# Learning Isn’t Linear: Desmos Final Projects

by Morika Tsujimura and Sonia von Gutfeld, Math

The last few weeks of eighth grade bring a familiar chorus every year—grumbles about the workload when summer feels so close, joyous celebrations of milestones and accomplishments and a rollercoaster of emotions as classmates reflect on their time together in anticipation of the changes high school will bring. The Desmos project is a culminating math assignment that brings out all three motifs in raucous harmony, a melding of adolescence and algebra.

Desmos is an approachable, web-based graphing program we use throughout the year. With this project, students find new, creative ways to apply this tool. First, students draw a Grace-themed design on graph paper that they then convert into equations and inequalities in order to reproduce the image on Desmos. Utilizing various types of functions learned over the year, students write equations for straight lines and curves and restrict them to the segments they need for their drawing. They also shade sections by using inequalities instead of equations. (You can view some of their projects on our Academic Excellence page.)

The Desmos project is first an opportunity for the eighth graders to review the types of functions learned (linear, quadratic, and exponential). Once students start engaging with the parameters of the project, however, there are many more layers to explore. Grappling with how to generate a recognizable image on the graph often generates the “aha” moment when the abstract relationship expressed in numbers and letters suddenly makes sense in a concrete way. Manipulating slopes of lines by trial and error might be how it starts, but applying the patterns of parallel and perpendicular lines empowers students to work with greater efficiency. Though they have already studied how the terms in an equation affect the shape of its graph, sometimes it takes trying to make a curve look more like a shoulder or a piece of the Grace quatrefoil for those understandings to fully click.

The project goes beyond reinforcing math content and allows students to hone the skills it takes to be a good student, and specifically a math student. From time management to breaking down tasks into smaller steps, to knowing when to ask for help, to sharing new knowledge with peers, all of the benefits of project-based learning come into play. In addition, the technology that allows students to see instantaneously how a line on the graph changes according to adjustments in an equation provides a low-stakes way to take risks. This is crucial to developing problem-solving skills and the courage to try new things. Many students ended up incorporating more detail than they had first imagined possible or learned and used equations beyond the requirements and, in turn, extended their mathematical understanding.

As teachers, we initially added the requirement that the drawings relate to Grace Church School in order to streamline the decision-making process and help students make personal meaning of the assignment. It has had some surprisingly sentimental outcomes, especially at the end of this singularly stressful school year. We read reflections on how their designs revealed favorite memories of early childhood, after school routines with friends, the experience of singing in the church — special connections that made Grace home. For our students to be able to express even a fraction of those emotions and their growth using a few dozen algebraic functions might be well worth the ups and downs of those final weeks of Middle School.