Last year I was introduced to Personal Polynomials over at the Global Math Project through a Facebook post by a fellow educator explaining how she uses this with her calculus students. She asks them to create their own personal polynomial, and then create a graph of its first and second derivatives.
Since my goal for this semester was to incorporate Inquiry-Based Learning Techniques into my College Algebra class, I decided to adapt the Personal Polynomial activity for my college algebra students. I considered just assigning the activity for homework, but I decided to take my students to the computer lab since I happen to have a small class this semester, and the lab was open at the time. Once in the lab, I went through an example on the board using my own personal polynomial, and then students got to work on their own.
The activity is below, or you can download it here.
Personal Polynomials & Function Questions/Looking Ahead
- Create your own personal polynomial by going to https://www.globalmathproject.org/personal-polynomial/ and entering your name. Take a screenshot of your personal polynomial and embed below.
- Obtain the formula for your personal polynomial, P(x), and plot it in Desmos (https://www.desmos.com/calculator) as it is easier to view as a whole here.
- Once you have your Desmos graph completed, complete the table below:
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Interval(s) where P(x) is increasing |
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Interval(s) where P(x) is decreasing |
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Identify all of the roots (zeros/x-intercepts) of P(x) |
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Domain and range of P(x) |
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Relative maxima and minima |
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Absolute maximum and minimum (if they exist) |
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All points where the slope is zero–identify the points where the graph comes to a “peak” or “valley” |
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End behavior as x→∞ and end behavior as x→-∞ |
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Is the graph continuous on its entire domain? What about your classmates’ graphs? |
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What are the x-values called in the points where the slope of the function is zero? |
One of the first things that I noticed is that even after I showed students how to take a screenshot and insert it into their document, and even after I showed them how to use the equation editor, many of them still did not know how to do so. And even if they did, many times, what they wrote on their paper is not how it ended up looking on the screen. I think this is part of a bigger systemic problem with the assumptions of digital natives’ knowledge of technology.
I used to teach the Introduction to Computers classes at my college and found that some students didn’t know how to turn on the computer, some of them didn’t know how to hold a mouse correctly, and the list goes on. I’ve seen enough of what many of our students don’t know to realize that many of them simply do not understand how to use technology. I say this because although I would love to conduct this activity with students in my online classes, I think it would end up being a complete and utter disaster.
But one of the things that surprised me the most is that the students could find the domain and range easily. Of course, it’s a polynomial. However, they could not make the connection of how to change the viewing window on Desmos to make sure that all of their relative extrema showed up on the screen.
On that note, one thing I noticed is that students had a difficult time understanding the maximum value of a function. For example, the maximum value of f(x) = -x2 + 2 is 2. But many of my students would say things such as x = 0, 0, or (0, 2). Perhaps I am a bit too stringent, but I really want my students to understand that 2 is the maximum value of the function, and 0 is the value at which the maximum value occurs. I still am not sure how to get students to understand this difference. Even when I tell students to write their responses in the form of “the maximum of ____ occurs at x = ____,” I still get many incorrect responses, and I don’t think it helps with understanding, either.
When I have the opportunity to do this activity with my students again, I think I’ll try making a video demo using my own personal polynomial so that students can watch it at their own pace and go back to watch certain steps and listen for certain vocabulary words. I’ll also plan to have separate tutorials for using the equation editor and adjusting the viewing window on Desmos. Hopefully, this will free up some time for me to walk around the classroom to help students with mathematics rather than with technology issues.
Overall, this was an interesting and enriching experience for everyone. Seeing students work on this activity was a far contrast from if students were to all work on the same worksheet in class and leaned over to one another just to ask for an answer. In fact, one of my favorite parts of this activity was seeing students help each other. I saw many students leaning over to one another to explain concepts, discuss each other’s graphs, and to provide technical support. And that’s what happens when mathematics gets personal.
