Carl’s Basketball Problem

With a simple set up, CAMI enters a rabbit hole of notice/wonder and number patterns.

Facilitator(s): Mark Trushkowsky
Date of Meeting: January 11, 2017
Problem: · docx
Further Reading: pdf

At our meeting, we worked on two tasks that I got last summer at a gathering of teachers from the K-12 system called NYC Twitter Math Camp.

The first is an activity teachers can use to develop group problem-solving norms with students.It came from Nicora Placa and is called Master Designer. The goal is to introduce three norms for working in groups:


The basic idea is that students get into groups, each with a set of paper tangrams and an upright folder to create a hidden workspace. One member of the group is chosen as the master designer who arranges their tangrams. The master designer then has to explain by telling the other group members how to create the design. The descriptions and questioning continue until the other members of the group have recreated the design.

Master Designer is not just about the recreation of the design – the heart of the activity is the reflection on the process. General consensus was the activity is something that teachers would try with students. Teachers felt like it was harder than they thought it would be. The idea of vocabulary came up as an important aspect of the activity. Groups needed to have (or develop) a shared vocabulary. Todd said this activity would be effective to help students practice using vocabulary, especially in geometry, especially since vocab is geometry is one of the things the TASC assesses. He also suggested it could help students appreciate the utility of having vocabulary when trying to give a description to someone else.


At the heart of the meeting was an adaptation of a great problem developed by Carl Oliver. I told the following story (asking three volunteers to help me model the situation in the middle of the room):

“When I was a kid our gym teacher has us do this passing drill. She would put us in groups and we’d play a game called ’rounds’. Let’s say we were a group of 4. The first round, I pass the pass to the person to my right and everyone does the same until the ball comes back to me. When it gets back to me, that is the end of the round. For the next round, I skip one person, and everyone else does the same until the ball again comes back to me. For the third round, I skip two people, pass the ball and everyone else does the same. Once again, the round ends when the ball gets back to me. The game ends when I would skip everyone else in the circle and pass the ball to myself.”

In summary:

  • In round 1, we just pass it around the circle.
  • In round 2, we skip one person and pass it around the circle again, until it comes back to me.
  • For each round, we keep skipping an additional player.
  • The game ends when I would pass it to myself.

I asked, “What questions do you have about the way the game works?” Once those were all answered, I asked everyone to create a visual representation of what we just did – it could be a picture, a chart or anything else that helps you see the situation.

Then I asked everyone to write down what questions they have and then we shared out:

I added one more, telling the following story:

“I remember one time, we were playing rounds in gym and I can’t remember how many people were in our group, but I remember we played all the rounds and ended up with 20 total passes. Then my gym teacher joined our circle, we went through all the rounds and we still ended up with 20 total passes.”

I added the question, “How many people were in the circle that had 20 passes before my gym teacher joined us?”

Todd added the question, “Is it possible to play this game with a certain number of people such that it will never end?”

<<If you were not at the meeting, this is a good place to stop and do a little work on your own. You can choose one of the questions above or write your own.>>


I then asked everyone to choose one question they’d like to work on.  I had everyone work on their own to make sure everyone could have a chance to make sense of things on their own first. I also reminded everyone about the visual representations they had each created, in case in case it was helpful.

One thing I had meant to try but forgot was to say was, “As you are working, if you have a lull – I’d like you to stop, pull back, look over everything on the page and write what you notice and what you wonder.”

A few people came in late, and to help them get started I used a visual representation with a table to recreate our acting it out with our initial group of four.


After 10 minutes of problem-solving I said, “If you want to keep working solo, you can, but I’d like you to take a few minutes and share what you’ve done/what you’re thinking with the people at your table.”

Towards the end of our meeting, I decided to stop everyone and have them begin to transfer any findings and discoveries they’d made to newsprint. I knew that folks were not finished exploring their questions, but I wanted to make sure we had a little time to share, especially since I thought seeing each other’s work would deepen the understandings folks were starting to build.

Gallery Walk

One the posters were done, we all walked around the room and wrote questions and comments on each other’s work.

Teacher Findings

Here are the posters with peer comments and questions:

Teaching Question

One question I had that we did not have time to discuss was what everyone thought about the fact that the problem was conveyed orally. At the end of the meeting, I did give a handout with the problem written out (follow docx link above), but I was curious to try to work a whole problem without text.

I tried to scaffold the process by giving everyone a few opportunities to understand the situation:

  1. Listening to my explaining the rules of “rounds”
  2. Watching me act it out with a group of 4 people
  3. Asking clarifying questions
  4. Creating a visual representation to (a) understand the situation, (b) raise additional clarifying questions, and (c) have something to refer back to once they choose a question to explore
  5. Share visual representations with each other
  6. Work in small groups

In addition to a handout with a problem on it, I also gave out this chart with some data for folks to notice/wonder about.

In attendance: Stephanie, Steven, Steven, Lionel, Meghan, Cynthia, Rachel, Solange, Tyler, Eric, Todd, Davida, Mark

Programs represented: Literacy Assistance Center, BMCC, Fifth Avenue Committee, NYC College of Technology, CUNY Adult Literacy PD Team, DOE, Literacy Partners, Touro College, District 79: Pathways to Graduation

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