the following graph in Desmos and asked them what did they notice. If they had to describe it to someone over the phone for the person at the other end of the line to draw, what would they say? The conversations I heard at first started with general statements like, "it wiggles in the middle" and "it looks like a parabola that someone dented and bent in the middle".
"It wiggles in the middle could look like a lot of different graphs," I replied to the one group. "Can you be more specific?"
At that point students started talking about x-intercepts and they also noticed that sometimes the graph crossed the x-axis and sometimes it touched the x-axis. The student who noted the parabola shape on the ends was starting to hint at end behavior. After a bit more discussion, we decided to play around with the exponents and to see how that impacted the behavior at the x-intercepts. We also talked about what we could do to change the end behavior of the function.
It is important to note that even before my students put pencil to paper we were discussing concepts and playing around with ideas, dynamically. At times students were discussing ideas in groups and other times we were putting our ideas together as an entire class. It takes time and sometimes patience to have students learn this way. But ultimately, I believe it leads to deeper understanding. Students come to realize that if they can't recall a concept (like, does it fall on the left and the right for end behavior), they have a way to reconstruct that concept, because they constructed it initially.
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