Dana’s Rectangle

Inspired by the work of the Navajo Math Circle, CAMI explores the area of rectangles and their borders, testing conjectures and making generalizations.

Facilitator(s): Eric Appleton
Date of Meeting: October 13, 2016
Problem: pdf · docx
Further Reading: pdf · pdf2

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.

Rachel shares the results of her group’s exploration

Lionel shares an expression for finding Dana’s rectangle
Presentation from Lionel, Deneise, Maggie and Brian

Eric points out two ways of seeing the border
Two different ways to calculate the area of the border

Testing Lionel’s 2nd algebraic equation

Ramon explains his equation
Ramon explains his equation
Ramon’s general expression

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.

This graph from Desmos.com plots the dimensions of the two rectangles we found (4×6) and (3×10). What do you notice? What do you wonder?


Here’s what the graph looks like if we add Lionel’s group’s equation: 2(x+2) + 2y = xy.

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.


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








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