Building off of Michael Fenton’s quadratics matching activity, I created a polynomial matching activity. I like it, but I don’t love it. I want to somehow push students to prove whay they make their choices, not just state them & I want to add more questions at the end that require students to connect the various representations of a function, so please leave suggestion in the comments!
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Crazy Math Teacher Lady 
I know this comment is months late, but today’s the first time I found your blog! I did a similar matching activity on piecewise functions with my classes in the spring, though I didn’t ask them to record their matches as long as they were discussing them with their partners. The following day, I asked them to choose any match from the set and “prove” to me that they were right. I explained that they needed to give me more than one reason and really convince me that they were right. As weak as that assignment sounds on its face, they blew me away with the detail and thought they put into their answers. They pretty much all filled up one page. There are a few reasons I think it worked. First, the students could pick a match that they felt comfortable working on and so I saw some of the students who had been struggling choose a simpler graph- no problem. Second, instead of asking the students to record their reasoning for every match, I asked for one really good answer and I suspect that made them realize that I was truly looking for a complete, detailed answer. Third, their confidence was boosted because the problems were ones they’d worked on the day before. You can check out my post on the topic here: http://iisanumber.blogspot.com/2013/05/my-kids-wrote-page-about-one-math.html
Thanks for the feedback! It motivates me to stick with this! I think your idea makes sense. So many times I tend to create more work for myself and end up making the lesson less effective. I seem to associate my working hard with student achievement, which I know is not the reality.
I expanded this activity to include polynomials with negative leading coefficients and added more with degree four and five, and more polynomials in factored form. I also added polynomials that have repeating zeros to show where they bounce or cross the x axis.
Dario- Could you share your resource? I will be doing a similar activity with my Algebra students and was thinking of making the changes you suggested.
I loved this activity so much I created a desmos card sort using your polynomial match activity. You can find it here:
https://teacher.desmos.com/activitybuilder/custom/5f8a100ba7931b2cdfc4e9fa