This art piece was a part of a traveling exhibit called Geometry of Hope that came to Austin Texas while I was teaching at ACE Academy.  Art teacher Martha Merry and I wanted to collaborate on on art and math field trip.  The materials are simple, the connection to the art is authentic and the mathematics is accessible, but also complex.  The slides and resource materials can be accessed from this pdf.

The artwork that inspires this activity can be constructed from two series of squares one white, one black where each of the 4 white squares is sized down from the previous by 1/2 the side length, and 1/4 the area.  Each of the 4 black squares are sized down from the corresponding white by square root of 2 on the side, 1/2 the area.  Similarly white squares are related to each other by 1/2 the side length and 1/4 the area.  Print the materials (on card stock if you can).

Teachers and students can be creative too and explore infinite series, partial sums, and more.  Sixty Texas teachers attended this session.  You can make your own squares or access a printable here.  The challenge sheet has some patterns and some “answers” at the bottom.  Digital version.

Surprisingly many patterns have the same area, 2/3 and an analysis of the fraction of black in the art piece itself offers an opportunity to “find” a variation that leads us to 3/4 as an answer.  We can also ask students to explore the sum of black in a pattern made with just the 8 pieces, or to consider what we calculate if the pattern continues infinitely.