I treat opening problems differently from one day to the next. Some days, the opener might be the focus of the whole lesson, and it will unfold into a 20+ minute mini-lesson. Other days, the opener is an instruction that gets kids started on their work for the day. Others, like today's opener, are of the quick, memory-jogging sort. On days like this, I don't worry too much about whether or not students complete the entire opener. As they arrive, I coach them to get started and to see what they can figure out. But I'm not going to worry too much about each of these being done perfectly.
So here is today's opener as presented on the agenda. Students have seen number pattern openers before, but what's unique about this one is that only the first is an arithmetic sequence growing at a linear rate. The rest are a sampling of other sorts of growth. This is not the last time students will see any of these sequences.
As students get started, I circulate to see how everyone is doing. As students get started on these problems, one trip around the room is enough for me to get a pretty comprehensive pulse of the class. Depending on what I see, I'll adapt my approach to debriefing this opener. Here are some possibilities for my next move:
In any case, the idea here is just to help kids get more and more comfortable with the nature of numbers. They're going to build their own patterns today, and this opener helps to activate their minds for that work.
About the Work
For today's activity, students will work in pairs. I introduce the task on the second slide of today's notes: everyone needs to find a partner, to get a sheet of 11x17 ledger paper, and to fold that paper into quadrants. I ask if everyone knows what that last instruction means, and when I see the first few students do it successfully, I ask that they hold up their paper for their classmates to see.
In each quadrant, each student pair will complete a different task. In between each task, they will pass their work to a neighbor, and complete the next part for someone else's pattern. I present these tasks to students with slides #3 through #7 of today's notes.
The first task is to "Draw the first four figures in a [new] dot pattern." I instruct students to "Make a pattern that grows by the same number of dots at each step," which ensures that each pattern can be modeled with a linear function. I tell students that creativity is the thing here, and that I'd like for them to try to make a dot pattern they haven't seen before, if possible. The amount of time I allot to this part of the activity depends on how engaged students are in developing their patterns.
After the first task, I ask everyone to write their names in the same quadrant as their work, and then to pass their work around the room in a counter-clockwise direction. In my classroom there are two pairs of student at each table. I ask that everyone makes sure that their work has moved to a different table. When everyone has another group's paper with a newly-minted pattern on it, we move on.
The next task is to make a table of values, and to write an algebraic rule. Actually, some students thought it was natural to add a table of values and rule after creating the original pattern, so it's up to me to make sure that no one does that during the first task. I tell everyone to do exactly what the instruction says, "no more, no less." At most, I give students five minutes to complete this part of the assignment.
After everyone writes their names in the current quadrant and another counter-clockwise shift, the next task is to "Sketch the 100th figure in this dot pattern," and then to determine the number of dots that are in that figure. This is where I'll focus the most of my attention when I assess today's work. The focus of yesterday's lesson was to think about the relationship between the structure of a dot pattern as it is drawn and the algebraic rule for the number of dots in figure n. This serves as my check-in for how well kids are getting that; it's an assessment of how well each student can think about structure and use it to understand each pattern. If students are unclear on what I'm asking for here, I'll show them how I'd sketch an example of how I'd think about the structure on one of the problems from yesterday's dot problem work (which is due today).
Up to Here, What Happens?
In some classes, students take the path of least resistance and just make minor adjustments to the patterns from yesterday. Here, it will be easy - and still useful - to get through all parts of today's work. In other classes, there's a real creative vibe, which leads to all sorts of interesting mathematical conversations about how to make a pattern grow. In classes where all students want to create something beautiful, it's worth it to slow down a bit, and even allow this activity to span part of tomorrow's class.
Look at this narrative video for a description of some conversations I had with kids as they tried to develop a heart-themed pattern or to grow the letter N. If groups are really flexing their creative muscle on making new dot patterns, I'll let them at it, and I won't worry too much about finishing everything today. That's the fun of teaching a lesson like this: it lends itself well to scrapping the script if great stuff is happening. Each student will get some practice exactly where they need it.
I dig a little deeper into some student work in my "A Variety of Student Work" reflection below.
Adaptations and Accommodations
A few tricks can be useful in implementing this lesson. Sometimes it works to set a timer for each section (I use http://www.online-stopwatch.com). This might come at the expense of fostering creativity but does help everyone to get stuff done. It's hard to recommend exact amounts of time for each section. In my experience, it's different for each group of kids.
If I have a good rapport with a high-achieving class, I'll challenge students to stump me with their pattern. Most often, the lesson that comes out of this challenge is that it's really hard to stump anyone with a linear pattern, but that's a pretty neat takeaway on its own.
Furthermore, if I'm really feeling adventurous, I'll allow for non-linear functions by omitting the detail about growing by the same amount of dots each time. This opens up a whole new set of possibilities, and it's a fun way to twist this whole lesson into something completely different.
If some pairs get a little behind and don't trade their work for one or more rotations, it's ok to let them go through consecutive parts on the same poster. What I really like about this activity is that it gets kids to ask questions exactly where they are. It makes a natural opportunity to wonder and to ask questions, in a low-pressure situation.
For the fourth quadrant of the chart, I introduce a new kind of function representation, called Function Diagrams. Rather than explaining them here, I'll refer you to the work of Henri Picciotto, who has developed an outline of how to integrate Function Diagrams into the curriculum. He explains connections between these diagrams and the kinds of algebraic thinking we want to develop in kids, and he offers some handouts to help you get started. One of Henri's handouts is the focus of tomorrow's lesson, and if there is time at the end of today's class, I'll distribute these as homework, so kids can get a head start.
To make the initial introduction, I review one of the problems from yesterday's work. I make a quick table of values that shows the number of dots in the first four figures in one of yesterday's dot patterns. By now, that's the easy part for almost everybody. Then I say that I'm going to show everyone a new way to think about patterns. I draw two vertical lines, I label them n and d(n), and then I number each line to from 0 to 15. As I draw, I point to the corresponding parts of the table and the function diagram.
To complete the diagram, I connect each "input" on the n line on the left to its "output" or "number of dots" on the right-hand d(n) line. What's beautiful about this representation is that it instantly makes sense to kids. They're eager to try it for themselves because it just makes sense, so now that it's their turn to make function diagrams in the last quadrant of the poster, student are ready to go.