One of the most important concepts to master in my precalculus course is the ability to easily translate between equations and graphs. The students must be able to see an equation, understand the basic shape of the graph of that equation and finally use transformations of the plane to visualize where the graph will be located in the plane, whether it is stretched out or squashed in and whether it is a reflection of another graph that they know. This is a lot to ask and is often difficult for students to grasp abstractly. When I began to think about how to help the students internalize these concepts, I realized that the topic naturally lends itself to a project in which the students are experimenting with graphs and technology in order to get instant feedback.
I wanted the students to create something. There is no greater challenge than to create something new. It would be easy enough to ask the students how the transformations would move the graph of an equation around. It is very difficult to determine exactly which transformation to use to make a graph move to the exact position and shape that you want it to. I conceived of the idea for the students to make a drawing by using only graphs by using an online graphing tool.
After an open invitation to the faculty from the art department, I began to think about how the precalculus project could be intertwined with the art department and I set up a meeting with Holly Hooper. I gave Holly the context for my project; showing her the parent functions and the transformations that my students would be working with. She immediately saw how these images resembled the cubist movement. She told me about the cubist movement and showed me student work that had been inspired by it. Slowly we cooked up an idea.
My students would come in to the art building, study student work and identify the parent functions in the existing art. Then, they would go off on their own to create their own pieces of art, inspired by their peers and inspired by cubism. They would submit their art, along with all the functions they used to create it, and a brief essay describing the math behind their work. These pieces would then be submitted to Holly’s students who would choose their favorites and render beautiful paintings-inspired by color theory. Math students and art students would collaborate during this stage to create something magnificent.
The learning objectives of the project are for the students to understand the graphs of the parent functions (the building blocks of more complicated functions) and for the students to be able to use transformations to “tweak” the graphs of the parent functions.
The technology that I chose to employ is an online graphing tool. It is ideal because the students can immediately see how changing the equation for a function transforms the graph-thereby reinforcing the concepts of transformations and instant feedback.
The students had a week to work on these projects providing ample time to work at their own pace. Each project was unique and varied in complexity. More advanced students pushed themselves by using sophisticated transformations and more complicated functions. Struggling students used more basic functions and only used one transformation at a time. It was amazing to see how quickly they learned how to manipulate the graphs and many students surprised me with the complexity of their work.
In the final essays, one student explained: “I liked this project because it gave me a deeper understanding of the subject. It was also fun to play around with the formulas and manipulate it….the project was really fun to do because we got to choose our own drawing and go at our own level and pace.” Dioulde Diallo.