I’m excited to share this really successful activity – perfect for Algebra 1 students! You’ll see in the below writeup how using Desmos and Pear Deck made this difficult task not only doable but also fun. This activity teaches students how to write equations of lines with a restricted domain.

*You’ll find the Pear Deck we used at the very end of this blog post. Please feel free to copy/modify and use with your classes.*

*Preview of Desmos portion of activity where students use movable points to visualize how to write equations of lines*

One of my colleagues (Jeff Bellestri) and I teamed up to create an engaging and fun way to teach 9th grade Algebra 1 students how to write equations of lines. Since students would be completing this activity the day before winter break, we decided to give it a holiday theme. My goal was to incorporate technology to create a healthy sense of competition and collaboration among students. As you read through the activity students completed, you’ll see how tech enabled students to visualize the process of writing linear equations and also required the participation and attention of every student in the classroom.

Going into the lesson, students already knew how to graph a line given its equation, but they had only just begin working the other way around—to write an equation given the graph of a line. The ultimate goal for this exploration was for students to come up with equations for all line segments making up the holiday tree below:

To complete this task, students needed knowledge beyond what they had learned. Though they couldn’t have completed the activity on their own, by guiding students through an active step-by-step exploration, they were able discover how to apply prior learned knowledge to execute the task at hand.

We used Pear Deck to engage all students in the initial exploration—how do you write the equation of various line segments (lines with a restricted domain)? Pear Deck is a teacher presentation tool that allows for real-time formative assessments, so it was the perfect thing to engage students in the process of discovering how to write these equations.

We began with something easy, the horizontal line pointed out in the screenshot below:

Using Pear Deck, students saw this image on their laptops and individually typed in their responses. After all students had submitted a response, we displayed the results of the class on the projector. There is great benefit in reviewing and discussing the results of the whole class. This enables the teachers to talk through why a certain answer is incorrect and to immediately address popular misconceptions. The problem that we typically run into when reviewing work on the board is that a student might feel ashamed of an incorrect answer. Pear Deck keeps these results anonymous, a point that’s very important to me. In an exploration, students should feel safe. Trial and error is an important part of the learning process. Students learn at different paces and technology can be an incredible tool to help differentiate the classroom.

The second step was to begin exploring domain. Since the blue line making up the base of the tree is only a line segment, we need to restrict the domain of the function to appropriately control its bounds. Students were asked to drag two red, vertical bars to explore this question:

The above screenshot captures an overlaid display of all student responses from the class. This is a powerful way to visualize where most students placed their bars.

Exploring domain is an advanced topic, well beyond the scope of an Algebra 1 class. Using Pear Deck, we were able to make this understanding so visual that students quickly caught on. It’s hard to explain how fantastic this is! Visualizing math concepts is often a major hurdle for students. It can all seem so abstract and the relationship between solving equations and drawing graphs is something that students often force to master. Again, technology can bring all of these understandings to life. By moving the red bars around on the graph above, students saw that finding the domain of the function involved looking at the x-bounds. Pear Deck allowed us to easily create this virtual manipulative for students, to give them a visual understanding of a difficult concept.

Putting the two pieces of information together, students discovered that they could write the equation of the blue lines as: y = 1 {-3 < x < 3}. (This is the notation that students would be using to enter their equations into Desmos, a free online grapher (and much more!).)

The next step was to find the equation of the green line pointed out in the screenshot below. The problem was that, thus far in the course, students had only learned how to write equations of lines given a slope and a y-intercept. Since the line segment being pointed out did not cross the y-axis, students needed a way to visualize where that line, if extended, would touch the axis. Again, this is where the power of Pear Deck comes in! By asking students to use the line drawing tool in Pear deck to extend the green line, students could clearly visualize the y-intercept of the line:

This is a screenshot of a student’s response. They used the black line tool, in Pear Deck, to extend the green line being pointed out. In doing so, they could easily visualize the y-intercept of this line.

The next step was asking students to type in the y-intercept and slope of the line they drew in the step above. At this point, students were really getting the hang of the activity and getting excited about being able to come up with the right answer. Pear Deck allowed us to create a healthy sense of competition in responding to each question. Because students were responding on their own screen, each student was held accountable to independently come up with an answer. The fact that we then displayed these responses on the project gave that added incentive to come up with the right answer and truly energized the lesson. After discussing the responses as a class, everyone was able to come up with the correct equation of the line: y = 2x + 7. But again, we needed to restrict the domain to only graph the desired segment of the line. Students dragged the red, vertical bars in Pear Deck to determine the bounds of that line.

After discussing these bounds, students were able to come up with the equation of the line segment: y = 2x + 7 {-3 < x < -1}

Given this foundation, students were ready to work independently. They worked directly in Desmos for this portion of the activity. Mr. Bellestri and I had already set things up so that the holiday tree would be the background image for the graph students would work in. This way, students could visualize their end goal and easily go back and forth between the desired image and the equation being typed. They also used a draggable line tool to extend their lines (as they had done in the Pear Deck activity) to easily find the y-intercept of each line. Below is a video showing the Desmos portion of the activity:

## Copy of Full Pear Deck Activity

*** On slide 12 of the Pear Deck, you will want to embed your own Desmos activity link for your class. To do that: 1. make a copy of this Desmos activity; 2. in the Pear Deck google slides, go to slide 12, open the Pear Deck sidebar, choose “website” and paste in your Desmos activity link*

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your link to the desmos peardeck tree isn’t working…..

Thank you for letting me know! I have updated the link, so it should be working now :)

what is the equation for the bottom of the tree?

To make the tree trunk, I used the equation: -.5<x<.5 {0<y<1} and set the fill opacity to .4