitive leap (which never happens at the same pace).

    That's also why part of why I do the problem of the week variants to give kids a chance to (hopefully) think longer about something offline.