Functions: When One Representation Sheds Light on Another
Lesson 8 of 19
Objective: SWBAT begin to explore how different representations of a function can help us to construct an understanding of a situation.
Opener: Bus Ride
About this Lesson
I'd like to start by framing this lesson in the context of the week. The lessons on Monday and Tuesday shared a couple of structures that we'll revisit once again today. For both of those lessons, our focus was on noticing and interpreting some of qualitative features of the graphs of different situations. Yesterday, we took a look at some of the parameters of linear and exponential functions and how they can change the graph of a function, taking a brief break from thinking about context. Now it's Thursday, and in this lesson students will continue to build upon the sort of work they'd done on Monday and Tuesday.
Bus Ride Opener
Today's opener is like the ones from Monday and Tuesday. Students are given a situation and asked to sketch a graph. Because we're doing this for the third time this week, my strategy is simple: I post the opener as class begins, circulate to see that everyone is getting started, and as I see that kids are either finishing up or floundering on this work, I encourage them to discuss it their small groups. After a few minutes, I project the blank graph at the front of the room and ask for volunteers to sketch what they've got on the board. As students share their work, a class conversation begins.
To make sure everyone is clear on a few initial points, I ask "Does it matter how far Lizzie lives from school?" or "Does it matter exactly how long it takes her to get home?" We all agree that, by not labeling the axes, we can say simply that Lizzie lives a certain distance from school, and that it takes her a certain amount of time to get home.
As will be a pattern today, I want to have some fun with interpreting some representations that don't work. For example, as you can see on this graph, one of my students drew a pretty straight line in blue. I've worked to build a classroom culture based on trust, and at this point in the year I can poke a little fun at someone's work if they see where I'm going. This is hard to write about here, because I can imagine how this might sound mean, but in the context of my classroom, it works. I'll trace this graph with my hand and offer my interpretation of it. "As you can see, this is good: Lizzie got on the bus at school, a certain distance from home and the bus started driving her toward home," starting from the y-intercept, I move along the graph. "Then the bus keeps moving, and moving, still moving, until it rolls right past Lizzie's house, where Lizzie leaps from the cruising bus, and just like that she's home!" Kids see the humor in my description, but they also note an important mathematical idea: the bus doesn't always move at the same speed, and the graph has to reflect that. I give the kid who made this graph first dibs on trying again, and now engagement is high: everyone wants to revise this work.
Now we're talking about how the steepness of the line indicates the speed at which Lizzie is moving, whether on the bus or no. We also talk about what's happening when the bus stops, which leads to a conversation about horizontal and vertical lines. I want students to notice - and often they will point this out on their own - that it's possible for time to pass while the bus is not moving (a horizontal line, with slope = 0), but it's impossible for the bus to move without any time passing (a vertical line, with undefined slope). So an appropriate graph will have four horizontal sections, each representing one of the bus stops.
A slightly more subtle point is that the slope of the line should change when Lizzie gets off the bus and walks for the final part of her trip home. Again, this is a point that I can usually count the kids making, but if they don't, I'm sure to bring it up. Similarly, if we're really creating a perfectly accurate graph, we have to account for the idea that even in between the bus stops, the bus probably isn't traveling at a constant speed. I make sure that everyone notices this idea, but depending on how much time and attention to detail we have, I might give it a pretty brief treatment.
Both here and in the graphing story: pointing to parts of the graph and asking for interpretation. This is laying foundations for doing the same with function rules next week. My goal is for students to be able to look any representation of a function - the graph, an equation, a table, a couple of points - and interpret what they see. Today's work cultivates that skill.
More Graphing Stories
Now it's time for our third Graphing Story of the week. As in the opener, students are seeing this lesson structure for the third time this week. When I tell them to find their Graphing Stories handouts, everyone is excited to hear that we're doing this again - it's that kind of high-interest, high-engagement activity. By now, they know the drill. Please take a look at Monday's lesson for a detailed overview of how I use Graphing Stories, and see Tuesday's lesson to get an full idea of what students have seen and done with this kind of activity.
Today, I use the "Distance from Camera" clip by Adam Poetzel. This one is a periodic function, and it's the first exposure many of my students are getting to this sort of function. I'm not about to teach trig today - I just want students to continue to get comfortable with the idea that a graph might look like this.
I show the video once, and if kids are puzzled, I provide the hint that they should "count how many times he goes around on that ride." When we watch again, students count (the rider takes 3.5 laps on the ride) and I wonder aloud, "How might we represent those three-and-a-half spins on the ride?"
Whatever ends up on kids' pages - and many of them get a pretty nice approximation of the graph - I like that they have a nice context for the 3.5 periods that show up on the solution. As we look at the correct graph at the end of the video, we note that, according to the graph, the rider starts out 3 feet away from the camera. I point out that, "it isn't so important that it's exactly 3 feet. What is more important is that each time he passes the camera from the front of the ride, he's returning to the same starting distance."
The final noteworthy part of this graph is what happens at the end. The rider walks away from the ride, and the graph accounts for this. I point to this little "tail" at the end of the graph and ask how many people included this feature. I give students a chance to interpret what they see aloud.
If there's time, we might watch a second Graphing Stories video today. I hope to have at least 15 minutes to run the final segment of today's class.
To conclude today's lesson, I return the "Four Situations" group quiz that students turned in at the end of Tuesday's lesson. At the start of tomorrow's class, we'll spend a little time making a table of values to see how that can help us figure out what a graph should look like. Today, as I describe in this narrative video, my goal is to think-aloud and model how to consider whether or not an algebraic rule might match given situation, and then to give students a little time to practice using those rules to answer questions.
To conclude the week, I end tomorrow's class by giving students another assignment like this one.