More Graphing Stories
Lesson 6 of 19
Objective: SWBAT sketch informal graphs that represent different real-world situations. Foundations are laid for students to make sense of how algebraic rules can be used to model these situations.
Each of the first two structures in today's lesson are repeated from this image, it's pretty normal for at least one student's graph to intersect the x-axis. Just as I'm repeating a class activity from yesterday, I can repeat a message when I see this. "Even though we haven't labeled the x- and y-axes with numbers," I say, before running a finger along the x-axis, "what is the temperature of the water when the graph hits this line?" It doesn't matter whether we're talking about Celsius or Fahrenheit here; 0 degrees means a frozen cup of water. I ask for a volunteer to tell the story of what's happening to a graph that goes that low, and if no one takes the opportunity, I do. "Here's what this graph says. The water starts out very hot when it's poured into the cup. Then, as it sits on the counter, it cools off." Now I raise my voice until I'm almost yelling about this, although I make sure to keep a smile, "then the water keeps cooling, IT KEEPS GETTING COLDER UNTIL IT'S FROZEN SOLID!"
Kids love it. They get it. Now they're interacting with this graph, and they're all eager to give it another try. I let a volunteer come up and revise our work. If they're not already scribbling new graphs in their notebooks, I encourage them to revise their work.
Now we can have a great conversation. It doesn't matter what the boiling point of water is (although I tell them if they ask), it just matters that the water is its hottest the moment it's poured. Then it cools, but it doesn't cool forever. It can only cool as far as "room temperature," and those are words that always come from a student. It doesn't matter what the temperature of the room is, exactly, just that the water isn't going to get any cooler than that.
- You'll see in the image of student work on the board (U4 L6 Cooling Tea Cup) that one of the graphs starts on the origin before rising and falling again. Can you figure out what this student was thinking? Ok here it is: he wanted his graph to show the temperature of the water from the time it was poured into the tea kettle and then heated. For any student who makes this move, I explain how important it is to read the problem. It's not wrong to consider that the water had to be heated at some point, but it's never a good idea to read a problem that isn't there. (Of course, what is wrong about this graph is that it starts at the origin, which would mean that a handful of ice cubes was thrown into the kettle, boiled, and...).
- Depending on the class, I pay more or less attention to the idea that the graph should be curved. If we haven't taken too long to discuss the graph and I have everyone's attention, I'll ask, "When is the water cooling at the fastest rate?" We discuss that the drop from boiling - and here I use specifics, 212 degrees - to, say 150 degrees happens pretty quickly, but that the drop from warm - say, 100 degrees - to room temperature is more gradual. With that kind of thinking in mind, the idea of a curve should emerge. On the other hand, if it feels unnatural to have this conversation now, I move on; we'll return to this graph and to the idea of a curved graph a little later this week.
Another Graphing Story
The second recurring structure from yesterday's lesson is today's "Graphing Stories" video. Please take a look at the Graphing Stories section of yesterday's lesson for a detailed description of how I enact this part of the lesson. Here are the my key points for using one of these videos:
- Keep it simple and stay quiet. Don't over-contextualize, editorialize, or say anything that might prevent kids from having their own great ideas.
- Pause at the beginning for scaling the axis, pause at the end and ask kids if they want to see it again. When they do, they're spending time really engaging in thinking about the qualitative aspects of a graph.
- Invite volunteers to sketch their graph on the axes that are shown right before the solution.
- Discuss/label/interpret key features of the answer. See if everyone agrees with it.
- Invite students to make a second draft on their Graphing Stories handout. Here is an example of a student's first and second drafts from yesterday's and today's lessons.
Today, I use Adam Poetzel's "Height of Waist Off Ground" video, found on graphingstories.com. This is a great second video to yesterday's first, because it's a simpler situation, and the graph has terrific "qualitative features" that are useful to interpret. When kids begin to comment about the "grown man on the playground," I have some fun with that, telling them how much I appreciate that this guy has put himself out there just so they can learn math.
Now that my students have another chance to practice making graphs - from both a general situation and from a video graphing story - I take this opportunity to assess what they know so far on today's group quiz. For this quiz, groups should work together to get ideas for how they'll graph a set of four situations, but everyone must turn in their own work.
About This Assignment: It's One of Many Great Resources!
For the group quiz, I use an activity from the Functions and Everyday Situations lesson at the Mathematics Assessment Project (MAP) website on map.mathshell.org. If you haven't already seen it, this is definitely a web site you should check out. The resources here are top notch, and you're bound to find more great ideas, activities, questioning techniques, and implementation strategies.
The particular activity I use for this group quiz is on pages 15 & 16 of this Functions and Everyday Situations pdf that is hosted on the MAP site.
How I Use It
This group quiz serves as a check-in on how well my students can graph a situation describe in words, and it is a pre-assessment for an upcoming lesson. The front of the handout (page 15) consists of four situations that should feel similar to the openers of today and yesterday's lessons. At minimum, I expect everyone to sketch these four graphs by the end of class. I instruct students not to label the axes, and to talk to each other about the situations and what kind of graph would be most accurate. I don't give instructions about how to think about these problems; that will come later this week.
If they finish the front with time to spare, students should flip to the back (page 16) of the assignment. As I describe in this narrative video, this assignment leads kids through a fantastic learning arc, as they go from general graphs to using algebraic rules that establish specific quantitative values for each situation.
Not everyone will finish both sides of this assignment, and that ok. This is a formative assessment, kids will do exactly as much as they can today, and you and I can have some great ideas for what to do next when we see their work. In a few days, the opener will be about making a table and using it to come up with a few points, and this assignment will serve as a springboard to that work.