At the 2017 NCTM Annual Meeting I was invited to do a short Wednesday-afternoon presentation on Function Dances in the NCTM Networking Lounge. (Here’s the handout from the presentation.)
The idea of function dances is to get students (or in this case teachers) moving around, acting as the independent and dependent variables in geometric transformations. My presentation’s emphasis was on geometric transformations as functions, on the relative motion (relative rate of change) of the dancers, on using restricted domains, and on composition of functions. To set the stage, I created ahead of time several websketches—documents created with Web Sketchpad (WSP)—on this web page. We took a brief look at these examples before we began our own dances.
Our first dances were two-person reflection dances. Here is the WSP example (which is also on the web page):
We divided into groups and took turns being the independent dancer, the dependent dancer, and an observer/choreographer who helped make sure the dancers were directly opposite each other and the same distance from the mirror. (Each “mirror” was a piece of clothesline, about 4 meters long, that we stretched out on the floor.) After practicing this dance for a few minutes, we talked about how the dependent dancer had to move relative to the independent dancer: always at the same speed, but sometimes in the exact same direction, sometimes in the exact opposite direction, and sometimes in other directions, all at the whim of the independent dancer. (We all agreed that it’s tough to be a dependent variable, because you have to make your motion follow exactly the rule of the dance.)
Next we formed a larger group and explored composite dances; the first one we tried was a reflection dance composed of a reflection across one clothesline followed by a reflection across a second clothesline parallel to the first. Here is the WSP example (also on the web page):
Making this work requires three dancers: the independent variable, the “intermediate” variable who is reflected across the first mirror, and the dependent variable who is the reflection of the intermediate variable across the second mirror. (Tap “Show Hippo” in the websketch above to see the intermediate variable.) We had to work at this one to get it right. In our subsequent discussion, we debated whether it would make a difference whether you first reflected across clothesline 1 and then across clothesline 2, or instead reflected across 2 and then 1. Would the dependent variable end up in the same place? We went back to the mirrors to figure out the answer to this one!
Though we were running out of time, we tried one more composition using two clotheslines, this time with the two clotheslines at different angles from each other so that they cross. (The WSP example is on page 2 of the second websketch above.) We found this easier to analyze by making the clotheslines perpendicular than having them cross at some other angle. As we worked on this dance, we debated about what would happen if the independent variable moves in a straight line: would the dependent variable also move in a straight line?
We were already over our (too-short) allotted time, and didn’t have a chance to discuss the details of how best to use an activity such as this with students. (I have used a related activity in several high-school classes with good student interest and valuable discussions.) But we used our time well: we formed concrete connections between our physical movements and the math, and we gained a new perspective in our understanding of transformations, variables, and functions.
We had enough fun in the networking lounge at NCTM that I am inclined to add short function-dance activities to several of the activities in our unit “Investigate Geometric Transformations as Functions.”