So I’m not great at coming up with activities. Not in the way people talk about. But I recently was in a moment in my Algebra 2 classes where we had discussed function notation and how we use it, and also we had introduced interval notation to discuss things like domain and range. I wanted to challenge students to test their understanding. So I came up with this activity!
Kids have to come in knowing:
(a) what a function is, what interval notation is
(b) what domain and range are conceptually, and how to write them in interval notation
(b) how to read/understand function notation
Here’s what I did.
Kids in each group got a giant whiteboard. In one color, they were asked to draw x- and y-axes and put tick marks so each axis went from -5 to 5.
Then they were asked to draw a function. The requirements: it had to be complicated and interesting. I made it into a small competition, with my subjective interpretation deciding which group won. They also had to be able to determine pretty clearly what the domain and range for their function were. They were told that their graphs would be given to other groups to stump them. So make ’em good!
Kids rose to the challenge. Here are three examples:
Cool, right? I had each group write what the domain and range was for their functions on a post-it note on the back of the whiteboard.
Then I assigned each group a different group’s graph. Everyone in class took a crisp photograph of the graph they were assigned. And then class was over.
That night, kids did problems #1-#4 in this sheet I created. I’m pretty proud of this sheet! (Here it is in .docx form to download/edit.)
The next day, kids in groups compared their answers to #1-#4 with each other. They made revisions. They checked to see if their domain/range matched the post it on the back (the post-it the original group made when they created the graph). Then they worked collaboratively on #5 and #6.
When they were all done, I went around and checked their answers. (I had filled out an answer sheet for all the graphs so this part could be smooth.) I had discussions with groups about misconceptions they had. These conversations helped me see precisely where kids were getting tied up.
That’s where we are right now. A great finishing activity to function notation, domain, and range. I was so so so happy with the strong work kids were doing with such tricky functions! It was incredible! I even found a few mistakes in my own answer key!
At the start of our next class, I’m going to project a few questions like these to draw together our understandings and talk through some larger things that I realized I needed to highlight from my smaller conversations with groups:
Overall, this was relatively simple to execute. It broke up the monotony of class. And I love what I got out of it in terms of student thinking/analyzing.
Some notes from doing this:
- I loved kids working on the whiteboards to create their functions, with the easy ability to erase and recreate parts of their graphs. And I’m glad the whiteboards are large. I only wish that the whiteboards had gridlines on them to make the graphs extra neat and easier to read.
- I wondered if, after a group themselves finishes drawing their graphs, they should be given the worksheet to fill out on their own work. (In addition to a new group.) Then the worksheets could be compared and discussions/debates could happen.
- That being said, I liked that the worksheets/questions were hidden from kids, so they felt like extensions beyond the domain/range.
- I thought a lot about how Desmos activity builder could probably be harnessed to make this happen… where kids create their own graphs to challenge classmates with… But even if kids don’t come up with their own graphs, a Desmos activity with well-created graphs could also be neat to have at my fingertips.
- It took kids about 20 minutes to draw the axes and come up with their graphs. And some took a little longer if they had a tough time identifying the domain/range for their post-it.
Continuous Everywhere but Differentiable Nowhere 
I love whiteboards. So do the students. I love your use of their cameras to take a unique, student-generated problem home. I agree that having the questions be concealed until they left for home brings more authenticity to the task. Maybe have another copy for groups when they arrive the next day and they use these to complete the same questions for their function before checking how they did on their classmates’ function? Or have them take home a picture of their function and do the worksheet twice – once for theirs and once for their classmates’ graph.
Oooh I’m into your ideas!
I bet @cluzniak could make this into a great debate opportunity!
Might be too expensive, but you could get D&D battle maps, like https://www.amazon.com/Magnetic-Erase-Battle-Grid-Fantasy/dp/B0757VTN84/ref=asc_df_B0757VTN84/
Alternatively, if you are using pieces of MDF with a whiteboard surface, you could use a long ruler and a razor knife to scratch in the grid. (Cheap, but time consuming.)
Great ideas, both!!! Thank you!
I thought about the classroom I am currently in and how practice almost always takes the form of repeatedly doing the same or similar types of problems. One of the common math misconceptions is doing and learning math is independent, which is not true because mathematicians ask other mathematicians for advice all the time. This activity works to change this misconception by having students working together in groups, which fosters collaboration and learning from one another. Another math misconception is there is only one way to do something, and this activity definitely challenges this misconception. In addition, students are given some autonomy and creative freedom for how they want their function to be. I think whiteboards might be better in terms of having students actually write and draw out their functions. Thinking about accessibility, not all students might be able to access Desmos outside of the classroom either. Overall, I could see myself implementing this activity in the future.
I really like how you let the kids apply what they know to create their own graphs. This is a really useful way to also quickly get a check on if the kids understand the topic and are applying the various concepts correctly. I think applying something like this with the sketch feature in Desmos would be another fun way to try and implement this activity although I would be concerned about how long it would take in class to sketch their own functions on the computer.
I enjoyed learning about this activity! I think that students develop more conceptual understanding when they create their own problems and have their peers solve them. It may be challenging for them to do-especially if we encourage them to make the problems as difficult as we can-but it will be very beneficial to student learning in the end!