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.


  • 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!





Mental Math

Continuing the process of letting some ideas breath on the blog this summer. Here’s another.

It’s a simple activity for those few unexpected extra minutes near the end of the period…or if I just want to hit them with some quick mental stimulation. I picked it up from my fourth grade teacher: mental math.

I simply call out a sequence of operations with a pause between each operation. For example, I might say “2 plus 5 (pause)…times 8 (pause)…minus 10 (pause)…divided 13…what’s the answer?

Students can’t allowed to say anything out loud and, obviously, any electronic device is prohibited. I don’t require that everyone plays (most do). The students must wait until I say ‘What’s the answer?’ before raising their hands. I call on a different student each time and if that person’s answer is wrong, someone else gets a chance. Because it’s a terribly simple idea, it’s always engaging. The trick is to make it challenging to the point where they get hooked and want more.

Some tidbits: I’ll usually start with one that’s pretty straightforward with long pauses – especially at the beginning of the year. Things get interesting when I start to call out the operations lightning fast or the sequence contains something like 15 operations. Make things fun by using numbers in the millions – or even billions. Also, depending on the class, the level of the math can vary from basic arithmetic to roots and exponents to evaluating trig functions. It’s endless. And fun.



I always hear about Trashketball. I thought I’d briefly share the version I play with my students.

I set up the room with 1, 2, and 3 point distances, one of each. I use duct tape and white out for this. 

I post a question. Each group works on it and I randomly select one student to answer. Before answering, the chosen student can confer with their group members, but he must able to explain or justify his answer. Here are the guidelines for the game:

  • If a student gets the question correct, he earns three shots to make a basket. All shots must be taken from the same location.
  • Each group is allowed to passed earned shots to another group member once during the game.
  • Groups can use any resource but a non-calculator, electronic device to aid them in answering the questions.
  • If a student gets the question incorrect, I earn one shot to make one basket. I can earn a maximum of three shots per question. Each shot is separate from any others and can be taken from any distance.
  • Groups cannot communicate with other groups. If they do, I earn a bonus shot.
  • The game is the entire class versus me. Whoever has the most points at the end of the class period wins.

Here is what we use to shoot.

The trash basket is the typical, run-of-mill classroom trash receptacle.

My record as of today is 83-0-1. No kidding. Despite what my record may reveal about my utter dominance, my students LOVE playing. This is due in part because each class desperately wants to hand me my first loss. Hey, students need motivation, right?

Oh, and how have I amassed such an impeccable record? You’ll have to ask my students.



The asymptote as more than an invisible line 

Suppose you ate Chinese food that was left on the kitchen counter for four days. You didn’t know, and now you’re sick. You go to the doc and get some medicine. You’re diligent and take the prescribed dosage each day, but what if you don’t? What if you take the meds one day and stop? 

With help from Lois and Max that was the problem I posed to my students recently during my exponential functions unit.

What initiated it was asymptotic behavior. I no longer want this powerful constraint to be “an invisible line that a graph approaches infinitely.” In a world of NYS Regents exams, I want that line to have meaning. Note: my students have not graphed exponential functions in my class (it’s part of the algebra 1 curriculum), but they have explored function transformations.

When the kids walk in, I have three containers in the front of the class. Thank you science department for the borrow.

From left to right: waste water, body water, replacement water

The bell ringer opens things up with an exponential functions exploration. I want them to relate the value of b with either growth or decay. This will help later with constructing a model for our medicine.

We discuss that over time the medicine gets diluted and you must consume it everyday to maintain its potency. Otherwise, you’re never going to feel better. I mention that we can model this situation with mathematics and represent it with the containers of water.

Then I put 5 drops of food coloring into the “body” container and mix it up. I originally wanted to use 5ml, but realized in preparation that this was way too much to see a difference in color for our trial. To represent the medicine leaving the body each day, I tell them that I will remove 1 cup (10%) of the body water and replace it with 1 cup of clean water. Before we start, I have them estimate, based on our fictional simulation, how much medicine would remain in their body after 2 months if they only consume one dose of meds.

We start diluting. I do it 7 times (to represent a week) and they calculate the amount of drops remaining in the container after each day. They graph. The decreasing nature of the graph is noted and I ask them to find an exponential model for this data.

They use the model extrapolate 2 months, with one week intervals, and graph the data.

At this point a few students start to get it. Bam. The asymptote is born.

Here’s the handout.


  • Pretty cool activity. Somewhat teacher centered, but highly engaging. The design was such that the intervals of estimation, graphing, and modeling helped to put their focus on the mathematics and not me.
  • My focus was asymptotes, but it also served as an introduction to exponential functions in algebra 2. I’m not sure the exponential relationship between the variables was really understood. To that end, I’m not convinced that this was the best activity for the job – at least how it’s designed right now. I should tweak things.
  • I was pleased with the bell ringer. There was some good debate around b being fraction and whether that’s why y2 was decreasing.
  • My goal after the demo was to go into graphing exponentials with transformations, so I included a series of questions that are typically found on NYS Regents exams. It tanked. I didn’t make a strong connection between the Desmos activity and this one. The 2nd class did slightly better because I bridged the two concepts a little better on day 2 (with an improved bell ringer), but for the goal I set out here, I still need to adjust my planning to better relate things like f(x) = 2^x and f(x+2) = 2^(x+2). They knew what transformation f(x+2) meant, just not how to interpret it as an exponential function.
  • Neither class answered the last writing prompt (stop & jot), mainly because I wasn’t feeling it during the lesson. Still thinking about how to wrap things up.
  • I started this on one of our shortened days, so I’m not sure if we could have done this in one day. Probably not.
  • I didn’t realize it until now, but this whole thing has a 3-act feel to it. I like!



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