This week, our focus turns to graphing, and analyzing the graphs of functional relationships. As relates to this unit, some of these relationships will be linear and exponential, but others will be more general, and we will spend plenty of time focusing on what the CCSS refers to as the "qualitative features" of such graphs.
One way that we'll approach this work is to open each class by sketching a graph of some real-life situation. For today's opener, I prompt students to "Sketch a graph of your height from the day you were born until right now."
After students have had a few moments to ask clarifying questions to get started on the task, I post these date vs height axes on the board. I tell everyone that they should feel free to label their axes with numbers, but that they don't have to. I also emphasize the word sketch: it can be hard to put perfectionism aside here, but we are certainly not looking for a perfect graph. We're just looking for something that reveals the basic relationship between height and age in our own lives.
As students begin to finish their graphs I ask that they share what they've got with a neighbor. "After you look at a colleagues work, feel free to make another draft of your graph," I say. We'll spend some real time practicing revision this week, and I'm introducing it here.
When the number of conversations happening in the room indicates that students have had enough time to create and share their graphs in small groups, I ask for a few volunteers to share what they've got on the board. The results might end up looking like this, as happened in one of my class this year. Here is my approach: after the first student sketches their graph on the board, I reserve all judgement and ask for another volunteer to draw what they've got. Often, the conversations will take off once there are two graphs on the board for us to compare, as these conversations happen, there's bound to be a third (or more) volunteer who asks to add their work (whether original or revised) to the mix.
My role depends on how well students address some key ideas on their own. In particular, I'm looking for the following:
Often, kids will come up with these ideas on their own. If one or more of these ideas fails to surface, I'll raise the idea. For example, to that first point, I like to point to the axes and ask if there are any numbers written here. When kids say that there are not, I point to the origin and say that I agree, except that the point (0,0) is here whether we label the axes or not. Then I ask the class to interpret what it would mean for the graph to start here, and whether or not that's possible.
I've allotted a little less than 10 minutes to all of this, but if it takes longer to have some great conversations, I'm ready to adjust the lesson elsewhere.
I take a few minutes to return work from the Sequences Gallery Walk that students turned in at the end of the previous class. If kids want to, I might go over a problem or two from the gallery walk. On the other hand, I might also just use a few of these problems as coming attractions. (If no one recognizes it on their own, I tell kids to search the web for 1, 1, 2, 3, 5, 8, ..., which can kick-start their own explorations into the Fibonacci Sequence.)
The gallery walk is about numbers and rules. Today's lesson is about graphs of real situations, and this week we'll be leaving a little less focused on quantities than on the qualities of graphs. Over the course of the unit, we'll bring the two together.
The next activity comes from an excellent little web site called Graphing Stories, and it naturally extends the work that began with today's opener. No explanation I could give would be as good as just checking it out for yourself; you should start by clicking on the link, reading the perfectly succinct "How this works" at the top left, and watching a few videos to get an idea of how things work.
What I'll share here are what I've found to be some winning strategies for using the site. To begin with, you should keep things simple. As suggested on the site, I distribute the Graphing Stories handout, and play a video. This year, I began with "Height," by Jean-Phillipe Choiniere. I pause at the 12-second mark and tell students to make sure that they've labeled and scaled their vertical axis as shown. Plenty of kids will have questions about what is supposed to go on the graph, and I work hard to stay quiet and let them see it on their own. I take a quick lap around the room to check that all the y-axes are ready to go, and then I press play again.
The video plays twice: once in real-time, and once at half-speed. When it's done, I press pause again before the solution is shown. I ask the class if they'd like to see it again, and now that they understand what's going on, they're excited to take another look. We watch again, then I pause while the blank graph is displayed ahead of the solution.
With the blank graph on the board, I ask for a volunteer or two to sketch their graph on the board. It's just like today's opener, except that the situation spans just 15 seconds instead of a teenager's lifetime, and the axes are specifically labeled. If they want to, I give everyone a chance to discuss what the work of their colleague. Next, when we watch the solution, I leave the student work on the board, so we can compare it to the actual solution.
This is such a high engagement activity that I can pretty much promise that amazing conversations will happen among your students. In keeping with the goal of interpreting the qualitative features of a graph, I point to some key points on the graph and ask, "what's going on here?" before eventually labeling a few moments from the video. We talk about whether or not we agree with the graph, which - perhaps by design - doesn't seem perfect. I choose to use this one first for just that reason: I want to see if students have any problems with some of the features of the graph. For example, at ~7 and ~12.5 seconds, it would appear that the adventurer's height is 0. But we don't see her hit the ground in this video, and this leads us to question whether or not "height" means "height above the ground" or something else. I let the conversations continue along these sorts of lines for as long they will.
To conclude, I tell everyone to write "Second Draft" as the title of the second graph on their handout. I instruct students to sketch a second draft of their height graph, making changes based on their conversations with classmates and seeing the solution, but not necessarily just copying the "answer" given to us by the video. Here's an example of the results.
Next, if there's time, we'll watch another. If not, we'll do this again tomorrow, so I tell kids not to lose this handout.
Now that we've had some time to delve into the features of another graph, we return the situation that opened today's class. I distribute today's exit slip and tell students that they have until the bell to sketch the second draft of their height since the day they were born. I make sure that they see the instruction at the bottom of their half-page: to explain how their graph has changed since the beginning of class.
Two things are important to me here. First of all, I'm really trying to cultivate a habit of revision among my students. I want it to feel normal to go back and try something again, so there's been specific revision time during the "Graphing Stories" part of today's class and right now. As kids work, I circulate and ask them to take another look at the first graph they sketched today. I prompt them to think about how their thinking has changed over the course of today's class.
Second, I want to be able to how well my kids can approach the idea of a sketch. I want to know how many of them insist of scaling their axes, whether or not they use points or a continuous line, and the extent to which they can just make a general graph. As you'll see on the student work I've provided here, students span each of the descriptions I've just described. Seeing where they're at helps me get ready for where I'll try to get my students to go next.