Today's warm-up builds on the skills from yesterday's lesson. The profit maximization warm-up includes data tables that are missing linear factors. The idea is that students can either find ways to fill in the linear columns or they can simply find a function rule for the quadratic data that is shown. The approach is consistent with my general intention to always push students, slightly. Yesterday, they likely became comfortable with the tables provided, so I want to push their thinking even more, past where they were yesterday.
As they work I want my students to be thinking: how can I apply what I already learned to this new problem? In my classroom, I find this to be a great way to encourage students to make sense of problems (MP1). When students express confusion about a new problem, I often ask them this question, “How can you turn this problem into one you have solved before?”
During today's warmup I expect some students will ask, “Is there more than one way to fill in the middle columns?” If they do, I will respond, “I don’t know. Great question. What do you think? Can you find more that one way?”
Other students will ask, “Does it matter what order I put the numbers in?” I will respond the same way. Students will get frustrated, but I find that the more consistently I respond this way, the more students will start to look for answers on their own. But, I definitely want to celebrate the fact that they are askin good questions. So, I will create a poster or white board of “good questions” and record questions as students ask them. This way I show that I value their questions, even if I do not answer them.
I generated these questions by keeping track of the questions that I most commonly asked students while circulating during the lesson. I find that I can come up with better questions when I am actually engaging with students and trying to understand their thinking, than when I am sitting at my desk preparing for class. I use these questions to close the lesson, giving all students the chance to think about the big ideas I pushed in my small group interactions.
To motivate the conversation, I gave each pair of students all of these questions on strips of paper with paper clips. Then, I ask students to sort the questions into piles based on how well they could answer the question. Finally, I ask the pairs to choose three questions that they understand best, and, to answer each question on white-boards.
As they work, I take advantage of the opportunity to give some formative feedback and to read one or two answers from each pair of students to the class. These questions can also be used throughout the rest of the unit as part of lesson closure activities. So, I ask students to keep the question strips in their folders.