End behavior of functions via Connecting Representations

As the second and final assignment for the Structured Inquiry course I’m taking with New Visions, I was asked to create a task using the Connection Representations instructional routine (#ConnectingReps). More on Instrutional Routines here.

The instructor, Kaitlin Ruggiero, mentioned that multiple choice questions are good starting points for developing these tasks. Adopting her suggestion, I used #4 from the June 2016 Algebra 2 Regents Exam. The question focuses on roots and end behavior of a function. (F.IF.8). I chose to narrow my focus to strictly end behavior.

Here are the first set of representations, graphs of several polynomial functions:

Here are the second set of representations, statements about the end behavior of each graph:

During rehearsal, I showed graphs A, B, and D and their corresponding end behavior statements. We followed the routine to match the representations. I then revealed graph C and had them come up with the statement, which is 4. Lastly, the class reflected on what they learned using meta-reflection prompts.

I don’t have a formal write-up of the activity, but here are the above images.

Reflections:

  • I designed this for my algebra 2 class. My gut is telling me that it may fit in well at the beginning of my rational and polynomial functions unit. I may also consider using it if/when we review domain and range.
  • Initial noticings about the graphs had more to do with the “inner behavior” rather than the end behavior. In other words, the class was drawn to the minima, maxima, and roots.
  • There was some blank stares when I revealed the statements. This will most likely happen with students, too. The mapping symbol (i.e. function arrow) can be confusing if you’ve never seen it before. But that was the point.
  • Most of the class chunked all of the “x approaches…” statements and realized that they were the same in each representation. Since two of the given graphs (A and B) had both ends going to either positive or negative infinity and the other graph (D) didn’t, this led them to conclude that graph D had to match with statement 2. From there they reasoned that since graph A is going up on both ends, it should match with statement 1. Similar reasoning was used to match graph C with statement 4.
  • By giving the class three graphs and three statements, the third match (C and 4) was kind of boring. I still made them justify why C and 4 matched, but it didn’t feel as meaningful.
  • In retrospect, I wouldn’t change any of the representations, but I would revise what I give the class and what I have them construct on their own in order to help them move between representations more fluidly:
    • Give only A and B and their matching statements (1 and 3). Students reason through the matches.
    • Then give graph C and have them construct the corresponding statement (which is statement 4).
    • As an extension, I would give statement 2 and have them sketch a graph that goes with it. All student graphs will be similar to graph D, but the “inner behavior” will be all over the place. Because of the infinite number of possible correct responses, we could show several graphs under the Elmo to guide this part of the routine. I could save these student-generated graphs for later analysis on other properties, including even/odd, roots, maxima/minima, etc.
  • I didn’t use the words chunk, change, or connect at any point during the rehearsal. This is somewhat disappointing since I want my students to use these terms to describe their reasoning during this routine. Mental note taken.
  • Instead of me selecting the next presenter, sometimes I should allow the student that just presented to choose. This is student-centric and I like it (when appropriate).
  • Compared to Contemplate then Calculate, I feel that Connecting Representations is a slightly more complex in nature. With that said, Connecting Representations uses matching, which is really user-friendly. Both routines emphasize mathematical structure, but it seems to me like Connecting Representations emphasizes structure between different representations while Contemplate then Calculate focuses on structure within a representation. Dylan Kane and Nicole Hansen hinted at this during TMC16.
  • Though I didn’t use Connecting Representations, this past spring I foreshadowed this work with my Sigma notation lesson. Given two representations (sigma notation and its expanded sum), students used reasoning to connect the two.
  • This in-depth experience with both of these routines will allow my students to surface and leverage mathematical structure through inquiry like never before. So exciting!

 

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Height Problem from Graphing Stories

Height Image from Graphing Stories

This week in my Precal class, we were working on increasing, decreasing, and constant functions. I showed them a Graphing Story. These problems are perfect for analyzing graphs of functions. I chose the one that asks the viewer to graph height vs. time based on a lady swinging on a rope in Costa Rica.

If you haven’t already, first watch it for yourself here.

I’ve done some of the other graphing stories before and I love them. We did this one, discussed various aspects of the graph, and eventually hit on increasing, decreasing and constant intervals of the function. Its a great activity, but an interesting debate arose regarding the given answer.

You notice that the answer (i.e. graph) has a relatively large increase during 7-10 seconds.

Height Graph

When I initially displayed the solution to my students, they debated whether this interval was accurate. They found it intriguing that, according to the solution, she swung roughly 32 meters higher than her original height on the platform. Some of them said yes, it was possible for her to swing 44 meters in the air. I mean, looking at the video her upswing does look VERY high – much higher than the platform she started jumped from. Some students said no, that the camera angle was playing tricks on us. It was such an awesome discussion, and to be honest, I told them I wasn’t sure about who was right, but I’d do my best to find out.

After school I spoke to a physics teacher at my school, Shane Coleman (who these students also have), about the problem to get his scientific opinion. He informed me that it is physically impossible for the lady to have swung as high as 44 meters, unless she was pushed hard, had extra momentum when she left the platform, or pumped her legs on the swing, none of which happened. He told me for the footage in the clip, her kinetic energy would not have exceeded her potential energy. In other words, she wouldn’t have been able to swing higher than her original height; it would have been less and less with every swing. I probably should have known this, because its sort of common sense, but whatever. So then I brought all this back to my Precalc kids and it made for a pretty good discussion while we ironed out the actual solution. Later that day, the students probed even more while in Shane’s class. Awesome.

Now here’s my question: have my students, Shane, and I missed something here or is the solution inaccurate? If so, my follow up question is what assumption(s) could we make about the swing that would make the given solution valid?

On a slightly different note, isn’t it also interesting that her height is 0 at two different instances, yet she never reaches the ground?

bp