On Sunday, I posted my plan for a set of cards that I wanted my students to sort. We started sorting them yesterday. I gave each group about 5 minutes to sort the cards. I asked them at that point to share out their strategies. They continued sorting until at least a few groups were finished. Most teams started with 5 groups, not 4 as I'd expected. They separated the equations by form and then separated the graphs by positive or negative slope. A few groups sorted by y-intercept (where they could decipher one). Only one group actually started down the path they eventually needed to be on. I interrupted to tell them that I actually needed 6 equal groups of cards. With that direction, about 2/3 of the groups figured out what to do, even if they had some trouble with how to accomplish it. I still had several groups in the proverbial woods, though. I asked, "I wanted six groups. Do you have six of anything?" Each time, I got a version of, "Oh, there are six graphs!" Sorting took off quickly then. It took quite some time for the groups to sort the cards into the six groups. I'm glad that I included both positive and negative 3/2 as a slope because I was able to catch some misconceptions there. It also helped that repeated a y-intercept for the same reason.
Once the students had six groups each containing one graph and its equation in three forms, I challenged them to match the numbers in the point-slope form (which they hadn't learned yet) to the features of the graph. They figured out the slope quickly and were likely aided by how many of the slopes were fractions. I was happy to hear many of them explain that they knew which number was the slope because they knew they would distribute the number outside the parentheses to the x, which would make it the slope in slope-intercept form. The "point" of "point-slope" form was an enigma. I pushed them for 20-30 minutes. My first class didn't quite get there. One boy saw it, explained it to me and his group, and then got picked up early. His explanation got completely lost in translation because his group member tried to explain it to the class and even had me confused as to what he was trying to communicate. I ended up giving it away to this class. I wish I'd had a few more minutes to push them to list ordered pairs to make the connection. The second class went better because I saw how a few of my lines of questioning didn't work and how others worked well. I had over half of the class get the meaning of the point without me giving it away and I was able to get those students to clearly explain it to their classmates.
The reason I wrote this post was to remind myself that being less helpful is hard, but good. The students were having incredibly good, productive discussions today, using their vocabulary and their reasoning to work on what was a challenging but possible task. We spent a full hour in groups having these discussions and sorting the cards. I was aiming to be done this part in about 15 minutes, but my students' engagement convinced me to hang in there until they had gotten all the way through, even though it look longer than I had anticipated. The first group to whine, "We can't get this!" was actually the first group to fully understand and articulate the meaning of point-slope form. Now, will they remember how their perseverance paid off? I hope so!
How do you scaffold exploration activities with questioning so that students are successful?