## 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

## Contemplate then Calculate

I’m taking a Structured Inquiry course this summer offered by New Visions for Public Schools. As part of the course, we are required to design a Contemplate then Calculate instructional routine and rehearse it with the class. David Wees and Kaitlin Ruggiero have both done some inspiring work with #CthenC routines and much of my motivation comes from them.

If you’re unaware of Contemplate then Calculate, the goal is to get students thinking about mathematical structure. The routine leverages student observations, honors their approaches, and highlights reflections on their own thinking. There is a public collection already created by David, Kaitlin, and the New Visions team.

Reflections:

• I initially planned for this to come at the end of an exponentials unit, but after some discussion, I think it would fit much better at the beginning of the unit – or at least before we discuss exponential equations. One thing I’m great at is having students get overly consumed with procedures, so my hope is to get them thinking instead about the structure of the equation and the relationships that exist within it before any mention of the term “exponential equation.” I want their mathematical insights to drive how we solve these equations…and the entire curriculum.
• To this end, I thought of four strategies that students may use, ranging from guess and check to equating exponents (excluding the one involving logarithms since the task would be presented before they appear). Interestingly, guess and check was not an approach that was adopted much during rehearsal (possibly because they were all math teachers).
• Since my phrasing of the number of solutions was ambiguous, Robin mentioned that students could use a graphical representation to address this. They could reason that only one solution exists because f(x) = 10^(x+5) -1 is one-to-one.
• I also considered how to represent the equation. Specifically, I thought about 100 = 10^(x-5),  10^(x-5) = 100, 100 – 10^(x-5) = 0, or using any other set of constants in place of 99 and 1. All of these alternatives have consequences for how students may approach the problem.
• During the rehearsal, I did a fairly poor job at annotating, which is a critical aspect of the activity because it models student thinking for the rest of the class. Although I anticipated all the strategies while planning, I rushed myself during the process and the result was unclear and unorganized.
• The reflection prompts should be tailored to the goals of the activity. The more specific they are, the more beneficial the reflection.
• I should omit the question (e.g. “find all values of x”) during the flash of the task. This move opens things up for more diverse observations and student thinking.
• An interesting extension was asking what if the 1 wasn’t there? How would this impact our strategy? This is a nice prelude to logarithms.
• Designing this activity made me think about mathematical structure like I never have before. Often times I take student thinking related to structure for granted. This activity helped me better value student understandings of mathematical structure – and how to leverage those understandings to enhance learning.
• One of the most beautiful aspects of this routine is that it goes beyond mere discovery learning. The goal isn’t for students to end up at the same strategy. The idea is to develop fluency and flexibility in their numeracy. It echoes the theme from my current book, Building Powerful Numeracy for Middle and High School Students by Pam Weber Harris.
• All of my planning materials can be found here, which include the anticipated strategies students will use, my planned annotations, the annotations I actually did during the rehearsal, and the slides.

bp