This past semester I added a fresh coat of paint to a unit on parametric equations by challenging students to develop their own animations on Desmos. For many years I have used a Desmos activity to introduce the idea of time as a parameter within equations – https://teacher.desmos.com/activitybuilder/custom/576ed1058e03e695283c88b8 – and Desmos invites students to think about the role of time by inserting the idea into ordered pairs easily. Here’s how I introduced the idea. If you get lost or want to look under the hood, here is the final Desmos graph I share with students as a resource: https://www.desmos.com/calculator/7zuubmunhl

## STEP 1: Develop Your Vision

Before we dive into Desmos, think about what you want to animate and the path you want it to take. For this demo I have a vision of a ball starting in quadrant 3 and following 3 linear paths until the end in quadrant 1. Sketching out the vision is helpful:

The animation here has 3 linear phases – see the points below, and you can certainly allow for non-linear paths as well. There are a few things for students to attend to here: how the coefficients of t allow for movement, and notice how (t-7) and (t-10) are used in the second and third points in order to “trigger” the animations at the right time, matching the domain of t given for each of the points. Take time to build this with students. Define the second point to begin when the path of the first ends, then include the parameter t.

Allow students to interact with the points and alter them to their liking. Or have students develop their own paths.

## STEP 2: Introduce an Image

Next, find an image you would like to animate. I used a soccer ball here, and added a cute sun in the sky later. Upload the image to Desmos, and drag the corners to adjust the size of the image.

Now, a little heavy lifting with Desmos. Students will need to attend to precision and symbols here – encourage students to work together to follow the steps and syntax.

The goal is to replace the center of the image with a conditional statement using the 3 paths we defined earlier. The center I used with my soccer ball is shown below, and note the structure: for each path, start by stating a period for t, followed by the parametric point, separated by a colon. Then, a comma will separate each of the 3 stages.

It’s helpful to share the graph with students so they can dissect the command and make sure the syntax they are using is working.

Defining the center in this manner will then invite us to create a slider for t, which we will do here. Click the endpoints of the slider to define the start and end of the time period you would like. Then play and let the oohs and aahs wash over the room.

## STEP 3: Explore the Space

Now it’s time for students to build their own creations. As students build, they may become inspired to investigate new ideas. In my class, some things which came up are:

- Non-linear paths: these can be defined within the points
- Rotations: t can be used to define the angle of an image
- Image dilations and appearances: the slider for t can also be used to define the height and width of an image, as well as the opactity
- Backgrounds: students can find a general image to serve as a background. I encouraged students to lower the opactity of and background image so that the animation pops on the screen.

## STEP 4: Gallery Walks

Allow students to share their creations with each other half-way through the project and ask questions about procedures. In tech-based lessons, students are often their best resource, and inspiration for a new idea can come from each other. Here are a few student creations from this first project attempt.