## 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!

bp

## Summation notation, but way more

I’ve been rethinking all of my lessons this year. My hope has been to get my students to reason more. To think independently. To not be sponges. I’d like to think it’s been working. Here’s a recent lesson on summation notation that showcases this shift.

To open things up, I gave them this.

Super accessible and relevant to summation notation. In the past, I would have chosen a bell ringer that was closely connected to a prior lesson (i.e. review) than the current one. I wanted to provide remediation. I’ve learned this year that a relevant bell ringer is pivotal to any lesson.

Here’s what came next.

Again, very accessible. Last year Jennifer Preissel mentioned the “Stop & Jot” idea as a simple way of getting kids to write and reflect more during a lesson. Here, I gave them five minutes to express, on their own, what they wondered and noticed about the expression. After, they shared with their groups and we discussed as a class. By including “left side” and “right side,” I wanted to focus student responses. There were comments like “the +2 happens in every parenthesis” and “the number next to the +2 is going up by one.” Their observations led us to the brink of directly relating sigma notation to its expanded sum. In the past, I would jump right into defining sigma, the upper and lower limits, argument, etc. There would have been no exploring or thinking on their own.

Next, I ask them to move on to another example with the hope of finding a relationship.

It worked like magic. They see the same pattern from the Stop & Jot and they start to generalize. They have no idea what the “E thing” is, but it’s beginning to settle in how the left and right sides relate to one another. They discuss all of this in their groups. I float around. Observing. Listening. In the past, I would show them how to find this sum and answer their questions. Again, no self-exploration and making meaning of what they see.

Now they are to dissect and interpret.

This lacks clarity. Some students knew to write their interpretation next to the arrows, but many did not. As a checkpoint, we came back together and discussed.

Next: remove the right side.

Things are flowing now. The scaffolds are working. They know the relationship and successfully express the sum. In the past: The students would probably be completing this problem, but instead of using their own insight to drive the work, they’d be following what I said was the correct procedure.

Finish it off.

We come back together one more time to debrief and to address any questions the groups haven’t already. To bring things full circle, I mention the task from the bell ringer. “Ohhh!”

Lastly, on the next page, the proper names are reveled.

We then have just enough time for an exit slip.

This lesson is heavy on notation and I didn’t want to bog them down with symbols. The goal was to find meaning first, then discuss representation. It succeeded. What I miss out on is working in reverse. Namely, using sigma notation to represent a given sum.

What I love most about this lesson has little to do with summation notation. It’s much bigger. It stems from the approach. Bottom up. Using their own insights to help them find meaning. Doing less and allowing them to put the pieces of the puzzle together. This lesson is a microcosm of how I try to teach nowadays, which is much different than in the past. It symbolizes my growth as a teacher, as a learner.

bp