Eric started the meeting by talking about the Navajo Math Circles, which is a joint project of the Navajo Nation and mathematicians from Math Teachers Circle Network. A recent documentary tells the story. This meeting’s problem is from an article about the Navajo Math Circle (see Further Reading pdf link above) by Tatiana Shubin, whose video Grid Power was the subject of this past July’s CAMI meeting.
Eric distributed graph paper to the tables and started with a Google slide presentation of the problem:
After participants had shown each other their drawings and discussed the possible rectangles, Eric shared another condition:
Participants looked back at their rectangles and talked among themselves. Eric then shared the following image:
The group looked at the image and started to raise some problems. The area of the border isn’t the same as the area of the inside rectangle! Brian came up to show how the area of the whole rectangle was 30 squares (5×6) and the area of the purple rectangle is 12 squares ( 3 x 4). That would make the area of the tan border 18 squares (5 x 6 – 3 x 4). Other teachers pointed out that you can just count the squares and see that they aren’t the same.
We agreed that this wasn’t Dana’s rectangle and Eric moved on to the central question for the session:
Three teams of participants worked together for about 45 minutes, then presented to the larger group.
We came up with two rectangles that could possible be Dana’s, but we weren’t able to prove that there weren’t other possible rectangles that would fit the conditions. We were able to show that there were many rectangles that wouldn’t work, but we were unable to generally prove that other sizes wouldn’t work.
Could this help us prove either that there are other possible solutions or that we can found them all?
After presentations, Eric shared that the mathematician who presented the problem at the Navajo Math Circle, Dave Auckly, uses it as an example of Diophantine equations, which are equations with solutions that have to be integers (whole numbers). In this problem, the possible dimensions of Dana’s rectangle are whole number solutions.
- Google slide presentation of the problem
- Desmos graph of the solutions (and one equation)
- more Navajo math circle problems
- Documentary about the Navajo Math Circle (PBS)
- Article about Navajo Math Circle – contains solution to rectangle problem
In attendance: Brian, Deneise, Eric, Linda, Lionel, Maggie, Mark, Rachel, Ramon, Solange, Stephanie, Vulcanus
Programs represented: BMCC, CUNY PD Team, CUNY Start, DOE, Literacy Partners, York College
Location: Literacy Partners