My Algebra 2 students have been working on describing key features of polynomial graphs. Our reporting standard for this unit is F.IF.C.7:
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Specifically, it is part c of this standard:
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
We focus on the key features in this standard because later in the year, we have another reporting standard where the main focus is using factorized polynomials to sketch graphs without a calculator. So in this unit we are using Desmos and working on vocabulary. The key features listed in our district’s SBAR (standards based assessment and reporting) rubric for this standard are:
- end behavior
- increasing/decreasing intervals
We have described key features previous to this for piecewise functions, absolute value functions, step functions, logarithmic functions, and exponential functions. From those, they are pretty comfortable with intercepts, domain and range. New to this function family are end behavior (which in Algebra 2 we describe informally without limits as rising or falling left/right), extrema, and increasing/decreasing intervals. They really struggle with types of extrema (local/relative vs. absolute) and with increasing/decreasing intervals.
Before their quiz, I wanted to give them a bit of a challenge to help them process some of these vocabulary terms more. In this activity, the numbers don’t really matter as much as the features themselves, so it really promotes understanding of the concepts. Originally, my thought was to make actual BINGO boards for each student with pictures of graphs on them, and then call out a key feature. They would be able to mark off a graph with that key feature. I started to do this, and made a Google Slides presentation with all the key feature descriptions I wanted to use, and then I started trying to make the BINGO boards and realized how much work it was going to take to make what I wanted. I wanted everyone to have a different board, and I was going to have to take screenshots of graphs that met all the features I had listed…so I went back to the drawing board. I did find this cool site though, where you can make randomized BINGO cards, so if you were doing this with vocabulary or something easier, check bingo baker out! I am realizing now that this may have worked if I had made the prompts a graph, and the spots on their boards the key features…next time!
I decided to reframe it as more of a challenge than a typical BINGO game, but with their goal to still get 5 in a row. I made up a Google Sheet BINGO board, with a blank cell beneath each key feature description. As the instructions state, students are to use Desmos to try to make a polynomial graph that meets any of the described key features. When they do, they copy and paste the Desmos graph’s link in the blank cell, then highlight that cell to show they’ve completed it.
As soon as they started working, my students asked, “can we use the same graph for more than one box?”, which for some reason I had not considered yet. I made the decision on the spot that they could, because I figured if they could make a graph that fit a whole bunch of the requirements, then good for them. Depending on what you want from your students, you might tell them that each box has to have a unique graph. My students used the strategy of making a random polynomial and then seeing how many of the boxes it worked for and putting the link in all those squares. I think they enjoyed the challenge of trying to make a polynomial that fit a large number of the squares. If you wanted them to focus more on the individual squares, then you may want to force them to have a unique graph for every square.
You can see that this student used the same link for several boxes:
The students had a fun time with this, and it was a perfect length activity for yet another 2 hr late start day that we had (our winter has been miserable – 7 days cancelled and 5 days shortened!). I think I will be more strict on encouraging them to fill as many squares as possible once they get a BINGO next time to push them further, although many students did that.
A student also let me know that I had two of the same end behavior squares – I fixed this so that they are now different, but you may notice the duplicate in the screenshots! (This is what happens when you are trying to come up with 24 different key feature descriptions and not paying careful attention and there’s only four possibilities for the end behavior of polynomials)
You can make a copy of my Google Sheet for this activity HERE. This will make a copy of the file to your own Google Drive which you can then edit as you wish 🙂