Today's activity is taken from the work of Henri Picciotto, who blogs and shares curriculum at his web site, www.mathedpage.org. Mr. Picciotto is a brilliant resource to teachers, for both the thinking and the activities he shares on his site. The "Function Diagrams" activity that students will complete today is great, in part, because it motivates students to think, work, and make observations without too much prompting from the teacher. For a complete description of the assignment and the thinking behind it, I recommend that you start by reading Mr. Picciotto's notes here.
It's possible that students got started on this assignment at the end of yesterday's class. If that's the case, I make adjustments to the agenda as per their progress. To keep things simple here, I'm sharing how I run this class with students who are seeing function diagrams for the first time today.
As students arrive, I greet them at the door with a copy of this "Nine Function Diagrams" document, which is hosted at Mr. Picciotto's web site. When the late bell rings, I explain what to do, as quickly as possible. I try to speak for less than 60 seconds. I post the document on the board, and I annotate the first function diagram by showing students what I'm asking them to do. Each line connecting the vertical x and y number lines in a diagram represents one ordered pair. I leave this example posted and the timer running on the front board as students get to work. On the timer, I give students 18 minutes and say, "When this expires, I expect everyone to be done making their tables of values."
Many students will be done with this part of the assignment in less than 18 minutes. As they finish, I describe the next part of the assignment to individuals or small groups. The purpose of the timer is to build some urgency among students. The initial task is simple enough for everyone: it's finding numbers on a number line, and copying them to a table. But there's a lot of work to do, and I want to make sure that everyone gets it done. Some students will need more than 18 minutes, and if they need the time for the right reasons, that's fine. But I'm trying to reach students who can and should be able to get it done, and used sparingly, the timer helps to accomplish that goal.
When the timer expires, I ask for everyone's attention and show them the next step. Those who already completed the first step have heard it. The next task is to write a rule for each table. I ask the class, "What do you have to do to x to get y in each function diagram?" My students have seen simple algebraic rules before, and this jogs their memories. Students have a pretty easy time seeing that they have to add 2 to x to get y in the first diagram. I simply show them, if they're unable to see it for themselves, how each rule should look. I say "y=x+2" and I write it on the board. That's all everyone needs to get started, and I set them to it.
There are similarities between these problems and the rules kids have been writing for number sequences. Actually, the the rules here are simpler: almost all of them consist of just one arithmetic step. On the other hand, there is less of an algorithm developed here than what we've made for number sequences. My role is to highlight the similarities while explaining the differences.
If the x column counts by 1's, the pattern is just like the sequences we've seen. In tables that skip numbers on the x column, kids might need a little help at first. When students have trouble finding a particular rule, I ask them to tell me what they've done so far. Everyone can explain why the first rule is y=x+2, so I ask what they have to do to get from the first x to the first y on whatever problem they're stuck on, and then we test whether or not that works for all pairs.
On problem (c), for example, a student might notice that you must subtract 2 to get from 4 to 2, but then see that that does not work for all pairs. I ask, "If subtraction isn't working, then what can we do? What other options do we have?" As students name other operations, I say, "Add what? Multiply by what? Divide by what?" Soon, we find that dividing by two will work.
Extension and Coaching Kids Where They're At
This assignment differentiates within itself. Making tables is an accessible entry point, writing rules is useful practice that prepares students for where this class goes next, and then the questions at the bottom of the page help students challenge students to make generalizations and explain what they're thinking. The wording of these questions is very difficult for many of my students - so I let those who get there to try it, but if they need the full time to write the rules, then that's fine too. There is a low barrier to entry here, but the ceiling - deeply understanding the answers to those questions - is high. It's exciting to watch kids make sense of these questions.
Even though some kids might not be able to handle the text-rich questions at the bottom of the page, it's natural for them to start to see connections between the diagrams. "This one is the opposite of that one" they'll say, for example, as they compare diagram (a) to (b) or (d) to (e). When that happens, it's teaching gold. Our job, as teachers, is to cultivate those conversations. Introducing a new representation like this always helps to draw in students at various levels of success. I've been impressed, over and over again, by which students suddenly feel successful when they try an assignment like this, because it gives them access to making sense of important ideas.
I'm squeezing this work into one class period, but Mr. Picciotto recommends spending two days on this activity. One more time, I recommend reading his description of the assignment here.
As I've mentioned above, I'm working to build urgency with many of my students. No matter how much my students complete on the "Nine Function Diagrams," I'll collect their work with a little less than 10 minutes left in class, which helps toward that goal. More importantly, I can quickly assess how much they've accomplished and what they know. I want to know who was able to write all nine rules correctly, and I can see their progress on the comprehension questions.
In the time that remains, I distribute tonight homework, Equivalent Line Segments, which is the focus of tomorrow's lesson. To introduce the assignment, I point out that students have been working with pairs of vertical parallel lines, and that now, "each problem on this assignment consists of a pair of horizontal lines." We read the instructions, and student have a few minutes to get started before the bell rings.