Quadratics feel important. This impression is no doubt influenced by the boated importance placed on calculus in secondary school. They represent the giant leap from linearity and pave the way for more elaborate functions; therefore, I often find myself musing on ways to have students meaningfully interact with the topic. Once the structure of the function is established, I’ve played around with interesting ways to help students visualize quadratic growth, connect that growth to the Cartesian plane, and build these functions to specifications; however, my introduction to quadratics in vertex-graphing form has always been a series of “What happens to the graph when I change the ___ value?” questions. These aren’t bad questions (and a quick setup of Desmos sliders helps visualize the effects), but they don’t exactly build up understanding from experience. Such was my introductory quadratics lesson for years, lukewarm but lacking the epiphany to address it.
Enter Ockham’s razor. Actually, enter Ben Orlin to call (or rather, re-call) my attention to the elegance that can exist in simplicity. I am a total a sucker for an elegant task, and sometimes I go searching for this elegance in novelty and intricacy. Yet, if we can get students tinkering on the cusp of familiarity, there is elegance to be had in simplicity. Take quadratics in vertex-graphing form. They might be a “new” topic, but they have several familiar features. That is, they are built from polynomial expressions, still operate on an input-output mechanic, and students who have worked with linear functions have a rudimentary way to explore their behaviour: Graph them.
I convinced myself that this is enough to orchestrate a meaningful introductory experience with quadratics without any bells and whistles, but I wanted some sort of gristle in the task. In other words, I needed something to become cumbersome and provide stress on the ways in which the students handled linear functions. These sorts of lurking obstructions are great opportunities to observe deep understanding. When the automatic directions break down, who can make (and justify) mathematical decisions to move thinking forward?
I created random groups of three and positioned them throughout the classroom. They were given no other materials (other than writing implements), and each watched expectantly as I circulated around the room writing something on each board. It was eerily quiet, which added to the effect, to be honest. At each station, I wrote the equation of a quadratic function in vertex-graphing form.
y = a(x – p)2 + q
I didn’t pre-select the functions; instead, the functions followed three simple rules:
- All “a” values come from the list: -2, -1, 1, 2
- All “q” values stay between -5 and 5 (integers only)
- All “p” values are unreasonably large–less that -7 or greater than 7 (again, integers only)
The result was a bunch of functions that looked like this:
- y = -(x + 8)2 – 3
- y = 2(x – 9)2 -2
- y = (x + 10)2 + 5
My prompt (given verbally) was simply: Please graph the function I’ve written at your station.
Stages of student thinking
After I called attention to one group who decided to use a table of values, I was able to sit back and observe as the groups got to working. I watched for misunderstandings in arithmetic first. Did the students know how to handle the parentheses? the exponent? the “a” coefficient? Then I watched as the values from the calculations emerged, each one more unwieldily than the last–exactly as I’d hoped.
Stage 1: Blind inputs
This is where I immediately observed the curse and blessing of familiarity. Most students understood the input-output mechanic of functions (great!), but had become trained to automatically use a standard set of inputs when they were graphing straight lines (even better!). The two most popular were:
- 1, 2, 3, 4, 5
- -2, -1, 0, 1, 2
The result was a collection of horrendous outputs that they had no desire to graph. This provided me the opportunity to ask: “What is causing these numbers to explode out of control?” Our attention immediately focused on the exponent and the “damage” it had inflicted. From there, I offered something along the lines of, “Well, what inputs would control this damage?”
Stage 2: Convenience inputs
This seemed to grant the students a newfound control over their functions. They could choose anything as an input, so why not choose something that makes their lives easier? Eventually, groups started choosing inputs that were close to the “p” value, resulting in much more reasonable output values for their graphs. However, groups still showed the tendency to think in one direction. That is, if their function was “y = -(x + 8)2 – 3″ then they would choose inputs of -8, -9, -10, -11, … etc. The result was one “arm” of a parabola. From here I offered something along the lines of: “If we are trying to get close to -8, why not choose -7?”
Stage 3: Symmetrical inputs
Of course, these offerings don’t always result in the intended impact, but it didn’t take long for many groups to start thinking in terms of how close their inputs were to a specific “best input”. This emergence of the “best input” was a key milestone of thinking, and after it was solidified around the room, we named it “the vertex” and continued to generalize our patterns. Eventually, only half of the inputs needed calculation. We could choose an input, determine its distance from the vertex, and then mirror the output. I started entering into a group’s activity, changing the “p” value and then asking which inputs they would choose. This rapid re-calibration was a signal to me that it was time to consolidate critical features of these new types of functions.
This sort of simplicity eventually gave the control back to the students. In my previous introductory lessons, the function would move, and the students would respond. Now, the students were intentionally choosing inputs to make the function behave in a specific way. All it took was a mixture of familiarity and novelty, and the willingness of a teacher to stand alongside them in that tension for a while.