A few months back, I got the chance to go into Desmos headquarters to work with some great people (see When Plans Go Awry for some information on that background to this post). tl;dr: We started an ambitious activity to represent “Completing the Square” graphically, but I wasn’t sure how good it would be, educationally.
Fast forward a couple of months, and despite my best intentions, I never got the chance to really work through the activity to the level of detail that I’d hoped, and yet here we were in my Algebra 1 class, ready to cover the process of completing the square. After a crazy weekend (too sick to work on Friday and Saturday, then recovering just in time to celebrate Mother’s Day with my mother, wife and daughter, before finishing all my grading and submitting grades for the latest grading period at about 1AM Sunday night), I had just enough time to put off last minute adjustments until last night.
Those last minute adjustments ended up being pretty minor, so I crossed my fingers and this morning during class, I had my students start up Desmos and dive in to the activity. (If you want to see it yourself, check out CTS Investigation and let me know what you think.) I tried to hold students back, since some will rush through whenever they get the chance, but this was one of those days that I really wish there was a pause option for Desmos activities.
A few observations from each of the first four slides:
- On the first screen, I had a graph that students were supposed to look at and then estimate the zeros. In retrospect, I should have made the graph an image. I didn’t think about the fact that students would be able to click on the zeros to get their values, which really defeats the purpose of estimating. Luckily, I was able to rectify that (mostly) with some extra instruction, but good learning experience! The responses I got from students (below) ended up with a great discussion of estimation, what counted as zeros, when negatives matter, how to write the zeros, and more.
- In the second screen, students had to adjust the c value of the equations to get the parabolas to “sit” on the x-axis. I was worried about this one because I wasn’t sure how clear the instructions were, but it turned out to be clear for everyone involved. I wonder if it would look nicer if the x-axis and y-axis were a little bit bolder in projection mode. It didn’t confuse anyone as far as I knew, but I think it would be a nice feature.
- In the third screen, students were shown a graph, and asked how to change the equation to make it sit on the x-axis. Of course, I intended them to change the c value, but some chose to change the b value instead/as well. The question did not ask the why questions, but the discussion after the next slide got into that a little bit.
- In the fourth screen, I got some great responses, especially in the discussion that happened afterwards. Students were given four graphs and their equations, and had to figure out which equation didn’t sit on the x-axis and why. Sometimes, the best responses come from the students you didn’t expect, and today was one of those days. The highlighted answer came from a student who often doesn’t talk, and usually isn’t a prolific writer, but was on fire today. I asked this student to clarify the response, and got an explanation that the square root of the last number should be half the middle number, and that was true for the first three equations, but not the last, which should have a 16 instead of a 9.
That’s all we had time for in class. We got some great discussion, and yet didn’t really focus too much on the main geometric tie that I was hoping could be made – the number of units you move the vertex to get to the x-axis is the same as the number that you add to the c value. It seemed that a few kids started to make that connection, but I need to tweak some of the questions that are being asked to be less open-ended if we want to make that connection. Or, maybe I need to think about the intent – what is the focus here? Because I have a bunch of students who made connections between the b and c values that are also valuable observations and should help with conceptual understanding.
Also, and I can’t believe I missed this – initially, the last equation in each “which one doesn’t belong” was the one that didn’t fit the pattern I wanted them to see, so I really need to mix that up. All told, though, I’m pretty happy with this introduction to completing the square by graphing, and students kept themselves pretty well engaged.