## What Whole Number Areas Are Possible With Squares on Dot Paper?

I am getting better at picking good presentations based on their titles and descriptions. And, at the NCTM 2015 Boston Conference, I had a pretty high batting average. I have already written about Susan Jo Russell's talk and about Math Fun Facts shared by Francis Edward Su -- talks I really enjoyed. I also attended a presentation on NCTM's theme of "From Principles to Actions" where the speakers spent a lot of time talking about students' mathematical identities.

As part of the presentation they had us working to solve a problem on dot paper. I really liked the problem, but I thought it could be a bit more open and general. Here is my version:

#### Using a 5 x 5 grid of dot paper, what integer areas can you form using single squares?

Examples are provided below for clarity. Note that the vertices of the squares must be on the dots and that the square need not be orthogonal to the border. Here is blank dot paper if you would like it.