The Mathematics

As is typical of programs in EDC's Center for Mathematics Education, the mathematics of CSSM has a low threshold and a high ceiling.  For example, we plan to begin the institute with this modification of a Balanced Assessment task, asking participants to keep track of questions that arise as they work on the problems.  

Below is a collection of rectangles.

• Which of the rectangles is the “squarest”?
• Arrange the rectangles in order of “square-ness” from most to least square.
• Devise a measure of “square-ness,” expressed algebraically, that allows you to order any collection of rectangles in order of “squareness.”
• Devise at least one more measure of “square-ness” and discuss the advantages and disadvantages of each of your measures.

We anticipate that many possible measures of “squareness” will arise as participants work on this problem.  If it does not arise, we will then present a measure that connects this simple problem with continued fractions, the Euclidean Algorithm, and some of the most fundamental ideas of number theory.

To find the squareness of this rectangle, for example:

We could “lop off squares”: 

And say that this rectangle contains two squares with some leftover.  To think about how much is left over, continue lopping off squares:

And say that this rectangle has squareness sequence

2, 1, 4, 3. 

As participants investigate this squareness sequence, questions like the following may arise:

• How does this measure of squareness compare to other measures we found?
• Are there some squareness sequences that can be written down, but that do not generate a rectangle?
• How can you compare the squareness of two rectangles just by looking at their squareness sequences?
• Under what circumstances will squareness sequences be finite?
• If one has two non-similar rectangles, can their squareness sequences be the same?

• Does computing the squareness sequences
• 1, 1, 1, 1
• 1, 1, 1, 1, 1
• 1, 1. 1, 1, 1, 1

and so on get us closer to computing 1, 1, 1, 1, 1, ….

• Can we characterize types of sequences?

It is also worth noting the connection to continued fractions: