I ran a session with some teachers the other day with a scenario I modified from Thinking Mathematically. They posed some really interesting questions that I wanted to record here.

The Scenario and Their Questions

On a sheet of grid paper, draw a rectangle of any size.

Color in the largest square possible.

In the remaining rectangular region, repeat the process by coloring in the largest square possible.

Keep repeating until you are done.

I asked them to play around with this on their own for a little over five minutes, paying close attention to things they noticed and things they wondered about. After sharing ideas with their groups, we made a list of their wonderings:

1. Can we determine the final number of regions given the dimensions of the original rectangle? (i.e. 5 in the example)

2. Can we determine the number of different sized regions given the dimensions of the original rectangle? (i.e. 2 in the example...3x3 and 1x1)

3. Can we determine the size of the final square to be colored in given the dimensions of the original rectangle? (i.e. 1x1 in the example)

4. Are prime numbers somehow important as dimensions of the original rectangle?

5. What happens with both dimensions of the original rectangle are odd? When both are even? When one is even and one is odd?

Groupwork and Status Treatments

The focus of the session was "facilitating groupwork" with a brief nod to complex instruction. Here are some of the things I tried that seemed to have positive results with this group:

1. Before we did anything else, I started by reviewing group norms...
- Everyone participates and everyone's ideas are valued
- Share and explain your ideas
- Listen to understand the ideas of others
- Ask and respect questions
- Help without telling others how to think
...and we spent five minutes at the end of the session by having participants journal about how they (personally) did in adhering to group norms. They identified things they did well and things they did not.

2. Before launching the scenario, I passed out grid paper with only this on it...
Participants identified at least one way they could contribute for the session and shared that with the group. At the very end of the session, I asked them to identify at least one way each member in their group had been smart and they shared these publicly with each other in their groups.

3. During their group work time, I identified what (from my perspective) seemed to be lower status group members. I looked for ways each of these members was helping the group to function well and highlighted them publicly as we wrapped up the group work phase.


There were two things that stood struck me during the session that I am still thinking about:

1. One participant said that, upon receiving the dilemma, was extremely underwhelmed. Her statement during our meta-debrief was something like, "I was expecting something significant and profound." But, she said during the course of the session she and her group became extremely interested in one of their questions. By the end of the session their group was high-fiving each other over their solution....over rectangles and squares. It left me wondering what sparked this excitement for them. The freedom to pose their own question? To pursue one they found interesting? I'm not sure.

2. One participant commented that she had a very difficult time with the session because she felt like she didn't have enough time to think to herself. We talked a little bit about my decision to not give too much individual time so that, when coming back together as a group, nobody has a finished idea to share...only the beginnings of one. But her point is a good one and something I struggle with as a teacher. It gives me a lot to think about.

All in all, it was a fun investigation.


Matt E
09/06/2013 9:00am

Love this!

Another wondering: Does this always yield the FEWEST number of squares into which the given rectangle can be dissected? (For example, in the case of a 5x6 rectangle, the answer is "No"!)

09/06/2013 9:59am

Great post! This is one of my favorite problems.I hope it's OK if I share my own GeoGebra worksheet that demonstrates this. http://analemma.wikispaces.com/Visual+GCD

The question of finding the minimal number of squares is fascinating and currently unsolved, but a friend of mine has obtained some extraordinary results. http://int-e.eu/~bf3/squares/

Seth Leavitt
09/07/2013 9:54am

In response to your 2nd miscellaneous wondering:

Abe Lincoln is reputed to have said, "You can fool some of the people all of the time, and all of the people some of the time, but you can not fool all of the people all of the time."

I've reinterpreted the sense of the quote to, "You can satisfy some of the people ..."

In groups of almost any size, especially when they are not extremely homogeneous, there is no pleasing everyone. So you pay your money and take your choice. It works for some. (Maybe even all, sometimes.) When you're in a classroom every day with students, you change it up. I'm satisfied with pleasing some of the people, all of the time.

10/23/2013 6:01am

Hi Bryan,

I'm considering presenting this activity to my students, in French.
Can you please allow me to translate it in French at my website?
I'll credit you and the book (bottom right of the page, like here:
http://profgra.org/lycee/activite_algo_loi_binomiale.html ).

Also in your article, «things they noticed and things they wondered» is only linked on the letter «t».



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