I started this investigation by telling students some of the key questions of the first unit:
*How can we describe a pattern?
*What kinds of patterns are there?
I ask them to get started answering these questions by working on the patchwork questions. I distribute extra graph paper and square dot paper. The big idea while circulating at first is to make sure that all students have a way to get started--which might be counting the squares and triangles in each of the figures provided or attempting to draw the next figure. If students get stuck, I suggest a data table, but only if they seem to have locked up.
Students take about 10 or 15 minutes to attempt to answer these questions for figure 6 and figure 10. After this amount of time, I randomly assign them a new partner (I give a number to each student randomly, and they find the other student with the same number.) I ask them to solidify their answers to these 4 questions.
It is essential that you not tell students whether their answers are right or wrong. I send this message from day 1, and I know I will need to repeat myself many times. Students are used to having the teacher tell them the right answer, or at least tell them whether or not they have the correct answer. When they complain, I take the time to thoroughly explain my reasoning to them:
"As soon as I tell you whether or not your answer is right, the conversation and the thinking will be over, because everybody at your table will just write down your answer and move on. If I don't tell you, you will need to keep thinking about your reasoning and your teammates will also have to keep thinking about the reasoning and you will all keep critically thinking. My goal for you over the course of the year is that you will get to the point where you never look to me for the right answer, but trust your reasoning and that of your peers to help you arrive at answers that you know are right."
It is possible to explain this to the whole class, but I find that it is more effective to explain it to individuals or pairs of students as it comes up. I know it takes a long time, but this time is an investment in a classroom culture where the practice standards are alive. The idea is that we as teachers are not the only ones who hold the practice standards with intention, but that all students are equally engaged and committed to the practice standards. In some sense, I consider myself a salesperson and I am trying to sell this way of learning to my studetns.
Once students are in new pairs, they need to commit to their answers to these 4 questions. Then I give them the actual task. I tell them that it is unlikely they will fully solve this today, which is totally fine. I give them several days before the deadline. This can become their first portfolio task.
When approximately 10 minutes were left in the class period, I interrupted students' work and asked them to write answers to these check out questions.
In some of my classes, we didn't have enough time to do the first two questions, so even though I normally like to have an academic focus for the check-out, I wanted to make sure I gave students a quick chance to give me a bit of feedback and tell me any important information that they wanted to share. Basically, I wanted to set the tone from the beginning of the year that they could give me critical feedback and that it would not be a big deal. To me, this is incredibly important to creating a collaborative and student-centered classroom culture. If students feel like they can only give feedback when formally asked to do so on some kind of class evaluation, they are unlikely to volunteer helpful feedback whenever it occurs to them. I have found that students often don't remember or aren't aware of big generalizations about feedback they could offer to improve a class, but if they are constantly asked to think about their ideas for improvement, they are more likely to volunteer them spontaneously. This is the tone I wanted to set from the beginning.