On the Thursday before Thanksgiving break I had the opportunity to try out the Desmos activity builder *Parabola Slalom* in Scott Davidson’s Math 2 class. This is how that happened:

**Tuesday:** Spent the day at a Desmos training and played Parabola Slalom for the first time. So much fun and potential for learning! If only I could do this with students. . .

**Wednesday morning:** I reached out to the teachers in my district who I knew were teaching quadratics. “Can I come do a lesson on Thursday or Friday with your kids?”

**Wednesday afternoon**: I heard from Scott Davidson, the Math 2 lead at a high school in my district. “Sure, but time is a little tight. Thursday Block we are reviewing and Friday students are taking a midterm on all things quadratic.”

From there the whirl-wind began. 15 hours until I’d be teaching a lesson with students I had never met. And some of that time needed to be spent on eating and sleeping.

This first steps I took were getting to know exactly where the students were at and what was expected of them. I looked through the list of topics they had been studying and at the midterm assessment they would be taking on Friday. About 7 of the 20 problems were things I thought I could address with *Parabola Slalom*. Their learning had focused on vertex form, and this was reflected on the exam. Students, however, were expected to consider which forms of quadratic functions would be helpful in various situations. Completing the square, quadratic formula, and imaginary solutions did not easily come up in this activity. Factoring as a skill did not come up, but I could discuss why factored form of a function was helpful. I asked Scott if I could have the first half of his block period. The second half he could do last minute review on topics I did not address. Scott is wonderful, trusting, and adventurous. “Sure,” he said. “Can we invite my PLC in to watch?” My dream question. “Yes, Yes, YES!” A couple teachers had prep. For the others we got 15 minutes of coverage one after the other. Fortunately Scott has a big room. The stage was set.

The next step is where you’ll begin if you are a teacher using a Desmos activity with your class. I went to the activity builder and played it again. You can try it here. On Tuesday at the training I had the opportunity to talk with other teachers about which slides would be good to stop and talk at, but now I had specific students at a specific spot in their learning, and I needed to rethink where the valuable conversations would be for this group. After playing it again, I printed the attached Teacher Guide. (see button in top right corner)

Teacher guides are a great place to start with any Desmos activity. They allow you to think through what you want to get out of the activity, the teaching moves you will make, which things you can skip and which things all students will engage with. This checklist is at the beginning of each teacher guide:

I had already done step one. Check! That felt good. On to step 2 – learning targets. Desmos activities are very flexible, and you might chose to use them at a wide variety of spots in a learning progression. I’d love to try *Parabola Slalom* as an introduction to vertex form sometime and let students explore what the happens as you change different numbers. If I did that, my goals and planning would be very different. For this day I settled on these as my goals:

Next, according to my checklist, it was time to think about how I wanted to lead discussion with these kids I did not know. When would I bring the class together using “Teacher Pacing” and “Pause Class”? What would I discuss on those screens? I felt a lot of pressure to make sure the lesson prepared the students for the next day’s exam, because Scott was trusting me with this time. I wanted to make sure I picked up the students that were really struggling and gave them one more opportunity to make sense of this equations of parabolas thing. I decided that screen 3 was the first place where equation of parabola came up, so I would stop and question to see where kids were at. For students that were confused, I planned to make a connection to the y=mx+b as an equation of a line.I decided to look for these examples to discuss:

- I wanted to look for a sample of student work that used an upside down graph and asked what in the equation made that happen.
- I wanted to find a student that had made a parabola with vertex other than (0,0) and discuss what about that equation caused the shift.
- I also knew I wanted them to realize that the square was important, so decided to watch for someone who entered an equation without the square and then pause to discuss the resulting graph.

After this discussion, I would pace them to work on slides 3 to 5. The Tip for Teachers provided by Desmos gave me some suggestions to help lead conversation on slide 5.

Having looked at their previous work, I anticipated that most students would give an answer in vertex form. I decided I would focus on those answers here with the whole group, and help struggling students focus on what vertex form looked like. I decided I would NOT give away yet what effect changing the a, h, and k would have, even though technically this should have been review. Instead I would encourage them to play with the form, trying in different numbers and seeing if they could figure out what changing each of them did to the graph.

I decided slide 6 and 9 were other places I wanted to have discussions. Slide 6 would give me a chance to talk about the usefulness of factored form. I considered taking time to explain why x-intercepts are easy to solve for from this form, depending on student response. This was a secondary goal for me and I didn’t want to overwhelm with too much information. Slide 9 would give me a chance to consolidate the learning, and I would chose to push kids all to 9 at some point to catch the struggling students up and clear misconceptions. In the discussion about slide 9 I would listen to what they had discovered about a, h, and k in vertex form, and make sure all students were hitting the learning targets. To support their arguments about why each answer was incorrect I might open a desmos calculator page, graph the given points, and put in the equations. With their groups, I would ask them to propose an equation that would work. From there I would open slides 3 – 11 let them work at their own pace, giving extra time to students who had misunderstanding I had uncovered.

The teacher guide encouraged me to plan something for students who finished quickly, but I felt like slide 10 and 11 would cover that. I only had an hour and the note at the beginning said the activity could take more than one regular class period. I decided I would have students log in so they could go back to the activity at home.

Last steps – I printed a copy of my teacher guide with notes for the teachers that would be coming in and out to observe. That way they could quickly see where we were in the lesson and what had happened so far. Excited to work with actual students tomorrow, I set my alarm and dropped into bed.

**Thursday morning** at 7:15 I arrived at Scott’s classroom. I connected my computer to the projector, logged in, and made a class code for the activity. I jotted the code down on my teacher guide and copied it to the whiteboard. I paused the activity so early students couldn’t begin before I was ready. Then I helped Scott set out laptops and soon students begin to come in.

The activity was even more fun than I imagined. Being with kids was such a rush, and watching the ah-ha moments gave me all the warm fuzzies. Man, I love teaching.

A few things that came up during the activity/lesson:

- Making the connection between ended up being a big thing for a few students. There was more good discussion to be had there about how an equation in two variables defines/creates as set of points that are arranged in a certain shape. Something to come back to.

- As we consolidated their understanding of vertex form, I wrote the equationon the board and asked students to predict where the graph would be and what it would look like. Students readily volunteered a correct description of the graph, which I sketched on the board. I wanted to emphasize the horizontal shift being opposite the sign, so I asked the class “Is there anything surprising to you about the graph that you wouldn’t have expected just looking at the equation?” A girl raised her hand. “The negative three.” Excited she was bringing up what I was looking for, I said “ And what was surprising about the negative three?” Her reply was a shock. “ Well usually when there is a negative the parabola is upside down.” Man, I had almost missed this misunderstanding and assumed we were on the same page! But with a poker face I said, “That’s true. Lots of times a negative means the parabola will be upside down. Does anyone have an idea how you can tell when the negative makes it upside down?” Students were ready and able to clarify that, and I let them be the voice that shared that it was only
being negative that made the parabola upside down. “So what happens if the negative is in a different place?” And I unlocked the screen and let them experiment. How many opportunities like this have I missed because I fail to ask a follow up question and really listen?*a*

- Several students brought up the fact that the squared term is what made the graph a parabola. This was overgeneralized by students, so that if they didn’t see a superscripted 2 written in the problem, they thought that it could not be a parabola. They did not expect to have a graph that was parabolic. I had not anticipated that misconception. And actually, I was lucky to have discovered it, because the students had not shared that confusion out loud. One of the visiting teachers brought it to my attention near the end of the hour, after she spent some time talking to a student. Scott then addressed this misunderstand with the class after I left.

One of the many great things about an activity builder is that it gives the teacher:

- an easy way to spot struggling students (the dashboard), and
- the time to actually have a conversation with that student about what they are thinking.

If we want students to think in our classrooms, we need to give them opportunity to be more than a consumer of information. We need to let them figure things out, propose hypotheses and hash out what is true. Desmos graphing calculator and well crafted Desmos Activity Builders are a wonderful way to open the door to that opportunity. Listen to what the students said about the lesson:

What thing do you know understand better after the activity? |

how to find the parabola and the equation |

how to graph and where the numbers go |

the vertex form |

how to make parabolas |

parabolas |

how to graph better |

Which letters in the function of the vertex and stuff. |

How to make parabolas. |

I understand the form used to graph parabolas |

I understood the vertex form much better. |

graphing and how to make an equation for the graph |

How to look at points on a graph and make an equation for the parabola. |

how to graph and move around parabolas |

vertex’s |

all of it |

vertex form |

how to plot |

I understand how to graph the parabola with the equation. |

how to make parabolas out of equations |

I now know how to solve for parabolas. |

parabolas |

HOW TO GRAPH PARABOLAS |

how to make the parabola wider or narrower. |

which numbers you put onto the graph |

I understand how to graph the parabolas and how to put the values into an equation. |

the way graphing equations work |

I already understood everything. I just got to put what I knew into practice. |

More about how squares and how each letter in the needed equation works |

The thing that I understand better is how to find what equations would go into the parabola. |

the vertex form and how to correctly make a parabola/solution |

* *

Any other things you would like to share about the activity? |

it was fun |

i liked the activity |

it wasn’t confusing as much as regular teaching |

It was a fun way to learn how to make parabolas. |

It helped me a good amount with solving parabolas, vertex, zero property and such. |

it really helped me a lot |

It was hard, but beneficial. |

It was very fun to work in groups and solve together |

It was fun. |

It was better to do something a little different instead of sitting around and doing the review |

This activity really helped me get a better understand parabolas and I now feel like I have a better chance of passing the mid-term then before. |

IT’S FUN AND EASY TO USE |

It was actually pretty fun |

This activity helped |

It was fun although it was math. |

Not really, other than it was informative and helpful to my test |

The activities really help you get practice and help you get better at finding the equations for the graphs. |

We should do this more often.The learning experience we were able to have as a class was very helpful. We are very into technology these days so it is easy to connect and study using a resource that we have access to. |