This year, my district is using a new curriculum program that has scheduled time for project-based learning at the end of each semester. During the fall semester, we didn’t have time for a project in Algebra 1 (we barely made it through the major standards on the curriculum calendar) and we didn’t really like the projects they had uploaded anyway. We wanted more creativity in our project, instead of having all our students do the same assignment and get the same answer. (That’s boring. Also one of the Algebra 1 teachers has 4 sections of Algebra 1 and would have to grade 100 of them. Awful.)
This semester, we found that we were going to have a couple of weeks at the end of the semester between the last standards and preparing for finals, so we decided we really did want to do a project. In the fall, we had seen an idea for a project in the NCTM Mathematics Teacher journal (two months in a row – apparently it’s the popular thing right now) that we thought would be perfect. The basic concept is that the students use graphs to draw a picture. This is a nice low-floor/high-ceiling activity, as students can use graphs whose shapes they are already familiar with or do additional research to learn about new types of graphs. They can also just type in a parent function and then play with the parameters until their graph looks like what they wanted, or they can use the algebra skills they’ve learned this year to write the equation they need. (Hint: no one chose the second option. I will say that I was ok with this, as I really wanted them to learn about transformations on graphs by playing with them, which is really the best way to make those connections.)
This semester in Algebra 1, our students have learned about linear and quadratic functions. When we taught quadratics, we did so with a very heavy emphasis on the relationship between the equation and the graph, so our students had a very solid foundation in graphing parabolas. I wish that we had emphasized graphing during the first semester when we were learning about linear functions as well. I know my own function fluency has come about largely because I’m so comfortable relating functions and their graphs, and I wish my students had that knowledge to fall back on. Things to consider for next year, I guess.
We created a worksheet for students to work through in the computer lab for about 2-3 days to remind them of how transformations on the different functions work. We also made sure to include circles so that students could have the shape to use in their artwork. I figured it would be really hard to make a lot of their ideas without circles, which turned out to be even more true than I expected.
Our introduction was very long, and included a “reference sheet” for students to fill out so they would have the information about all the transformations in one place. We wanted to make it shorter, because it really did seem repetitive and took longer than the time I had booked in the computer lab, but I felt like some of our students would need to go through and analyze each transformation separately. I did have a few students who skipped to the reference sheet and used sliders in Desmos to see what was happening under each transformation. These students tended to be the ones that I knew I could trust to do that and still make the connections between the numbers and the effect on the graph (or were working with a student who would help them see this).
Download our introduction sheet and reference page: Desmos Intro Exploration I’m not really very happy with this, as I said, but feel free to make changes to suit your needs.
We gave the students 3 requirements:
- Your drawing must contain at least 12 equations.
- Your drawing must use at least 3 different types of functions.
- Your drawing must be creative, artistic, and original.
(That last one is another reason we taught them how to graph circles – lines and parabolas only give you two types of functions. Also, yes, I realize that a circle is not a function.)
We required the students to tell us what they planned on drawing before their first day in the computer lab to work. This was partly to prevent them from using a graph someone else had created (my rationale is that I made them commit to an idea before they had a chance to look online at what other people had created) and partly just because I wanted to make sure my students had a starting point for their first day. I was gone that day, so I didn’t want them telling the substitute that they didn’t know what they were doing.
We intentionally left our assignment very open-ended. Again, this allows us to make sure there was a low point of entry for the assignment, but also allows students to take the project as far as they wanted. We assumed we would have some students who would do the bare minimum and some who would just go crazy with it, and we were right.
I spent two days in the computer lab doing the introduction pages linked above. My students didn’t finish and could have used another day on this activity. I finally told most of them to skip to the reference sheet, as that was more important anyway. None of them got to the last page where they were supposed to make a face.
The next day the class was back in my classroom where we went through the reference sheet as a class so I could make sure they had all the information they needed. I also made them commit to an idea on this day.
The fourth day the students were in the computer lab ready to start working, and I wasn’t there. A couple of students finished the intro pages instead of working on their projects, but most seemed to get started ok.
The next week, I gave them 3 more days in the computer lab (Monday/Wednesday/Friday), so they had 4 total days that were intended to be dedicated to the project itself. I was also really glad I broke this time up instead of doing 3 straight days – they needed that time to process and troubleshoot before getting back into the lab. (The other 2 days we reviewed factoring to prepare for the final exam, so it really helped to break up the review days as well.)
Now I get to show you all the pretty graphs. Let’s be honest, that’s really the whole point of this blog post.
My students know me well – one student graphed a TARDIS.
I also ended up with 3 different versions of Captain America’s shield.
One of the other Algebra teachers shared her student’s submission with me because she knew I would appreciate it.
This student got to learn about sine curves to make her stem.
This student took the gold medal for number of equations – 45. Not to mention this teddy bear is adorable.
Hurdles to Overcome
We had a few issues related to getting Desmos to do what we wanted it to. Shading concentric circles different colors proved to be an issue that is far beyond the capabilities of students in Algebra 1 to solve, and with the number of students drawing Captain America shields, well, let’s just say it was a problem. I had trouble figuring out how to make it happen, and had to ask the Desmos staff for help (on Twitter). They responded with assistance, and stated that “For art purposes we would like to make that a bit easier.” I just think it’s awesome that they consider art to be such an important component of their platform. Who says you can’t be creative in math?!
The Best Part
I’ve spent 10 months working with these students. When they don’t get something right away. I know how quickly they shut down and talk to their friends instead of continuing to try. I fully expected to spend every day in the computer lab watching them chat instead of work because they “don’t get it”. I knew going into this assignment that I was going to refuse to be helpful, and they were going to hate it. For the day that I was gone, I even left my sub specific instructions NOT to attempt to help them, and not to listen to their complaining.
This didn’t happen. (I mean, sure, nearly every student needed help at some point, but it wasn’t constant, and they didn’t whine.) I think they helped each other a lot. When they did ask me questions, they were specific questions – “How do I get this line to [blah blah]?” or “I want this parabola to look like this [pointing]. How do I do that?” or “I need to put hair on this face. How can I do that?”
Seriously, I cannot even begin to tell you how much time I have spent trying to convince them that specific questions are the best thing ever. I’m also a huge fan of pointing at the coordinate plane to show me what you mean. If you can visualize it in your head and explain it in words, I can help you translate it into math notation.
They asked each other, they used Google, they pulled up other people’s graphs and used them as a model to figure out what was wrong with their own equations, they used their reference sheets (not as much as I would like, but they did)… When they did finally call me over, we would talk through what they wanted to accomplish, and how to notate it. I asked a lot of questions and tried to avoid giving answers. I would give them a parent function, say, “Change these numbers until you get it where you want it,” and walk away. And they did it. They figured it out. I am so impressed by and proud of the level of perseverance that I saw in the computer lab over that week.
Even better, the results of this project have been so much fun to grade. Math teachers don’t get to grade art projects very often.