I found this technique in an article in a Middle School Mathematics Teacher Journal by NCTM. This journal revolved around ways to connect my two favorite subjects. As most teachers of this age realize, these students are all at different developmental levels. Although a few students may be prepared for abstract quadratics, the majority would rather clean their rooms. (Sorry, I just glimpsed my son's shambles he calls his room.)
I plan on playing with more of these equations over the summer, but as a launching point, we used nice perfect quadratics. Each part of the quadratic is transferred to a set pattern. We used the yellow for addition, light pink was our a, dark pink was our b. Three strands on the weaving frame would be for each of the parts of our quadratic. This translates x^2 to six threads of light pink, + is 1 strand of yellow, ab becomes 3 strands of light pink and 3 strands of dark pink, another + gives us 1 strand of yellow, and finally b^2 becomes 6 strands of dark pink. Our frame was large enough to get this repeat, from left to right, wrapped vertically on there twice. Once these strands are in place, the horizontal weaves begin. (The technical terms are warp and w??p, but I will have to learn those terms before they come easily to my typing.)
With the vertical strings set, now we use the same pattern working it from closest to us and upwards. After Brianne struggled through this weaving using a box and oodles of stick pins, I purchased a lap frame. Hopefully, I will be able to gain enough frames to let the each student in the 8th grade create one during the 1st quarter. It would be a great concrete introduction into a formula which is used so often in their classes!