Here are today's three opening problems. These are variations on a style of problem that's popular in textbooks and on exams, and I think there is a good reason for this. These problems require students to think flexibly, and to demonstrate understanding of the concept of mean and median.
I allow students about five minutes to work on these problems in their notebooks, and as I circulate, I encourage conversation at each table. After about five minutes, I ask for volunteers to come to the board to share their solutions. The most important thing I want students to understand here is that there are infinite right answers to each of the problems. So after one student writes a solution on the board, we can verify that it's right (because finding the median or mean of list is easy enough, once the data's in front of us), and then I ask if everyone got the same answer. Now, students begin to propose their other solutions, and we see that there are many possibilities.
This also allows us to talk briefly about constraints. Each of these problems can be rephrased in non-statistical terms. Instead of asking for four different numbers with a mean of 8, we can instead find a set of four numbers with a sum of 32, for example. Eliciting this sort of thinking from students is important, because it helps them make sense of the problem. If they're already thinking this way, I can see that quickly, and if they were not, they'll now be excited to come up with a few answers.
Once that happens, I might take a few minutes to add further constraints. I'll tell students to complete a problem in which I give them two or more of the numbers in the set.
Also, if no one else beats me to it, I like to write an absurd answer or two on the board. For example, here are four different numbers with a median of 5:
-2000001, -2000000, 2000010, 3000000.
Yesterday's Problem Set, Problem #5
There is a problem set due today. Almost everyone needs a little help with the fifth problem. That's why today's opener is what it is, because it sets kids up for this kind of thinking. This problem is difficult for a few reasons, all of which show why it's so necessary to spend some time on this.
Just like in the opener, there are many answers to this question. My kids have a hard time with the idea that there's not just one cut-and-dry answer to a math problem. When I answer their insistent, "Am I right?" with a "Maybe, let's see..." I'm trying to involve them in a process. The idea that their teacher can't just look at something and say whether something is right or wrong is new to some kids, and I hope that it sheds some light on the good work that we're all going to do together in this class.
Moving from an abstract representation like a box plot to a concrete list of numbers is hard. It's also a highly-transferable skill with applications across all of mathematics, and I think that this sort of thinking is foundational to the highest levels of mathematical success. (This is a great topic of conversation in math department meetings, that will lead to ideas and insights if you discuss it with colleagues.) Many of my students want to start on this problem by drawing a number line - they want to follow the steps of making a box plot - but that's already done. It's difficult for some kids to even understand what I'm asking for here. I'm ready to explicitly identify the process that we're inverting here. I'll show the data set that we used for Problem #3, of numbers separated by commas on a slide, and say, "remember that on this problem, I gave you this data set, and you made a box plot. Now, it's in reverse. I'm giving you the box plot, and your task is to make the data set. How many possible answers are there?" If I need to, I'll write a flowchart on the board, representing the sentence I just said.
Once kids get going on this problem, they enjoy it, but there can be a high barrier to entry. This exercise requires patience on both my part and theirs, but the moment when it clicks is great. It's a great opportunity for critique and revision. I can look at a solution with a small group of students and say, "the minimum and maximum are great, but what happens with the first quartile? How can you change that?"
And Whatever Else Needs to Get Done
The problem set is due by the end of class. Some students need a few hints, and I'll help them out if necessary. My ideal is for students to be demonstrating mastery on this assignment, but I recognize that if they're not yet ready to do that, then our work is develop mastery now.
Other students were successful with completing this as homework, so for them I have a few more tasks prepared, as described in the next section.
As students work, I return Linear Practice #2, which they completed yesterday. Just as they did with Linear Practice #1, I tell them to record their results on the data collection sheet that they received on Monday.
With a few minutes left in class, I display the last slide of today's class notes. I tell students that they can choose their homework for the weekend, either in the textbook or online at Delta Math, but that I expect everyone to do at least 30 minutes of homework this weekend.
I collect problem sets from everyone who finished, and I say that anyone who needs the weekend to finish up can take, but that I'll want this problem set first thing on Monday.